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Research article Special Issues

Revisiting European migration industries amid the Covid-19 pandemic: Changes in the recruiting practices of Bulgarian agencies

  • Workforce mobility in the European Union (EU) has faced enormous challenges regarding the outbreak of Covid-19 in 2020. Member-states reacted by imposing border closures and eliciting travel restrictions and sanitary measures, some of which continued well into 2022, with far-reaching consequences in all areas of mobility experience. In a time of constant changes in the regulations of the internal market, the work of labour market intermediaries deserves special attention. Recent studies have shown that, being a relatively new EU member-state, Bulgaria managed to develop an important migration industry comprising a complex network of public and private actors. We focus on an interpretative analysis of ten in-depth interviews conducted in 2017 with practitioners working in such agencies, and ten follow-up interviews five years later. The comparative analysis of these intermediaries' reflections on the mobility policies and practices before and during the pandemic allows us to examine the processes of structural and functional adaptation in the industry in multiple dimensions: assessment of changes in mobility practices, perceptions of employers' and migrants' needs and expectations, and the challenges facing their agencies vis-à-vis the role of other actors in the field. The conclusion underlines trends and transformations which are of interest for the developing recruitment sectors in other Central and East European countries, as well as for the dynamics of the East-West labour mobility throughout the entire EU.

    Citation: Boris Popivanov, Siyka Kovacheva. Revisiting European migration industries amid the Covid-19 pandemic: Changes in the recruiting practices of Bulgarian agencies[J]. AIMS Geosciences, 2023, 9(2): 409-425. doi: 10.3934/geosci.2023022

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  • Workforce mobility in the European Union (EU) has faced enormous challenges regarding the outbreak of Covid-19 in 2020. Member-states reacted by imposing border closures and eliciting travel restrictions and sanitary measures, some of which continued well into 2022, with far-reaching consequences in all areas of mobility experience. In a time of constant changes in the regulations of the internal market, the work of labour market intermediaries deserves special attention. Recent studies have shown that, being a relatively new EU member-state, Bulgaria managed to develop an important migration industry comprising a complex network of public and private actors. We focus on an interpretative analysis of ten in-depth interviews conducted in 2017 with practitioners working in such agencies, and ten follow-up interviews five years later. The comparative analysis of these intermediaries' reflections on the mobility policies and practices before and during the pandemic allows us to examine the processes of structural and functional adaptation in the industry in multiple dimensions: assessment of changes in mobility practices, perceptions of employers' and migrants' needs and expectations, and the challenges facing their agencies vis-à-vis the role of other actors in the field. The conclusion underlines trends and transformations which are of interest for the developing recruitment sectors in other Central and East European countries, as well as for the dynamics of the East-West labour mobility throughout the entire EU.



    Several important evolution equations arising in kinetic theory, mathematical physics and probability can be written in the form

    (t+XY)f=Q(f,Xf,X,Y,t), (1.1)

    where (X,Y,t):=(x1,...,xm,y1,...,ym,t)Rm×Rm×R=RN+1, N=2m, m1, and the coordinates X=(x1,...,xm) and Y=(y1,...,ym) are, respectively, the velocity and the position of the system. In its simplest form,

    Q(f,Xf,X,Y,t)=XXf=ΔXf,

    the equation in (1.1) was introduced and studied by Kolmogorov in a famous note published in 1934 in Annals of Mathematics, see [25]. In this case Kolmogorov noted that the equation in (1.1) is an example of a degenerate parabolic operator having strong regularity properties and he proved that the equation has a fundamental solution which is smooth off its diagonal. In fact, in this case the equation in (1.1) is hypoelliptic, see [24].

    In kinetic theory, f represents the evolution of a particle distribution

    f(X,Y,t):UX×UY×R+R,UX, UYRm,

    subject to geometric restrictions and models for the interactions and collisions between particles. In this case the left-hand side in (1.1) describes the evolution of f under the action of transport, with the free streaming operator. The right-hand side describes elastic collisions through the nonlinear Boltzmann collision operator. The Boltzmann equation is an integro- (partial)-differential equation with nonlocal operator in the kinetic variable X. The Boltzmann equation is a fundamental equation in kinetic theory in the sense that it has been derived rigorously, at least in some settings, from microscopic first principles. In the case of so called Coulomb interactions the Boltzmann collision operator is ill-defined and Landau proposed an alternative operator for these interactions, this operator is now called the Landau or the Landau-Coulomb operator. This operator can be stated as in (1.1) with

    Q(f,Xf,X,Y,t)=X(A(f)Xf+B(f)f), (1.2)

    where again A(f)=A(f)(X,Y,t) and B(f)=B(f)(X,Y,t) are nonlocal operators in the variable X. In this case the equation in (1.1) is a nonlinear, or rather quasilinear, drift-diffusion equation with coefficients given by convolution like averages of the unknown. As mentioned above the Landau equation is considered fundamental because of its close link to the Boltzmann equation for Coulomb interactions.

    In the case of long-range interactions, the Boltzmann and Landau-Coulomb operators show local ellipticity provided the solution enjoys some pointwise bounds on the associated hydrodynamic fields and the local entropy. Indeed, assuming certain uniform in (Y,t)UY×I bounds on local mass, energy, and entropy, see [30,33], one can prove that

    0<Λ1IA(f)(X,Y,t)ΛI,|B(f)(X,Y,t)|Λ,

    for some constant Λ1 and for (X,Y,t)UX×UY×I, i.e., under these assumptions the operator Q in (1.2) and in the Landau equation becomes locally uniformly elliptic. As a consequence, and as global well posedness for the Boltzmann equation and the construction of solutions in the large is an outstanding open problem, the study of conditional regularity for the Boltzmann and Landau equations has become a way to make progress on the regularity issues for these equations. We refer to [11,13,14,15,28,33,36,37] for more on the connections between Kolmogorov-Fokker-Planck equations, the Boltzmann and Landau equation, statistical physics and conditional regularity.

    Based on the idea of conditional regularity one is lead to study the local regularity of weak solutions to the equation in (1.1) with

    Q(f,Xf,X,Y,t)=X(A(X,Y,t)Xf)+B(X,Y,t)Xf, (1.3)

    assuming that A is measurable, bounded and uniformly elliptic, and that B is bounded. In [20], see also [21,22,23] for subsequent developments, the authors extended, for equations as in (1.1) assuming (1.3), the De Giorgi-Nash-Moser (DGNM) theory, which in its original form only considers elliptic or parabolic equations in divergence form, to hypoelliptic equations with rough coefficients including the one in (1.1) assuming (1.3). [20] has spurred considerable activity in the field, see below for a literature review, as the results proved give the correct scale- and translation-invariant estimates for local Hölder continuity and the Harnack inequality for weak solutions.

    In this paper we consider equations as in (1.1) with

    Q(f,Xf,X,Y,t)=X(A(Xf,X,Y,t)), (1.4)

    subject to conditions on A which allow A to be a nonlinear function of Xf. In this case we refer to the equations in (1.1) as nonlinear Kolmogorov-Fokker-Planck type equations with rough coefficients. Our contributions is twofold. First, we establish higher integrability (Theorem 1.1) and local boundedness (Theorem 1.2) of weak sub-solutions, weak Harnack and Harnack inequalities (Theorem 1.3), and Hölder continuity with quantitative estimates (Theorem 1.4), for the equation

    (t+XY)u=X(A(Xu,X,Y,t)). (1.5)

    Second, we establish existence and uniqueness, in certain bounded X, Y and t dependent domains, for a Dirichlet problem involving the equation in (1.5) also allowing for boundary data and a right hand side (Theorem 1.5). In the linear case, if A(X,Y,t) is a uniformly elliptic positive definite matrix with bounded measurable coefficients, then A(ξ,X,Y,t)=A(X,Y,t)ξ satisfies the hypothesis we impose on the symbol A, and in this case the equation in (1.5) reduces to the equation

    (t+XY)u=X(A(X,Y,t)Xu). (1.6)

    Concerning regularity, our results therefore generalize [20,22,23], to nonlinear Kolmogorov-Fokker-Planck type equations with rough coefficients.

    To the best of our knowledge, nonlinear equations of the form in (1.5) have so far not been investigated in the literature, and the purpose of this paper is to contribute to the regularity and existence theory for these equations. We believe that generalizations of the De Giorgi-Nash-Moser (DGNM) theory to nonlinear Kolmogorov-Fokker-Planck type equations with rough coefficients are relevant and interesting. We also believe that our treatment of the Dirichlet problem is new and enlightening.

    We consider equations as in (1.5) subject to conditions on A. Concerning the symbol A our baseline assumption is that A belongs to the class M(Λ), where Λ[1,) is a constant. In our treatment of the Dirichlet problem we will need to impose stronger conditions on A and we will assume that A belongs to the class R(Λ). In the following denotes the standard Euclidean scalar product in Rm.

    Definition 1. Let Λ[1,). Then A is said to belong to the class M(Λ) if A=A(ξ,X,Y,t):Rm×Rm×Rm×RRm is continuous with respect to ξ, measurable with respect to X,Y and t, and

    (i)|A(ξ,X,Y,t)|Λ|ξ|,(ii)A(ξ,X,Y,t)ξΛ1|ξ|2,(iii)A(λξ,X,Y,t)=λA(ξ,X,Y,t)λR{0}, (1.7)

    for almost every (X,Y,t)RN+1 and for all ξRm.

    Definition 2. Let Λ[1,). Then A is said to belong to the class R(Λ) if A=A(ξ,X,Y,t):Rm×Rm×Rm×RRm is continuous with respect to ξ, measurable with respect to X,Y and t, and

    (i)|A(ξ1,X,Y,t)A(ξ2,X,Y,t)|Λ|ξ1ξ2|,(ii)(A(ξ1,X,Y,t)A(ξ2,X,Y,t))(ξ1ξ2)Λ1|ξ1ξ2|2,(iii)A(λξ,X,Y,t)=λA(ξ,X,Y,t)λR{0}, (1.8)

    for almost every (X,Y,t)RN+1 and for all ξ1,ξ2,ξRm.

    Remark 1.1. Note that (1.8)-(iii) implies that A(0,X,Y,t)=0 for a.e. (X,Y,t)RN+1. Hence we deduce from (1.8)-(i),(ii) and (iii) that R(Λ)M(Λ).

    We will often use the notation (Z,t)=(X,Y,t)RN+1 to denote points. The natural family of dilations for our operators and equations, (δr)r>0, on RN+1, is defined by

    δr(X,Y,t)=(rX,r3Y,r2t), (1.9)

    for (X,Y,t)RN+1, r>0. Our classes of operators are closed under the group law

    (˜Z,˜t)(Z,t)=(˜X,˜Y,˜t)(X,Y,t)=(˜X+X,˜Y+Y+t˜X,˜t+t), (1.10)

    where (Z,t), (˜Z,˜t)RN+1. Note that

    (Z,t)1=(X,Y,t)1=(X,Y+tX,t), (1.11)

    and hence

    (˜Z,˜t)1(Z,t)=(˜X,˜Y,˜t)1(X,Y,t)=(X˜X,Y˜Y(t˜t)˜X,t˜t), (1.12)

    whenever (Z,t), (˜Z,˜t)RN+1. Given (Z,t)=(X,Y,t)RN+1 we let

    (Z,t)=(X,Y,t):=|(X,Y)|+|t|12,|(X,Y)|=|X|+|Y|1/3. (1.13)

    Given r>0 and (˜Z,˜t)=(˜X,˜Y,˜t)RN+1, we let

    Qr:={(X,Y,t):|X|<r,|Y|<r3,r2<t<0},Qr(˜Z,˜t):=(˜Z,˜t)Qr. (1.14)

    We refer to Qr(˜Z,˜t) as a cylinder centered at (˜Z,˜t) and of radius r.

    We here state the regularity part of our results, Theorem 1.1–Theorem 1.4. These theorem are derived under the assumption that the symbol A belongs to the class M(Λ) introduced in Definition 1. For the notions of weak sub-solutions, super-solutions and solutions, we refer to Definition 3 below. For the definitions of function spaces used we refer to the bulk of the paper.

    Theorem 1.1 (Higher integrability). Let (Z0,t0)=(X0,Y0,t0)RN+1,0<r1<r01 and let u be a non-negative weak sub-solution to (1.5) in an open set of RN+1 containing Qr0(Z0,t0) in the sense of Definition 3 below. Then for any q[2,2+1/m) and s[0,1/3), we have*

    *Ws,1Y denotes the fractional Sobolev space.

    uLq(Qr1(Z0,t0))c1(2+1mq)1uL2(Qr0(Z0,t0)), (1.15)
    uL1t,XWs,1Y(Qr1(Z0,t0))c2(13s)1uL2(Qr0(Z0,t0)). (1.16)

    Here

    c1=(1+1r0r1)c,c2=r1+2m0(1+1r0r1)c,

    where

    c=c(m,Λ)(1+1(r0r1)2+|X0|+r0(r0r1)r21+1(r0r1)r1),

    for some constant c(m,Λ)1.

    Theorem 1.2 (Local boundedness). Let (Z0,t0)=(X0,Y0,t0)RN+1,0<r<r01 and let u be a non-negative weak sub-solution to (1.5) in an open set of RN+1 containing Qr0(Z0,t0) in the sense of Definition 3 below. Then for any p>0, there exists a constant c=c(m,Λ)1 and θ=θ(m)>1 such that

    supQr(Z0,t0)uc(1+|X0|r2(r0r)3)θpuLp(Qr0(Z0,t0)). (1.17)

    Theorem 1.3 (Harnack inequalities). Let u be a non-negative weak super-solution to (1.5) in an open set of RN+1 containing Q1 in the sense of Definition 3 below. Then there exists ζ>0 and c1, both depending only on m and Λ such that

    ( (1.18)

    where r_0 = {1}/{20} and \tilde Q_{{r_0}/{2}}^{-}: = Q_{{r_0}/{2}}(0, 0, -{19}r_0^2/8) . Furthermore, if u is a non-negative weak solution to (1.5) in an open set of \mathbb R^{N+1} containing Q_1 , then

    \begin{equation} \sup\limits_{\tilde Q_{{r_0}/{4}} ^{-}} u \leq c \inf\limits_{Q_{{r_0}/{4}}} u, \end{equation} (1.19)

    where \tilde Q_{{r_0}/{4}}^{-}: = Q_{{r_0}/{4}}(0, 0, -{19}r_0^2/8) .

    Theorem 1.4 (Hölder continuity). Let u be a weak solution to (1.5) in an open set of \mathbb R^{N+1} containing Q_2 in the sense of Definition 3 below. Then there exists \alpha\in(0, 1) and c\geq 1 , both depending only on m and \Lambda such that

    \begin{equation} \frac{|u(X_1, Y_1, t_1))-u(X_2, Y_2, t_2)|}{\|(X_2, Y_2, t_2)^{-1}\circ(X_1, Y_1, t_1)\|^\alpha}\leq c\|u\|_{L^2(Q_2)}, \end{equation} (1.20)

    whenever (X_1, Y_1, t_1), (X_2, Y_2, t_2)\in Q_1, \, (X_1, Y_1, t_1)\neq (X_2, Y_2, t_2) .

    We here state the existence and uniqueness part of our results, Theorem 1.5. Throughout the paper we let U_X\subset\mathbb R^m be a bounded Lipschitz domain and let V_{Y, t}\subset \mathbb R^{m}\times \mathbb R be a bounded domain with boundary which is C^{1, 1} -smooth, i.e., C^{1} with respect to Y as well as t . Let N_{Y, t} denote the outer unit normal to V_{Y, t} . We establish existence and uniqueness of weak solutions to a formulation of the Dirichlet problem

    \begin{equation} \begin{cases} \nabla_X\cdot(A(\nabla_X u, X, Y, t))-(\partial_t+X\cdot\nabla_Y)u = g^* &\text{in} \ U_X\times V_{Y, t}, \\ u = g & \text{on} \ \partial_{\mathcal K}(U_X\times V_{Y, t}). \end{cases} \end{equation} (1.21)

    Here

    \begin{eqnarray} \quad\partial_{{\mathcal K}}(U_X\times V_{Y, t}): = (\partial U_X\times V_{Y, t})\cup\{(X, Y, t)\in \overline{U_X}\times \partial V_{Y, t}\mid (X, 1)\cdot N_{Y, t} < 0\}. \end{eqnarray} (1.22)

    \partial_{\mathcal K}(U_X\times V_{Y, t}) will be referred to as the Kolmogorov boundary of U_X\times V_{Y, t} , and the Kolmogorov boundary serves, in our context, as the natural substitute for the parabolic boundary used in the context of the Cauchy-Dirichlet problem for uniformly elliptic parabolic equations. In particular, we study weak solutions in the sense of Definition 4. For the definition of the functional setting we refer to Section 2. We believe that the following result is of independent interest in particularly as we allow the symbol A to depend nonlinearly on \nabla_Xu .

    Theorem 1.5 (Existence and uniqueness). Let (g, g^*)\in W(U_X\times V_{Y, t})\times L_{Y, t}^2(V_{Y, t}, {H}_X^{-1}(U_X)) and assume that A belongs to the class R(\Lambda) introduced in Definition 2. Then there exists a unique weak solution u to the problem in (1.21) in the sense of Definition 4 below. Furthermore, there exists a constant c , depending only on m , \Lambda and U_X\times V_{Y, t} , such that

    \begin{equation} \begin{split} ||u||_{W(U_X\times V_{Y, t})}&\leq c\bigl (||g||_{W(U_X\times V_{Y, t})}+||g^*||_{L_{Y, t}^2(V_{Y, t}, {H}_X^{-1}(U_X))}\bigr ). \end{split} \end{equation} (1.23)

    As mentioned, the equation in (1.6), possibly also allowing for lower order terms, has attracted considerable attention in recent years. Anceschi-Cinti-Pascucci-Polidoro-Ragusa [2,12,34] proved local boundedness of weak sub-solutions of (1.6) and some versions thereof. Their approach is based on the Moser's iteration technique, the use of fundamental solutions and a Sobolev type inequality is crucial. It is worth noting that while the results in these papers are stated assuming only bounded and measurable coefficients, an implicit regularity assumption on the coefficients is imposed as the authors use a stronger notion of weak solutions assuming also (\partial_t+X\cdot\nabla_Y)u\in L^2_{\mathrm{loc}} . It is unclear for what assumptions on the coefficients such weak solutions can be constructed. Bramanti-Cerutti- Manfredini-Polidoro-Ragusa [8,32,35] proved L^p estimates, interior Sobolev regularity and local Hölder continuity of weak solutions of (1.6) imposing additional assumptions on the coefficients beyond bounded, measurable and elliptic. In fact it was only recently that Golse-Imbert- Mouhot-Vasseur [20] proved local boundedness, Harnack inequality and local Hölder continuity of (true) weak solutions of (1.6) based on De-Giorgi and Moser's iteration technique. Still, it seems unclear to us how the authors actually resolve questions concerning the existence of weak solutions unless smooth coefficients are assumed qualitatively. However, subsequent developments have appeared in [22,23]. A weak Harnack inequality for weak super-solutions of (1.6) has been obtained by Guerand-Imbert [22] and this has been generalized by Anceschi-Rebucci [3]. In [23], Guerand-Mouhot revisited the theory for the linear equation in (1.6), also allowing for lower order terms, and gave lucid, novel and short proofs of the De Giorgi intermediate-value Lemma, weak Harnack and Harnack inequalities, and the Hölder continuity with quantitative estimates. [23] is an essentially self-contained account of the linear theory. Local Hölder continuity results are also proved in Wang-Zhang [38,39,40] for various linear analogues of (1.6). We emphasize that all results mentioned concern linear equations. Zhu [41] proved local boundedness and local Hölder continuity of weak solutions of (1.6) when the drift term \partial_t+X\cdot\nabla_Y is replaced by \partial_t+b(X)\cdot \nabla_Y for some nonlinear function b .

    Boundary value problems for equations as in (1.6) but in non-divergence form were studied by Manfredini [31] who proved existence of strong solutions for the Dirichlet problem assuming Hölder continuous coefficients. Lanconelli-Lascialfari-Morbidelli [26,27] considered a quasilinear case, still in non-divergence form, allowing the coefficients to depend not only on (X, Y, t) but also the solution u , and as a function of (X, Y, t) the coefficients are assumed to be with Hölder continuous. In fact, functional analytic approaches to weak solutions to Kramers equation and Kolmogorov- Fokker-Planck equations have only recently been developed. Albritton-Armstrong-Mourrat- Novack [1] have developed a functional analytic approach to study well-posedness of Kramers equation, and its parabolic analogue

    \begin{equation} \partial_t u-\Delta_X u+X\cdot\nabla_X u+X\cdot\nabla_Y u+b\cdot\nabla_X u = g^*, \end{equation} (1.24)

    for suitable g^* . Equation (1.24) is often referred to as the kinetic Fokker-Planck equation. Litsgård-Nyström [29] studied existence and uniqueness results for the (linear) Dirichlet problem associated with (1.6), with rough coefficients A . In particular, in [29] Theorem 1.5 is proved in the case when A(\xi, X, Y, t) = A(X, Y, t)\xi . However, existence and uniqueness for (1.5) do not seem to have been studied in the literature so far. It is important to note that Theorem 1.5 states, similar to [29], the existence of a unique weak solution u to the problem in (1.21) in the sense of Definition 4 below. The latter is, as it assumes no knowledge of underlying traces, trace spaces and extension operators in the functional setting considered, a weaker formulation of the Dirichlet problem compared to what one usually aims for. Indeed, this is one way to formulate a weak form of the Dirichlet problem which circumvents a largely open problem in the context of kinetic Fokker-Planck equations, linear as well as non-linear, and that is the problem of a well defined trace operator and trace inequality. We refer to Section 6 for more.

    The regularity part of our results is modelled on the approach of Golse-Imbert-Mouhot-Vasseur [20] and the work of Guerand-Mouhot [23]. In fact, as can be seen from the very formulations of our regularity results, this part of our work is strongly influenced by [23] and armed with Theorem 1.1 and Theorem 1.2 we can to large extent refer to the corresponding arguments in [23] for the proofs of Theorem 1.3 and Theorem 1.4. The new difficulties in our case stem from the nonlinearity of A in \nabla_X u . However, as we learn from the regularity theory for quasi-linear parabolic PDEs, see [16] for example, a careful development of the De Giorgi-Nash-Moser theory tends to be robust enough to handle the type of non-linearities considered in this paper. The higher integrability result in Theorem 1.1 is proved by combining the energy estimate in Lemma 3.1 with a Sobolev regularity estimate and here it is important that A has linear growth in \nabla_X u . In particular, in the proof of Theorem 1.1 one is lead, after preliminaries and the use of an appropriate cut-off function, to conduct estimates for a (global) weak sub-solution u_1 to the equation

    \begin{equation} (\partial_t+X\cdot\nabla_Y)u_1\leq \nabla_X\cdot A(\nabla_X u_1, X, Y, t)+g^\ast, \ g^\ast: = -(\nabla_X\cdot F_1+F_0)\text{ in } \mathbb R^{N+1}, \end{equation} (1.25)

    where F_1, F_0 are in L^2(\mathbb R^{N+1}) and u_1, F_1, F_0 are supported in Q_{r_0}(0, 0, 0) . To close the argument, as u_1 is only a weak sub-solution, it seems important to replace it by a function which actually solves an equation. In particular, to make this operational one needs to construct a weak solution v to

    \begin{equation} (\partial_t+X\cdot\nabla_Y)v = \nabla_X\cdot A(\nabla_X v, X, Y, t)+g^\ast, \end{equation} (1.26)

    such that v bounds u_1 from above. One approach to Sobolev regularity estimates is then attempt to use an approach based on Bouchut [7] which implies a Sobolev embedding

    \begin{equation} H_{X, Y, t}^{{1}/{3}}( \mathbb R^{N+1})\to L_{X, Y, t}^q( \mathbb R^{N+1}), \quad q: = \frac{6(2m+1)}{6m+1} > 2. \end{equation} (1.27)

    To get hold of the H_{X, Y, t}^{{1}/{3}}(\mathbb R^{N+1}) norm of v one uses a result of Bouchut [7] which gives control of D_Y^{1/3}v, \ D_t^{1/3}v given energy estimates. To be able to bound u_1 from above by v as in (1.26) one seems to need Theorem 1.5 and the comparison principle that we prove in Theorem 5.1 below. As the result of Bouchut [7] requires a solution which exists globally in time one can make this approach operational using Theorem 1.5 to prove Theorem 1.1 with the cylinders in (1.14) replaced by centered cylinders. An alternative approach to Sobolev regularity estimates, which in the end gives Theorem 1.1 as stated, is to first observe that if u_1 satisfies (1.25), then one deduces that the weak formulation of (1.25) induces a positive distribution. One is therefore lead to prove estimates for v satisfying

    \begin{equation} (\partial_t+X\cdot\nabla_Y)v = \nabla_X\cdot A(\nabla_X v, X, Y, t)+g^\ast-\mu, \end{equation} (1.28)

    where \mu is now a positive measure. Due to the structure of g^\ast , Sobolev regularity estimates can then be deduced using a semi-classical approach via the fundamental solution associated to the linear equation (\partial_t+X\cdot\nabla_Y)f = \Delta_X f originally studied by Kolmogorov, see Lemma 10 in [23]. In the end, we follow this approach and here it is again important that A has linear growth in \nabla_X u . Armed with the Sobolev regularity estimates the proofs of Theorem 1.1–Theorem 1.4 can be completed along the lines of the corresponding arguments in the linear case. Finally, to prove the existence and uniqueness result in Theorem 1.5 we use a variational approach and proceed along the lines of [1,4,29]. In particular, our argument is similar to the proof of Theorem 1.1 in [29].

    In Section 2 we introduce the functional setting and the notion of weak solutions. Section 3 is devoted to a number of preliminary technical results to be used in the proofs of Theorem 1.1– Theorem 1.4. Theorem 1.1–Theorem 1.4 are proved in Section 4, and in the proof of Theorem 1.3 and Theorem 1.4 we for brevity mainly refer to the corresponding arguments in [23]. Theorem 1.5 is proved in Section 5. In Section 6 we mention a number of challenging problems for future research which we hope will inspire the community to look further into the topic of nonlinear Kolmogorov-Fokker-Planck type equations.

    We denote by {H}_X^1(U_X) the Sobolev space of functions g\in L_{}^2(U_X) whose distributional gradient in U_X lies in (L^2(U_X))^m , i.e.,

    \begin{eqnarray*} \label{fspace-} {H}_X^1(U_X): = \{g\in L_{X}^2(U_X)\mid \nabla_Xg\in (L^2(U_X))^m\}, \end{eqnarray*}

    and we set

    ||g||_{{H}_X^1(U_X)}: = \bigl (||g||_{L^2(U_X)}^2+||\, |\nabla_Xg|\, ||_{L^2(U_X)}^2\bigr )^{1/2}, \ g\in {H}_X^1(U_X).

    We let {H}_{X, 0}^1(U_X) denote the closure of C_0^\infty(U_X) in the norm of {H}_X^1(U_X) and we recall, as U_X is a bounded Lipschitz domain, that C^\infty(\overline{U_X}) is dense in {H}_X^1(U_X) . In particular, equivalently we could define {H}_X^1(U_X) as the closure of C^\infty(\overline{U_X}) in the norm ||\cdot||_{{H}_X^1(U_X)} . Note that as {H}_{X, 0}^1(U_X) is a Hilbert space it is reflexive, hence ({H}_{X, 0}^1(U_X))^\ast = H_X^{-1}(U_X) and (H_X^{-1}(U_X))^\ast = {H}_{X, 0}^1(U_X) , where ()^\ast denotes the dual. Based on this we let {H}_X^{-1}(U_X) denote the dual to {H}_{X, 0}^1(U_X) acting on functions in {H}_{X, 0}^1(U_X) through the duality pairing \langle \cdot, \cdot\rangle: = \langle \cdot, \cdot\rangle_{H_X^{-1}(U_X), H_{X, 0}^{1}(U_X)} . We let L^2_{Y, t}(V_{Y, t}, H_{X, 0}^{1}(U_X)) be the space of measurable function u:V_{Y, t}\to H_{X, 0}^{1}(U_X) equipped with the norm

    ||u||^2_{L_{Y, t}^2(V_{Y, t}, H_X^1(U_X))}: = \iint_{V_{Y, t}}||u(\cdot, Y, t)||_{{H}_X^1(U_X)}^2\, {{\text{d}}} Y{{\text{d}}} t.

    L^2_{Y, t}(V_{Y, t}, H_{X}^{-1}(U_X)) is defined analogously. In analogy with the definition of {H}_X^1(U_X) , we let W(U_X\times V_{Y, t}) be the closure of C^\infty(\overline{U_X\times V_{Y, t}}) in the norm

    \begin{align} ||u||_{W(U_X\times V_{Y, t})}&: = \bigl (||u||_{L_{Y, t}^2(V_{Y, t}, H_X^1(U_X))}^2+||(\partial_t+X\cdot\nabla_Y)u||_{L_{Y, t}^2(V_{Y, t}, {H}_X^{-1}(U_X))}^2\bigr )^{1/2}. \end{align} (2.1)

    In particular, W(U_X\times V_{Y, t}) is a Banach space and u\in W(U_X\times V_{Y, t}) if and only if

    \begin{eqnarray} u\in L_{Y, t}^2(V_{Y, t}, H_X^1(U_X))\quad\mbox{and}\quad (\partial_t+X\cdot\nabla_Y)u\in L_{Y, t}^2(V_{Y, t}, H_X^{-1}(U_X)). \end{eqnarray} (2.2)

    Note that the dual of L_{Y, t}^2(V_{Y, t}, H_{X, 0}^1(U_X)) , denoted by (L_{Y, t}^2(V_{Y, t}, H_{X, 0}^1(U_X)))^\ast , satisfies

    (L_{Y, t}^2(V_{Y, t}, H_{X, 0}^1(U_X)))^\ast = L_{Y, t}^2(V_{Y, t}, H_X^{-1}(U_X)),

    and, as mentioned above,

    (L_{Y, t}^2(V_{Y, t}, H_X^{-1}(U_X)))^\ast = L_{Y, t}^2(V_{Y, t}, H_{X, 0}^1(U_X)).

    Finally, the spaces L_{Y, t, \mathrm{loc}}^2(V_{Y, t}, H_{X, \mathrm{loc}}^1(U_X)) , L_{Y, t, \mathrm{loc}}^2(V_{Y, t}, {H}_{X, \mathrm{loc}}^{-1}(U_X)) , and W_{\mathrm{loc}}(U_X\times V_{Y, t}) are defined in the natural way. The topological boundary of U_X\times V_{Y, t} is denoted by \partial(U_X\times V_{Y, t}) . Let N_{Y, t} denote the outer unit normal to V_{Y, t} . We define a subset \partial_{{\mathcal K}}(U_X\times V_{Y, t})\subset\partial(U_X\times V_{Y, t}) , the Kolmogorov boundary of U_X\times V_{Y, t} , as in (1.22). We let C^\infty_{{\mathcal K}, 0}(\overline{U_X\times V_{Y, t}}) and C^\infty_{X, 0}(\overline{U_X\times V_{Y, t}}) be the set of functions in C^\infty(\overline{U_X\times V_{Y, t}}) which vanish on \partial_{{\mathcal K}}(U_X\times V_{Y, t}) and \{(X, Y, t)\in \partial{U_X}\times \overline{V_{Y, t}}\} , respectively. We let W_0(U_X\times V_{Y, t}) and W_{X, 0}(U_X\times V_{Y, t}) denote the closure in the norm of W(U_X\times V_{Y, t}) of C^\infty_{{\mathcal K}, 0}(\overline{U_X\times V_{Y, t}}) and C^\infty_{X, 0}(\overline{U_X\times V_{Y, t}}) , respectively.

    We here introduce the notion of weak solutions.

    Definition 3. Let g^*\in L_{Y, t}^2(V_{Y, t}, {H}_X^{-1}(U_X)) . A function u\in W_{\mathrm{loc}}(U_X\times V_{Y, t}) is said to be a weak sub-solution (or super-solution) to the equation

    \begin{align} (\partial_t+X\cdot\nabla_Y)u-\nabla_X\cdot(A(\nabla_X u, X, Y, t))+g^* = 0 \text{ in } \ U_X\times V_{Y, t}, \end{align} (2.3)

    if for every V_X\times V_Y\times J\Subset U_X\times V_{Y, t} , and for all non-negative \phi\in L_{Y, t}^2(V_Y\times J, H_{X, 0}^1(V_X)) , we have

    \begin{align} &\iiint_{V_X\times V_Y\times J}A(\nabla_X u, X, Y, t)\cdot\nabla_X\phi\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ &+\iint_{V_Y\times J}\ \langle g^\ast(\cdot, Y, t)+ (\partial_t+X\cdot\nabla_Y)u(\cdot, Y, t), \phi(\cdot, Y, t)\rangle\, {{\text{d}}} Y {{\text{d}}} t\leq 0\quad (\text{ or }\geq ). \end{align} (2.4)

    We say that u\in W_{\mathrm{loc}}(U_X\times V_{Y, t}) is a weak solution to the Eq (2.3) if equality holds in (2.4) without a sign restriction on \phi .

    Note that if u is a weak sub-solution (or super-solution) of (2.3) in the sense of Definition 3 above, with g^\ast\equiv 0 , then

    \begin{align} &\iint_{V_X\times V_Y}u(X, Y, t_2)\phi(X, Y, t_2)\, {{\text{d}}} X {{\text{d}}} Y-\iint_{V_X\times V_Y}u(X, Y, t_1)\phi(X, Y, t_1)\, {{\text{d}}} X {{\text{d}}} Y\\ &-\int_{t_1}^{t_2}\iint u(\partial_t+X\cdot\nabla_Y)\phi\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t+\int_{t_1}^{t_2}\iint A(\nabla_X u, X, Y, t)\cdot\nabla_X\phi\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ &\leq 0\quad (\text{ or }\geq ), \end{align} (2.5)

    whenever \phi\in C^\infty((t_1, t_2), C^\infty_0(V_X\times V_Y)) , is non-negative function. Furthermore, equality holds in (2.5) for every weak solution u of (2.3) without a sign restriction on \phi .

    Remark 2.1. Assume g^\ast\equiv 0 . (i) From Definition 3, it is clear that, if u is a weak sub-solution (resp. super-solution or solution) of (2.3) in U_X\times V_{Y, t} , then for any k\in \mathbb R , the function v = (u-k) is also weak sub-solution (resp. super-solution or solution) of (2.3) in U_X\times V_{Y, t} .

    (ii) Using the homogeneity property (iii) of A , it follows that, (a) for any c\geq 0 , cu is a weak sub-solution (resp. super-solution or solution) of (2.3) in U_X\times V_{Y, t} , provided u is a weak sub-solution (resp. super-solution or solution) of (2.3) in U_X\times V_{Y, t} and (b) u is a weak solution of (2.3) in U_X\times V_{Y, t} if and only if -u is a weak solution of (2.3) in U_X\times V_{Y, t} .

    Theorem 1.5 is a statement concerning existence and uniqueness of weak solutions to a formulation of the Dirichlet problem in (1.21). In particular, we study weak solutions in the following sense.

    Definition 4. Consider (g, g^\ast)\in W(U_X\times V_{Y, t})\times L_{Y, t}^2(V_{Y, t}, {H}_X^{-1}(U_X)) . Given (g, g^\ast) , u is said to be a weak solution to the problem in (1.21) if

    \begin{eqnarray} u\in W(U_X\times V_{Y, t}), \quad (u-g)\in W_0(U_X\times V_{Y, t}), \end{eqnarray} (2.6)

    and if

    \begin{equation} \begin{split} &\iiint_{U_X\times V_{Y, t}}\ A(\nabla_Xu, X, Y, t)\cdot \nabla_X\phi\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ &+\iint_{V_{Y, t}}\ \langle g^*(\cdot, Y, t)+(\partial_t+X\cdot\nabla_Y)u(\cdot, Y, t), \phi(\cdot, Y, t)\rangle\, {{\text{d}}} Y {{\text{d}}} t = 0, \end{split} \end{equation} (2.7)

    for all \phi\in L_{Y, t}^2(V_{Y, t}, H_{X, 0}^1(U_X)) and where \langle \cdot, \cdot\rangle = \langle \cdot, \cdot\rangle_{H_X^{-1}(U_X), H_{X, 0}^{1}(U_X)} is the duality pairing in H_X^{-1}(U_X) . If in (2.7), = is replaced by \leq (\geq) whenever \phi\geq 0 , then u is said to be a weak sub- (super-) solution of (1.21) respectively.

    In this section we prove a number of technical results to be used in the proof of Theorem 1.1–Theorem 1.4. Throughout the rest of the paper, we use the notation s^{+}: = \max\{s, 0\} for s\in \mathbb R . Moreover, from Sections 3 and 4, we assume that the symbol A belongs to the class M(\Lambda) introduced in Definition 1.

    Lemma 3.1. Let Z_0 = (X_0, Y_0, t_0)\in \mathbb R^{N+1} , 0 < r_1 < r_0 , be such that Q_{r_0}(Z_0, t_0)\Subset U_X\times V_{Y, t} . Let u be a weak sub-solution of the Eq (1.5) in U_X\times V_{Y, t} in the sense of Definition 3. Then

    \begin{align} &\sup\limits_{t_0-r_{1}^2 < t < t_0}\iint_{Q^t(Z_0, r_1)} u^2(X, Y, t)\, {{\text{d}}} X {{\text{d}}} Y+\Lambda^{-1}\iiint_{Q_{r_1}(Z_0, r_1)}|\nabla_X u|^2\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ &\leq c c_{0, 1}\iiint_{Q_{r_0}(Z_0, t_0)}u(X, Y, t)^2\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t, \end{align} (3.1)

    where Q^t(Z_0, r): = \{(X, Y):(X, Y, t)\in Q_{r}(Z_0, t_0)\} for r > 0 , c = c(m, \Lambda)\geq 1 and

    c_{0, 1}: = \frac{1}{(r_0-r_1)^2}+\frac{r_0+|X_0|}{(r_0 -r_1)r_1^{2}}+\frac{1}{(r_0 -r_1)r_1}+1.

    Proof. Let t_1: = t_0-r_0^{2} and t_2: = t_0 . Considering l_1 , l_2 , such that t_1 < l_1 < l_2 < t_2 , we introduce for \epsilon > 0 the function \theta_{\epsilon}\in W^{1, \infty}((t_1, t_2)) by

    \begin{equation} \theta_{\epsilon}(t): = \begin{cases} 0\text{ if }t_1\leq t\leq l_1-\epsilon, \\ 1+\frac{t-l_1}{\epsilon}, \text{ if }l_1-\epsilon < t\leq l_1, \\ 1\text{ if }l_1 < t\leq l_2, \\ 1-\frac{t-l_2}{\epsilon}\text{ if }l_2\leq t\leq l_2+\epsilon, \\ 0\text{ if }l_2+\epsilon < t\leq t_2. \end{cases} \end{equation} (3.2)

    Let \psi\in[0, 1] be smooth in Q_{r_0}(Z_0, t_0) such that \psi\equiv 1 on Q_{r_1}(Z_0, t_0) and \psi\equiv 0 outside Q_{r_0}(Z_0, t_0) satisfying

    |\nabla_X\psi|\leq\frac{c}{r_0- r_1}, \quad |\nabla_Y\psi|\leq\frac{c}{(r_0-r_{1})r_{1}^2}, \quad |\partial_t\psi|\leq\frac{c}{(r_0-r_{1})r_{1}},

    for some constant c = c(m)\geq 1 .

    Consider the function \phi(X, Y, t) = 2u(X, Y, t)\psi^2(X, Y, t)\theta_{\epsilon}(t) . We intend to test (2.5) with \phi and the following deductions are formal. However, as u is a weak sub-solution of the Eq (1.5) in U_X\times V_{Y, t} in the sense of Definition 3, we know that u\in W_{\mathrm{loc}}(U_X\times V_{Y, t}) and as W(U_X\times V_{Y, t}) is defined as the closure of C^\infty(\overline{{U_X\times V_{Y, t}}}) in the norm introduced in (2.1) our deduction can be made rigorous a posteriori. Testing (2.5) with \phi(X, Y, t) , letting \epsilon\to 0 , and then adding

    \iiint u^{2}\partial_t (\psi^2)\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t

    on both sides of the resulting inequality, we deduce that

    \begin{align} &I(l_2)-I(l_1)+ 2\iiint A(\nabla_X u, X, Y, t)\cdot\nabla_X(u\psi^2)\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ &\leq \iiint u^{2}(\partial_t+X\cdot\nabla_Y) \psi^2\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t, \end{align} (3.3)

    where

    \begin{align*} I(t): = \iint\psi^2(X, Y, t)u^2(X, Y, t)\, {{\text{d}}} X {{\text{d}}} Y. \end{align*}

    Using (1.7), (3.3) yields

    \begin{align} &I(l_2)-I(l_1)+2\iiint\psi^2 A(\nabla_X u, X, Y, t)\cdot\nabla_X u\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ &\leq \iiint u^{2}(\partial_t+X\cdot\nabla_Y) \psi^2\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ &-4\iiint u\psi A(\nabla_X u, X, Y, t))\cdot\nabla_X\psi\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ &\leq\iiint u^2\{(\partial_t+X\cdot\nabla_Y)\psi^2+4\Lambda^{3}(\psi+|\nabla_X\psi|)^2\}\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ &+\iiint\Lambda^{-1}\psi^2|\nabla_X u|^2\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t. \end{align} (3.4)

    Furthermore, using (1.7)- (i), (ii) we can continue the above estimate and conclude that

    \begin{align} &I(l_2)-I(l_1)+\Lambda^{-1}\iiint_{Q^t(Z_0, r_0)\times (l_1, l_2)}\psi^2 |\nabla_X u|^2\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ &\leq\iiint u^2\{(\partial_t+X\cdot\nabla_Y)\psi^2+4\Lambda^{3}(\psi+|\nabla_X\psi|)^2\}\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t. \end{align} (3.5)

    Using the properties of \psi and first letting l_1\to t_1 , and then letting l_2\to t_2 in (3.5), we obtain

    \begin{align} &\Lambda^{-1}\iiint_{Q_{r_1}(Z_0, t_0)}|\nabla_X u|^2\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ &\leq\iiint_{Q_{r_0}(Z_0, t_0)} u^2\{(\partial_t+X\cdot\nabla_Y)\psi^2+4\Lambda^{3}(\psi+|\nabla_X\psi|)^2\}\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ &\leq c c_{0, 1}\iiint_{Q_{r_0}(Z_0, t_0)}u(X, Y, t)^2\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t, \end{align} (3.6)

    where c = c(m, \Lambda)\geq 1 and

    c_{0, 1}: = \frac{1}{(r_0-r_1)^2}+\frac{r_0+|X_0|}{(r_0 -r_1)r_1^{2}}+\frac{1}{(r_0 -r_1)r_1}+1.

    Again using the properties of \psi and first letting l_1\to t_1 in (3.5), then taking supremum over l_2\in[t_0-r_{1}^2, t_0) and noting that for such l_2 , \psi\equiv 1 , we also have

    \begin{align} &\sup\limits_{t_0-r_{1}^2 < t < t_0}\iint_{Q^t(Z_0, r_0)} u^2(X, Y, t)\, {{\text{d}}} X {{\text{d}}} Y \end{align} (3.7)
    \begin{align} &\leq\iiint_{Q_{r_0}(Z_0, r_0)} u^2\{(\partial_t+X\cdot\nabla_Y)\psi^2+4\Lambda^{3}(\psi+|\nabla_X\psi|)^2\}\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ &\leq c c_{0, 1}\iiint_{Q_{r_0}(Z_0, t_0)}u(X, Y, t)^2\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t. \end{align} (3.8)

    This completes the proof.

    Lemma 3.2. Let u be a weak sub-solution of the Eq (1.5) in U_X\times V_{Y, t} in the sense of Definition 3. Let k\in \mathbb R . Then (u-k)^+ is also a weak sub-solution of the Eq (1.5) in U_X\times V_{Y, t} in the sense of Definition 3.

    Proof. By Remark 2.1, it is enough to prove that u^+ is a weak sub-solution of (1.5). Let \epsilon > 0 and \phi\in L_{Y, t}^2(V_Y\times J, H_{X, 0}^1(V_X)) be a non-negative test function in (2.4). Then \frac{u^+}{(u^+ +\epsilon)}\phi\in L_{Y, t}^2(V_Y\times J, H_{X, 0}^1(V_X)) is also a non-negative test function in (2.4). Using \frac{u^+}{(u^+ +\epsilon)}\phi as a test function in (2.4), we obtain

    \begin{align} &\iiint_{V_X\times V_Y\times J}A(\nabla_X u, X, Y, t)\cdot\nabla_X\phi \frac{u^+}{(u^+ +\epsilon)}\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ &+\epsilon\iiint_{V_X\times V_Y\times J}A(\nabla_X u, X, Y, t)\cdot\frac {\nabla_X u^+}{(u^+ +\epsilon)^2}\phi\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ &+\iint_{V_Y\times J}\ \langle (\partial_t+X\cdot\nabla_Y)u(\cdot, Y, t), \frac{u^+}{(u^+ +\epsilon)}\phi(\cdot, Y, t)\rangle\, {{\text{d}}} Y {{\text{d}}} t\leq 0. \end{align} (3.9)

    Letting \epsilon\to 0 , we obtain

    \begin{align} &\iint_{V_Y\times J}\langle(\partial_t+X\cdot\nabla_Y)u^+(\cdot, Y, t), \phi(\cdot, Y, t)\rangle\, {{\text{d}}} Y {{\text{d}}} t\\ &+\iiint_{V_X\times V_Y\times J}A(\nabla_X u^+, X, Y, t)\cdot\nabla_X\phi\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ &+\lim\inf\limits_{\epsilon\to 0}\epsilon\iiint_{V_X\times V_Y\times J}\frac{A(\nabla_X u^+, X, Y, t)\cdot\nabla_X u^+}{(u^+ +\epsilon)^2}\phi\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\leq 0. \end{align} (3.10)

    However, by (1.7)- (ii)

    \begin{align} \iiint_{V_X\times V_Y\times J}\frac{A(\nabla_X u^+, X, Y, t)\cdot\nabla_X u^+}{(u^+ +\epsilon)^2}\phi\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\geq 0. \end{align} (3.11)

    Hence,

    \begin{align} &\iint_{V_Y\times J}\langle(\partial_t+X\cdot\nabla_Y)u^+(\cdot, Y, t), \phi(\cdot, Y, t)\rangle\, {{\text{d}}} Y {{\text{d}}} t\\ &+\iiint_{V_X\times V_Y\times J}A(\nabla_X u^+, X, Y, t)\cdot\nabla_X\phi\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\leq 0. \end{align} (3.12)

    This proves that u^+ is a weak sub-solution.

    The following result follows from [23,Lemma 10].

    Lemma 3.3. Let f\geq 0 be locally integrable such that

    \begin{equation} (\partial_t+X\cdot\nabla_Y-\Delta_X)f = \nabla_X\cdot F_1+F_2-\mu, \end{equation} (3.13)

    where F_1, F_2\in L^1\cap L^2(\mathbb R^{2m}\times \mathbb R_-) and \mu\in M^1(\mathbb R^{2m}\times \mathbb R_-) is a non-negative measure with finite mass in \mathbb R^{2m}\times \mathbb R_- such that F_1, F_2 and \mu have compact support, in the time variable, included in (-\tau, 0] . Then for any p\in[2, 2+{1}/{m}) and \sigma\in[0, {1}/{3}) we have

    \begin{equation} \|f\|_{L^p( \mathbb R^{2m}\times \mathbb R_-)}\leq c\Big(2+\frac{1}{m}-p\Big)^{-1}(\|F_1\|_{L^2( \mathbb R^{2m}\times \mathbb R_-)}+\|F_2\|_{L^2( \mathbb R^{2m}\times \mathbb R_-)}) \end{equation} (3.14)

    and

    \begin{align} \|f\|_{L_{t, X}^1 W_Y^{\sigma, 1}( \mathbb R^{2m}\times \mathbb R_-)}\leq& c\Big(\frac{1}{3}-\sigma\Big)^{-1}(\|F_1\|_{L^1( \mathbb R^{2m}\times \mathbb R_-)}+\|F_2\|_{L^1( \mathbb R^{2m}\times \mathbb R_-)})\\ &+c\Big(\frac{1}{3}-\sigma\Big)^{-1}\|\mu\|_{M^1( \mathbb R^{2m}\times \mathbb R_-)}, \end{align} (3.15)

    for some constant c = c(\tau) .

    The lemmas stated so far will be sufficient for our proof of Theorem 1.1 and Theorem 1.2.

    Lemma 3.4 (Weak Poincaré inequality). Let \epsilon\in(0, 1) and \sigma\in(0, \frac{1}{3}) . Then every non-negative weak sub-solution u of (1.5) in Q_5 in the sense of Definition 3 satisfies

    \begin{equation} \Big\|(u-u_{Q_1^{-}})^+\Big\|_{L^1(Q_1^{+})}\leq c\Big(\frac{1}{\epsilon^{m+2}}\|\nabla_X u\|_{L^1(Q_5)}+\epsilon^{\sigma}\big(\frac{1}{3}-\sigma\big)^{-1}\|u\|_{L^2(Q_5)}\Big), \end{equation} (3.16)

    for some constant c = c(m, \Lambda)\geq 1 , where Q_1^{-}: = Q_1(0, 0, -1) and u_{Q_1^{-}}: = \frac{1}{|Q_1^{-}|}\int_{Q_1^{-}}u .

    Proof. Using that u is a non-negative weak sub-solution of (1.5), Theorem 1.1 and the property 1.7- (i) , the conclusion of the lemma follows from the lines of the proof of [23,Proposition 13,pages 8-10].

    Lemma 3.5 (Intermediate value lemma). Let \delta_1, \delta_2\in(0, 1) be given. Then there exists constants \theta = c(m, \Lambda)(\delta_1\delta_2)^{10m+15} , r_0 = \frac{1}{20} , and \nu\geq c(m, \Lambda)(\delta_1\delta_2)^{5m+8} , such that the following holds. Let u:Q_1\to \mathbb R be a weak sub-solution of (1.5) in Q_5 in the sense of Definition 3, assume that u\leq 1 in Q_\frac{1}{2} , and that

    \begin{equation} |\{u\leq 0\}\cap Q_{r_0}^{-}|\geq\delta_1|Q_{r_0}^{-}|\quad{{and}}\quad|\{u\geq 1-\theta\}\cap Q_{r_0}|\geq\delta_2|Q_{r_0}|, \end{equation} (3.17)

    where Q_{r_0}^{-}: = Q_{r_0}(0, 0, -2r_0^{2}) . Then

    \begin{equation} \Big|\{0 < u < 1-\theta\}\cap Q_\frac{1}{2}\Big|\geq\nu|Q_\frac{1}{2}|. \end{equation} (3.18)

    Proof. Using Lemma 3.1, Lemma 3.2 and Lemma 3.4, the result follows from the lines of the proof of [23,Theorem 3,pages 11-12].

    Lemma 3.6 (Measure to pointwise upper bound). Given \delta\in(0, 1) and r_0 = \frac{1}{20} , there exists a positive constant \gamma: = \gamma(\delta) = c(m, \Lambda)\delta^{2(1+\delta^{-10m-16})} > 0 such that the following holds. Let u be a weak sub-solution of (1.5) in Q_1 in the sense of Definition 3, assume that u\leq 1 in Q_\frac{1}{2} and that

    \begin{equation} |\{u\leq 0\}\cap Q_{r_0}^{-}|\geq\delta|Q_{r_0}^{-}|, \end{equation} (3.19)

    where Q_{r_0}^{-}: = Q_{r_0}(0, 0, -2r_0^{2}) . Then

    {{ u\leq 1-\gamma \;in \;Q_\frac{r_0}{2} }}.

    Proof. Using Remark 2.1, Theorem 1.2 and Lemma 3.5, the result follows by proceeding along the lines of the proof of [23,Lemma 16,page 12].

    In this section we prove Theorem 1.1–Theorem 1.4. We first note that since our class of operators is closed under the group law defined in (1.10), and by our definition of Q_{r_0}(Z_0, t_0) , we can throughout the second without loss of generality assume that (Z_0, t_0) = 0 . Note that Q_{r_0} = Q_{r_0}(0, 0) = V_X\times V_Y\times J where V_X = B(0, r_0) , V_Y = B(0, r_0^3) , J = (-r_0^2, 0) , and where B(0, \rho) denotes the standard Euclidean ball with center at 0 and radius \rho in \mathbb R^m .

    As discussed in subsection 1.7, since u is a weak sub-solution of (1.5), there exists a non-negative measure \bar{\mu} such that

    (\partial_t+X\cdot\nabla_Y)u = \nabla_X\cdot(A(\nabla_X u, X, Y, t))-\bar{{{\mu}}}.

    We define r_2: = \frac{r_0+r_1}{2} . Let \phi_1\in [0, 1] be smooth such that \phi_1\equiv 1 in Q_{r_1}(Z_0, t_0) and \phi_1\equiv 0 outside Q_{r_2}(Z_0, t_0) satisfying

    \begin{equation} |\nabla_X\phi_1|\leq\frac{c}{r_0- r_2}, \quad |\nabla_Y\phi_1|\leq\frac{c}{(r_0-r_{2})r_{2}^2}, \quad |\partial_t\phi_1|\leq\frac{c}{(r_0-r_{2})r_{2}}, \end{equation} (4.1)

    for some constant c = c(m)\geq 1 . Then we observe that v = u\phi_1 is a weak solution of

    \begin{equation} (\partial_t+X\cdot\nabla_Y-\Delta_X)v = \nabla_X\cdot F_1+F_2-\mu\quad\text{ in }\quad \mathbb R^{N+1}, \end{equation} (4.2)

    where

    F_1 = A(\nabla_X u)\phi_1-\phi_1\nabla_X u-u\nabla_X\phi_1,
    F_2 = -A(\nabla_X u)\cdot\nabla_X\phi_1+u(\partial_t+X\cdot\nabla_Y)\phi_1\quad \text{ and }\quad {{\mu}} = \bar{{{\mu}}}\phi_1.

    By Lemma 3.3, we have

    \begin{equation} \|v\|_{L^q( \mathbb R^{2m}\times \mathbb R_-)}\leq c\Big(2+\frac{1}{m}-q\Big)^{-1}(\|F_1\|_{L^2( \mathbb R^{2m}\times \mathbb R_-)}+\|F_2\|_{L^2( \mathbb R^{2m}\times \mathbb R_-)}) \end{equation} (4.3)

    and

    \begin{equation} \|v\|_{L_{t, X}^1 W_Y^{s, 1}( \mathbb R^{2m}\times \mathbb R_-)}\leq c\Big(\frac{1}{3}-s\Big)^{-1}(\|F_1\|_{L^1( \mathbb R^{2m}\times \mathbb R_-)}+\|F_2\|_{L^1( \mathbb R^{2m}\times \mathbb R_-)}+\|\mu\|_{M^1( \mathbb R^{2m}\times \mathbb R_-)}), \end{equation} (4.4)

    for some uniform constant c and for every q\in[2, 2+\frac{1}{m}) and s\in[0, \frac{1}{3}) . Using (4.1), (1.7)- (i) , Lemma 3.1 and that 0 < r_1 < r_0\leq 1 , it follows that

    \begin{equation} \|F_1\|_{L^2( \mathbb R^{2m}\times \mathbb R_-)}+\|F_2\|_{L^2( \mathbb R^{2m}\times \mathbb R_-)}\leq c_1, \end{equation} (4.5)

    where

    \begin{equation} c_1 = c(m, \Lambda)\Big(1+\frac{1}{r_0-r_1}\Big)\Big(1+\frac{1}{(r_0-r_1)^2}+\frac{|X_0|+r_0}{(r_0-r_1)r_1^{2}}+\frac{1}{(r_0-r_1)r_1}\Big). \end{equation} (4.6)

    Using (4.5) in (4.3), the estimate (1.15) follows. To obtain the estimate (1.16), let \phi_2\in [0, 1] be smooth such that \phi_2\equiv 1 in Q_{r_2}(Z_0, t_0) and \phi_2\equiv 0 outside Q_{r_0}(Z_0, t_0) satisfying (4.1). Choosing \phi_2 as a test function in (4.2) and proceeding similarly as in the proof of energy estimate in Lemma 3.1, we get

    \|\mu\|_{M^1(Q_{r_2}(Z_0, t_0))}\leq \|\phi_2\mu\|_{M^1( \mathbb R^{2m}\times \mathbb R_-)}\leq r_0^{1+2m}c_1\|u\|_{L^2(Q_{r_0}(Z_0, t_0))},

    where c_1 is given by (4.6). The last estimate, combined with (4.4) and (4.5), yields the estimate (1.16).

    As mentioned, we can, without loss of generality, assume that (Z_0, t_0) = (0, 0, 0) . For n\in \mathbb N\cup\{0\} , we define

    r_n = r_\infty+(r_0-r_\infty)2^{-n}, \quad T_n = -r_n^{2}, \quad k_n = \frac{1}{2}(1-2^{-n}), \quad u_n = (u-k_n)^{+},

    and

    A_n: = \sup\limits_{t\in (T_n, 0)}\iint_{B(0, r_n)\times B(0, r_n^3) }u_n^{2}(\cdot, \cdot, t)\, {{\text{d}}} X {{\text{d}}} Y.

    By Lemma 3.2 we know that u_n is a weak sub-solution of (1.5). Thus applying Lemma 3.1 we obtain

    \begin{align} A_n&\leq c\, c_{n-1, n}\iiint_{Q_{r_{n-1}}}u_n^{2}\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t, \quad \forall n\geq 1, \end{align} (4.7)

    where c = c(m, \Lambda)\geq 1 and

    \begin{equation} c_{n-1, n}: = \frac{1}{(r_{n-1}-r_n)^2}+\frac{r_{n-1}}{(r_{n-1} -r_{n})r_n^{2}}+\frac{1}{(r_{n-1} -r_{n})r_n}+1\leq \frac{2^{2n}}{r_\infty^{2}(r_0-r_\infty)^2}. \end{equation} (4.8)

    Now we will estimate the integral in the right hand side of (4.7). Let q = 2+\frac{1}{2m} . By Hölder's inequality we have

    \begin{equation} \begin{split} \iiint_{Q_{r_{n-1}}}u_n^{2}\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t&\leq \Big(\iiint_{Q_{r_{n-1}}}u_n^{q}\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\Big)^\frac{2}{q}\Big|\{u_n > 0\}\cap Q_{r_{n-1}}\Big|^{1-\frac{2}{q}}. \end{split} \end{equation} (4.9)

    Since k_n > k_{n-1} , we get u_n\leq u_{n-1} . Using this fact, that 0 < r_\infty < r_0\leq 1 , and Theorem 1.1, we get

    \begin{equation} \begin{split} \Big(\iiint_{Q_{r_{n-1}}}u_{n}^q\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\Big)^\frac{2}{q}&\leq \Big(\iiint_{Q_{r_{n-1}}}u_{n-1}^q\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\Big)^\frac{2}{q}\\ &\leq c^2\, A_{n-2}\\ &\leq\Big(\frac{c(m, \Lambda)2^{3n}}{r_\infty^{2}(r_0-r_\infty)^3}\Big)^2\, A_{n-2}, \end{split} \end{equation} (4.10)

    for every n\geq 2 , where we have used that

    c = c(m, \Lambda)\Big(1+\frac{1}{r_{n-2}-r_{n-1}}\Big)c_{n-2, n-1}\leq \frac{c(m, \Lambda) 2^{3n}}{r_\infty^{2}(r_0-r_\infty)^3},

    with c_{n-2, n-1} is as defined in (4.8).

    Next, we observe that

    \begin{align*} \iiint_{Q_{r_{n-1}}}u_{n-1}^2\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t&\geq \iiint_{\{u_{n-1}\geq 2^{-n-1}\}\cap Q_{r_{n-1}}}u_{n-1}^2\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ &\geq 2^{-2n-2}\Big|\{u_{n-1}\geq 2^{-n-1}\}\cap Q_{r_{n-1}}\Big|. \end{align*}

    Moreover,

    \Big|\{u_n > 0\}\cap Q_{r_{n-1}}\Big|\leq\Big|\{u_n\geq k_n-k_{n-1}\}\cap Q_{r_{n-1}}\Big| = \Big|\{u_n\geq 2^{-n-1}\}\cap Q_{r_{n-1}}\Big|.

    Combining the preceding two estimates and using 0 < r_\infty < r_0\leq 1 , we get

    \begin{equation} \Big|\{u_n > 0\}\cap Q_{r_{n-1}}\Big|\leq 2^{2n+2} A_{n-1}. \end{equation} (4.11)

    Using the estimates (4.10) and (4.11) in (4.9), we get

    \begin{equation} \iiint_{Q_{r_{n-1}}}u_n^{2}\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\leq c(m, \Lambda)\Big(\frac{2^{4n}}{r_{\infty}^2(r_0-r_{\infty})^3}\Big)^2 A_{n-2}^{2-\frac{2}{q}}, \quad\forall n\geq 2, \end{equation} (4.12)

    where we have also used that A_{n-1}\leq A_{n-2} . Note that the latter is true since u_{n-1}\leq u_{n-2}, \, r_{n-1} < r_{n-2} and T_{n-2} < T_{n-1} for every n\geq 2 . Using (4.12) in (4.7), we obtain

    A_n\leq c(m, \Lambda)\frac{2^{12n}}{r_{\infty}^6 (r_0-r_{\infty})^8}A_{n-2}^{\alpha},

    where \alpha = 2-\frac{2}{q} > 1 , since q > 2 . Therefore, defining S_n: = A_{2n} , we get

    S_n\leq \beta^n S_{n-1}^{\alpha}\quad\forall n\geq 1,

    where

    \beta = c(m, \Lambda)\frac{2^{24}}{r_{\infty}^6 (r_0-r_{\infty})^8}.

    Recursively we get

    \begin{equation} \begin{split} S_n&\leq \beta^{n+(n-1)\alpha+\ldots+\alpha^{n-1}}S_1^{\alpha^{n-1}}\\ &\leq \Big(\beta^\frac{\alpha^2}{(\alpha-1)^2}S_1\Big)^{\alpha^{n-1}}\\ &\leq \Big(c(m, \Lambda)c_{0, 1}\beta^\frac{\alpha^2}{(\alpha-1)^2}\|u\|^{2}_{L^2(Q_{r_0})}\Big)^{\alpha^{n-1}}, \end{split} \end{equation} (4.13)

    where we have used (4.7) and the estimate

    n+\alpha(n-1)+\ldots+\alpha^{n-1}\leq\frac{\alpha^{n+1}}{(\alpha-1)^2}.

    Let

    v: = \frac{1}{\sqrt{2c(m, \Lambda)c_{0, 1}\, \beta^\frac{\alpha^2}{(\alpha-1)^2}}}\frac{u}{\|u\|_{L^2(Q_{r_0})}}.

    We observe that

    \gamma: = c(m, \Lambda)c_{0, 1}\, \beta^\frac{\alpha^2}{(\alpha-1)^2}\|v\|_{L^2(Q_{r_0})}^2 = \frac{1}{2} < 1.

    Note that, by the property (ⅱ) in Remark 2.1, v is again a weak sub-solution of (1.5). Thus the estimate (4.13) holds by replacing u with v . This fact combined with \gamma < 1 gives v\leq \frac{1}{2} a.e. in Q_{r_\infty} . As a consequence we get

    \sup\limits_{Q_{r_\infty}}\, u\leq {\sqrt{2c(m, \Lambda)c_{0, 1}\, \beta^\frac{\alpha^2}{(\alpha-1)^2}}}\, {\|u\|_{L^2(Q_{r_0})}}\leq c\Big(\frac{1}{r_{\infty}^2 (r_0-r_{\infty})^3}\Big)^\frac{\theta}{2}{\|u\|_{L^2(Q_{r_0})}},

    for some c = c(m, \Lambda)\geq 1 and \theta = \theta(m) > 1 . Now, arguing similarly as in the proof of [23,Proposition 12,pages 7-8], the result follows.

    Using Remark 2.1, Theorem 1.2 along with Lemma 3.6, and following the lines of the proof of [23,Theorem 5,pages 13-14], the result follows.

    Using Remark 2.1, Lemma 3.6, and following the lines of the proof of [23,Theorem 7,pages 14-15], the result follows.

    The purpose of the section is to prove Theorem 1.5. As g\in W(U_X\times V_{Y, t}) we can in the following assume, without loss of generality, that g\equiv 0 .

    In domains of the form U_X\times U_Y\times I instead of U_X\times V_{Y, t} , one may attempt different approaches to prove Theorem 1.5, and perhaps the most natural first approach is to add the term \epsilon\Delta_Y to the operator and to instead consider the problem

    \begin{equation} \begin{cases} \nabla_X\cdot(A(\nabla_Xu_\epsilon, X, Y, t))+\epsilon\Delta_Yu_\epsilon-(\partial_t+X\cdot\nabla_Y)u_\epsilon = g^* &\text{in} \ U_X\times U_Y\times I, \\ u_\epsilon = 0 & \text{on} \ \partial_p(U_X\times U_Y\times I). \end{cases} \end{equation} (5.1)

    Here \partial_p(U_X\times U_Y\times I) is now the (standard) parabolic boundary of U_X\times U_Y\times I , i.e.,

    \partial_p(U_X\times U_Y\times I): = (\partial(U_X\times U_Y)\times \overline{I})\cup ((U_X\times U_Y)\times \{0\}).

    The existence and uniqueness of weak solutions to (5.1) is classical and one easily deduces that

    \begin{align} &\||\nabla_Xu_\epsilon|\|^2_{L^2(U_X\times U_Y\times I)}+\epsilon \||\nabla_Yu_\epsilon|\|^2_{L^2(U_X\times U_Y\times I)}\\ &\leq c \|g^*\|_{L_{Y, t}^2(U_Y\times I, {H}_X^{-1}(U_X))}\times\||u_\epsilon|+|\nabla_Xu_\epsilon|\|_{L^2(U_X\times U_Y\times I)}, \end{align} (5.2)

    for some positive constant c , independent of \epsilon . By the standard Poincaré inequality, applied on U_X to u_\epsilon(\cdot, Y, t) with (Y, t) fixed, we have

    \begin{align} \|u_\epsilon\|_{L^2(U_X\times U_Y\times I)}\leq c\||\nabla_Xu_\epsilon|\|_{L^2(U_X\times U_Y\times I)}. \end{align} (5.3)

    Hence, using Cauchy-Schwarz we can conclude that

    \begin{align} &\|u_\epsilon\|^2_{L^2(U_X\times U_Y\times I)}+\||\nabla_Xu_\epsilon|\|^2_{L^2(U_X\times U_Y\times I)}+\epsilon \||\nabla_Yu_\epsilon|\|^2_{L^2(U_X\times U_Y\times I)}\\ &\leq c\|g^*\|^2_{L_{Y, t}^2(U_Y\times I, {H}_X^{-1}(U_X))}, \end{align} (5.4)

    for a constant c which is independent of \epsilon . The idea is then to let \epsilon\to 0 and in this way construct a solution to the problem in (2.7). To make this operational, already in the linear case, A(\xi, X, Y, t) = A(X, Y, t)\xi , one seems to need some uniform estimates up to the Kolmogorov boundary \partial_{\mathcal K}(U_X\times U_Y\times I) to get a solution in the limit. In addition, in the nonlinear case considered in this paper we also need to ensure that \nabla_Xu_\epsilon\to \nabla_Xu pointwise a.e. as \epsilon\to 0 and how to achieve this is even less clear. One approach is to try to adapt the techniques of Boccardo and Murat [6] but it seems unclear how to make this approach operational in our case due to the presence of the term \epsilon\Delta_Yu_\epsilon in the approximating equation.

    In this paper we will instead prove Theorem 1.5 by using a variational approach recently explored in Albritton-Armstrong-Mourrat-Novack [1] and Litsgård-Nyström [29]. We will prove that the solution to (1.21) can be obtained as the minimizer of a uniformly convex functional. The fact that a parabolic equation can be cast as the first variation of a uniformly convex integral functional was first discovered by Brezis-Ekeland [9,10] and for a modern treatment of this approach, covering uniformly elliptic parabolic equations of second order in the more general context of uniformly monotone operators, we refer to [4] which in turn is closely related to [19], see also [18].

    To make the approach operational we will use a variational representation of the mapping \xi \mapsto A(\xi, X, Y, t) , for each (X, Y, t)\in \mathbb R^{N+1} , that we learned from [4] and [5] and we refer to these papers for more background. Indeed, by [5,Theorem 2.9], there exists \tilde A \in L^\infty_{\mathrm{loc}}(\mathbb R^m\times \mathbb R^m\times \mathbb R^{N+1}) satisfying the following properties, for \Gamma : = 2\Lambda + 1 and for each (X, Y, t)\in \mathbb R^{N+1} . First, the mapping

    \begin{equation} (\xi, \eta) \mapsto \tilde A(\xi, \eta, X, Y, t) - \frac 1 {2\Gamma}(|\xi|^2 + |\eta|^2) \quad \text{is convex}. \end{equation} (5.5)

    Second, the mapping

    \begin{equation} (\xi, \eta) \mapsto \tilde A(\xi, \eta, X, Y, t) - \frac {\Gamma} 2(|\xi|^2 + |\eta|^2) \quad \text{is concave}. \end{equation} (5.6)

    Third, for every \xi, \eta \in \mathbb R^{m} , we have

    \begin{equation} \tilde A(\xi, \eta, X, Y, t)\ge \xi\cdot \eta, \end{equation} (5.7)

    and

    \begin{equation} \tilde A(\xi, \eta, X, Y, t) = \xi\cdot \eta \iff \eta = A(\xi, X, Y, t). \end{equation} (5.8)

    Note that the choice of \tilde A is in general not unique. Note also that (5.5) and (5.6) imply, in particular that

    \begin{align} \frac 1{2\Gamma}|\xi_1-\xi_2|^2&\leq \frac 1 2\tilde A(\xi_1, \eta, X, Y, t)+\frac 1 2\tilde A(\xi_2, \eta, X, Y, t)\\ &-\tilde A(\frac 1 2\xi_1+\frac 1 2\xi_2, \eta, X, Y, t)\leq \frac {\Gamma} 2|\xi_1-\xi_2|^2. \end{align} (5.9)

    To ease the notation we will in the following at instances use the notation

    W: = W(U_X\times V_{Y, t}), \quad W_0: = W_{0}(U_X\times V_{Y, t}),

    and we let

    {\mathcal L} u: = \nabla_X\cdot(A(\nabla_X u, X, Y, t))-(\partial_t+X\cdot\nabla_Y)u.

    Given an arbitrary pair (f, {\bf{j}}) such that

    \begin{eqnarray} \quad f\in L_{Y, t}^2(V_{Y, t}, H_X^1(U_X))\quad\mbox{ and }\quad {\bf{j}}\in L^2(V_{Y, t}, L^2(U_X)))^m, \end{eqnarray} (5.10)

    we introduce

    \begin{equation} {\mathcal J}[f, {\bf{j}}] : = \iiint_{U_X\times V_{Y, t}} (\tilde A(\nabla_X f, {\bf{j}}, X, Y, t)-\nabla_Xf\cdot {\bf{j}}) \, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t. \end{equation} (5.11)

    Using this notation, and given an arbitrary pair (f, f^\ast) such that

    \begin{eqnarray} f\in L_{Y, t}^2(V_{Y, t}, H_X^1(U_X))\quad\mbox{ and }\quad f^\ast, \ f^\ast+(\partial_t+X\cdot\nabla_Y)f\in L_{Y, t}^2(V_{Y, t}, H_X^{-1}(U_X)), \end{eqnarray} (5.12)

    we set

    \begin{equation} J[f, f^*] : = \inf \iiint_{U_X\times V_{Y, t}} {\mathcal J}[f, {\bf{g}}] \, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t, \end{equation} (5.13)

    where the infimum is taken with respect to the set

    \begin{equation} \bigl\{ {\bf{g}} \in (L^2(V_{Y, t}, L^2(U_X)))^m \mid {\nabla_X \cdot {\bf{g}}} = f^* +(\partial_t+X\cdot\nabla_Y)f \bigr\}. \end{equation} (5.14)

    The condition

    \begin{equation*} {\nabla_X \cdot {\bf{g}}} = f^* +(\partial_t+X\cdot\nabla_Y)f, \end{equation*}

    appearing in (5.14), should be interpreted as stating that

    \begin{equation} - \iiint_{U_X\times V_{Y, t}} {\bf{g}} \cdot \nabla_X \phi \, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t = \iint_{V_{Y, t}} \langle f^*(\cdot, Y, t) +(\partial_t+X\cdot\nabla_Y)f(\cdot, Y, t), \phi\rangle \, {{\text{d}}} Y{{\text{d}}} t, \end{equation} (5.15)

    for all \phi \in L^2(V_{Y, t}, H^1_{X, 0}(U_X)) . Finally, for g^*\in L_{Y, t}^2(V_{Y, t}, {H}_X^{-1}(U_X)) fixed we introduce

    \begin{equation} \mathcal{A}(g^\ast): = \{ (f, {\bf{j}}) \in W_{0}\times( L^2(V_{Y, t}, L^2(U_X)))^m \mid \nabla_X\cdot {\bf{j}} = g^\ast+(\partial_t+X\cdot\nabla_Y)f \}. \end{equation} (5.16)

    Lemma 5.1. Let g^*\in L_{Y, t}^2(V_{Y, t}, {H}_X^{-1}(U_X)) be fixed and let \mathcal{A}(g^\ast) be the set introduced in (5.16). Then \mathcal{A}(g^\ast) is non-empty.

    Proof. Take f\in W_{0} and consider the equation

    \begin{align} &\Delta_Xv(X, Y, t) = (g^\ast(X, Y, t)+(\partial_t+X\cdot\nabla_Y)f(X, Y, t))\in H_X^{-1}(U_X), \end{align} (5.17)

    for {{\text{d}}} Y{{\text{d}}} t -a.e. (Y, t)\in V_{Y, t} . By the Lax-Milgram theorem this equation has a (unique) solution v(\cdot) = v(\cdot, Y, t)\in H^1_{X, 0}(U_X) and

    \begin{align} ||\nabla_Xv||_{L_{Y, t}^2(V_{Y, t}, L^2(U_X))}\leq c||g^\ast+(\partial_t+X\cdot\nabla_Y)f||_{L_{Y, t}^2(V_{Y, t}, {H}_X^{-1}(U_X))} < \infty, \end{align} (5.18)

    as f\in W_0 . In particular,

    \begin{equation} (f, \nabla_Xv)\in \mathcal{A}(g^\ast), \end{equation} (5.19)

    and hence \mathcal{A}(g^\ast) is non-empty.

    Lemma 5.2. The functional {\mathcal J} introduced in (5.11) is uniformly convex on \mathcal{A}(g^*) .

    Proof. Note that if (f, {\bf{j}})\in \mathcal{A}(g^*) and (\tilde f, \tilde {\bf{j}})\in \mathcal{A}(0) , then (f+\tilde f, {\bf{j}}+\tilde {\bf{j}})\in \mathcal{A}(g^*) and (f-\tilde f, {\bf{j}}-\tilde {\bf{j}})\in \mathcal{A}(g^*) . Consider (f, {\bf{j}})\in \mathcal{A}(g^*) . We first consider the term

    \begin{equation*} \label{e.uu} -\iiint_{U_X\times V_{Y, t}} \nabla_Xf\cdot {\bf{j}} \, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t. \end{equation*}

    We have

    \begin{align} -\iiint_{U_X\times V_{Y, t}} \nabla_Xf\cdot {\bf{j}} \, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t & = \iiint_{U_X\times V_{Y, t}} f\nabla_X\cdot {\bf{j}} \, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ & = \iint_{V_{Y, t}} \langle g^\ast(\cdot, Y, t)+(\partial_t+X\cdot\nabla_Y)f(\cdot, Y, t)), f(\cdot, Y, t)\rangle \, {{\text{d}}} Y {{\text{d}}} t\\ & = \iint_{V_{Y, t}} \langle g^\ast(\cdot, Y, t), f(\cdot, Y, t)\rangle \, {{\text{d}}} Y {{\text{d}}} t\\ &+\iint_{V_{Y, t}} \langle (\partial_t+X\cdot\nabla_Y)f(\cdot, Y, t)), f(\cdot, Y, t)\rangle \, {{\text{d}}} Y {{\text{d}}} t. \end{align} (5.20)

    Recall that W_0 = W_0(U_X\times V_{Y, t}) is the closure in the norm of W(U_X\times V_{Y, t}) of C^\infty_{{\mathcal K}, 0}(\overline{U_X\times V_{Y, t}}) . In particular, there exists \{f_j\} , f_j\in C^\infty_{{\mathcal K}, 0}(\overline{U_X\times V_{Y, t}}) such that

    ||f-f_j||_W\to 0\mbox{ as }j\to \infty,

    and consequently

    ||(\partial_t+X\cdot\nabla_Y)(f-f_j)||_{L_{Y, t}^2(V_{Y, t}, {H}_X^{-1}(U_X))}\to 0\mbox{ as }j\to \infty.

    Using this we see that

    \begin{equation} \begin{split} &\iint_{V_{Y, t}} \langle(\partial_t+X\cdot\nabla_Y)f(\cdot, Y, t), f(\cdot, Y, t)\rangle \, {{\text{d}}} Y{{\text{d}}} t\\ &\geq \liminf\limits_{j\to\infty} \iint_{V_{Y, t}} \langle(\partial_t+X\cdot\nabla_Y)f_j(\cdot, Y, t), f_j(\cdot, Y, t)\rangle \, {{\text{d}}} Y{{\text{d}}} t. \end{split} \end{equation} (5.21)

    However, using that f_j\in C^\infty_{{\mathcal K}, 0}(\overline{U_X\times V_{Y, t}}) we see that

    \begin{equation} \begin{split} &\iint_{V_{Y, t}} \langle(\partial_t+X\cdot\nabla_Y)f_j(\cdot, Y, t), f_j(\cdot, Y, t)\rangle \, {{\text{d}}} Y{{\text{d}}} t\\ & = \iiint_{U_X\times V_{Y, t}} (\partial_t+X\cdot\nabla_Y)f_j f_j \, {{\text{d}}} X{{\text{d}}} Y{{\text{d}}} t\\ & = \frac 1 2\iiint_{U_X\times V_{Y, t}} (\partial_t+X\cdot\nabla_Y)f_j^2 \, {{\text{d}}} X{{\text{d}}} Y{{\text{d}}} t\\ & = \frac 12 \int_{U_X}\iint_{\partial V_{Y, t}} f_j^2(X, 1)\cdot N_{Y, t} \, {{\text{d}}} \sigma_{Y, t}{{\text{d}}} X\geq 0, \end{split} \end{equation} (5.22)

    by the divergence theorem and the definition of the Kolmogorov boundary. Hence,

    \begin{align} -\iiint_{U_X\times V_{Y, t}} \nabla_Xf\cdot {\bf{j}} \, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t & = \iiint_{U_X\times V_{Y, t}} f\nabla_X\cdot {\bf{j}} \, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ &\geq \iint_{V_{Y, t}} \langle g^\ast(\cdot, Y, t), f(\cdot, Y, t)\rangle \, {{\text{d}}} Y {{\text{d}}} t. \end{align} (5.23)

    Using this, and observing that,

    \begin{align} &-\frac 1 2\iiint_{U_X\times V_{Y, t}} \bigl (\nabla_X(f+\tilde f)\cdot( {\bf{j}}+\tilde {\bf{j}}) +\nabla_X(f-\tilde f)\cdot( {\bf{j}}-\tilde {\bf{j}})-2\nabla_Xf\cdot {\bf{j}}\bigr )\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ & = -\iiint_{U_X\times V_{Y, t}} \nabla_X\tilde f\cdot\tilde {\bf{j}}\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t, \end{align} (5.24)

    we can conclude that

    \begin{align} &-\frac 1 2\iiint_{U_X\times V_{Y, t}} \bigl (\nabla_X(f+\tilde f)\cdot( {\bf{j}}+\tilde {\bf{j}}) +\nabla_X(f-\tilde f)\cdot( {\bf{j}}-\tilde {\bf{j}})-2\nabla_Xf\cdot {\bf{j}}\bigr )\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\geq 0, \end{align} (5.25)

    over the set \mathcal{A}(g^\ast) . Hence it suffices to prove that

    \begin{equation*} \iiint_{U_X\times V_{Y, t}} \tilde A(\nabla_Xf, {\bf{j}}, X, Y, t)\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t \end{equation*}

    is uniformly convex over the set \mathcal{A}(g^\ast) . With (f, {\bf{j}})\in \mathcal{A}(g^*) and (\tilde f, \tilde {\bf{j}})\in \mathcal{A}(0) as above, (5.5) implies that

    \begin{equation*} \frac 1 2 \tilde A(\nabla_X (f+\tilde f), {\bf{j}} + \tilde {\bf{j}}, \cdot) + \frac 1 2 \tilde A(\nabla_X (f-\tilde f), {\bf{j}} - \tilde {\bf{j}}, \cdot) - \tilde A(\nabla_X f, {\bf{j}}, \cdot) \ge \frac 1 {2\Gamma} \left( |\nabla_X\tilde f|^2 + |\tilde {\bf{j}}|^2 \right) . \end{equation*}

    We also have

    \begin{align*} \|(\partial_t+X\cdot\nabla_Y) \tilde f\|_{L_{Y, t}^2(V_{Y, t}, H^{-1}(U_X))} \le \|\tilde {\bf{j}}\|_{L^2(U_X\times V_{Y, t})}. \end{align*}

    Thus

    \begin{align*} & \iiint_{U_X\times V_{Y, t}}\biggl (\frac 1 2 \tilde A(\nabla_X (f+\tilde f), {\bf{j}} + \tilde {\bf{j}}, \cdot) + \frac 1 2 \tilde A(\nabla_X (f-\tilde f), {\bf{j}} - \tilde {\bf{j}}, \cdot) - \tilde A(\nabla_X f, {\bf{j}}, \cdot)\biggr )\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\notag\\ &\geq \frac{1}{4\Gamma} \left( ||\nabla_X\tilde f||^2_{L^2(U_X\times V_{Y, t})}+\|(\partial_t+X\cdot\nabla_Y) \tilde f\|^2_{L_{Y, t}^2(V_{Y, t}, H^{-1}(U_X))} + ||\tilde {\bf{j}}||^2_{L^2(U_X\times V_{Y, t})} \right)\notag\\ &\geq \frac{1}{4c\Gamma} \left( ||\tilde f||^2_{W(U_X\times V_{Y, t})}+\||\tilde {\bf{j}}||^2_{L^2(U_X\times V_{Y, t})} \right), \end{align*}

    by using the (standard) Poincaré inequality. Hence {\mathcal J} is uniformly convex on \mathcal{A}(g^*) .

    As the functional {\mathcal J} is uniformly convex over \mathcal{A}(g^\ast) there exists a unique minimizing pair (f_1, {\bf{j}}_1)\in \mathcal{A}(g^\ast) such that

    \begin{align*} (f_1, {\bf{j}}_1): = &\mathop {{\rm{arg}}\;{\rm{min}}}\limits_{(f, {\bf{j}})\in \mathcal{A}(g^\ast)} {\mathcal J}[f, {\bf{j}}]\notag\\ = &\mathop {{\rm{arg}}\;{\rm{min}}}\limits_{(f, {\bf{j}})\in \mathcal{A}(g^\ast)} \iiint_{U_X\times V_{Y, t}} (\tilde A(\nabla_Xf, {\bf{j}}, X, Y, t)-\nabla_Xf\cdot {\bf{j}}) \, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t. \end{align*}

    Note that

    \begin{align*} \min\limits_{(f, {\bf{j}})\in \mathcal{A}(g^\ast)} {\mathcal J}[f, {\bf{j}}] = \min\limits_{f\in W_0} J[f, g^*]. \end{align*}

    Moreover, by construction of \tilde A , see (5.7), we have

    \begin{equation} J[f_1, g^\ast] \ge 0. \end{equation} (5.26)

    Lemma 5.3. There is a one-to-one correspondence between weak solutions in the sense of Eq (2.7) to {\mathcal L} u = g^\ast in U_X\times V_{Y, t} , such that u\in W_0 , and null minimizers of J[\cdot, g^\ast] .

    Proof. To prove the lemma we need to prove that for every f \in W_{0} , we have

    \begin{align*} f { \;{\rm{solves}}\; {\mathcal L} u = g^\ast \;{\rm{in}} \;{\rm{the}} \;{\rm{weak}}\; {\rm{sense}} \;{\rm{in}} \;U_X\times V_{Y, t} } \iff J[f, g^\ast] = 0. \end{align*}

    Indeed, the implication " \implies " is clear since if f solves {\mathcal L} u = g^\ast in the weak sense, then

    \begin{equation*} (f, A(\nabla_X f, X, Y, t)) \in \mathcal{A}(g^\ast) \quad \text{ and } \quad {\mathcal J}[f, A(\nabla_X f, X, Y, t)] = 0 = J[f, g^\ast]. \end{equation*}

    Conversely, if J[f, g^\ast] = 0 , then f = f_1 and

    \begin{equation} {\mathcal J}[f_1, {\bf{j}}_1] = \iiint_{U_X\times V_{Y, t}} (\tilde A(\nabla_Xf_1, {\bf{j}}_1, X, Y, t)-\nabla_Xf_1\cdot {\bf{j}}_1) \, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t = 0. \end{equation} (5.27)

    Using (5.8), we see that the identity (5.27) implies that

    \begin{equation*} {\bf{j}}_1 = A(\nabla f_1, \cdot, \cdot, \cdot) \quad \text{a.e. in } U_X\times V_{Y, t}, \end{equation*}

    and by the definition of the set \mathcal{A}(g^\ast) ,

    \begin{equation*} \nabla_X\cdot {\bf{j}}_1 = g^\ast+(\partial_t+X\cdot\nabla_Y)f_1. \end{equation*}

    Hence f_1 indeed solves

    \begin{equation*} \nabla_X\cdot A(\nabla f_1, \cdot, \cdot, \cdot) -(\partial_t+X\cdot\nabla_Y)f_1 = g^\ast \end{equation*}

    in the weak sense. I.e., we recover that f = f_1 is indeed a weak solution of {\mathcal L} u = g^\ast . In particular, the fact that there is at most one solution to {\mathcal L} u = g^* is clear.

    Using (5.26) and Lemma 5.3 we see that to complete the proof of Theorem 1.5 it remains to prove that

    \begin{equation} J[f_1, g^*] \le 0. \end{equation} (5.28)

    In order to do so, we introduce the perturbed convex minimization problem defined, for every f^* \in L_{Y, t}^2(V_{Y, t}, H_X^{-1}(U_X)) , by

    \begin{equation*} G(f^*) : = \inf\limits_{f \in W_{0}}\bigl ( J[f, f^*+g^* ] {- \iint_{ V_{Y, t}} \langle f^*(\cdot, Y, t), f(\cdot, Y, t)\rangle\, {{\text{d}}} Y{{\text{d}}} t\bigr ).} \end{equation*}

    As

    \begin{equation*} G(0) = \inf\limits_{f \in W_{0}} J[f, g^* ], \end{equation*}

    we see that to prove (5.28) is suffices to prove that G(0) \le 0 .

    Lemma 5.4. G is a convex, locally bounded from above and lower semi-continuous functional on L_{Y, t}^2(V_{Y, t}, H_X^{-1}(U_X)) .

    Proof. For every pair (f, {\bf{j}}) \in \mathcal A(f^*+g^*) , we have

    \begin{equation*} \nabla_X \cdot {\bf{j}} = f^*+g^* +(\partial_t+X\cdot\nabla_Y)f, \end{equation*}

    and thus

    \begin{align*} {\mathcal J}[f, {\bf{j}}] & = \iiint_{U_X\times V_{Y, t}} (\tilde A(\nabla_Xf, {\bf{j}}, X, Y, t)-\nabla_Xf\cdot {\bf{j}}) \, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\notag\\ & = \iiint_{U_X\times V_{Y, t}} \tilde A(\nabla_Xf, {\bf{j}}, X, Y, t) \, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t{+\iint_{V_{Y, t}} \langle (f^*+g^*)(\cdot, Y, t), f(\cdot, Y, t)\rangle\, {{\text{d}}} Y{{\text{d}}} t}\notag\\ &+\iint_{V_{Y, t}} \langle (\partial_t+X\cdot\nabla_Y)f(\cdot, Y, t), f(\cdot, Y, t)\rangle\, {{\text{d}}} Y{{\text{d}}} t. \end{align*}

    Hence

    \begin{align*} &{\mathcal J}[f, {\bf{j}}] {-\iint_{V_{Y, t}} \langle f^*(\cdot, Y, t), f(\cdot, Y, t)\rangle\, {{\text{d}}} Y{{\text{d}}} t} \notag\\ & = \iiint_{U_X\times V_{Y, t}} \tilde A(\nabla_Xf, {\bf{j}}, X, Y, t) \, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t+\iint_{V_{Y, t}} \langle g^*(\cdot, Y, t), f(\cdot, Y, t)\rangle\, {{\text{d}}} Y{{\text{d}}} t\\ &+\iint_{V_{Y, t}} \langle (\partial_t+X\cdot\nabla_Y)f(\cdot, Y, t), f(\cdot, Y, t)\rangle\, {{\text{d}}} Y{{\text{d}}} t. \end{align*}

    Taking the infimum over all (f, {\bf{j}}) satisfying the affine constraint (f, {\bf{j}}) \in \mathcal A(f^*+g^*) we obtain the quantity G(f^*) , i.e., G(f^*) can be expressed as

    \begin{equation*} G(f^*) = \inf\limits_{(f, {\bf{j}}):\ (f, {\bf{j}}) \in \mathcal A(f^*+g^*)}\bigl ( {\mathcal J}[f, {\bf{j}}] { - \iint_{V_{Y, t}} \langle f^*(\cdot, Y, t), f(\cdot, Y, t)\rangle\, {{\text{d}}} Y{{\text{d}}} t} \bigr ). \end{equation*}

    In particular, G(f^*) can be expressed as the infimum of

    \begin{align} &\iiint_{U_X\times V_{Y, t}} \tilde A(\nabla_Xf, {\bf{j}}, X, Y, t) \, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t+\iint_{V_{Y, t}} \langle g^*(\cdot, Y, t), f(\cdot, Y, t)\rangle\, {{\text{d}}} Y {{\text{d}}} t\\ &+\iint_{V_{Y, t}} \langle (\partial_t+X\cdot\nabla_Y)f(\cdot, Y, t), f(\cdot, Y, t)\rangle\, {{\text{d}}} Y{{\text{d}}} t \end{align} (5.29)

    with respect to (f, {\bf{j}}) such that (f, {\bf{j}}) \in \mathcal A(f^*+g^*) . We now recall the argument in (5.21) and (5.22). In particular, given f\in W_0 there exists \{f_j\} , f_j\in C^\infty_{{\mathcal K}, 0}(\overline{U_X\times V_{Y, t}}) such that

    \begin{equation} ||f-f_j||_W\to 0\mbox{ as }j\to \infty, \end{equation} (5.30)

    and consequently

    ||(\partial_t+X\cdot\nabla_Y)(f-f_j)||_{L_{Y, t}^2(V_{Y, t}, {H}_X^{-1}(U_X))}\to 0\mbox{ as }j\to \infty.

    Using (5.21) and (5.22) we have

    \begin{equation} \label{apauu} \begin{split} &\iint_{V_{Y, t}} \langle(\partial_t+X\cdot\nabla_Y)f(\cdot, Y, t), f(\cdot, Y, t)\rangle \, {{\text{d}}} Y{{\text{d}}} t\notag\\ & = \lim\limits_{j\to\infty} \frac 12 \int_{U_X}\iint_{\partial V_{Y, t}} f_j^2|(X, 1)\cdot N_{Y, t}| \, {{\text{d}}} \sigma_{Y, t}{{\text{d}}} X. \end{split} \end{equation}

    Obviously we get the same limit in (5.31) independent of what sequence \{f_j\} chosen as long as (5.30) holds. Now consider f, g\in W_0 and let \{f_j\} , f_j\in C^\infty_{{\mathcal K}, 0}(\overline{U_X\times V_{Y, t}}) , \{g_j\} , g_j\in C^\infty_{{\mathcal K}, 0}(\overline{U_X\times V_{Y, t}}) , be such that

    \begin{equation} ||f-f_j||_W+||g-g_j||_W\to 0\mbox{ as }j\to \infty, \end{equation} (5.31)

    Then

    \begin{equation} ||(\tau f+(1-\tau)g)-(\tau f_j+(1-\tau)g_j)||_W\to 0\mbox{ as }j\to \infty, \end{equation} (5.32)

    for all \tau\in [0, 1] . Hence

    \begin{align} &\iint_{V_{Y, t}} \langle(\partial_t+X\cdot\nabla_Y)(\tau f+(1-\tau)g)(\cdot, Y, t), (\tau f+(1-\tau)g)(\cdot, Y, t)\rangle \, {{\text{d}}} Y{{\text{d}}} t\\ & = \lim\limits_{j\to\infty} \frac 12 \int_{U_X}\iint_{\partial V_{Y, t}} (\tau f_j+(1-\tau)g_j)^2|(X, 1)\cdot N_{Y, t}| \, {{\text{d}}} \sigma_{Y, t}{{\text{d}}} X\\ &\leq\lim\limits_{j\to\infty} \frac 12 \int_{U_X}\iint_{\partial V_{Y, t}} \tau f_j^2|(X, 1)\cdot N_{Y, t}| \, {{\text{d}}} \sigma_{Y, t}{{\text{d}}} X\\ &+\lim\limits_{j\to\infty} \frac 12 \int_{U_X}\iint_{\partial V_{Y, t}} (1-\tau)g_j^2|(X, 1)\cdot N_{Y, t}| \, {{\text{d}}} \sigma_{Y, t}{{\text{d}}} X, \end{align} (5.33)

    and we deduce that

    \begin{align} &\iint_{V_{Y, t}} \langle(\partial_t+X\cdot\nabla_Y)(\tau f+(1-\tau)g)(\cdot, Y, t), (\tau f+(1-\tau)g)(\cdot, Y, t)\rangle \, {{\text{d}}} Y{{\text{d}}} t\\ &\leq\tau\iint_{V_{Y, t}} \langle(\partial_t+X\cdot\nabla_Y)f(\cdot, Y, t), f(\cdot, Y, t)\rangle \, {{\text{d}}} Y{{\text{d}}} t\\ &+(1-\tau)\iint_{V_{Y, t}} \langle(\partial_t+X\cdot\nabla_Y)g(\cdot, Y, t), g(\cdot, Y, t)\rangle \, {{\text{d}}} Y{{\text{d}}} t. \end{align} (5.34)

    In particular, we can conclude that the mapping

    f\to \iint_{V_{Y, t}} \langle(\partial_t+X\cdot\nabla_Y)(\tau f+(1-\tau)g)(\cdot, Y, t), (\tau f+(1-\tau)g)(\cdot, Y, t)\rangle \, {{\text{d}}} Y{{\text{d}}} t

    is convex on W_0 . Using this, and (5.5), we see that the expression in (5.29) is convex as a function of (f, f^*, {\bf{j}}) and this proves that G is convex. Furthermore, using (5.17)–(5.19) we can conclude that the infimum of the expression in (5.29) is finite, hence G(f^*) < \infty . In particular, the function G is locally bounded from above. These two properties imply that G is lower semi-continuous, see [17,Lemma 2.1 and Corollary 2.2].

    We denote by G^* the convex dual of G , defined for every

    h \in (L_{Y, t}^2(V_{Y, t}, H_X^{-1}(U_X)))^\ast = L_{Y, t}^2(V_{Y, t}, H_{X, 0}^1(U_X)),

    as

    \begin{equation*} G^*(h) : = \sup\limits_{f^* \in L_{Y, t}^2(V_{Y, t}, H_X^{-1}(U_X))} \bigl( -G(f^*) + \iint_{V_{Y, t}} \langle f^*(\cdot, Y, t), h(\cdot, Y, t)\rangle\, {{\text{d}}} Y{{\text{d}}} t \bigr). \end{equation*}

    Let G^{**} be the bidual of G . Since G is lower semi-continuous, we have that G^{**} = G (see [17,Proposition 4.1]), and in particular,

    \begin{equation*} G(0) = G^{**}(0) = \sup\limits_{h \in L_{Y, t}^2(V_{Y, t}, H_{X, 0}^1(U_X))} \bigl( -G^*(h) \bigr) . \end{equation*}

    In order to prove that G(0) \le 0 , it therefore suffices to show that

    \begin{equation} G^*(h) \ge 0\mbox{ for all } h \in L_{Y, t}^2(V_{Y, t}, H_{X, 0}^1(U_X)). \end{equation} (5.35)

    To continue we note that we can rewrite G^*(h) as

    \begin{equation} \begin{split} G^*(h) = \sup\limits_{(f, {\bf{j}}, f^*)} &\bigg\{ \iiint_{U_X\times V_{Y, t}} -(\tilde A(\nabla_X f, {\bf{j}}, \cdot, \cdot, \cdot)-(\nabla_X f\cdot {\bf{j}})) \, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ &+\iint_{V_{Y, t}} {\langle f^*(\cdot, Y, t), (h(\cdot, Y, t)+f(\cdot, Y, t))\rangle}\, {{\text{d}}} Y {{\text{d}}} t\bigg \}, \end{split} \end{equation} (5.36)

    where the supremum is taken with respect to

    (f, {\bf{j}}, f^*)\in W_{0}\times (L_{Y, t}^2(V_{Y, t}, L^2_X(U_X)))^m\times L_{Y, t}^2(V_{Y, t}, H_X^{-1}(U_X)),

    subject to the constraint

    \begin{equation} \nabla_X\cdot {\bf{j}} = f^* + g^* +(\partial_t+X\cdot\nabla_Y)f. \end{equation} (5.37)

    Furthermore, note that for every h \in L_{Y, t}^2(V_{Y, t}, H_{X, 0}^1(U_X)) , we have G^*(h) \in \mathbb R \cup \{+\infty\} .

    Lemma 5.5. Consider h \in L_{Y, t}^2(V_{Y, t}, H_{X, 0}^1(U_X)) . Then

    \begin{equation} G^*(h) < +\infty \quad \implies \quad h \in W\cap L_{Y, t}^2(V_{Y, t}, H_{X, 0}^1(U_X)). \end{equation} (5.38)

    Proof. To prove the lemma we need to prove that (\partial_t+X\cdot\nabla_Y)h \in L_{Y, t}^2(V_{Y, t}, H_{X}^{-1}(U_X)) . Using that we take a supremum in the definition of G^\ast we can develop lower bounds on G^\ast by restricting the set with respect to which we take the supremum. Here, for f\in W_0 , we choose to restrict the supremum to (f, {\bf{j}}, f^*) where {\bf{j}} = {\bf{j}}_0 is a solution of \nabla_X\cdot {\bf{j}}_0 = g^* and f^* : = -(\partial_t+X\cdot\nabla_Y)f . Recall from (5.19) that such a {\bf{j}}_0 \in (L_{Y, t}^2(V_{Y, t}, L^2_X(U_X)))^m exists. With these choices for {\bf{j}} and f^* , the constraint (5.37) is satisfied, and we obtain that

    \begin{align*} G^*(h)\geq \sup\limits_{f\in W_0} &\biggl\{ \iiint_{U_X\times V_{Y, t}} -(\tilde A(\nabla_X f, {\bf{j}}_0, \cdot, \cdot, \cdot)-(\nabla_X f\cdot {\bf{j}}_0)) \, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\notag\\ &-\iint_{V_{Y, t}} {\langle (\partial_t+X\cdot\nabla_Y)f(\cdot, Y, t), (h(\cdot, Y, t) + f(\cdot, Y, t))\rangle}\, {{\text{d}}} Y{{\text{d}}} t\biggr \}. \end{align*}

    Consider f\in C^\infty_{{\mathcal K}, 0}(\overline{U_X\times V_{Y, t}})\subset W_0 . Then, again arguing as in (5.21), (5.22),

    \begin{align*} -\iint_{V_{Y, t}} \langle (\partial_t+X\cdot\nabla_Y)f(\cdot, Y, t), f(\cdot, Y, t)\rangle\, {{\text{d}}} Y{{\text{d}}} t\leq 0. \end{align*}

    Furthermore, restricting to f\in C^\infty_{0}({U_X\times V_{Y, t}})\subset C^\infty_{{\mathcal K}, 0}(\overline{U_X\times V_{Y, t}}) yields by the same argument that

    \begin{align*} -\iint_{V_{Y, t}} \langle (\partial_t+X\cdot\nabla_Y)f(\cdot, Y, t), f(\cdot, Y, t)\rangle\, {{\text{d}}} Y{{\text{d}}} t = 0. \end{align*}

    Hence we have the lower bound

    \begin{align*} G^*(h)\geq \sup &\biggl\{ \iiint_{U_X\times V_{Y, t}} -(\tilde A(\nabla_X f, {\bf{j}}_0, \cdot, \cdot, \cdot)-(\nabla_X f\cdot {\bf{j}}_0)) \, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\notag\\ &-\iint_{V_{Y, t}} {\langle (\partial_t+X\cdot\nabla_Y)f(\cdot, Y, t), h(\cdot, Y, t)\rangle}\, {{\text{d}}} Y{{\text{d}}} t\biggr \}, \end{align*}

    where the supremum now is taken with respect to f\in C_0^\infty(U_X\times V_{Y, t})\subset C^\infty_{{\mathcal K}, 0}(\overline{U_X\times V_{Y, t}}) . Moreover, as G^*(h) < +\infty , we have that

    \begin{align*} &-\iint_{V_{Y, t}} \langle (\partial_t+X\cdot\nabla_Y)f(\cdot, Y, t), h(\cdot, Y, t)\rangle\, {{\text{d}}} Y{{\text{d}}} t\notag\\ &\leq \iiint_{U_X\times V_{Y, t}} (\tilde A(\nabla_X f, {\bf{j}}_0, \cdot, \cdot, \cdot)-(\nabla_X f\cdot {\bf{j}}_0)) \, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t +G^*(h) < \infty, \end{align*}

    for every f\in C_0^\infty(U_X\times V_{Y, t}) fixed. Note that by replacing f with -f in the above argument we also obtain a lower bound. In particular,

    \sup \ \biggl |\iint_{V_{Y, t}} \langle (\partial_t+X\cdot\nabla_Y)h(\cdot, Y, t), f(\cdot, Y, t)\rangle\, {{\text{d}}} Y{{\text{d}}} t\biggr | < \infty,

    where the supremum is taken over f \in C_0^\infty(U_X\times V_{Y, t}) such that ||f||_{L_{Y, t}^2(V_{Y, t}, H_{X, 0}^1(U_X))}\leq 1 . Using that C_0^\infty(U_X\times V_{Y, t}) is dense in L_{Y, t}^2(V_{Y, t}, H_{X, 0}^1(U_X)) we can conclude that

    (\partial_t+X\cdot\nabla_Y)h\in L_{Y, t}^2(V_{Y, t}, H_X^{-1}(U_X))

    and this observation proves (5.38).

    Lemma 5.5 gives at hand that in place of (5.35), we have reduced the matter to proving that

    \begin{equation} \qquad G^*(h) \ge 0\mbox{ for all }h\in W\cap L_{Y, t}^2(V_{Y, t}, H_{X, 0}^1(U_X)). \end{equation} (5.39)

    Furthermore, note that for \tilde h\in W\cap C_{X, 0}^\infty(\overline{U_X\times V_{Y, t}}) we have

    \begin{equation} G^*(h) \geq G^*(\tilde h) - \|f^*\|_{L^2_{Y, t}(V_{Y, t}, H_X^{-1}(U_X))} \| h-\tilde h \|_{L^2_{Y, t}(V_{Y, t}, H^1_{X}(U_X))}. \end{equation} (5.40)

    As we are to establish a lower bound on G^* , we may restrict to taking the supremum over f^* such that

    \begin{equation} \|f^*\|_{L_{Y, t}^2(V_{Y, t}, H_X^{-1}(U_X))}\leq 1. \end{equation} (5.41)

    In Lemma 5.6 below we prove that

    G^*(h) \ge 0\mbox{ for all }h\in W\cap C_{X, 0}^\infty(\overline{U_X\times V_{Y, t}}).

    By combining this with (5.40) and (5.41) we see that

    G^*(h) \geq G^*(\tilde h) - \| h-\tilde h \|_{L^2_{Y, t}(V_{Y, t}, H^1_{X, 0}(U_X))}\geq - \| h-\tilde h \|_{L^2_{Y, t}(V_{Y, t}, H^1_{X, 0}(U_X))},

    for all \tilde h \in W\cap C^\infty_{X, 0}(\overline{U_X\times V_{Y, t}}) . Furthermore, by the definitions of W , and L_{Y, t}^2(V_{Y, t}, H_{X, 0}^1(U_X)) , we can choose a sequence h_j\in W\cap C^\infty_{X, 0}(\overline{U_X\times V_{Y, t}}) such that

    \lim\limits_{j\rightarrow \infty} \| h-h_j \|_{L^2_{Y, t}(V_{Y, t}, H^1_{X, 0}(U_X))} = 0.

    Hence the proof that G^*(h)\geq 0 , and hence the final piece in the proof of existence in Theorem 1.5, is to prove the following lemma.

    Lemma 5.6.

    \begin{equation} \qquad G^*(h) \ge 0\;{{for\; all}}\;h\in W\cap C^\infty_{X, 0}(\overline{U_X\times V_{Y, t}}). \end{equation} (5.42)

    Proof. To start the proof of the lemma we first note that we have, as f\in W_0 , that

    (\partial_t+X\cdot\nabla_Y)f \in L_{Y, t}^2(V_{Y, t}, H_X^{-1}(U_X)),

    and hence we can replace f^* by f^* - (\partial_t+X\cdot\nabla_Y)f in the variational formula (5.36) for G^* to get

    \begin{align*} G^*(h) \geq \sup\limits_{(f, {\bf{j}}, f^*)} &\biggl\{ \iiint_{U_X\times V_{Y, t}} -(\tilde A(\nabla_X f, {\bf{j}}, \cdot, \cdot, \cdot)-(\nabla_X f\cdot {\bf{j}})) \, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\notag\\ &+\iint_{V_{Y, t}} {\langle (f^* - (\partial_t+X\cdot\nabla_Y)f)(\cdot, Y, t), (h(\cdot, Y, t) + f(\cdot, Y, t))\rangle}\, {{\text{d}}} Y{{\text{d}}} t\biggr \}, \end{align*}

    where the supremum now is taken with respect to

    \begin{align} (f, {\bf{j}}, f^*)\in (W\cap C^\infty_{{\mathcal K}, 0}(\overline{U_X\times V_{Y, t}}))\times (L_{Y, t}^2(V_{Y, t}, L^2_X(U_X)))^m\times L_{Y, t}^2(V_{Y, t}, H_X^{-1}(U_X)), \end{align} (5.43)

    subject to the constraint

    \begin{equation} \nabla_X\cdot {\bf{j}} = f^* + g^*. \end{equation} (5.44)

    Next using that f\in C^\infty_{{\mathcal K}, 0}(\overline{U_X\times V_{Y, t}}) , h\in C^\infty_{X, 0}(\overline{U_X\times V_{Y, t}}) , we have

    \begin{align*} &\iint_{V_{Y, t}} -\langle (\partial_t+X\cdot\nabla_Y)f(\cdot, Y, t), (h(\cdot, Y, t)+f(\cdot, Y, t))\rangle\notag\\ & = \iint_{V_{Y, t}} \langle (\partial_t+X\cdot\nabla_Y)h(\cdot, Y, t), f(\cdot, Y, t)\rangle\, {{\text{d}}} Y{{\text{d}}} t\notag\\ &\quad {-\int_{U_X}\iint_{\partial V_{Y, t}} \bigl (\frac 1 2 f^2+fh)(X, 1)\cdot N_{Y, t}\, {{\text{d}}}\sigma_{Y, t}{{\text{d}}} X.} \end{align*}

    Using the identity in the last display we see that

    \begin{align} G^*(h)\geq \sup\limits_{(f, {\bf{j}}, f^*)} &\biggl\{ \iiint_{U_X\times V_{Y, t}} -(\tilde A(\nabla_X f, {\bf{j}}, \cdot, \cdot, \cdot)-(\nabla_X f\cdot {\bf{j}})) \, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ &+\iint_{V_{Y, t}} \langle f^*, (h(\cdot, Y, t)+f(\cdot, Y, t))\rangle +\langle (\partial_t+X\cdot\nabla_Y)h(\cdot, Y, t), f(\cdot, Y, t)\rangle\, {{\text{d}}} Y{{\text{d}}} t\\ &-\int_{U_X}\iint_{\partial V_{Y, t}} \bigl (\frac 1 2 f^2+fh)(X, 1)\cdot N_{Y, t}\, {{\text{d}}}\sigma_{Y, t}{{\text{d}}} X\biggr \}, \end{align} (5.45)

    where the supremum still is with respect to (f, {\bf{j}}, f^*) as in (5.43) subject to (5.44). Now, by arguing exactly as in the passage between displays (3.23) and (3.26) in [29], using the properties of \tilde A , we can conclude that it suffices to prove that \tilde G^*(h)\geq 0 where

    \begin{align*} \tilde G^*(h) : = \sup\limits_{(\tilde f, {\bf{j}}, f^*, b)} &\biggl\{ \iiint_{U_X\times V_{Y, t}} -(\tilde A(\nabla_X \tilde f, {\bf{j}}, \cdot, \cdot, \cdot)-(\nabla_X \tilde f\cdot {\bf{j}})) \, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\notag\\ &+\iint_{V_{Y, t}} \langle f^*, (h(\cdot, Y, t)+\tilde f(\cdot, Y, t))\rangle +\langle (\partial_t+X\cdot\nabla_Y)h(\cdot, Y, t), \tilde f(\cdot, Y, t)\rangle\, {{\text{d}}} Y{{\text{d}}} t\notag\\ &-\int_{U_X}\iint_{\partial V_{Y, t}} \bigl (\frac 1 2 b^2+bh)(X, 1)\cdot N_{Y, t}\, {{\text{d}}}\sigma_{Y, t}{{\text{d}}} X\biggr \}, \end{align*}

    and where the supremum is taken with respect to all (\tilde f, {\bf{j}}, f^*, b) in the set

    \begin{align*} (W\cap C^\infty_{X, 0}(\overline{U_X\times V_{Y, t}}))\times (L_{Y, t}^2(V_{Y, t}, L^2_X(U_X)))^m\times L_{Y, t}^2(V_{Y, t}, H_X^{-1}(U_X))\times C^\infty_{{\mathcal K}, 0}(\overline{U_X\times V_{Y, t}}), \end{align*}

    subject to the condition stated in [29], i.e., that

    \Gamma(\tilde f, b): = ||\tilde f||_{L_{Y, t}^2(V_{Y, t}, H_X^1(U_X))}+||b||_{L_{Y, t}^2(V_{Y, t}, H_X^1(U_X))}\leq\Gamma

    for some large but fixed \Gamma\geq 1 . However, this implies that \tilde f: = -h is an admissible function. With this choice of \tilde f , we then let {\bf{j}}: = A(-\nabla_X h, X, Y, t) \in (L_{Y, t}^2(V_{Y, t}, L^2_X(U_X)))^m and then

    f^* = \nabla_X\cdot {\bf{j}} -g^*\in L_{Y, t}^2(V_{Y, t}, H_X^{-1}(U_X)).

    Using this we deduce that

    \begin{align*} \tilde G^*(h) \geq \sup\limits_{b} &\biggl\{-\int_{U_X}\iint_{\partial V_{Y, t}} \frac 1 2 (b+h)^2(X, 1)\cdot N_{Y, t}\, {{\text{d}}}\sigma_{Y, t}{{\text{d}}} X\biggl \}, \end{align*}

    where supremum now is taken with respect to b\in C^\infty_{{\mathcal K}, 0}(\overline{U_X\times V_{Y, t}}) . Using Lemma 5.7 below it follows that

    \begin{align*} \sup\limits_{b} &\biggl\{-\int_{U_X}\iint_{\partial V_{Y, t}} \frac 1 2 (b+h)^2(X, 1)\cdot N_{Y, t}\, {{\text{d}}}\sigma_{Y, t}{{\text{d}}} X\biggl \}\geq 0. \end{align*}

    The proof of the lemma is therefore complete.

    Lemma 5.7. Assume that h\in W(U_X\times V_{Y, t})\cap C^\infty_{X, 0}(\overline{U_X\times V_{Y, t}}) . Then

    \begin{align} \sup\limits_{b\in W\cap C^\infty_{{\mathcal K}, 0} (U_X\times V_{Y, t})} -\iiint_{U_X\times\partial V_{Y, t}} {(b+h)^2}(X, 1)\cdot N_{Y, t}\, {{\text{d}}}\sigma_{Y, t}{{\text{d}}} X\geq 0. \end{align} (5.46)

    Lemma 5.7 is Lemma 3.7 in [29] and in the next subsection we supply parts of the proof for completion.

    Let \psi(s)\in C^\infty(\mathbb R) be such that 0\leq \psi \leq 1 ,

    \begin{equation*} \psi \equiv 1\ \mbox{on }[ 0, 1], \ \psi \equiv 0\ \mbox{on }[ 2, \infty), \end{equation*}

    |\psi'|\leq 2 and such that \sqrt{1-\psi^2}\in C^\infty(\mathbb R) . Based on \psi we introduce for r , 0\leq r < \infty

    \begin{equation} \psi_r(X, Y, t) : = \psi\biggl( r\, \frac{\big((X, 1)\cdot N_{Y, t}\big)^+}{1+|X|^2} \biggr), \end{equation} (5.47)

    where we use the notation s^+: = \max\lbrace s, 0 \rbrace for s\in \mathbb R . As h is smooth, and U_X and V_{Y, t} are bounded domains, we have

    \begin{equation} \iiint_{U_X\times \partial V_{Y, t}} h^2|(X, 1)\cdot N_{Y, t}|{{\text{d}}} X{{\text{d}}} \sigma_{Y, t} < \infty. \end{equation} (5.48)

    Let, for any r\geq 0 ,

    \begin{equation} {b_r : = (\psi_r-1)h.} \end{equation} (5.49)

    As in the proof of Lemma 3.7 in [29] it follows that

    \begin{equation} b_r\in W(U_X\times V_{Y, t}). \end{equation} (5.50)

    By construction, b_r vanishes on \partial_{\mathcal K}(U_X\times V_{Y, t}) . Together with (5.50), this yields that b_r\in W\cap C^\infty_{{\mathcal K}, 0}(\overline{U_X\times V_{Y, t}}) . Furthermore,

    \begin{equation*} \begin{split} -\iiint_{U_X\times\partial V_{Y, t}} {(b_r+h)^2}(X, 1)\cdot N_{Y, t}\, {{\text{d}}}\sigma_{Y, t}{{\text{d}}} X & = -\iiint_{U_X\times\partial V_{Y, t}} \psi_r^2h^2(X, 1)\cdot N_{Y, t}\, {{\text{d}}}\sigma_{Y, t}{{\text{d}}} X. \end{split} \end{equation*}

    Letting r\to \infty we see that

    \begin{align*} &\lim\limits_{r\rightarrow \infty} -\iiint_{U_X\times\partial V_{Y, t}} \psi_r^2h^2(X, 1)\cdot N_{Y, t}\, {{\text{d}}}\sigma_{Y, t}{{\text{d}}} X\\ & = \iiint_{U_X\times\partial V_{Y, t}} h^2\big((X, 1)\cdot N_{Y, t}\big)^+\, {{\text{d}}}\sigma_{Y, t}{{\text{d}}} X\geq 0. \end{align*}

    Retracing the argument we see by (5.26) and (5.28) that

    \begin{equation} J[f_1, g^*] = 0\mbox{ for some }f_1\in W_0. \end{equation} (5.51)

    Using Lemma 5.3 we can conclude that f_1 is the unique weak solution f_1\in W_0 to {\mathcal L} u = g^\ast in U_X\times V_{Y, t} in the sense of Eq (2.7). This completes the proof of existence and uniqueness part of Theorem 1.5. The quantitative estimate follows in the standard way.

    Assume that u\in W(U_X\times V_{Y, t}) is a weak sub-solution to the equation

    \begin{equation} \nabla_X\cdot(A(\nabla_X u, X, Y, t))-(\partial_t+X\cdot\nabla_Y)u = g^* \text{ in } \ U_X\times V_{Y, t}. \end{equation} (5.52)

    By definition this means in particular that u\in W(U_X\times V_{Y, t}) . Given u we now let v\in W(U_X\times V_{Y, t}) be the unique weak solution to the problem

    \begin{equation} \begin{cases} \nabla_X\cdot(A(\nabla_X v, X, Y, t))-(\partial_t+X\cdot\nabla_Y)v = g^* &\text{in} \ U_X\times V_{Y, t}, \\ v = u & \text{on} \ \partial_{\mathcal K}(U_X\times V_{Y, t}), \end{cases} \end{equation} (5.53)

    in the sense that

    \begin{eqnarray} v\in W(U_X\times V_{Y, t}), \ (v-u)\in W_0(U_X\times V_{Y, t}), \end{eqnarray} (5.54)

    and in the sense that (2.7) holds for all \phi\in L_{Y, t}^2(V_{Y, t}, H_{X, 0}^1(U_X)) . By Theorem 1.5 v exists and is unique. We want to prove that u\leq v a.e. in U_X\times V_{Y, t} . To achieve this we let \epsilon > 0 be arbitrary and we use the test function \phi = (u-v-\epsilon)^+ . Then \phi is a non-negative admissible test function and \phi = 0 on \partial_{\mathcal K}(U_X\times V_{Y, t}) . Hence,

    \begin{align} &\iiint_{U_X\times V_{Y, t}}A(\nabla_X u, X, Y, t)\cdot\nabla_X\phi\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ &+\iint_{V_{Y, t}}\ \langle g^\ast(\cdot, Y, t)+ (\partial_t+X\cdot\nabla_Y)u(\cdot, Y, t), \phi(\cdot, Y, t)\rangle\, {{\text{d}}} Y {{\text{d}}} t\leq 0, \end{align} (5.55)

    and

    \begin{align} &\iiint_{U_X\times V_{Y, t}}A(\nabla_X v, X, Y, t)\cdot\nabla_X\phi\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ &+\iint_{V_{Y, t}}\ \langle g^\ast(\cdot, Y, t)+ (\partial_t+X\cdot\nabla_Y)v(\cdot, Y, t), \phi(\cdot, Y, t)\rangle\, {{\text{d}}} Y {{\text{d}}} t = 0. \end{align} (5.56)

    Subtracting these relations, we get

    \begin{align} &\iiint_{U_X\times V_{Y, t}}(A(\nabla_X v, X, Y, t)-A(\nabla_X u, X, Y, t))\cdot\nabla_X\phi\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ &+\iint_{V_{Y, t}}\langle\partial_t+X\cdot\nabla_Y)(v-u)(\cdot, Y, t), \phi(\cdot, Y, t)\rangle\, {{\text{d}}} Y {{\text{d}}} t\geq 0. \end{align} (5.57)

    Using the property (1.8)- (ii) , we now first note that

    \begin{align} &\iiint_{U_X\times V_{Y, t}}(A(\nabla_X v, X, Y, t)-A(\nabla_X u, X, Y, t))\cdot\nabla_X\phi\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ &\leq -\Lambda^{-1}\iiint_{U_X\times V_{Y, t}}|\nabla_X(u-v-\epsilon)^+|^2\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t. \end{align} (5.58)

    Second, again using the definition of W(U_X\times V_{Y, t}) and that (v-u)\in W_0(U_X\times V_{Y, t}) , we see that we see that there exists a sequence \{f_j\} , f_j\in C_{{\mathcal K}, 0}^\infty(\overline{U_X\times V_{Y, t}}) such

    \begin{align} &\iint_{V_{Y, t}}(\partial_t+X\cdot\nabla_Y)(v-u)(\cdot, Y, t), \phi(\cdot, Y, t)\rangle\, {{\text{d}}} Y {{\text{d}}} t\\ & = -\lim\limits_{j\to\infty} \iiint_{U_X\times V_{Y, t}}(\partial_t+X\cdot\nabla_Y)(f_j-\epsilon)^+(f_j-\epsilon)^+\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\\ & = -\frac 12\lim\limits_{j\to\infty} \iint_{U_X\times \partial V_{Y, t}}((f_j-\epsilon)^+)^2\, (X, 1)\cdot N_{Y, t}{{\text{d}}} \sigma_{Y, t}\leq 0, \end{align} (5.59)

    as f_j = 0 on \partial_{\mathcal K}(U_X\times V_{Y, t}) . Hence, combining (5.57)–(5.59) we conclude that

    \begin{align} \iiint_{U_X\times V_{Y, t}}|\nabla_X(u-v-\epsilon)^+|^2\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\leq 0. \end{align} (5.60)

    Finally, using, for a.e. (Y, t)\in V_{Y, t} , the Poincaré inequality on U_X we deduce from (5.60) that

    \begin{align} \iiint_{U_X\times V_{Y, t}}|(u-v-\epsilon)^+|^2\, {{\text{d}}} X {{\text{d}}} Y {{\text{d}}} t\leq 0. \end{align} (5.61)

    Hence (u-v-\epsilon)^+ = 0 a.e. in U_X\times V_{Y, t} and hence u\leq v+\epsilon a.e. in U_X\times V_{Y, t} . We can conclude that we have proved the following theorem.

    Theorem 5.1. Let u\in W(U_X\times V_{Y, t}) be a weak sub-solution to the equation in (5.52) in the sense of Definition 4. Given u , let v\in W(U_X\times V_{Y, t}) be the unique weak solution to the problem in (5.53) in the sense of Definition 4. Then u\leq v a.e. in U_X\times V_{Y, t} . Similarly, if u\in W(U_X\times V_{Y, t}) is a weak super-solution to the equation in (5.52) in the sense of Definition 4, then v\leq u a.e. in U_X\times V_{Y, t} .

    In this paper we have initiated the study of weak solutions, and their regularity, for what we call nonlinear Kolmogorov-Fokker-Planck type equations. We believe that there are many directions to pursue in this field and in the following we formulate a number of problems.

    Let p , 1 < p < \infty , be given and let A = A(\xi, X, Y, t): \mathbb R^m\times \mathbb R^m\times \mathbb R^m\times \mathbb R\to \mathbb R^m be continuous with respect to \xi , and measurable with respect to X, Y and t . Assume that there exists a finite constant \Lambda\geq 1 such that

    \begin{align} \Lambda^{-1}|\xi|^p\leq A(\xi, X, Y, t)\cdot\xi\leq \Lambda|\xi|^p \end{align} (6.1)

    for almost every (X, Y, t)\in \mathbb R^{N+1} and for all \xi\in\mathbb{R}^m . Given A and p we introduce the operator \mathcal{L}_{A, p} through

    \begin{eqnarray} \mathcal{L}_{A, p}u: = \nabla_X\cdot (A(\nabla_X u(X, Y, t), X, Y, t))-(\partial_t+X\cdot\nabla_Y)u(X, Y, t). \end{eqnarray} (6.2)

    This defines a class of strongly degenerate nonlinear parabolic PDEs modelled on the classical PDE of Kolmogorov and the p -Laplace operator, and to our knowledge there is currently no literature devoted to these operators. The results established in this paper concern \mathcal{L}_{A, 2} assuming that A\in M(\Lambda) or A\in R(\Lambda) . We see a number of interesting research problems.

    Problem 1: Establish existence and uniqueness of weak solutions to the Dirichlet problem

    \begin{align} \begin{cases} \mathcal{L}_{A, p}u = g^*, \quad & \text{ in } U_X\times V_{Y, t}, \\ u = g , \quad & \text{ on }\partial_{\mathcal K}(U_X\times V_{Y, t}). \end{cases} \end{align} (6.3)

    Problem 2: Prove higher integrability, local boundedness, Harnack inequalities and local Hölder continuity of weak solutions for the equation \mathcal{L}_{A, p}u = 0 in the case p\neq 2 . This is a challenging problem and the first step is probably to figure out how to replace the result of Bouchut [7], or the use of the fundamental solution constructed by Kolmogorov, in this case. The problem is already very interesting for the prototype

    \begin{align} \nabla_X\cdot(|\nabla_X u(X, Y, t)|^{p-2}\nabla_X u(X, Y, t))-(\partial_t+X\cdot\nabla_Y)u(X, Y, t) = 0. \end{align} (6.4)

    Problem 3: Consider the equation in (6.4). Prove bounds for \nabla_Xu and local Hölder continuity of \nabla_Xu . Note that this must be a difficult problem in the nonlinear setting due to the lack of ellipticity in the variable Y . Again, the right place to start is probably to (simply) consider the equation

    \nabla_X\cdot(A(\nabla_Xu))-(\partial_t+X\cdot\nabla_Y)u = 0,

    where A(\xi) has linear growth, i.e., a nonlinear p = 2 case.

    Finally, we discuss the very formulation of the Dirichlet problem. Consider the geometry of U_X\times V_{Y, t} and let \Gamma: = \partial U_X\times V_{Y, t} and

    \begin{align} \Sigma^+&: = \{(X, Y, t)\in \overline{U_X}\times \partial V_{Y, t}\mid (X, 1)\cdot N_{Y, t} > 0\}, \\ \Sigma_0&: = \{(X, Y, t)\in \overline{U_X}\times \partial V_{Y, t}\mid (X, 1)\cdot N_{Y, t} = 0\}, \\ \Sigma^-&: = \{(X, Y, t)\in \overline{U_X}\times \partial V_{Y, t}\mid (X, 1)\cdot N_{Y, t} < 0\}. \end{align} (6.5)

    Using this notation \partial_{\mathcal K}(U_X\times V_{Y, t}) = \Gamma\cup \Sigma^- . Recall that W(U_X\times V_{Y, t}) is defined as the closure of C^\infty(\overline{U_X\times V_{Y, t}}) in the norm

    \begin{align} ||u||_{W(U_X\times V_{Y, t})}&: = \bigl (||u||_{L_{Y, t}^2(V_{Y, t}, H_X^1(U_X))}^2+||(\partial_t+X\cdot\nabla_Y)u||_{L_{Y, t}^2(V_{Y, t}, {H}_X^{-1}(U_X))}^2\bigr )^{1/2}. \end{align} (6.6)

    Assuming u\in W(U_X\times V_{Y, t}) , it is relevant to define and study the trace of u to \Gamma\cup \Sigma^+\cup \Sigma_0\cup\Sigma^- . We let = B^{2, 2}_{1/2}(\partial U_X) denote the Besov space defined as the trace space of H_X^1(U_X) to \partial U_X (this space is often denoted H^{1/2}(\partial U_X) in the literature). It is well known, that if U_X is a bounded Lipschitz domain, then there exists a bounded continuous non-injective operator T:H_X^1(U_X)\to B^{2, 2}_{1/2}(\partial U_X) , called the trace operator, and a bounded continuous operator E: B^{2, 2}_{1/2}(\partial U_X)\to H_X^1(U_X) called the extension operator. The trace space of L_{Y, t}^2(V_{Y, t}, H_X^1(U_X)) on \Gamma is therefore L_{Y, t}^2(V_{Y, t}, B^{2, 2}_{1/2}(\partial U_X)) and

    ||u||_{L_{Y, t}^2(V_{Y, t}, B^{2, 2}_{1/2}(\partial U_X))}\leq c||u||_{W(U_X\times V_{Y, t})}.

    The trace to \Sigma^+\cup \Sigma_0\cup\Sigma^- is less clear. Indeed, recall that the space W_{X, 0}(U_X\times V_{Y, t}) is defined as the closure in the norm of W(U_X\times V_{Y, t}) of C_{X, 0}^\infty(\overline{U_X\times V_{Y, t}}) . In particular, given f\in W_{X, 0}(U_X\times V_{Y, t}) there exists \{f_j\} , f_j\in C_{X, 0}^\infty(\overline{U_X\times V_{Y, t}}) such that

    ||f-f_j||_{W(U_X\times V_{Y, t})}\to 0\mbox{ as }j\to \infty,

    and consequently,

    ||(\partial_t+X\cdot\nabla_Y)(f-f_j)||_{L_{Y, t}^2(V_{Y, t}, {H}_X^{-1}(U_X))}\to 0\mbox{ as }j\to \infty.

    Using this we see that

    \begin{align} &\iint_{V_{Y, t}} \langle(\partial_t+X\cdot\nabla_Y)f(\cdot, Y, t), f(\cdot, Y, t)\rangle \, {{\text{d}}} Y{{\text{d}}} t\\ & = \lim\limits_{j\to\infty} \iint_{V_{Y, t}} \langle(\partial_t+X\cdot\nabla_Y)f_j(\cdot, Y, t), f_j(\cdot, Y, t)\rangle \, {{\text{d}}} Y{{\text{d}}} t\\ & = \frac 12\lim\limits_{j\to\infty} \iint_{\Sigma^+\cup \Sigma_0\cup\Sigma^-}f_j^2(X, Y, t)\, (X, 1)\cdot N_{Y, t}{{\text{d}}} \sigma_{Y, t}. \end{align} (6.7)

    The first obstruction to a trace inequality is that (X, 1)\cdot N_{Y, t}{{\text{d}}} \sigma_{Y, t} is a signed measure on \Sigma^+\cup \Sigma_0\cup\Sigma^- . Assuming that f\in W_{0}(U_X\times V_{Y, t}) we deduce that

    \begin{align} &\iint_{V_{Y, t}} \langle(\partial_t+X\cdot\nabla_Y)f(\cdot, Y, t), f(\cdot, Y, t)\rangle \, {{\text{d}}} Y{{\text{d}}} t \end{align} (6.8)
    \begin{align} & = \frac 12\lim\limits_{j\to\infty} \iint_{\Sigma^+\cup \Sigma_0}f_j^2(X, Y, t)\, (X, 1)\cdot N_{Y, t}{{\text{d}}} \sigma_{Y, t}. \end{align} (6.9)

    Hence, in this case

    \begin{align} \lim\limits_{j\to\infty} \iint_{\Sigma^+\cup \Sigma_0}f_j^2(X, Y, t)\, (X, 1)\cdot N_{Y, t}{{\text{d}}} \sigma_{Y, t}\leq c||f||_{W(U_X\times V_{Y, t})}, \end{align} (6.10)

    and we see that we can extract a subsequence of \{f_j\} converging in L^2(K, (X, 1)\cdot N_{Y, t}{{\text{d}}} \sigma_{Y, t}) whenever K is a compact subset of \Sigma^+ . At the expense of additional notation the roles of \Sigma^+ and \Sigma^- can be interchanged in this argument. This observation highlights the difficulty concerning the possibility of a trace inequality and concerning the identification of the trace space for W(U_X\times V_{Y, t}) . This explains why we in this paper, as in [29], have used the weaker formulation of the Dirichlet problem introduced.

    Problem 4: What function space is the space of traces, to \Gamma\cup \Sigma^+\cup \Sigma_0\cup\Sigma^- , of W(U_X\times V_{Y, t}) ?

    The authors declare no conflict of interest.



    [1] European Commission, Guidelines concerning the exercise of the free movement of workers during COVID-19 outbreak, 2020. Available from: https://eur-lex.europa.eu/legal-content/EN/TXT/?uri = CELEX%3A52020XC0330%2803%29.
    [2] Lynggaard K, Jensen M, Kluth M (2022) Governments' responses to the Covid-19 pandemic in Europe: Navigating the Perfect Storm, Basingstoke: Palgrave Macmillan.
    [3] Alfano V, Ercolano S, Pinto M (2022) Fighting the COVID pandemic: National policy choices in non-pharmaceutical interventions. J Policy Model 1: 22–40. https://doi.org/10.1016/j.jpolmod.2021.11.003 doi: 10.1016/j.jpolmod.2021.11.003
    [4] Gammeltoft-Hansen T, Nyberg-Sørensen N (2013) The migration industry and the commercialization of international migration, London and New York: Routledge.
    [5] McCollum D, Findlay A (2018) Oiling the wheels? Flexible labour markets and the migration industry. J Ethn Migr Stud 44: 558–574. https://doi.org/10.1080/1369183X.2017.1315505 doi: 10.1080/1369183X.2017.1315505
    [6] Tagliacozzo S, Pisacane L, Kilkey M (2021) The interplay between structural and systemic vulnerability during the COVID-19 pandemic: Migrant agricultural workers in informal settlements in Southern Italy. J Ethn Migr Stud 47: 1903–1921. https://doi.org/10.1080/1369183X.2020.1857230 doi: 10.1080/1369183X.2020.1857230
    [7] Szytniewski B, van der Haar M (2022) Mobility power in the migration industry: Polish workers' trajectories in the Netherlands. J Ethn Migr Stud 48: 4694–4711. https://doi.org/10.1080/1369183X.2022.2061931 doi: 10.1080/1369183X.2022.2061931
    [8] Shubin S, McCollum D (2021) Migrant subjectivities and temporal flexibility of East-Central European labour migration to the United Kingdom. Popul Space Place 27: e2508. https://doi.org/10.1002/psp.2508 doi: 10.1002/psp.2508
    [9] Maeva M, Zahova S (2013) Ethography of migrations: Bulgarians in the Mediterranean, Sofia: Paradigma.
    [10] Minchev V, Markova E, Misheva M, et al. (2012) Bulgarian emigration: theories, policies, empirical studies. Sofia: IKOPIS.
    [11] Rangelova R, Vladimirova K (2004) Migration from central and eastern Europe: the case of Bulgaria. South-East Eur Rev Labour Soc Affairs 3: 7–30.
    [12] Vladimirova K (2020) Migration and Economic Development: Opportunities and Challenges for Bulgaria. Naselenie 3: 93–115.
    [13] Alfano V, Capasso S, Goel RK (2021) EU accession: A boon or bane for corruption? J Econ Finan 45: 1–21. https://doi.org/10.1007/s12197-020-09522-8 doi: 10.1007/s12197-020-09522-8
    [14] Charron N, Lapuente V, Bauhr M (2021) Sub-national Quality of Government in EU Member States: Presenting the 2021 European Quality of Government Index and its relationship with Covid-19 indicators. Gothenburg: QoG. Available from: https://www.gu.se/sites/default/files/2021-05/2021_4_%20Charron_Lapuente_Bauhr.pdf.
    [15] Dorn D, Zweimüller J (2021) Migration and labor market integration in Europe. J Econ Persp 35: 49–76. https://doi.org/10.1257/jep.35.2.49 doi: 10.1257/jep.35.2.49
    [16] Garrote Sanchez D, Kreuder J, Testaverde M (2021) Migration in Bulgaria: Current Challenges and Opportunities. Discussion Paper 2109. Washington DC: The World Bank. Available from: https://documents1.worldbank.org/curated/en/914401640249485571/pdf/Migration-in-Bulgaria-Current-Challenges-and-Opportunities.pdf.
    [17] National Statistical Institute of the Republic of Bulgaria, Migration and migration behaviour: Sample survey accompanying Census 2021, 2023. Available from: https://www.nsi.bg/sites/default/files/files/pressreleases/Census2021_Migration_SampleSurvey.pdf.
    [18] Kovacheva S, Popivanov B, Burcea M (2019) (Self-) Reflecting on International Recruitment: Views on the Role of Recruiting Agencies in Bulgaria and Romania. Soc Inclusion 7: 49–59. https://doi.org/10.17645/si.v7i4.2287 doi: 10.17645/si.v7i4.2287
    [19] Salt J, Stein J (1997) Migration as a Business: The Case of Trafficking. Int Migr 35: 467–494. https://doi.org/10.1111/1468-2435.00023 doi: 10.1111/1468-2435.00023
    [20] Castles S, Miller M (2003) The Age of Migration: International Population Movements in the Modern World, Third Edition, New York: The Guilford Press.
    [21] Faist T (2014) Brokerage in Cross-Border Mobility: Social Mechanisms and the (Re)Production of Social Inequalities. Soc Incl 4: 38–52.
    [22] Cranston S (2018) Calculating the migration industries: Knowing the successful expatriate in the global mobility industry. J Ethn Migr Stud 4: 626–643.
    [23] Sporton D (2013) 'They control my life': The role of local recruitment agencies in East European migration to the UK. Pop Space Place 5: 443–458.
    [24] Jones K (2014) 'It was a whirlwind. A lot of people made a lot of money': The role of agencies in facilitating migration from Poland into the UK between 2004 and 2008. Central East Eur Migr Rev 2: 105–125.
    [25] Seale H, Dyer CEF, Abdi I, et al. (2020) Improving the impact of non-pharmaceutical interventions during COVID-19: examining the factors that influence engagement and the impact on individuals. BMC Infect Dis 20: 607. https://doi.org/10.1186/s12879-020-05340-9 doi: 10.1186/s12879-020-05340-9
    [26] Maloney W, Taskin T (2020) Determinants of Social Distancing and Economic Activity during COVID-19: A Global View. Policy Research Working Paper 9242. Washington, DC: The World Bank.
    [27] Kantor BN, Kantor J (2020) Non-pharmaceutical interventions for pandemic COVID-19: a cross-sectional investigation of US general public beliefs, attitudes, and actions. Front Med 7: 384. https://doi.org/10.3389/fmed.2020.00384 doi: 10.3389/fmed.2020.00384
    [28] Dryhurst S, Schneider CR, Kerr J, et al. (2020) Risk perceptions of COVID-19 around the world. J Risk Res 23: 994–1006. https://doi.org/10.1080/13669877.2020.1758193 doi: 10.1080/13669877.2020.1758193
    [29] Gauvin L, Bajardi P, Pepe E, et al. (2021) Socio-economic determinants of mobility responses during the first wave of COVID-19 in Italy: from provinces to neighbourhoods. J R Soc Interface 18: 20210092. https://doi.org/10.1098/rsif.2021.0092 doi: 10.1098/rsif.2021.0092
    [30] Grasso M, Klicperová-Baker M, Koos S, et al. (2021) The impact of the coronavirus crisis on European societies. What have we learnt and where do we go from here? Eur Soc 23: S2–S32. https://doi.org/10.1080/14616696.2020.1869283
    [31] Fasani F, Mazza J (2020) A vulnerable workforce: migrant workers in the COVID-19 pandemic. Luxembourg: EC Publications Office. Available from: https://data.europa.eu/doi/10.2760/316665
    [32] Ulceluse M, Bender F (2022) Two-tier EU citizenship: Disposable Eastern European workers during the COVID-19 pandemic. Organization 3: 449–459. https://doi.org/10.1177/13505084211061229 doi: 10.1177/13505084211061229
    [33] Deneva N (2021) Essential Workers with Dangerous Bodies: The COVID-19 Crisis and East European Migrants. Soc Probl 2: 538–560.
    [34] Brändle V (2022) Well Informed? EU Governments' Digital Information Campaigns for (Potential) Migrants. J Immigr Refug Stud. https://doi.org/10.1080/15562948.2022.2046896
    [35] Collins F (2021) Geographies of migration I: Platform migration. Prog Hum Geog 45: 866–877. https://doi.org/10.1177/0309132520973445 doi: 10.1177/0309132520973445
    [36] Hooper K (2022) Reassessing Recruitment Costs in a Changing World of Labor Migration. Policy Brief. Washington: Migration Policy Institute.
    [37] Leiblfinger M, Prieler V, Rogoz M, et al. (2021) Confronted with COVID-19: Migrant live-in care during the pandemic. Global Soc Policy 21: 490–507. https://doi.org/10.1177/14680181211008340 doi: 10.1177/14680181211008340
    [38] Heindlmaier A, Kobler C (2022) Essential, lonely and exploited: why mobile EU workers' labour rights are not enforced. J Ethn Migr Stud, 1–20. https://doi.org/10.1080/1369183X.2022.2102971
    [39] Faraday F (2021) The Empowerment of Migrant Workers in a Precarious Situation: An overview. World Bank, Washington, DC. Available from: https://www.knomad.org/sites/default/files/2021-12/KNOMAD%20Working%20Paper%2039-Empowerment%20of%20Worker-Dec%202021.pdf
    [40] Ortlieb R, Weiss S (2018) How do labor market intermediaries help young Eastern Europeans find work? Youth Labor in Transition: Inequalities, Mobility, and Policies in Europe, New York: Oxford Academic, 443–460.
    [41] Iossa A, Selberg N (2022) Socio-Legal Aspects of Labour Market Segmentation in the Agri-Food Sector in Sweden: Spatio-Temporal Dimensions. Eur J Migr Law 24: 241–264. https://doi.org/10.1163/15718166-12340128 doi: 10.1163/15718166-12340128
    [42] Blauberger M, Grabbe C, Servent A (2023) EU free movement of people: fully recovered or suffering from long COVID? J Eur Public Policy 30: 696–720. https://doi.org/10.1080/13501763.2022.2140818 doi: 10.1080/13501763.2022.2140818
    [43] Glaser B, Strauss A (1967) The discovery of grounded theory: Strategies for qualitative research, Chicago, IL: Aldine.
    [44] Kovacheva S, Popivanov B (2021) On the Road. Migration in the Lived Experiences of Bulgarian Migrants in the EU, Sofia: St. Kliment Ohridski University Press.
    [45] Hooper K, Benton M (2022) The Future of Remote Work: Digital Nomads and the Implications for Immigration Systems. Washington, DC: Migration Policy Institute. Available from: https://www.migrationpolicy.org/sites/default/files/publications/mpi-remote-work-2022_final.pdf.
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