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Research article

Effect of Hydrological Properties on the Energy Shares of Reflected Waves at the Surface of a Partially Saturated Porous Solid

  • Received: 22 October 2016 Accepted: 23 January 2017 Published: 25 February 2017
  • In the present study, the reflection of inhomogeneous waves is investigated at the stress-free plane surface based on multiphase poroelasticity theory. The porous medium is considered as dissipative due to the presence of viscosity in pores fluid. Four inhomogeneous (i.e. different direction of propagation and attenuation) reflected waves (three longitudinal and one shear) exists due to an incident wave. By using the appropriate boundary conditions, closed-form analytical expressions for the reflection coeffcients are derived at the stress-free surface. These reflection coeffcients are used to drive the analytical expressions for the energy shares of various reflected inhomogeneous waves. In mathematical framework, the conservation of incident energy is confirmed by considering an interaction energy between two dissimilar waves. It validates that the numerical calculations are analytically correct. Finally, a numerical example is considered to study the effects of viscous cross-coupling, porosity, saturation of gas, pore-characteristics and wave frequency on the energy shares of various reflected inhomogeneous waves and depicted graphically.

    Citation: Mahabir Barak, Manjeet Kumari, Manjeet Kumar. Effect of Hydrological Properties on the Energy Shares of Reflected Waves at the Surface of a Partially Saturated Porous Solid[J]. AIMS Geosciences, 2017, 3(1): 67-90. doi: 10.3934/geosci.2017.1.67

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  • In the present study, the reflection of inhomogeneous waves is investigated at the stress-free plane surface based on multiphase poroelasticity theory. The porous medium is considered as dissipative due to the presence of viscosity in pores fluid. Four inhomogeneous (i.e. different direction of propagation and attenuation) reflected waves (three longitudinal and one shear) exists due to an incident wave. By using the appropriate boundary conditions, closed-form analytical expressions for the reflection coeffcients are derived at the stress-free surface. These reflection coeffcients are used to drive the analytical expressions for the energy shares of various reflected inhomogeneous waves. In mathematical framework, the conservation of incident energy is confirmed by considering an interaction energy between two dissimilar waves. It validates that the numerical calculations are analytically correct. Finally, a numerical example is considered to study the effects of viscous cross-coupling, porosity, saturation of gas, pore-characteristics and wave frequency on the energy shares of various reflected inhomogeneous waves and depicted graphically.


    Well-posedness of final value problems for a large class of parabolic differential equations was recently obtained in a joint work of the author and given an ample description for a broad audience in [5], after the announcement in [4]. The present paper substantiates the indications made in the concise review [21], namely, that the abstract parts in [5] extend from V-elliptic Lax–Milgram operators A to those that are merely V-coercive-despite that such A may be non-injective.

    As an application, the final value heat conduction problem with the homogeneous Neumann condition is shown to be well-posed.

    The basic analysis is made for a Lax–Milgram operator A defined in H from a V-coercive sesquilinear form a in a Gelfand triple, i.e., three separable, densely injected Hilbert spaces VHV having norms , || and , respectively. Hereby V is the form domain of a; and V the antidual of V. Specifically there are constants Cj>0 and kR such that all u,vV satisfy vC1|v|C2v and

    |a(u,v)|C3uv,a(v,v)C4v2k|u|2. (1)

    In fact, D(A) consists of those uV for which a(u,v)=(f|v) for some fH and for all vV, and Au=f; hereby (u|v) denotes the inner product in H. There is also an extension AB(V,V) given by Au,v=a(u,v) for u,vV. This is uniquely determined as D(A) is dense in V.

    Both a and A are referred to as V-elliptic if the above holds for k=0; then AB(V,V) is a bijection. One may consult the book of Grubb [12] or that of Helffer [14], or [5], for more details on the set-up and basic properties of the unbounded, but closed operator A in H. Especially A is self-adjoint in H if and only if a(v,w)=¯a(w,v), which is not assumed; A may also be nonnormal in general.

    In the framework of such a triple (H,V,a), the general final value problem is this: for given data fL2(0,T;V) and uTH, determine the uD(0,T;V) such that

    {tu+Au=fin D(0,T;V),u(T)=uTin H. (2)

    By definition of Schwartz' vector distribution space D(0,T;V) as the space of continuous linear maps C0(]0,T[)V, cf. [28], the above equation means that for every scalar test function φC0(]0,T[) the identity u,φ+Au,φ=f,φ holds in V.

    As is well known, a wealth of parabolic Cauchy problems with homogeneous boundary conditions have been treated via triples (H,V,a) and the D(0,T;V) set-up in (2); cf. the work of Lions and Magenes [24], Tanabe [30], Temam [31], Amann [2] etc.

    The theoretical analysis made in [4,5,21] shows that, in the V-elliptic case, the problem in (2) is well posed, i.e., it has existence, uniqueness and stability of a solution uX for given data (f,uT)Y, in certain Hilbertable spaces X, Y that were described explicitly. Hereby the data space Y is defined in terms of a particular compatibility condition, which was introduced for the purpose in [4,5]. More precisely, there is even a linear homeomorphism XY, which yields well-posedness in a strong form.

    This has seemingly closed a gap in the theory, which had remained since the 1950's, even though the well-posedness is decisive for the interpretation and accuracy of numerical schemes for the problem (the work of John [19] was pioneering, but also Eldén [8] could be mentioned). In rough terms, the results are derived from a useful structure on the reachable set for a general class of parabolic differential equations.

    The main example treated in [4,5] is the heat conduction problem of characterising the u(t,x) that in a C-smooth bounded open set ΩRn with boundary Γ=Ω fulfil the equations (for Δ=2x1++2xn),

    {tu(t,x)Δu(t,x)=f(t,x)for t]0,T[xΩ,u(t,x)=g(t,x)for t]0,T[xΓ,u(T,x)=uT(x)for xΩ. (3)

    An area of interest of this could be a nuclear power plant hit by a power failure at t=0: after power is regained at t=T>0, and the reactor temperatures uT(x) are measured, a calculation backwards in time could possibly settle whether at some t0<T the temperatures u(t0,x) could cause damage to the fuel rods.

    However, the Dirichlet condition u=g at the boundary Γ is of limited physical importance, so an extension to, e.g., the Neumann condition, which represents controlled heat flux at Γ, makes it natural to work out an extension to V-coercive Lax–Milgram operators A.

    In this connection it should be noted that when A is V-coercive (that is, satisfies (1) only for some k>0), it is possible that 0σ(A), the spectrum of A, for example because λ=0 is an eigenvalue of A. In fact, this is the case for the Neumann realisation ΔN, which has the space of constant functions C1Ω as the null space. Well-posedness is obtained for the heat problem (3) with a replacement of the Dirichlet condition by the homogeneous Neumann condition in Section 4 below.

    At first glance, it may seem surprising that the possible non-injectivity of the coercive operator A is inconsequential for the well-posedness of the final value problem (2). In particular this means that the backward uniqueness-u(T)=0 in H implies u(t)=0 in H for 0t<T-of the equation u+Au=f will hold regardless of whether A is injective or not. This can be seen from the extensions of the abstract theory given below; in particular when the results are applied in Section 4 to the case A=ΔN.

    The point of departure is to make a comparison of (2) with the corresponding Cauchy problem for the equation u+Au=f. For this it is classical to seek solutions u in the Banach space

    X=L2(0,T;V)C([0,T];H)H1(0,T;V),uX=(T0u(t)2dt+sup0tT|u(t)|2+T0(u(t)2+u(t)2)dt)1/2. (4)

    In fact, the following result is essentially known from the work of Lions and Magenes [24]:

    Proposition 1. Let V be a separable Hilbert space with VH algebraically, topologically and densely, and let A denote the Lax–Milgram operator induced by a V-coercive, bounded sesquilinear form on V, as well as its extension AB(V,V). When u0H and fL2(0,T;V) are given, then there is a uniquely determined solution u belonging to X, cf. (4), of the Cauchy problem

    {tu+Au=fin D(0,T;V),u(0)=u0in H. (5)

    The solution operator (f,u0)u is continuous L2(0,T;V)HX, and problem (5) is well-posed.

    Remarks on the classical reduction from the V-coercive case to the elliptic one will follow in Section 3. The stated continuity of the solution operator is well known to the experts. But for the reader's convenience, in Proposition 7 below, the continuity is shown by explicit estimates using Grönwall's lemma; these may be of independent interest.

    Whilst the below expression for the solution hardly is surprising at all, it has seemingly not been obtained hitherto in the present context of V-coercive Lax–Milgram operators A and general triples (H,V,a):

    Proposition 2. The unique solution u in X provided by Proposition 1 is given by Duhamel's formula,

    u(t)=etAu0+t0e(ts)Af(s)dsfor0tT. (6)

    Here each of the three terms belongs to X.

    As shown in Section 3 below, it suffices for (6) to reinforce the classical integration factor technique by injectivity of the semigroup etA.

    In fact, it is exploited in (6) and throughout that A generates an analytic semigroup ezA in B(H). As a consequence of the analyticity, the family of operators ezA was shown in [5] to consist of injections on H in case A is V-elliptic. This extends to general V-coercive A, as accounted for in Proposition 5 below. Hence etA also in the present paper has the inverse etA:=(etA)1 for t>0.

    For t=T, the Duhamel formula (6) obviously yields a bijection u(0)u(T) between the initial and terminal states (for fixed f), as one can solve for u0 by means of the inverse eTA. In particular backwards uniqueness of the solutions to u+Au=f holds in the large class X.

    Returning to the final value problem (2) it would be natural to seek solutions u in the same space X. This turns out to be possible only when the data (f,uT) are subjected to substantial further conditions.

    To formulate these, it is noted that the above inverse etA enters the theory through its domain, which in the algebraic sense simply is a range, namely D(etA)=R(etA); but this domain has the structural advantage of being a Hilbert space under the graph norm u=(|u|2+|etAu|2)1/2.

    For t=T the domains D(eTA) have a decisive role in the well-posedness result below, where condition (9) is a fundamental clarification for the final value problem in (2) and the parabolic problems it represents.

    Another ingredient in (9) is the yield yf of the source term f:]0,T[V, i.e.

    yf=T0e(Tt)Af(t)dt. (7)

    Hereby it is used that etA extends to an analytic semigroup in V, as the extension AB(V,V) is an unbounded operator in the Hilbertable space V satisfying the necessary estimates (cf. Remark 4; and also [5,Lem. 5] for the extension). So yf is a priori a vector in V, but in fact yf lies in H as Proposition 2 shows it equals the final state of a solution in C([0,T],H) of a Cauchy problem having u0=0.

    These remarks on yf make it clear that in the following main result of the paper-which relaxes the assumption of V-ellipticity in [4,5] to V-coercivity-the difference in (9) is a member of H:

    Theorem 1.1. Let A be a V-coercive Lax–Milgram operator defined from a triple (H,V,a) as above. Then the abstract final value problem (2) has a solution u(t) belonging the space X in (4), if and only if the data (f,uT) belong to the subspace

    YL2(0,T;V)H (8)

    defined by the condition

    uTT0e(Tt)Af(t)dt  D(eTA). (9)

    In the affirmative case, the solution u is uniquely determined in X and

    uXc(|uT|2+T0f(t)2dt+|eTA(uTT0e(Tt)Af(t)dt)|2)1/2=c(f,uT)Y, (10)

    whence the solution operator (f,uT)u is continuous YX. Moreover,

    u(t)=etAeTA(uTT0e(Tt)Af(t)dt)+t0e(ts)Af(s)ds, (11)

    where all terms belong to X as functions of t[0,T], and the difference in (9) equals eTAu(0) in H.

    The norm on the data space Y in (10) is seen at once to be the graph norm of the composite map

    L2(0,T;V)HΦHeTAH (12)

    given by (f,uT) uTyf  eTA(uTyf) and Φ(f,uT)=uTyf.

    In fact, the solvability criterion (9) means that eTAΦ must be defined at (f,uT), so the data space Y is its domain. Being an inverse, eTA is a closed operator in H, and so is eTAΦ; hence Y=D(eTAΦ) is complete. Now, since in (10) the Banach space V is Hilbertable, so is Y.

    Thus the unbounded operator eTAΦ is a key ingredient in the rigorous treatment of the final value problem (2). In control theoretic terms, the role of eTAΦ is to provide the unique initial state given by

    u(0)=eTAΦ(f,uT)=eTA(uTyf), (13)

    which is steered by f to the final state u(T)=uT; cf. the Duhamel formula (6).

    Criterion (9) is a generalised compatibility condition on the data (f,uT); such conditions have long been known in the theory of parabolic problems, cf. Remark 7. The presence of e(Tt)A and the integral over [0,T] makes (9) non-local in both space and time. This aspect is further complicated by the reference to D(eTA), which for larger final times T typically gives increasingly stricter conditions:

    Proposition 3. If the spectrum σ(A) of A is not contained in the strip {zC|kzk}, whereby k is the constant from (1), then the domains D(etA) form a strictly descending chain, that is,

    HD(etA)D(etA)for 0<t<t. (14)

    This results from the injectivity of etA via well-known facts for semigroups reviewed in [5,Thm. 11] (with reference to [26]). In fact, the arguments given for k=0 in [5,Prop. 11] apply mutatis mutandis.

    Now, (6) also shows that u(T) has two radically different contributions, even if A has nice properties. First, for t=T the integral equals yf, which can be anywhere in H. Indeed, fyf is a continuous surjection yf:L2(0,T;V)H. This was shown for k=0 via the Closed Range Theorem in [5,Prop. 5], and for k>0 surjectivity follows from this case as e(Ts)Af(s)=e(Ts)(A+kI)eskf(s) in (7), whereby A+kI is V-elliptic and feskf is a bijection on L2(0,T;V).

    Secondly, etAu(0) solves u+Au=0, and for u(0)0 and V-elliptic A it is a precise property in non-selfadjoint dynamics that the ``height'' h(t)=|etAu(0)| is

    strictly positive(h>0),strictly decreasing(h<0),strictlyconvex(h>0).

    Whilst this holds if A is self-adjoint or normal, it was emphasized in [5] that it suffices that A is just hyponormal (i.e., D(A)D(A) and |Ax||Ax| for xD(A), following Janas [18]). Recently this was followed up by the author in [20], where the stronger logarithmic convexity of h(t) was proved equivalent to the formally weaker property of A that, for xD(A2),

    2((Ax|x))2(A2x|x)|x|2+|Ax|2|x|2. (15)

    For V-coercive A only the strict decrease may need to be relinquished. Indeed, the strict positivity h(t)>0 follows by the injectivity of etA in Proposition 5 below. Moreover, the characterisation in [20,Lem. 2.2] of the log-convex C2-functions f(t) on [0,[ as the solutions of the differential inequality ff(f)2 and the resulting criterion for A in (15) apply verbatim to the coercive case; hereby the differential calculus in Banach spaces is exploited in a classical derivation of the formulae for u(t)=etAu(0),

    h(t)=(Au(t)|u(t))|u(t)|, (16)
    h(t)=(A2u(t)|u(t))+|Au(t)|2|u(t)|((Au(t)|u(t)))2|u(t)|3. (17)

    But it is due to the strict positivity |etAu(0)|>0 for t0 in the denominators that the expressions make sense, so injectivity of etA also enters crucially at this point. Similarly the singularity of || at the origin poses no problems for the mere differentiation of h(t). Therefore it is likely that the natural formulas for h, h have not been rigorously proved before [21]. These remarks also shed light on the usefulness of Proposition 5 below.

    However, the stiffness intrinsic to strict convexity, hence to log-convexity, corresponds well with the fact that u(T)=eTAu(0) in any case is confined to a dense, but very small space, as by the analyticity

    u(T)nND(An). (18)

    For u+Au=f, the possible uT will hence be a sum of some arbitrary yfH and a stiff term eTAu(0). Thus uT can be prescribed in the affine space yf+D(eTA). As any yf0 will shift D(eTA)H in an arbitrary direction, u(T) can be expected anywhere in H (unless yfD(eTA) is known). So neither (18) nor u(T)D(eTA) can be expected to hold if yf0-not even if |yf| is much smaller than |eTAu(0)|. Hence it seems best for final value problems to consider inhomogeneous problems from the outset.

    Remark 1. To give some backgound, two classical observations for the homogeneous case f=0, g=0 in (3) are recalled. First there is the smoothing effect for t>0 of parabolic Cauchy problems, which means that u(t,x)C(]0,T]ׯΩ) whenever u0L2(Ω). (Rauch [27,Thm. 4.3.1] has a version for Ω=Rn; Evans [9,Thm. 7.1.7] gives the stronger result uC([0,T]ׯΩ) when fC([0,T]ׯΩ), g=0 and u0C(¯Ω) fulfill the classical compatibility conditions at {0}×Γ-which for f=0, g=0 gives the C property on [ε,T]×¯Ω for any ε>0, hence on ]0,T]ׯΩ). Therefore u(T,)C(¯Ω); whence (3) with f=0, g=0 cannot be solved if uT is prescribed arbitrarily in L2(Ω). But this just indicates an asymmetry in the properties of the initial and final value problems.

    Secondly, there is a phenomenon of L2-instability in case f=0, g=0 in (3), which perhaps was first described by Miranker [25]. The instability is found via the Dirichlet realization of the Laplacian, ΔD, and its L2(Ω)-orthonormal basis e1(x),e2(x), of eigenfunctions associated to the usual ordering of its eigenvalues 0<λ1λ2, which via Weyl's law for the counting function, cf. [6,Ch. 6.4], gives

    λj=O(j2/n) for j. (19)

    This basis gives rise to a sequence of final value data uT,j(x)=ej(x) lying on the unit sphere in L2(Ω) as uT,j=ej=1 for jN. But the corresponding solutions to uΔu=0, i.e. uj(t,x)=e(Tt)λjej(x), have initial states u(0,x) with L2-norms that because of (19) grow rapidly with the index j,

    uj(0,)=eTλjej=eTλj. (20)

    This L2-instability cannot be removed, of course, but it only indicates that the L2(Ω)-norm is an insensitive choice for problem (3). The task is hence to obtain a norm on uT giving better control over the backward calculations of u(t,x)-for the inhomogeneous heat problem (3), an account of this was given in [5].

    Remark 2. Almog, Grebenkov, Helffer, Henry [1,10,11] studied the complex Airy operator Δ+ix1 recently via triples (H,V,a), leading to Dirichlet, Neumann, Robin and transmission boundary conditions, in bounded and unbounded regions. Theorem 1.1 is expected to apply to final value problems for those of their realisations that satisfy the coercivity condition in (1). However, Δ+ix1 has empty spectrum on Rn, cf. the fundamental paper of Herbst [15], so it remains to be seen for which of the regions in [1,10,11] there is a strictly descending chain of domains as in (14).

    As indicated in the introduction, it is central to the analysis of final value problems that an analytic semigroup of operators, like etΔD, always consists of injections. This shows up both at the technical and conceptual level, that is, both in the proofs and in the objects that enter the theorem.

    A few aspects of semigroup theory in a complex Banach space B is therefore recalled. Besides classical references by Davies [7], Pazy [26], Tanabe [30] or Yosida [32], a more recent account is given in [3].

    The generator is Ax=limt0+1t(etAxx), where x belongs to the domain D(A) when the limit exists. A is a densely defined, closed linear operator in B that for some ω0, M1 satisfies the resolvent estimates (Aλ)nB(B)M/(λω)n for λ>ω, nN.

    The corresponding C0-semigroup of operators etAB(B) is of type (M,ω): it fulfils that etAesA=e(s+t)A for s,t0, e0A=I and limt0+etAx=x for xB; whilst

    etAB(B)Meωt for 0t<. (21)

    Indeed, the Laplace transformation (λIA)1=0etλetAdt gives a bijection of the semigroups of type (M,ω) onto (the resolvents of) the stated class of generators.

    To elucidate the role of injectivity, recall that if etA is analytic, u=Au, u(0)=u0 is uniquely solved by u(t)=etAu0 for every u0B. Here injectivity of etA is equivalent to the important geometric property that the trajectories of two solutions etAv and etAw of u=Au have no confluence point in B for vw.

    Nevertheless, the literature seems to have focused on examples of semigroups with non-invertibility of etA, like [26,Ex. 2.2.1]; these necessarily concern non-analytic cases. The well-known result below gives a criterion for A to generate a C0-semigroup ezA that is defined and analytic for z in the open sector

    Sθ={zC|z0, |argz|<θ}. (22)

    It is formulated in terms of the spectral sector

    Σθ={0}{λC||argλ|<π2+θ}. (23)

    Proposition 4. If A generates a C0-semigroup of type (M,ω) and ωρ(A), the following properties are equivalent for each θ]0,π2[:

    (ⅰ) The resolvent set ρ(A) contains ω+Σθ and

    sup{|λω|(λIA)1B(B)|λω+Σθ, λω}<. (24)

    (ⅱ) The semigroup etA extends to an analytic semigroup ezA defined for zSθ with

    sup{ezωezAB(B)|z¯Sθ}<whenever 0<θ<θ. (25)

    In the affirmative case, etA is differentiable in B(B) for t>0 with derivative (etA)=AetA, and for every η such that α(A)<η<ω one has

    supt>0etηetAB(B)+supt>0tetηAetAB(B)<, (26)

    whereby α(A)=supσ(A) denotes the spectral abscissa of A (here α(A)<ω, as 0Σθ).

    In case ω=0, the equivalence is just a review of the main parts of Theorem 2.5.2 in [26]. For general ω0, one can reduce to this case, since A=ωI+(AωI) yields the operator identity etA=etωet(AωI), where et(AωI) is of type (M,0) for some M. Indeed, the right-hand side is easily seen to be a C0-semigroup, which since etω=1+tω+o(t) has A as its generator, so the identity results from the bijectiveness of the Laplace transform. In this way, (i)(ii) follows straightforwardly from the case ω=0, using for both implications that ezA=ezωez(AωI) holds in Sθ by unique analytic extension.

    Since ωρ(A) implies α(A)<ω, the above translation method gives etA=etηet(AηI), where et(AηI) is of type (M,0) whenever α(A)<η<ω. This yields the first part of (26), and the second now follows from this and the case ω=0 by means of the splitting A=ηI+(AηI) for α(A)<η<η.

    The reason for stating Proposition 4 for general type (M,ω) semigroups is that it shows explicitly that cases with ω>0 only have other estimates on R+ or in the closed subsectors ¯Sθ-but the mere analyticity in Sθ is unaffected by the translation by ωI. Hence one has the following improved version of [5,Prop. 1]:

    Proposition 5. If a C0-semigroup etA of type (M,ω) on a complex Banach space B has an analytic extension ezA to Sθ for some θ>0, then ezA is injective for every zSθ.

    Proof. Let ez0Au0=0 hold for some u0B and z0Sθ. Analyticity of ezA in Sθ carries over by the differential calculus in Banach spaces to the map f(z)=ezAu0. So for z in a suitable open ball B(z0,r)Sθ, a Taylor expansion and the identity f(n)(z0)=Anez0Au0 for analytic semigroups (cf. [26,Lem. 2.4.2]) give

    f(z)=n=01n!(zz0)nf(n)(z0)=n=01n!(zz0)nAnez0Au00. (27)

    Hence f0 on Sθ by unique analytic extension. Now, as etA is strongly continuous at t=0, we have u0=limt0+etAu0=limt0+f(t)=0. Thus the null space of ez0A is trivial.

    Remark 3. The injectivity in Proposition 5 was claimed in [29] for z>0, θπ/4 and B a Hilbert space (but not quite obtained, as noted in [5,Rem. 1]; cf. the details in Lemma 3.1 and Remark 3 in [20]). A local version for the Laplacian on Rn was given by Rauch [27,Cor. 4.3.9].

    As a consequence of the above injectivity, for an analytic semigroup etA we may consider its inverse that, consistently with the case in which etA forms a group in B(B), may be denoted for t>0 by etA=(etA)1. Clearly etA maps D(etA)=R(etA) bijectively onto H, and it is an unbounded, but closed operator in B.

    Specialising to a Hilbert space B=H, then also (etA)=etA is analytic, so Z(etA)={0} holds for its null space by Proposition 5; whence D(etA) is dense in H. Some further basic properties are:

    Proposition 6. [5,Prop. 2] The above inverses etA form a semigroup of unbounded operators in H,

    esAetA=e(s+t)Afor t,s0. (28)

    This extends to (s,t)R×],0], whereby e(t+s)A may be unbounded for t+s>0. Moreover, as unbounded operators the etA commute with esAB(H), that is, esAetAetAesA for t,s0, and have a descending chain of domains, HD(etA)D(etA) for 0<t<t.

    Remark 4. The above domains serve as basic structures for the final value problem (3). They apply for A=A that generates an analytic semigroup ezA in B(H) defined in Sθ for θ=arccot(C3/C4)>0. Indeed, this was shown in [5,Lem. 4] with a concise argument using V-ellipticity of A; the V-coercive case follows easily from this via the formula ezA=ekzez(A+kI) that results for z0 from the translation trick after Proposition 4; and then it defines ezA by the right-hand side for every zSθ. (A rather more involved argument was given in [26,Thm. 7.2.7] in a context of uniformly strongly elliptic differential operators.)

    To clarify a redundancy in the set-up, it is remarked here that in Proposition 1 the solution space X is a Banach space, which can have its norm in (4) rewritten in the following form, using the Sobolev space H1(0,T;V)={uL2(0,T;V)|tuL2(0,T;V)},

    uX=(u2L2(0,T;V)+sup0tT|u(t)|2+u2H1(0,T;V))1/2. (29)

    Here there is a well-known inclusion L2(0,T;V)H1(0,T;V)C([0,T];H) and an associated Sobolev inequality for vector functions ([5] has an elementary proof)

    sup0tT|u(t)|2(1+C22C21T)T0u(t)2dt+T0u(t)2dt. (30)

    Hence one can safely omit the space C([0,T];H) in (4) and remove sup[0,T]|u| from X. Similarly T0u(t)2dt is redundant in (4) because C2, so an equivalent norm on X is given by

    |||u|||X=(T0u(t)2dt+T0u(t)2dt)1/2. (31)

    Thus X is more precisely a Hilbertable space, as V is so. But the form given in (4) is preferred in order to emphasize the properties of the solutions.

    As a note on the equation u+Au=f with uX, the continuous function u:[0,T]H fulfils u(t)V for a.e. t]0,T[, so the extension AB(V,V) applies for a.e. t. Hence Au(t) belongs to L2(0,T;V).

    The existence and uniqueness statements in Proposition 1 are essentially special cases of the classical theory of Lions and Magenes, cf. [24,Sect. 3.4.4] on t-dependent V-elliptic forms a(t;u,v). Indeed, because of the fixed final time T]0,[, their indicated extension to V-coercive forms works well here: since ue±tku and fe±tk are all bijections on L2(0,T;V) and L2(0,T;V), respectively, the auxiliary problem v+(A+kI)v=ektf, v(0)=u0 has a solution vX according to the statement for the V-elliptic operator A+kI in [24,Sect. 3.4.4], when k is the coercivity constant in (1); and since multiplication by the scalar ekt commutes with A for each t, it follows from the Leibniz rule in D(0,T;V) that the function u(t)=ektv(t) is in X and satisfies

    u+Au=f,u(0)=u0. (32)

    Moreover, the uniqueness of a solution uX follows from that of v, for if u+Au=0, u(0)=0, then it is seen at once that v=ektu solves v+(A+kI)v=0, v(0)=0; so that v0, hence u0.

    In the V-elliptic case, the well-posedness in Proposition 1 is a known corollary to the proofs in [24]. For coercive A, the above exponential factors should also be handled, which can be done explicitly using

    Lemma 3.1 (Grönwall). When φ, k and E are positive Borel functions on [0,T], and E(t) is increasing, then validity on [0,T] of the first of the following inequalities implies that of the second:

    φ(t)E(t)+t0k(s)φ(s)dsE(t)exp(t0k(s)ds). (33)

    The reader is referred to the proof of Lemma 6.3.6 in [17], which actually covers the slightly sharper statement above. Using this, one finds in a classical way a detailed estimate on each subinterval [0,t]:

    Proposition 7. The unique solution uX of (5), cf. Proposition 1, fulfils in terms of the boundedness and coercivity constants C3, C4 and k of a(,), for 0tT,

    t0u(s)2ds+sup0st|u(s)|2+t0u(s)2ds(2+2C23+C4+1C24e2kt)(C4|u0|2+t0f(s)2ds). (34)

    For t=T, this entails boundedness L2(0,T;V)HX of the solution operator (f,u0)u.

    Proof. As uL2(0,T;V), while f and Au and hence also u=fAu belong to the dual space L2(0,T;V), one has in L1(0,T) the identity

    tu,u+a(u,u)=f,u. (35)

    Here a classical regularisation yields t|u|2=2tu,u, cf. [31,Lem. Ⅲ.1.2] or [5,Lem. 2], so by Young's inequality and the V-coercivity,

    t|u|2+2(C4u2k|u|2)2|f,u|C14f2+C4u2. (36)

    Integration of this yields, since |u|2 and t|u|2=2tu,u are in L1(0,T),

    |u(t)|2+C4t0u(s)2ds|u0|2+C14t0f(s)2ds+2kt0|u(s)|2ds. (37)

    Ignoring the second term on the left, it follows from Lemma 3.1 that, for 0tT,

    |u(t)|2(|u0|2+C14t0f(s)2ds)exp(2kt); (38)

    and since the right-hand side is increasing, one even has

    sup0st|u(s)|2(|u0|2+C14t0f(s)2ds)exp(2kt). (39)

    In addition it follows in a crude way, from (37) and an integrated version of (38), that

    C4t0u(s)2ds(|u0|2+C14t0f(s)2ds)(1+t0(e2ks)ds)=e2kt(|u0|2+C14t0f(s)2ds). (40)

    Moreover, as u solves (5), clearly tu2(f+Au)22f2+2Au2, and since AC3 holds for the norm in B(V,V), the above estimates entail

    t0u(s)2ds2t0f(s)2ds+2C23t0u(s)2ds2(C4+C23C4e2kt)(|u0|2+C14t0f(s)2ds). (41)

    Finally the stated estimate (34) follows from (39), (40) and (41).

    As a preparation, a small technical result is recalled from Proposition 3 in [5], where a detailed proof can be found:

    Lemma 3.2. When A generates an analytic semigroup on the complex Banach space B and wH1(0,T;B), then the Leibniz rule

    te(Tt)Aw(t)=(A)e(Tt)Aw(t)+e(Tt)Atw(t) (42)

    is valid in D(0,T;B).

    In Proposition 2, equation (6) is of course just the Duhamel formula from analytic semigroup theory. However, since X also contains non-classical solutions, (6) requires a proof in the present context-but as noted, it suffices just to reinforce the classical argument by the injectivity of etA in Proposition 5:

    Proof of Proposition 2. To address the last statement first, once (6) has been shown, Proposition 1 yields etAu0X for f=0. For general (f,u0) one has uX, so the last term containing f also belongs to X.

    To obtain (6) in the above set-up, note that all terms in tu+Au=f are in L2(0,T;V). Therefore e(Tt)A applies for a.e. t[0,T] to both sides as an integration factor, so as an identity in L2(0,T;V),

    t(e(Tt)Au(t))=e(Tt)Atu(t)+e(Tt)AAu(t)=e(Tt)Af(t). (43)

    Indeed, on the left-hand side e(Tt)Au(t) is in L1(0,T;V) and its derivative in D(0,T;V) follows the Leibniz rule in Lemma 3.2, since uH1(0,T;V) as a member of X.

    As C([0,T];H)L2(0,T;V)L1(0,T;V), it is seen in the above that e(Tt)Au(t) and e(Tt)Af(t) both belong to L1(0,T;V). So when the Fundamental Theorem for vector functions (cf. [31,Lem. Ⅲ.1.1], or [5,Lem. 1]) is applied and followed by use of the semigroup property and a commutation of e(Tt)A with the integral, using Bochner's identity, cf. Remark 5 below, one finds that

    e(Tt)Au(t)=eTAu0+t0e(Ts)Af(s)ds=e(Tt)AetAu0+e(Tt)At0e(ts)Af(s)ds. (44)

    Since e(Tt)A is linear and injective, cf. Proposition 5, equation (6) now results at once.

    Remark 5. It is recalled that for fL1(0,T;B), where B is a Banach space, it is a basic property that for every functional φ in the dual space B, one has Bochner's identity: T0f(t)dt,φ=T0f(t),φdt.

    As all terms in (6) are in C([0,T];H), it is safe to evaluate at t=T, which in view of (7) gives that u(T)=eTAu(0)+yf. This is the flow map

    u(0)u(T). (45)

    Owing to the injectivity of eTA once again, and that Duhamel's formula implies u(T)yf=eTAu(0), which clearly belongs to D(eTA), this flow is inverted by

    u(0)=eTA(u(T)yf). (46)

    In other words, not only are the solutions in X to u+Au=f parametrised by the initial states u(0) in H (for fixed f) according to Proposition 1, but also the final states u(T) are parametrised by the u(0). Departing from this observation, one may give an intuitive

    Proof of Theorem 1.1. If (2) is solved by uX, then u(T)=uT is reached from the unique initial state u(0) in (46). But the argument for (46) showed that uTyf=eTAu(0)D(eTA), so (9) is necessary.

    Given data (f,uT) that fulfill (9), then u0=eTA(uTyf) is a well-defined vector in H, so Proposition 1 yields a function uX solving u+Au=f and u(0)=u0. By the flow (45), this u(t) has final state u(T)=eTAeTA(uTyf)+yf=uT, hence satisfies both equations in (2). Thus (9) suffices for solvability.

    In the affirmative case, (11) results for any solution uX by inserting formula (46) for u(0) into (6). Uniqueness of u in X is seen from the right-hand side of (11), where all terms depend only on the given f, uT, A and T>0. That each term in (11) is a function belonging to X was seen in Proposition 2.

    Moreover, the solution can be estimated in X by substituting the expression (46) for u0 into the inequality in Proposition 7 for t=T. For the norm in (31) this gives

    |||u|||2X(2+2C23+C4+1C24e2kT)max(C4,1)(|u0|2+T0f(s)2ds)c(|eTA(uTyf)|2+f2L2(0,T;V)). (47)

    Here one may add |uT|2 on the right-hand side to arrive at the expression for (f,uT)Y in Theorem 1.

    Remark 6. It is easy to see from the definitions and proofs that Pu=(tu+Au,u(T)) is a bounded operator XY. The statement in Theorem 1.1 means that the solution operator R(f,uT)=u (is well defined and) satisfies PR=I, but by the uniqueness also RP=I holds. Hence R is a linear homeomorphism YX.

    In the sequel Ω stands for a C smooth, open bounded set in Rn, n2 as described in [12,App. C]. In particular Ω is locally on one side of its boundary Γ=Ω. For such Ω, the problem is to characterise the u(t,x) satisfying

    {tu(t,x)Δu(t,x)=f(t,x) in ]0,T[×Ω,γ1u(t,x)=0 on ]0,T[×Γ,rTu(x)=uT(x) at {T}×Ω. (48)

    While rTu(x)=u(T,x), the Neumann trace on Γ is written in the operator notation γ1u=(νu)|Γ, whereby ν is the outward pointing normal vector at xΓ. Similarly γ1 is used for traces on ]0,T[×Γ.

    Moreover, Hm(Ω) denotes the usual Sobolev space that is normed by um=(|α|mΩ|αu|2dx)1/2, which up to equivalent norms equals the space Hm(¯Ω) of restrictions to Ω of Hm(Rn) endowed with the infimum norm.

    Correspondingly the dual of e.g. H1(¯Ω) has an identification with the closed subspace of H1(Rn) given by the support condition in

    H10(¯Ω)={uH1(Rn)|suppu¯Ω}. (49)

    For these matters the reader is referred to [16,App. B.2]. Chapter 6 and (9.25) in [12] could also be references for this and basic facts on boundary value problems; cf. also [9,27].

    The main result in Theorem 1.1 applies to (48) for V=H1(¯Ω), H=L2(Ω) and VH10(¯Ω), for which there are inclusions H1(¯Ω)L2(Ω)H10(¯Ω), when gL2(Ω) via eΩ (extension by zero outside of Ω) is identified with eΩg belonging to H10(¯Ω). The Dirichlet form

    s(u,v)=nj=1(ju|jv)L2(Ω)=nj=1Ωju¯jvdx (50)

    satisfies |s(v,w)|v1w1, and the coercivity in (1) holds for C4=1, k=1 since s(v,v)=v21v20.

    The induced Lax–Milgram operator is the Neumann realisation ΔN, which is selfadjoint due to the symmetry of s and has its domain given by D(ΔN)={uH2(Ω)|γ1u=0}. This is a classical but non-trivial result (cf. the remarks prior to Theorem 4.28 in [12], or Section 11.3 ff. there; or [27]). Thus the homogeneous boundary condition is imposed via the condition u(t)D(ΔN) for 0<t<T.

    By the coercivity, A=ΔN generates an analytic semigroup of injections ezΔN in B(L2(Ω)), and the bounded extension ˜Δ:H1(¯Ω)H10(¯Ω) induces the analytic semigroup ez˜Δ on H10(¯Ω); both are defined for zSπ/4. As previously, (etΔN)1=etΔN.

    The action of ˜Δ is (slightly surprisingly) given by ˜Δu=div(eΩgradu) for each uH1(¯Ω), for when wH1(Rn) coincides with v in Ω, then (50) gives

    ˜Δu,v=s(u,v)=nj=1RneΩ(ju)¯jwdx=nj=1j(eΩju),wH1(Rn)×H1(Rn)=nj=1j(eΩju),vH10(¯Ω)×H1(¯Ω). (51)

    To make a further identification one may recall the formula j(uχΩ)=(ju)χΩνj(γ0u)dS, valid for uC1(Rn) when χΩ denotes the characteristic function of Ω, and γ0, S the restriction to Γ and the surface measure at Γ, respectively; a proof is given in [16,Thm. 3.1.9]. Replacing u by ju for some uC2(¯Ω), and using that ν(x) is a smooth vector field around Γ, we obtain that j(eΩju)=eΩ(2ju)(γ0νjju)dS. This now extends to all uH2(¯Ω) by density and continuity, and by summation one finds that in D(Rn),

    ˜Δu=Δ(eΩgradu)=eΩ(Δu)(γ1u)dS. (52)

    Clearly the last term vanishes for uD(ΔN); whence Δ(eΩgradu) identifies in Ω with the L2-function Δu for such u. But for general u in the form domain H1(¯Ω), none of the terms on the right-hand side make sense.

    The solution space for (48) amounts to

    X0=L2(0,T;H1(Ω))C([0,T];L2(Ω))H1(0,T;H10(¯Ω)),uX0=(T0u(t)2H1(Ω)dt=(T0+supt[0,T]Ω|u(x,t)|2dx+T0tu(t)2H10(¯Ω)dt)1/2. (53)

    The corresponding data space is here given in terms of yf=T0e(Tt)Δf(t)dt, cf. (7), as

    Y0={(f,uT)L2(0,T;H10(¯Ω))L2(Ω)|uTyfD(eTΔN)},(f,uT)Y0=(T0f(t)2H10(¯Ω)dt=(T0+Ω(|uT(x)|2+|eTΔN(uTyf)(x)|2)dx)1/2. (54)

    With this framework, Theorem 1.1 at once gives the following new result on a classical problem:

    Theorem 4.1. Let A=ΔN be the Neumann realization of the Laplacian in L2(Ω) and ˜Δ=Δ(eΩgrad) its extension H1(¯Ω)H10(¯Ω). When uTL2(Ω) and fL2(0,T;H10(¯Ω)), then there exists a solution uX0 of

    tuΔ(eΩgradu)=f,rTu=uT (55)

    if and only if the data (f,uT) are given in Y0, i.e. if and only if

    uTT0e(Ts)˜Δf(s)dsbelongs toD(eTΔN)=R(eTΔN). (56)

    In the affirmative case, u is uniquely determined in X0 and satisfies the estimate uX0c(f,uT)Y0. It is given by the formula, in which all terms belong to X0,

    u(t)=etΔNeTΔN(uTT0e(Tt)˜Δf(t)dt)+t0e(ts)˜Δf(s)ds. (57)

    Furthermore the difference in (56) equals eTΔNu(0) in L2(Ω).

    Besides the fact that ˜Δ=Δ(eΩgrad) appears in the differential equation (instead of Δ), it is noteworthy that there is no information on the boundary condition. However, there is at least one simple remedy for this, for it is well known in analytic semigroup theory, cf. [26,Thm. 4.2.3] and [26,Cor. 4.3.3], that if the source term f(t) is valued in H and satisfies a global condition of Hölder continuity, that is, for some σ]0,1[,

    sup{|f(t)f(s)||ts|σ|0s<tT}<, (58)

    then the integral in (6) takes values in D(A) for 0<t<T and At0e(ts)Af(s)ds is continuous ]0,T[H.

    When this is applied in the above framework, the additional Hölder continuity yields u(t)D(ΔN)={uH2(Ω)|γ1u=0} for t>0, so the homogeneous Neumann condition is fulfilled and ˜Δu identifies with Δu, as noted after (52). Therefore one has the following novelty:

    Theorem 4.2. If uTL2(Ω) and f:[0,T]L2(Ω) is Hölder continuous of some order σ]0,1[, and if uTyf fulfils the criterion (56), then the homogeneous Neumann heat conduction final value problem (48) has a uniquely determined solution u in X0, satisfying u(t){uH2(Ω)|γ1u=0} for t>0, and depending continuously on (f,uT) in Y0. Hence the problem is well posed in the sense of Hadamard.

    It would be desirable, of course, to show the well-posedness in a strong form, with an isomorphism between the data and solution spaces.

    Remark 7. Grubb and Solonnikov [13] systematically treated a large class of initial-boundary problems of parabolic pseudo-differential equations and worked out compatibility conditions characterising well-posedness in full scales of anisotropic L2-Sobolev spaces (such conditions have a long history in the differential operator case, going back at least to work of Lions and Magenes [24] and Ladyzenskaya, Solonnikov and Ural'ceva [22]). Their conditions are explicit and local at the curved corner Γ×{0}, except for half-integer values of the smoothness s that were shown to require so-called coincidence, which is expressed in integrals over the Cartesian product of the two boundaries {0}×Ω and ]0,T[×Γ; hence coincidence is also a non-local condition. Whilst the conditions of Grubb and Solonnikov are decisive for the solution's regularity, condition (9) in Theorem 1.1 is in comparison crucial for the existence question.

    Remark 8. Injectivity of the linear map u(0)u(T) for the homogeneneous equation u+Au=0, or equivalently its backwards uniqueness, was proved much earlier for problems with t-dependent sesquilinear forms a(t;u,v) by Lions and Malgrange [23]. In addition to some C1-regularity properties in t, they assumed that (the principal part of) a(t;u,v) is symmetric and uniformly V-coercive in the sense that a(t;v,v)+λv2αv2 for certain fixed λR, α>0 and all vV. In Problem 3.4 of [23], they asked whether backward uniqueness can be shown without assuming symmetry (i.e., for non-selfadjoint operators A(t) in the principal case), more precisely under the hypothesis a(t;v,v)+λv2αv2. The present paper gives an affirmative answer for the t-independent case of their problem.

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    1. Jon Johnsen, 2022, Chapter 62, 978-3-030-87501-5, 621, 10.1007/978-3-030-87502-2_62
    2. Jon Johnsen, 2020, Chapter 16, 978-3-030-36137-2, 259, 10.1007/978-3-030-36138-9_16
    3. Jon Johnsen, Isomorphic Well-Posedness of the Final Value Problem for the Heat Equation with the Homogeneous Neumann Condition, 2020, 92, 0378-620X, 10.1007/s00020-020-02602-8
    4. Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan, On a final value problem for a nonlinear fractional pseudo-parabolic equation, 2021, 29, 2688-1594, 1709, 10.3934/era.2020088
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