Citation: Kaimin Cheng. Permutational behavior of reversed Dickson polynomials over finite fields[J]. AIMS Mathematics, 2017, 2(2): 244-259. doi: 10.3934/Math.2017.2.244
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For sufficiently differentiable maps ˜u:T2×[0,T]→~R2 and ˜ν:T2×[0,T]→S2, with T2 a torus and S2 the unit sphere, we have shown in reference [12] that the system
ut+(u⋅∇)u−Δu+∇π=−∇⋅(∇ν⊤∇ν)−∇ν⊤Δνt,∇⋅u=0,Δνt+Δ((u⋅∇)ν)−Δ2ν=νt+(u⋅∇)ν+|∇ν|2ν−Δν, |
reasonably describes the dynamics over T2 of oriented (i.e., polarized or spin) fluids, a representation in which we account for second-neighbor director interactions in a minimalistic way, the one giving us sufficient amount of regularity to allow existence of a certain class of weak solutions.
In the balance of microstructural actions governing the evolution of ν, an hyper-stress behaving like ∇2ν accounts for second-neighbor interactions; it enters the equation through its double divergence, which generates the term Δ2ν. A viscous-type contribution (namely ∇ν⊤Δνt) affects Ericksen's stress in the balance of macroscopic momentum, an equation in which π is pressure, i.e., the reactive stress associated with the volume-preserving constraint ∇⋅u=0.
We have explicitly underlined in reference [12] the terms neglected in the previous balance equations with respect to a complete representation of second-neighbor director interactions, and their contribution to the Ericksen stress.
Also, to tackle the analysis of such balances, in reference [12] we considered transient states foreseeing |ν|≤1 (i.e., a polarized fluid not in saturation conditions) and replaced the nonlinear term |∇ν|2ν with its approximation 1ε2(1−|ν|2)ν, ε a positive parameter. Eventually, we established just local existence of a certain class of weak solutions.
The description of such fluids falls within the general model-building framework of the mechanics of complex materials (a format involving manifold-valued microstructural descriptors) in references [25] and [26] (see also [27,28]). By following that format, if we derive balance equations by requiring invariance of the sole external power of actions under isometric changes in observers even just for first-neighbor interactions, since the infinitesimal generator of SO(3) action over S2 is −ν×, we find the possible existence of a conservative self-action proportional to ν, i.e., something like λν, with λ≥0.
Consequently, we consider here a relaxed version of the balances above by accounting for |ν|≤1 and introducing the self-action λν. Then, we write
ut+(u⋅∇)u−Δu+∇π=−∇⋅(∇ν⊤∇ν)−∇ν⊤Δνt, | (1.1) |
∇⋅u=0, | (1.2) |
Δνt+Δ((u⋅∇)ν)−Δ2ν=νt+(u⋅∇)ν−Δν+1ε2(1−|ν|2)ν−λν,|ν|≤1, | (1.3) |
with initial conditions
u|t=0=u0,ν|t=0=ν0. | (1.4) |
We tackle its analysis by filtering the balance of macroscopic momentum by (I−Δ)−1. In the process, we define the regularized velocity
w:=(I−Δ)−1u, |
and approximate the filtered version of equation (1.1) by considering that ∇⋅(I−Δ)−1(u⊗u)≈∇⋅(w⊗w). Then, we apply the inverse filter (I−Δ) (and we write once again π and ν for pressure and director field, respectively). The resulting system reads
wt−Δwt+(w⋅∇)w−Δw+∇π=−∇⋅(∇ν⊤∇ν)−∇ν⊤Δνt, | (1.5) |
∇⋅w=0, | (1.6) |
Δνt+Δ((w⋅∇)ν)−Δ2ν=νt+(w⋅∇)ν−Δν+1ε2(1−|ν|2)ν−λν,|ν|≤1. | (1.7) |
For it, we prove global existence of weak solutions (defined as in reference [12]).
The obtained regularity could allow us to obtain a uniqueness result. Also, the granted global existence of weak solutions can be used for analyzing possible weak or strong attractors, which we may foresee in appropriate state spaces. All these aspects will be matter of a forthcoming work.
For p≥1, by Lp=Lp(T2) we indicate the usual Lebesgue space with norm ‖⋅‖p. When p=2, we use the notation ‖⋅‖:=‖⋅‖L2 and denote by (⋅,⋅) the related inner product. Moreover, with k a nonnegative integer and p≥1, we denote by Wk,p:=Wk,p(T2) the usual Sobolev space with norm ‖⋅‖k,p (using ‖⋅‖k when p=2). We write W−1,p′:=W−1,p′(T2), p′=p/(p−1), for the dual of W1,p(T2) with norm ‖⋅‖−1,p′.
Let X be a real Banach space with norm ‖⋅‖X. We will use the customary spaces Wk,p(0,T;X), with norm denoted by ‖⋅‖Wk,p(0,T;X). In particular, W0,p(0,T;X)=Lp(0,T;X) are the standard Bochner spaces.
(Lp)n:=L2(T2,Rn), p≥1, is the function space of vector-valued L2-maps. Similarly, (Wk,p)n:=(Wk,p(T2))n is the usual Sobolev space of vector-valued maps with components in Wk,p, while (Hs)n is the space of vector-valued maps with components in Hs=Ws,2∩{w:∇⋅w=0}. We also define the following spaces:
H:=closure of C∞0(T2,R2)∩{w|∇⋅w=0} in (L2)2 ,Hs:=closure of C∞0(T2,R2)∩{w|∇⋅w=0} in (Ws,2)2,Hs:={ν∈(Ws,2)3}. |
This last space is the usual Sobolev space of vector fields with components Ws,2-functions. Again H:=H0. By H−s we indicate the space dual to Hs. We denote by ⟨⋅,⋅⟩H−s,Hs the duality pairing between H−s and Hs. We will also assume that the vector fields u and w have null average on T2. In particular, under such an assumption, Poincaré's inequality holds true.
Here and in the sequel, we denote by c (or ˉc) positive constants, which may assume different values.
We'll make use of the following well-known inequalities (see, e.g., [1,2,15,18,21,22,32]): Ladyzhenskaya's,
‖v‖L4≤C‖v‖12‖∇v‖12,v∈H1, | (2.1) |
Agmon's,
‖v‖L∞≤C‖v‖12‖Δv‖12,v∈H2. | (2.2) |
In the sequel (especially to get estimates in Hs, with s non-integer) we'll also make use of commutator-type estimates as the one in the following lemma concerning the operator Λs, s∈R+ (see, e.g., [19,20,30], see also [6,31]), with Λ:=(−Δ)1/2.
Lemma 2.1. For s>0 and 1<r≤∞, and for smooth enough u and v, we get
‖Λs(uv)‖Lr≤c(‖u‖Lp1‖Λsv‖Lq1+‖v‖Lp2‖Λsu‖Lq2), | (2.3) |
where 1/r=1/p1+1/q1=1/p2+1/q2 and c is a suitable positive constant.
We also recall the following result about product-laws in Sobolev spaces ([16, Theorem 2.2], see also [29])
Lemma 2.2. Let s0, s1, s2∈R. The product estimate
‖fg‖H−s0≤c‖f‖Hs1‖g‖Hs2 | (2.4) |
holds, provided that
s0+s1+s2≥n2, where n is the space dimension, | (2.5) |
s0+s1≥0, | (2.6) |
s0+s2≥0, | (2.7) |
s1+s2≥0, | (2.8) |
If in (2.5) the equality sign holds,inequalities (2.6)–(2.8) must be strict. | (2.9) |
Set T2:=2πZ2/L. T2 is the torus defined by T2:=(R2/T2). We can expand w∈Hs(T2) in Fourier series as
w(x)=∑k∈T⋆2ˆwkeik⋅x, |
with k=(k1,k2)∈Z2 the wave-number, |k|=√|k1|2+|k2|2. The Fourier coefficients for w are defined by ˆwk:=1(2π)2∫T2w(x)e−ik⋅xdx. The norm in Hs is given by
‖w‖2Hs=∑|k|2s|ˆwk|2, |
and the inner product (⋅,⋅)Hs=(Λs⋅,Λs⋅) is characterized by
(w,v)Hs=∑|k|≥1|k|2sˆwk⋅¯ˆvk, |
where the over-bar denotes, as usual, complex conjugation. Consider the inverse Helmholtz operator
G:=(I−Δ)−1, | (2.10) |
taking values
Gw(x):=∫T2G(x,y)w(y))dy, | (2.11) |
where G(x,y) is the associated Green function (see, e.g., [5,7,8,9,10]). For w∈Hs, take the Fourier expansion w=∑k∈T⋆2ˆwkeik⋅x, so that, by inserting this expression in (2.11), we get
Gw:=∑k∈T⋆211+|k|2ˆwkeik⋅x. | (2.12) |
G is self-adjoint. It commutes with differential operators (see, e.g., [4,5,7]). We get also
(Gv,w)=(G1/2v,G1/2w)L2=(v,w)H−1and(G1/2v,G1/2v)L2=‖v‖H−1. | (2.13) |
We set
fε(ν):=1ε2(1−|ν|2)ν,ε>0. |
Then, we rewrite the filtered balances as
wt−Δwt+(w⋅∇)w−Δw+∇π=−∇⋅(∇ν⊤∇ν)−∇ν⊤Δνt, | (3.1) |
∇⋅w=0, | (3.2) |
Δνt+Δ((w⋅∇)ν)−Δ2ν=νt+(w⋅∇)ν−Δν+fε(ν)−λν,|ν|≤1, | (3.3) |
with initial conditions
w|t=0=w0,ν|t=0=ν0. | (3.4) |
To keep the notation compact, here and in the sequel we omit the dependence of w and ν on ε.
Definition 3.1 (Regular weak solution). For a given T>0, a pair (w,ν) is a regular weak solution of (3.1)–(3.3) if (w,ν)∈L∞(0,T;H32×H52), (∂tw,∂tν)∈L2(0,T;H1×H32), and
∫T0((wt(s),v(s))+(∇wt(s),∇v(s))+((w(s)⋅∇)w(s),v)+(∇w(s),∇v(s)))ds=∫T0((∇ν⊤∇ν(s),∇v(s))+(∇ν⊤∇νt(s),∇v(s)))ds, | (3.5) |
holds true for every v∈C∞0((0,T)×T2), and
∫T0((∇νt(s),∇h(s))+(∇[(w(s)⋅∇)ν(s)],∇h(s))+(Δν(s),Δh(s)))ds=∫T0((νt(s),h(s))+((w(s)⋅∇))ν(s),h(s))+(fε(ν(s)),h(s))+(∇ν(s),∇h(s))−λ(ν(s),h(s)))ds | (3.6) |
for every h(t,x)=ψ(t)ϕ(x), with ϕ∈H52, ψ∈C∞0(0,T), and |ν(x,t)|≤1 a.e. in (0,T)×T2.
In the following, we'll always refer to "regular weak solutions" simply as "weak solutions", for the sake of brevity.
Theorem 3.1. Assume (w0,ν0)∈H32×H52, with |ν0(x)|≤1 for a.e. x∈T2. Then, systems (3.1)–(3.3), supplied with (3.4), admits a weak solution (w,ν) which is defined for any fixed time T≥0.
The chosen regularity for the initial data allows the reader to compare easily the result here with what we got in reference [12], realizing our passage from local (short time) to global (large fixed time) existence. Also, by renouncing to a certain amount of solution regularity (i.e., considering a weaker class) we could accept data (w0,ν0)∈H1×H2, obtaining for them once again an existence result (see Remark 4.1 below).
Remark 3.1. For the integral ∫T2∇((w⋅∇)ν)⋅∇ω dx, with w∈H1, ω∈H1, and ν∈H2, we get
∫T2∇((w⋅∇)ν)⋅∇ω dx=∫T2∂j(wi∂iνk)∂jωk dx=∫T2∂jwi∂iνk∂jωk dx+∫T2wi∂ijνk∂jωk dx. |
The first term on the right-hand side of the above identity is such that
∫T2∂jwi∂iνk∂jωk dx=∫T2∂iνk∂jωk∂jwi dx=∫T2(∇ν⊤∇ω)⋅∇w dx, |
and for the second term we find
∫T2wi∂ijνk∂jωk dx=∫T2∂ijνk∂jωkwi dx=∫T2∇(∇ν)⊤∇ω⋅w dx. |
For the second term on the right-hand side of (3.1), we compute
∇ν⊤Δνt=∇⋅(∇ν⊤∇νt)−∇(∇ν)⊤∇νt. | (3.7) |
We introduce Galerkin's approximating functions {(wn,νn)}, prove a maximum principle, by which the constraint |νn|≤1 is verified, and compute some a-priori estimates. The Aubin-Lions compactness theorem [24] allows us to get convergence of a subsequence. Actually, we apply Galerkin's procedure originally used for the standard Navier-Stokes equations, by adapting it to systems (3.1)–(3.3). (Further details about such a scheme appear in references [23, §2], [13, Appendix A], [7,11]).
Note: In the sequel, for the sake of conciseness we often avoid writing explicitly the integration measure in some integrals, every time we find it appropriate.
We apply directly Galerkin's method only to the velocity field w (this scheme is also known as "semi-Galerkin formulation"; see, e.g, [13]).
For any positive integer i, let us denote by (ωi,πi)∈H2×W1,2 the unique solution of the following Stokes problem:
Δωi+∇πi=−λiωi,in T2,∇⋅ωi=0,in T2, | (4.1) |
with ∫T2πidx=0, for i=1,2,… and 0<λ1≤λ2≤…λn… with λn→+∞, as n→∞. Functions {ωi}+∞i=1 determine an orthonormal basis in H made of the eigenfunctions pertaining to (4.1).
Let Pn:H3/2→Hn:=H3/2∩span{ω1,ω2,…,ωn} be the orthonormal projection of H3/2 on its finite dimensional subspace Hn. Take T>0. For every positive integer n, we look for an approximate solution (wn,νn)∈C1(0,T;Hn)×L∞(0,T,H32)∩L2(0,T,H52) to systems (3.1)–(3.3) with
wn(t,x)=n∑i=1ϕni(t)ωi(x),ϕni to be determined. | (4.2) |
Consider the following problem defined a.e. in (0,T)×T2:
(wnt(t)−Δwnt(t),vn)H12+((wn(t)⋅∇)wn(t),vn)H12+(∇wn(t),∇vn)H12=((∇(νn)⊤∇νn)(t),∇vn)H12+((∇(νn)⊤Δνt)(t),vn)H12,∀vn∈Hn, | (4.3) |
(I−Δ)[∂tνn(t)−Δνn(t)+((wn(t)⋅∇)νn(t))]=−fε(νn(t))+λνn, | (4.4) |
|νn|≤1, | (4.5) |
wn(x,0)=wn0(x):=Pn(w0)(x),νn(0,x)=ν0(x), for x∈T2, | (4.6) |
where w0∈H32 and ν0∈H52, with |ν0(x)|≤1 a.e. in T2.
Instead of exploiting test functions in L2, we take directly the formulation in H1/2, for it provides the needed regularity, The pertinent analysis develops in two steps:
Step A: Let ¯wn∈C1(0,T;Hn) be a given velocity field of the form ¯wn(t,x)=∑ni=1¯ϕni(t)ωi(x), with ¯ϕni assigned. For
(I−Δ)[νnt(t)−Δνn(t)+(¯wn(t)⋅∇)νn(t)]=−fε(νn(t))+λνn(t),a.e.in$(0,T)×T2$, |
with νn(0,x)=ν0(x),for$x∈T2$, we actually look for a vector field
νn∈L∞(0,T;H32)∩L2(0,T;H52),νt∈L2(0,T;H1) |
solving a.e. on (0,T)×T2 the following system:
νnt(t)−Δνn(t)+(¯wn(t)⋅∇)νn(t)=−G(fε(νn(t)))+λG(νn(t)), | (4.7) |
νn(0,x)=ν0(x),for x∈T2, | (4.8) |
where G is once again the inverse Helmholtz operator G=(I−Δ)−1 introduced in (4.1). Since G has Fourier symbol corresponding to the inverse of two spatial derivatives, the right-hand side part of (4.7) results to be regularized (i.e., the terms −Gfε(νn) gains two additional spatial derivatives with respect to fε(νn); the same occurs for Gνn(t)). Thus, this new expression can be rewritten equivalently as a semilinear parabolic equation in the unknown νn. The existence of such νn is guaranteed by the classical theory of parabolic equations (see, e.g., [17]), which also provides higher regularity results (see [17, Theorem 6, Ch. 7.1]). They allow us to use the regularity of initial data ν0∈H52 to get νn∈L∞(0,T;H32)∩L2(0,T;H52) and νnt∈L2(0,T;H1) (by interpolation we also have that νt∈C(0,T;H1)). The following lemma (see [12, Lemma 4.1] and also [14, Lemma 2.1]) guarantees the constraint |ν|≤1.
Lemma 4.1 (Weak maximum principle). Let ν0∈H52 be such that |ν0(x)|≤1 for a.e. x∈T2. Take ˉwn∈C(0,T;Hn). Then, there exists a weak solution νn∈L∞(0,T;H32)∩L2(0,T;H52) to the problems (4.7)–(4.8). Moreover, fixed ϵ>0 large enough in the definition of fϵ, every such weak solution verifies |νn(x,t)|≤1 a.e. on T2×[0,T].
In performing the next calculations, we could relax hypotheses by assuming that ν0∈H1 and is such that |ν0(x)|≤1 a.e. x∈T2, with ˉwn∈C(0,T;Hn). Then, there would exist a weak solution ν∈L∞(0,T;H1)×L2(0,T;H2), with |ν(x,t)|≤1 a.e. in T2×(0,T). However, for the sake of simplicity, we still use the same regularity assumptions previously introduced, and we denote by ν and w the quantities νn and ˉwn, respectively for the sake of conciseness.
Proof. Existence of the solution νn∈L∞(0,T;H32)∩L2(0,T;H52) to (4.7)–(4.8) has been already mentioned above.
Define φ(x,t)=(|ν(x,t)|2−1)+, where z+=max{z,0} for each z∈R. Assume there exists a measurable subset B⊂T2 with positive measure |B|>0 such that |ν(x,t)|>1 a.e. in B×(t1,t2], 0≤t1<t2≤T, and |ν(x,t)|=1 a.e. in ∂B×(t1,t2]. By taking φν as a test function against (4.7), we get
12∫T2∂t(|ν|2)φ+∫T2(w⋅∇)|ν|2φ+∫T2∇ν⋅(φν)−1ϵ2∫T2G(|ν|2−1)ν⋅φν+λ∫B(Gν)⋅(φν)=0, |
which is equivalent to
12∫B∂t(|ν|2)φ+∫B(w⋅∇)|ν|2φ+∫B∇ν⋅∇(φν)−1ϵ2∫BG1/2(φν)⋅G1/2(φν)+λ∫B(G1/2ν)⋅(G1/2ν)=0. | (4.9) |
With ‖⋅‖ indicating ‖⋅‖L2(B), we can also write
12∫B∂t(|ν|2)φ=12∫B∂t(|ν|2−1)φ=14ddt‖φ‖2,∫B(w⋅∇)|ν|2φ=∫B(w⋅∇)(|ν|2−1)φ=∫B(w⋅∇)φ⋅φ=0,∫B∇ν⋅∇(φν)=12∫B∇(|ν|2)⋅∇φ+∫B|∇ν|2φ=12∫B∇(|ν|2−1)∇φ+∫B|∇ν|2φ=12‖∇φ‖2+∫B|∇ν|2φ≥12‖∇φ‖2≥0. |
Then, Eq. (4.9) becomes
ddt‖φ‖2+2‖∇φ‖2+4∫B|∇ν|2φ−4ϵ2∫BG1/2(φν)⋅G1/2(φν)+4λ∫B(G1/2ν)⋅G1/2(φν)=0. | (4.10) |
Since φ(t2)≥φ(t1) (here, φ(t1)=0), by integrating in time over (t1,t2], we get
2∫t2t1‖∇φ‖2+4∫t2t1(∫B|∇ν|2φ−4ϵ2∫B(G1/2ν)⋅G1/2(φν)+4λ∫B(G1/2ν)⋅G1/2(φν))≤0. |
In principle, B may have more than one connected component with positive measure. However, these components are finite in number for ¯B is compact. Thus, previous inequality can be rewritten as
∑i(2∫t2t1∫Bi|∇φ|2+4∫t2t1(∫Bi|∇ν|2φ−4ϵ2∫BiG1/2(φν)⋅G1/2(φν)+4λ∫Bi(G1/2ν)⋅G1/2(φν)))≤0. |
Then, there exists at least one connected component Bj, with |Bj|>0, on which
2∫t2t1∫Bj|∇φ|2+4∫t2t1(∫Bj|∇ν|2φ−4ϵ2∫BjG1/2(φν)⋅G1/2(φν)+4λ∫BjG1/2(ν)⋅G1/2(φν))≤0, |
and hence, by(2.13), we have
2∫t2t1‖∇φ‖2L2(Bj)+4∫t2t1∫Bj|∇ν|2φ≤4ϵ2∫t2t1∫BjG1/2(φν)⋅G1/2(φν)−4λ∫t2t1∫BjG1/2(ν)⋅G1/2(φν))≤cϵ2∫t2t1‖φν‖2H−1(Bj)+4λ|∫t2t1∫BjG1/2(ν)⋅G1/2(φν)|. | (4.11) |
Since |ν(x,t)|=1 a.e. on ∂B×(t1,t2), we get φ(x,t)=0 a.e. on ∂B×(t1,t2) and, in particular, φ(x,t)=0 a.e. on ∂Bj×(t1,t2). Assume that Bj is the closure of an open set. By using the Poincaré inequality on left-hand side first term of (4.11), along with the control (2.4) (see also [29]), we obtain
‖φν‖2H−1≤‖φ‖2H1(Bj)‖ν‖2L2(Bj)≤c‖φ‖2H1(Bj)‖ν‖2L∞(0,T;L2(T2))=˜c‖φ‖2H1(Bj), |
and
4λ|∫BG1/2(ν)⋅G1/2(φν)|≤4λ‖ν‖‖φν‖H−1≤4λ‖ν‖2‖φ‖H1≤cλ(∫Bj(‖ν‖2−1)dx+1)‖φ‖H1≤cλ(∫Bj(‖ν‖2−1)2dx)12‖φ‖H1+cλ‖φ‖H1≤cλ‖φ‖‖φ‖H1+ˉcλ‖φ‖2H1≤ˆcλ‖φ‖2H1. | (4.12) |
Hence, the inequality
C∫t2t1‖φ‖2L2(Bj)+∫t2t1‖∇φ‖2L2(Bj)+4∫t2t1∫Bj|∇ν|2φ≤(˜cϵ2+ˆcλ)∫t2t1‖φ‖2H1(Bj), |
where C is the constant involved in the Poincaré inequality, holds true. Then, we find
c∫t2t1‖φ‖2H1(Bj)+4∫t2t1∫Bj|∇ν|2φ≤(˜cϵ2+ˆcλ)∫t2t1‖φ‖2H1(Bj), | (4.13) |
which gives an absurd by assuming that ϵ is sufficiently large as λ is small.
The general case, when Bj is not the closure of an open set, follows the same line of the argument in reference [12].
Step B: Let νn∈L∞(0,T;H32)∩L2(0,T;H52) be the vector field just determined in the previous step. We search the approximating velocity field wn∈C1(0,T;Hn) satisfying the equation
(wnt(t),vn)H12+(∇wnt(t),∇vn)H12+(∇wn(t),∇vn)H12+((¯wn(t)⋅∇)wn(t),vn)H12,=((∇(νn)⊤∇νn),∇vn)H12+(∇(νn)⊤Δνt,vn)H12,∀vn∈Hn, |
with
wn(x,0)=wn0(x)=Pn(w0)(x), for x∈T2, |
where both νn and ¯wn are given. Thanks to the Cauchy-Lipschitz theorem, we can prove existence of a unique maximal solution wn of the above problem.
In the sequel, as short-hand notation, we use the same symbol ‖⋅‖Lp(0,T;Lk) for both the norm in Lp(0,T;Lk) and the one in Lp(0,T;(Lk)n). We employ the same convention also for Lp(0,T;Ws,k) and Lp(0,T;(Ws,k)n) (also Lp(0,T;Hs) and Lp(0,T;(Hs)n)).
Proof of Theorem 3.1. First, we deduce a priori estimates. Then, we apply a compactness criterion proving that the limiting pair (ˆw,ˆν) is actually a weak solution to (3.1)–(3.3), supplemented by (1.4). Only for the sake of conciseness we use (w,ν) instead of (wn,νn).
Step 1: Energy a priori estimates. Consider Eq. (3.3), to which we apply the operator G=(I−Δ)−1, and take the L2-product with test ν, obtaining
12ddt‖ν‖2+‖∇ν‖2≤1ε2∫T2|G((1−|ν|2)ν)||ν|dx+λ∫T2|G12ν|2dx≤cε2‖(1−|ν|2)ν‖H−2‖ν‖+cλ‖ν‖2H−1≤cε2‖(1−|ν|2)ν‖‖ν‖+cλ‖ν‖2≤(cε2+cλ)‖ν‖2, | (4.14) |
where the constraint |ν|2≤1 plays a role. Given T>0, the Gronwall Lemma implies ν∈L∞(0,T;L2)∩L2(0,T;H1).
By taking the L2-product of (3.1) with w, we compute
12ddt(‖w‖2+‖∇w‖2)+‖∇w‖2=∫T2(∇ν⊤∇ν)⋅∇wdx−∫T2∇ν⊤Δνt⋅wdx=∫T2(∇ν⊤∇ν)⋅∇wdx+∫T2(∇ν⊤∇νt)⋅∇wdx+∫T2∇(∇ν)⊤∇νt⋅wdx, | (4.15) |
where, to get the second equality, we have used relation (3.6), integrating by parts the first term obtained.
By multiplying (3.3) by νt and integrating over T2, we obtain
12ddt(‖∇ν‖2+‖Δν‖2)+‖νt‖2+‖∇νt‖2=−∫T2∇((w⋅∇)ν)⋅∇νtdx−∫T2(w⋅∇)ν)⋅νtdx−∫T2fε(ν)⋅νtdx+λ∫T2ν⋅νtdx. | (4.16) |
Remark 1 implies that the first term in the right-hand side of (4.16) can be rewritten as
∫T2∇((w⋅∇)ν)⋅∇νt dx=∫T2(∇ν⊤∇νt)⋅∇wdx+∫T2∇(∇ν)⊤∇νt⋅wdx. |
Then, by summing up (4.15) and (4.16), we infer
12ddt(‖w‖2+‖∇w‖2+‖∇ν‖2+‖Δν‖2)+‖∇w‖2+‖νt‖2+‖∇νt‖2≤∫T2|∇w||∇ν|2dx+∫T2|w||∇ν||νt|dx+∫T2|fϵ(ν)||νt|dx+λ∫T2|ν||νt|dx=:4∑i=1Ii. | (4.17) |
For the terms Ii, i=1,2,3, we have the following bounds
I1≤‖∇w‖‖∇ν‖2L4≤c‖∇w‖‖∇ν‖‖Δν‖≤cε‖∇w‖2‖∇ν‖2+Cε‖Δν‖2, | (4.18) |
I2≤‖w‖L4‖∇ν‖L4‖νt‖≤c‖w‖12|∇w‖12‖∇ν‖12‖Δν‖12‖νt‖≤cδ‖w‖‖∇w‖‖∇ν‖‖Δν‖+δ‖νt‖2≤ˉcδ‖∇w‖‖∇ν‖‖Δν‖+δ‖νt‖2≤¯cϵδ2‖∇w‖2‖∇ν‖2+cϵ‖Δν‖2+δ‖νt‖2, | (4.19) |
I3≤2ε2‖ν‖‖νt‖≤cε4δ‖ν‖2+δ‖νt‖2, | (4.20) |
and
I4≤λ‖ν‖‖νt‖≤cλδ‖ν‖2+λδ‖νt‖2. | (4.21) |
Estimates above, together with inequalities (4.14) and (4.17), allow us to write
ddt(‖w‖2+‖∇w‖2+‖∇ν‖2+‖Δν‖2)+‖∇w‖2+(1−(λ+2)δ)‖νt‖2+‖∇νt‖2≤cε,δ,λ‖ν‖2+cε‖Δν‖2+cε,δ‖∇w‖2‖∇ν‖2. | (4.22) |
In the present case the penalisation parameter ε>0 is constant, so we omit such a term along with δ and λ in the next calculations. From (4.22) we obtain
ddt(‖w‖2+‖∇w‖2+‖∇ν‖2+‖Δν‖2)≤c(‖ν‖2+‖Δν‖2+‖∇w‖2‖∇ν‖2)≤c‖ν‖2+c(‖∇w‖2+‖Δν‖2)(1+‖∇ν‖2). |
Set y=(‖w‖2+‖∇w‖2+‖∇ν‖2+‖Δν‖2). The differential inequality
y′≤c‖ν‖2+y(1+‖∇ν‖2) |
implies
(‖w‖2+‖∇w‖2+‖∇ν‖2+‖Δν‖2)(t)≤(‖w0,∇w0,∇ν0,Δν0‖2)e∫t0(1+‖∇ν(s)‖2)ds+‖ν‖L∞(0,T;L2)∫t0e∫ts(1+‖∇ν(ℓ)‖2)dℓds. | (4.23) |
Since (w,ν) stands for (wn,νn), as a consequence of the above estimates, for any fixed T>0, it follows that ‖wn,∇wn‖2L∞(0,T;H)+‖∇νn,Δνn‖2L∞(0,T;H) is uniformly bounded with respect to n∈N. The control (4.22) implies νt∈L2(0,T;H1).
Step 2: Further a priori estimates. We take the H1/2-inner product of (3.1) and (3.2) with w and νt, respectively, as in the case of Eqs (4.15) and (4.16). After integration by parts, we obtain
12ddt(‖w‖2H12+‖∇w‖2H12)+‖∇w‖2H12=−((w⋅∇)w),w)H12+((∇ν⊤∇ν),∇w)H12+((∇ν⊤∇νt),∇w)H12+(∇(∇ν)⊤∇νt,w)H12, | (4.24) |
and
12ddt(‖∇ν‖2H12+‖Δν‖2H12)+‖νt‖2H12+‖∇νt‖2H12=−((∇ν⊤∇νt),∇w)H12−(∇(∇ν)⊤∇νt,w)H12−((w⋅∇)ν),νt)H12−(fε(ν),νt)H12+λ(ν,νt)H12. | (4.25) |
From Eqs (4.24) and (4.25), we get
12ddt(‖w‖2H12+‖∇w‖2H12+‖∇ν‖2H12+‖Δν‖2H12)+‖∇w‖2H12+‖νt‖2H12+‖∇νt‖2H12≤|((w⋅∇)w),w)H12|+|((∇ν⊤∇ν),∇w)H12|+|((w⋅∇)ν),νt)H12|+|(fε(ν),νt)H12|+λ|(ν,νt)H12|=:5∑i=1Li. | (4.26) |
For the terms Li, i=1,…,5 we actually use the norm induced by (⋅,⋅)˙H12=(Λ12(⋅),Λ12(⋅)) instead of the full norm H12, although we still keep the same norm notation ‖⋅‖H12. Previous evaluation of the lower-order terms in the steps already described motivates our notational choice. Also, for the velocity vector filed w, the norm ‖w‖˙H12 is equivalent to the full norm ‖w‖H12.
Consider L1. Since ∫T2(w⋅∇)Λ12w⋅Λ12wds=0, we get
L1≤∫T2|(Λ12w⋅∇)w||Λ12w|ds≤‖Λ12w‖2L4‖∇w‖≤‖Λ12w‖‖Λ12∇w‖‖∇w‖≤c‖w‖H12‖∇w‖‖∇w‖H12≤cϵ‖w‖2H12‖∇w‖2+ϵ‖∇w‖2H12. | (4.27) |
Then, by exploiting (2.3), with s=1/2, r=2 and p1=p2=q1=q2=4, we find
L2≤‖∇ν⊤∇ν‖H12‖∇w‖H12≤c‖Λ12∇ν‖L4‖∇ν‖L4‖∇w‖H12≤c(‖Λ12∇ν‖12‖Λ12Δν‖12‖∇ν‖12‖Δν‖12)‖∇w‖H12≤cϵ(‖∇ν‖2H12‖∇ν‖2+‖Δν‖2H12‖Δν‖2)+ϵ‖∇w‖2H12, | (4.28) |
L3≤‖(w⋅∇)ν‖H12‖νt‖H12≤(‖Λ12w‖L4‖∇ν‖L4+‖w‖L4‖Λ12∇ν‖L4)‖νt‖H12≤cϵ(‖∇w‖2‖∇ν‖2H12+‖w‖2H12‖Δν‖2)+ϵ‖νt‖2H12, | (4.29) |
L4=|(Λ12fε(ν),Λ12νt)|≤c‖fε(ν)‖H1‖νt‖H12≤cϵε4(‖ν‖2+‖∇ν‖2)+ϵ‖νt‖2H12, | (4.30) |
and
L5=λ|(Λ12ν,Λ12νt)|≤cλ‖ν‖H12‖νt‖H12≤cλϵ‖ν‖2H12+ϵ‖νt‖2H12, | (4.31) |
after using Hölder's, Ladyzhenskaya's, and Young's inequalities as well as the continuous embedding W1/2,2(T2)⊂L4(T2).
By using the estimates (4.27)–(4.30) along with (4.26), and absorbing the parameter ε−4 in a generic constant c, we obtain
12ddt(‖w‖2H12+‖∇w‖2H12+‖∇ν‖2H12+‖Δν‖2H12)+(1−cϵ)‖∇w‖2H12+‖νt‖2H12+‖∇νt‖2H12≤c‖w‖2H12(1+‖∇w‖2+‖∇ν‖+‖Δν‖2)+c‖∇ν‖2H12(1+‖w‖2+‖∇ν‖2+‖Δν‖2)+c‖Δν‖2H12‖Δν‖2 |
with ϵ>0 small enough in a way that the coefficient ˉc:=(1−cϵ) is positive. Fix T>0. By Grönwall's lemma, we get
‖w(t)‖2H12+‖∇w(t)‖2H12+‖∇ν(t)‖2H12+‖Δν(t)‖2H12+ˉc∫t0(‖∇w‖2H12+‖νt‖2H12+‖∇νt‖2H12)ds≤βexp{c∫t0[(1+‖w‖2+‖∇w‖2+‖∇ν‖2+‖Δν‖2)]ds} |
for any 0<t≤T, with
β=c(‖w0‖2H12+‖∇w0‖2H12+∇ν0‖2H12+‖Δν0‖2H12), |
and the quantity on the right-hand side of the above inequality is bounded, for 0<t≤T, thanks to Eq. (4.23) and the hypotheses on initial data.
Until here, we mainly used the notation (w,ν) in place of (wn,νn) but, in view of extracting a convergent subsequence, in the last part of the proof we'll employ the (wn,νn) notation. Step 3: Estimate for wnt. In order to extract a convergent subsequence of {(un,νn)}, we exploit the classical Aubin-Lions lemma; to this end we have first to provide a suitable control on wnt. The next calculations also fixes a minor issue present in the analogous control in reference [12], where we estimate acceleration in L1(0,T;H−1).
Consider Eq. (3.1). For φ∈H1, ∫T2φdx=0, with ‖∇φ‖=1. Then, we get
⟨wnt−Δwnt,φ⟩H−1,H1≤|((wn⋅∇)φ,wn)|+|(∇wn,∇φ)|+∫T2|∇νn|2|∇φ|dx+c∫T2|∇νn||∇νnt||∇φ|dx+c|(∇(∇νn)⊤∇νnt,φ)|≤‖wn‖2L4‖∇φ‖+‖∇wn‖‖∇φ‖+‖∇νn‖2L4‖∇φ‖+c‖∇νn‖L4‖∇νnt‖L4‖∇φ‖+c‖∇(∇νn)⊤∇νnt‖H−1‖∇φ‖≤c(‖wn‖‖∇wn‖+c‖∇wn‖+‖∇νn‖‖Δνn‖+‖∇νn‖H12‖∇νnt‖H12+‖Δνn‖H12‖∇νnt‖H12), | (4.32) |
after using the estimates performed in previous steps along with Hölder's, Ladyzhenskaya's, and Poincaré's inequalities. In the last inequality above, we have also exploited the continuous embedding W1/2,2(T2)⊂L4(T2) and the Sobolev product laws (see, e.g., [?,3,29]) to get the estimate
‖∇(∇νn)⊤∇νnt‖H−1≤‖Δνn‖‖∇νnt‖H12≤‖Δνn‖H12‖∇νnt‖H12. |
Hence, we find
∫T0‖wnt‖2H1ds≤c[(1+‖wn‖2L∞(0,T;L2))‖∇wn‖2L2(0,T;L2)+‖Δνn‖2L2(0,T;L2)+(‖∇νn‖2L∞(0,T;H12)+‖Δνn‖2L∞(0,T;H12))‖∇νnt‖2L2(0,T;H12)]. |
As a final step in our argument, to extract a convergent subsequence from {(wn,νn)}, we can use the Aubin-Lions lemma following the same line as in the proof of [12, Theorem 3.1, Step 3]. Also, passage to the limit in weak formulation follows the same path exploited in reference [12]. So, we can conclude stating existence.
Remark 4.1. By assuming initial data (w0,ν0)∈H1×H2, we can still reproduce the same calculations of Step 1, while Step 2 would require higher-order estimates, which are not available in the present setting. However, by using an approach similar to the one in Step 3, we could obtain a weaker control on wt by using Eq. (3.1) and providing a uniform estimate on
‖Δwnt‖H−2=sup‖φ‖˙H2=1|⟨Δwnt,φ⟩H−2,H2| |
Indeed, also in this case, the worst term to be controlled is |(∇(∇νn)⊤∇νnt,φ)|. For it, we get
∫T0|(∇(∇νn)⊤∇νnt,φ)|2ds≤∫T0‖(∇(∇νn)⊤∇νnt‖2H−2‖φ‖2H2ds≤∫T0‖Δνn‖2‖∇νnt‖2ds≤‖Δνn‖2L∞(0,T;L2)∫T0‖∇νnt‖2ds, |
on the basis of inequalities (4.22), (4.23), and the product law (2.4). Then, to conclude about the existence of weak solutions, we can use again the same idea behind limiting and convergence procedures in [12,Theorem 3.1,Step 3].
We thank Tommaso Ruggeri for a discussion on this matter at the end of a talk delivered in 2018 by one of us (PMM) in the Accademia Peloritana dei Pericolanti at Messina (Italy). This work is part of activities in the research group in "Theoretical Mechanics" of the "Centro di Ricerca Matematica Ennio De Giorgi" of the Scuola Normale Superiore in Pisa. We acknowledge the support of INDAM groups GNAMPA and GNFM.
The authors declare no conflicts of interest in this paper.
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