Asymptotic profiles for the third grade fluids equations

International Journal of Non-Linear Mechanics 71 (2015) 132–151

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Asymptotic profiles for the third grade fluids equations Olivier Coulaud Université Paris Sud, Department of Mathematics, Faculté des Sciences d'Orsay, Bâtiment 425, 91405 Orsay Cedex, France

art ic l e i nf o

a b s t r a c t

Article history: Received 6 October 2013 Accepted 21 March 2014 Available online 29 December 2014

We study the long time behaviour of the solutions of the third grade fluids equations in dimension 2. Introducing scaled variables and performing several energy estimates in weighted Sobolev spaces, we describe the first order of an asymptotic expansion of these solutions. It shows in particular that, under smallness assumptions on the data, the solutions of the third grade fluids equations converge to selfsimilar solutions of the heat equations, which can be computed explicitly from the data. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Fluid mechanics Third grade fluids Asymptotic expansion

1. Introduction The study of the behaviour of the non-Newtonian fluids is a significant topic of research in mathematics, but also in physics or biology. Indeed, these fluids, the behaviour of which cannot be described with the classical Navier–Stokes equations, are found everywhere in the nature. For examples, blood, wet sand or certain kind of oils used in industry are non-Newtonian fluids. In this paper, we investigate the behaviour of a particular class of non-Newtonian fluids that is the third grade fluids, which are a particular case to the Rivlin–Ericksen fluids (see [29,30]). The constitutive law of such fluids is defined through the Rivlin–Ericksen tensors, given recursively by A1 ¼ ∇u þ ð∇uÞt ; Ak ¼ ∂t Ak  1 þu  ∇Ak  1 þ ð∇uÞt Ak  1 þ Ak  1 ∇u; where u is a divergence free vector field of R2 or R3 which represents the velocity of the fluid. The most famous example of a Rivlin–Ericksen fluid is the class of the Newtonian fluids, which are given through the stress tensor

σ ¼ pI þ νA1 ; where ν 4 0 is the kinematic viscosity and p is the pressure of the fluid. Introduced into the equations of conservation of momentum, this stress tensor leads to the well known Navier–Stokes equations. In this paper, we consider a larger class of fluids, for which the stress tensor is not linear in the Rivlin–Ericksen tensors, but a polynomial function of degree 3. As introduced by Fosdick and Rajagopal in [13], the stress tensor that we consider is defined by

σ ¼ pI þ νA1 þ α1 A2 þ α2 A21 þ β
A1
2 A1 ;

E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijnonlinmec.2014.11.018 0020-7462/& 2014 Elsevier Ltd. All rights reserved.

where ν 4 0 is the kinematic viscosity, p is the pressure, α1 4 0, α2 A R and β Z 0.

We assume in this paper that the density of the fluid is constant in space and time and equals 1. Actually, the value of the density is not significant, since we can replace the parameters ν, α1, α2 and β by dividing them by the density. Introduced into the equations of conservation of momentum, the tensor σ leads to the system     ∂t u  α1 Δu  νΔu þ curl u  α1 Δu 4 u 

    2  ðα1 þ α2 Þ A  Δu þ 2div LLt  β div
A
A þ ∇p ¼ 0; div u ¼ 0; u jt ¼ 0 ¼ u 0 ;

ð1:1Þ t

where L ¼ ∇u, AðuÞ ¼ ∇u þ ð∇uÞ and 4 denotes the classical vectorial product of R
3 .
For matrices A; B A Md ðRÞ, we define A : B ¼ Pd

2 i;j ¼ 1 Ai;j Bi;j and A ¼ A : A. If the space dimension is 2, we use the convention u ¼ ðu1 ; u2 ; 0Þ and curl u ¼ ð0; 0; ∂1 u2  ∂2 u1 Þ. Notice also that if α1 þ α2 ¼ 0 and β ¼ 0, we recover the equations of motion of second grade fluids, which are another class of non-Newtonian fluids, introduced earlier by Dunn and Fosdick in 1974 (see [10,15] or [9]). If in addition α1 ¼ 0, then one recovers the classical Navier– Stokes equations. The system of Eq. (1.1) has been studied in various cases, on bounded domains of Rd , d ¼2, 3 or in the whole space Rd (see [1–5,22,26]). On a bounded domain Ω of Rd with Dirichlet boundary conditions, Amrouche and Cioranescu have shown the existence of local solutions to (1.1) when the initial data belong to the Sobolev space H 3 ðΩÞd (see [1]). In addition, these solutions are unique. For this study, the authors have assumed the restriction jα1 þ α2 j r ð24νβÞ1=2 ; which is justified by thermodynamics considerations. The proof of their result is obtained via a Galerkin method with functions belonging to the eigenspaces of the operator curl ðI  α1 ΔÞ. In

O. Coulaud / International Journal of Non-Linear Mechanics 71 (2015) 132–151

dimension 3, a slightly different method has been applied by Bresch and Lemoine, who used Schauder's fixed point theorem to extend the result of [1] to the case of initial data belonging to the Sobolev spaces W 2;r ðΩÞ3 , with r 4 3. They have shown in [3] the local existence of unique solutions of (1.1) in the space C 0 ð½0; T; W 2;r ðΩÞ3 Þ, where T 4 0. In addition, if the data are small enough in the space W 2;r ðΩÞ3 , the solutions are global in time. Notice also that the existence of such solutions holds without restrictions on the parameters of the system (1.1). In the case of third grade fluids filling the whole space Rd , d¼ 2,3, Busuioc and Iftimie have established the existence of global solutions with initial data belonging to H 2 ðRd Þd , without restrictions on the parameters or on the size of the data (see [4]). In this study, the authors used a Friedrichs scheme and performed a priori estimates in H2 which allow to show the existence of solutions of (1.1) in the 2 þ d d space L1 loc ðR ; H ðR Þ Þ. Besides, these solutions are unique if d¼2. Later, Paicu has extended the results of [4] to the case of initial data belonging to H 1 ðRd Þd , assuming additional restrictions on the parameters of the equation; the uniqueness is not known in this space (see [26]). The method that he used is slightly different from the one used in [4]. Indeed, although Paicu also considered a Friedrichs scheme, the convergence of the approximate solutions to a solution of (1.1) is done via a monotonicity method. Notice that Theorem 1.1 of this paper shows the existence of solutions of the equations of third grade fluids on R2 for initial data in weighted Sobolev spaces (see Section 3). In what follows, we consider a third grade fluid filling the whole space R2 . Actually, the equations that we consider are not exactly the system (1.1) but the one satisfied by w ¼ curl u ¼ ∂1 u2  ∂2 u1 . In dimension 2, the vorticity equations of the third garde fluids are given by     ∂t w  α1 Δw  νΔw þ u  ∇ w  α1 Δw  

  

 2 2  β div
A
∇w  β div ∇
A
4 A ¼ 0; div u ¼ 0; wjt ¼ 0 ¼ w0 ¼ curl u0 :

ð1:2Þ

Notice that the parameter α2 does no longer appear in (1.2) and thus does not play any role in the study of these equations. Indeed, due to the divergence free property of u, a short computation shows that   curl A  Δu þ2 div LLt ¼ 0, or equivalently there exists q such that  A  Δu þ 2 div LLt ¼ ∇q. This phenomenon is very particular to the dimension 2 and does not occur in dimension 3. Notice also that the previous system is autonomous in w. Indeed, the vector field u depends on w and can be recovered from w via the Biot–Savart law, which is a way to get a divergence free vector field such that curl u ¼ w. The motivation for considering the vorticity equations instead of the equations of motion comes from the fact that, due to spectral reasons, we have to study the behaviour of the solutions of (1.2) in weighted Lebesgue spaces. Indeed, in what follows, we will consider scaled variables, which make appear a differential operator whose essential spectrum can be ”pushed to the left” by taking a convenient weighted Lebesgue space. We will see that the rate of convergence of the solutions of (1.2) is linked to the spectrum of this operator. Unfortunately, the weighted Lebesgue spaces are not suitable for the equations of motions and are not preserved by the system (1.1). Anyway, one can obtain the asymptotic profiles of the solutions of the equations of motion (1.1) from the study of the asymptotic behaviour of the solutions of the vorticity equations (see Corollary 1.1 below). We also emphasize that the system (1.2) allows to consider solutions whose velocity fields are not bounded in L2. In this paper, we establish the existence and uniqueness of solutions of (1.2) in weighted Sobolev spaces, but the main aim is the study of the asymptotic behaviour of these solutions when t goes to infinity. More precisely, we want to describe the first order asymptotic profiles of the solutions of (1.2). We consider a fluid of third

133

grade which fills R2 without forcing term applied to it. In this case, as it is expected, the solutions of (1.2) tend to 0 as t goes to infinity. Our motivation is to show that these solutions behave like those of the Navier–Stokes equations. In our case, we will show that the solutions of (1.2) behave asymptotically like solutions of the heat equations, up to a constant that we can compute from the initial data. The methods that we use in the present paper are based on scaled variables and energy estimates in several functions spaces. This work is inspired by several older results obtained for other fluid mechanics equations. The first and second order asymptotic profiles have been described for the Navier–Stokes equations in dimensions 2 and 3 by Gallay and Wayne (see [18–21]). In dimension 2, they have shown in [18,20] that the first order asymptotic profiles of the Navier–Stokes equations are given up to a constant by a smooth Gaussian function called the Oseen vortex sheet. More precisely, for a solution w of the vorticity Navier–Stokes equations (that is the system 1.2 with α1 ¼ β ¼ 0), for every 2 r p r þ 1, the following property holds: R





wðtÞ  R2 w0 ðxÞ dxG p: ffiffi ¼ Oðt  3=2 þ 1=p Þ when t- þ 1;

t t Lp where G is the Oseen vortex sheet GðxÞ ¼

1  jxj2 =4 e : 4π

ð1:3Þ

The methods that they used in [18] are very different from the ones that we develop in this paper. Although they also considered scaled variables, the convergence to the asymptotic profiles is not obtained through energy estimates. Indeed, using dynamical systems arguments, they established the existence of a finite-dimensional manifold which is locally invariant by the semiflow associated to the Navier–Stokes equations. Then, they showed that, under restrictions on the size of the data, the solutions of the Navier–Stokes equations behave asymptotically like solutions on this invariant manifold. The description of the asymptotic profiles is thus obtained by the description of the dynamics of the Navier–Stokes equations on the invariant manifold. Later, the smallness assumption on the data has been removed (see [20]). In [24], Jaffal-Mourtada describes the first order asymptotics of second grade fluids, under smallness assumptions on the initial data in weighted Sobolev spaces. She has shown that the solutions of the second grade fluids equations converge also to the Oseen vortex sheet. In this paper, we apply the methods used by Jaffal-Mourtada, namely scaled variables and energy estimates. According to these results, one can say that the fluids of second grade behave asymptotically like Newtonian fluids. In this paper, we show that, under the same smallness assumptions on the initial data, the same behaviour occurs for the third grade fluids equations. We emphasize that the rate of convergence that we obtain is better than the one obtained in [24]. Actually, we show that we can choose the rate of convergence as close as wanted to the optimal one, assuming that the initial data are small enough. Since second grade fluids are a particular case of third grade fluids, we establish an improvement of the rate obtained in [24]. Actually, the main difference between third and second grade fluids equations 

 in dimension 2 is the presence of 2 the additional term β div
A
A in the third grade fluids equations. Sometimes, this term helps to obtain global estimates, like in [4] or [26], but introduces additional difficulties when one looks for estimates in H3 or in more regular Sobolev spaces (see [1,2] or [5]). Here, we have to establish estimates in weighted Sobolev spaces with H2 regularity for the vorticity w, which is harder than doing estimates in H3 for u. We next introduce scaled variables. In order to simplify the notations, we assume that ν ¼ 1. Let T 4 1 be a positive constant which is introduced in order to avoid restrictions on the size of the parameter α1 and which will be made more precise later. We consider the solution w of (1.2) and define W and U such that curl U ¼ W pffiffiffiffiffiffiffiffiffi ffi through the change of variables X ¼ x= t þT and τ ¼ log ðt þ TÞ.

134

O. Coulaud / International Journal of Non-Linear Mechanics 71 (2015) 132–151

We set  8 1 x > > p ffiffiffiffiffiffiffiffiffi ffi p ffiffiffiffiffiffiffiffiffi ffi U log ðt þTÞ; ; uðt; xÞ ¼ > < t þT t þT  1 x > > > : wðt; xÞ ¼ t þ T W log ðt þ TÞ; pffiffiffiffiffiffiffiffiffiffi : t þT For τ Z log ðTÞ, we have   ( Uðτ; XÞ ¼ eτ=2 u eτ  T; eτ=2 X ;   Wðτ; XÞ ¼ eτ w eτ  T; eτ=2 X :

ð1:4Þ

WðτÞ ¼ ηG þ f ðτÞ;

ð1:5Þ

These variables, called scaled or self-similar variables, have been introduced in order to study the long time asymptotic of solutions of parabolic equations and particularly to show the convergence to selfsimilar solutions (seeffi[11,12,14] or [25]), that is to say under the form  pffiffiffiffiffiffiffiffiffi ð1=ðt þ TÞÞF x= t þ T . Scaled variables have been used to deal with the asymptotic behaviour of many equations, not necessarily parabolic ones (see [6,7,24,16] or [17]). For instance, in [16], Gallay and Raugel have described the first and second order asymptotic profiles in weighted Sobolev spaces for damped wave equations, using scaled variables. In [17], they use scaled variables to show a stability result of hyperbolic fronts for the same equations. For sake of simplicity, we set Ai;j ¼ ∂j U i þ ∂i U j . Considering selfsimilar variables, one can see that W and its corresponding divergence free vector field U satisfy the system     ∂τ W  α1 e  τ ΔW  LðWÞ þ U  ∇ W  α1 e  τ ΔW þ α1 e  τ ΔW    

 

2 X 2 þ α1 e  τ  ∇ΔW  βe  2τ div
A
∇W  β e  2τ div ∇
A
4 A ¼ 0; 2

div U ¼ 0; W jτ ¼ τ 0 ¼ W 0 ;

ð1:6Þ   where τ 0 ¼ log ðTÞ, W 0 ðXÞ ¼ eτ0 w0 eτ0 =2 X and L is the linear differential operator defined by LðWÞ ¼ ΔW þ W þ

X  ∇W: 2

Notice that the initial time of the system (1.6) is log ðTÞ. By choosing T sufficiently large, one can consider α1 e  τ as small as wanted. This fact allows to study the behaviour of the solutions of (1.6) without restrictions on the size of α1. Formally, we see that most of the terms of the system (1.6) tend to 0 as time goes to infinity. The purpose of the present paper is to show that the solutions of (1.6) asymptotically behave like solutions of ∂τ W 1 ¼ LðW 1 Þ:

ð1:7Þ

In order to describe the solutions of the system (1.7), we have to study the spectrum of the linear differential operator L in appropriate functions spaces. The form of the previous system and the definition of L lead to consider weighted Lebesgue spaces. For m A N, we define n o  m=2 L2 ðmÞ ¼ u A L2 ðR2 Þ : 1 þ jxj2 u A L2 ðR2 Þ ; equipped with the norm Z 1=2
2  m
1 þ jxj2
uðxÞ
dx : kukL2 ðmÞ ¼ R2

The spectrum of L in L2 ðmÞ is given in [18, Appendix A]. It is composed of the discrete spectrum  k σ d ðLÞ ¼  : k A f0; 1; …; m  2g ; 2 and the continuous spectrum  m1 : σ c ðLÞ ¼ λ A C : ReðλÞ r  2

In particular, the eigenvalue 0 is simple and the Oseen vortex G given by (1.3) is an eigenfunction of L associated to 0. Of course, G is a solution of (1.7) and we will show that the solutions of (1.6) behave like G when the time goes to infinity. To this end, we decompose the solutions W of (1.6) as follows: where η A R will be made more precise later and f ðτÞ is a rest which will tend to 0 as τ goes to infinity. In order to get a good rate of convergence for f, we shall ”push” the continuous spectrum of L to the left by choosing an appro2 priate weighted Lebesgue space. For  this reason, we work in L ð2Þ, 1 so that σ c ðLÞ ¼ λ A C : ReðλÞ r  2 . Since the second eigenvalue of L in L2 ð2Þ is  12, the best result that we expect is f ðτÞ ¼ Oðe  τ=2 Þ

in L2 ð2Þ

when τ- þ 1:

Notice that choosing a weighted space L2 ðmÞ with m 4 2 would be useless for describing the first order asymptotics only. Indeed, if we take m 4 2, the second eigenvalue would still be  12 and the rate of convergence could not be better than e  τ=2 . For later use, we define the divergence free vector field V such that curl V ¼ G. It is obtained by the Biot–Savart law and given by ! 2 1  e  jX j =4  X 2 VðXÞ ¼ : ð1:8Þ X1 2π jX j2 In particular, for every X A R2 , one has VðXÞ  X ¼ 0;

VðXÞ  ∇GðXÞ ¼ 0

and

VðXÞ  ∇ΔGðXÞ ¼ 0:

Before stating the main theorem of this paper, we have to define some additional functions spaces. For m A N, we set n o H 1 ðmÞ ¼ u A L2 ðmÞ : ∂j u A L2 ðmÞ; j A f1; 2g ; n o H 2 ðmÞ ¼ u A H 1 ðmÞ : ∂j u A H 1 ðmÞ; j A f1; 2g ; equipped with the norms  1=2 kukH1 ðmÞ ¼ kuk2L2 ðmÞ þ k∇uk2L2 ðmÞ and   1=2



2 ¼ kuk2H1 ðmÞ þ ∇2 u L2 ðmÞ ;

kukH2 ðmÞ



2 P  2 P where j∇uj2 ¼ 2i ¼ 1 ð∂i uÞ2 and
∇2 u
¼ 2i;j ¼ 1 ∂i ∂j u . The following theorem describes the first order asymptotic profile of W in H 2 ð2Þ, if one assumes that the initial data W0 are small enough in the weighted Sobolev space H 2 ð2Þ. Theorem 1.1. Let θ be a constant such that 0 o θ o 1. There exist two positive constants γ 0 ¼ γ 0 ðα1 ; β Þ and T 0 ¼ T 0 ðα1 Þ Z 1 such that, for all W 0 A H 2 ð2Þ satisfying the condition

2 α2

2

2

α





kW 0 k2H1 þ 1 ΔW 0 L2 þ jX j2 W 0 2 þ 21 jX j2 ΔW 0 2 r γ ð1  θÞ6 ; L L T T ð1:9Þ for some  T Z T 0 and 0 o γ r γ 0 , there exist a unique global solution W A C 0 ½τ0 ; þ 1Þ; H 2 ð2Þ of (1.6) and a positive constant C ¼ Cðα1 ; β ; θÞ such that, for all τ Z τ0 ,

 



1  α1 e  τ Δ WðτÞ  ηG 22 r C γ e  θτ ; ð1:10Þ L ð2Þ R where η ¼ R2 W 0 ðXÞ dX, τ0 ¼ log ðTÞ and the parameters α1 and β are fixed and given in (1.1). Remark 1.1. The smallness assumption (1.9) is not optimal. By working harder, it is possible to get γ ð1  θÞp with p o6 on the right-hand side of the inequality. Remark 1.2. Notice that Theorem 1.1 establishes an improvement of [24, Theorem 1.1] concerning the first order asymptotics of the second grade fluids equations. Indeed, the above theorem holds

O. Coulaud / International Journal of Non-Linear Mechanics 71 (2015) 132–151

135

also with β ¼ 0 and consequently describes the first order asymptotic profiles of the solutions of the second grade fluids equations. The improvement comes from the fact that one can choose θ as close as wanted to 1, which is the optimal rate. In [24], the constant θ cannot be bigger than 12.

2. Biot–Savart law and auxiliary lemmas

Theorem 1.1 implies the following result in the unscaled variables. In particular, it gives a description of the asymptotic profiles of the solutions of the equations of motion (1.1).

kuk ¼ kukL2 ;

Corollary 1.1. Let θ be a constant such that 0 o θ o 1. There exist two positive constants γ 0 ¼ γ 0 ðα1 ; β Þ and T 0 ¼ T 0 ðα1 ; βÞ Z1 such that, for all w0 A H 2 ð2Þ satisfying the condition

2

2 1

T kw0 k2L2 þT 2 k∇w0 k2L2 þ jxj2 w0 L2 þ α1 T 3 Δw0 L2 T

2 α2

þ 1 jxj2 Δw0 L2 r γ ð1  θÞ6 ; T

ð1:11Þ

o γ r γ 0 , there exists a unique global solution for some  T ZT 0 and 0  w A C 0 ½0; þ1Þ; H 2 ð2Þ of (1.2) such that, for all 1 r p r 2, there exists a positive constant C ¼ Cðα1 ; β ; θÞ such that, for all t Z 0,

 



 x

1  α1 Δ wðtÞ  η G pffiffiffiffiffiffiffiffiffi

r C γ ðt þ T Þ  1  θ=2 þ 1=p ; ffi

p t þT t þT L R where η ¼ R2 w0 ðxÞ dx. Moreover, for all 2 o q o þ 1, there exists a positive constant C ¼ Cðα1 ; β; θ; qÞ such that, for all t Z 0,

 



 η x

1  α1 Δ uðtÞ  pffiffiffiffiffiffiffiffiffi

r C γ ðt þ T Þ  1=2  θ=2 þ 1=q ; ffiV pffiffiffiffiffiffiffiffiffiffi

q t þT t þT L where V is obtained from G via the Biot–Savart law and defined by (1.8). Theorem 1.1 describes the asymptotic behaviour of the solutions of (1.6) in H 2 ð2Þ at the first order. Since the solutions of the Navier– Stokes equations also converge to the Oseen vortex sheet, we can say that the fluids of third grade behave asymptotically like Newtonian fluids. Notice that the functions space H 2 ð2Þ is suitable for the first order asymptotics because it “pushes” the continuous spectrum of L far enough to get 0 as an isolated eigenvalue. If we had to describe the asymptotics of (1.6) at the second order, we should work in a space where L has at least two isolated eigenvalues. Due to the forms of σc and σd, the second order asymptotics must be studied in functions space with polynomial weight of degree at least 3, in order to get the two isolated eigenvalues 0 and  12. Notice also that as the system (1.2) and our change of variables preserve the total mass. We have, for all τ Z τ0 and t Z0, Z Z Z Z η ¼ w0 ðxÞ dx ¼ wðt; xÞ dx ¼ W 0 ðXÞ dX ¼ Wðτ; XÞ dX: R2

R2

R2

R2

The plan of this paper is as follows. In Section 2, we recall classical results concerning the Biot–Savart law and give several technical lemmas. In Section 3, we introduce a regularized system, which is close to (1.6) and depends on a small parameter ε 4 0. 2 Actually, we add the regularizing term εΔ W to the system (1.6) and show the existence of unique regular solutions W ε to this new system. In Section 4, using energy estimates in various functions spaces, we show that W ε satisfies the inequality (1.10) of Theorem 1.1, and thus tends to the Oseen vortex sheet G when τ goes to infinity. In Section 5, we let ε go to 0 and show that W ε tends in a sense to a solution W of (1.6). Additionally, this solution satisfies the inequality (1.10) of Theorem 1.1 and consequently tends also to the Oseen vortex sheet. Finally, we establish the uniqueness of W, which enables us to say that every solution of (1.6) satisfying the assumption (1.9) converges to the Oseen Vortex sheet when τ goes to infinity.

In this section, we state several technical lemmas which are useful to prove Theorem 1.1. These lemmas concern the Biot–Savart law and state several inequalities involving weighted Lebesgue norms. In what follows, we use the notation

and C denotes a positive constant which can depend on the fixed constants α1 and β. The first lemma will be useful in Section 4 to obtain estimates in Sobolev spaces of negative order. We define, for s A R, the operator ð  ΔÞs , given by 

 2s b ; ð  ΔÞs u ¼ F
ξ
u b (also denoted F ðuÞ) is the Fourier transform of u, given by where u Z b ðξ Þ ¼ uðxÞe  ixξ dx; u R2

and F denotes the inverse Fourier transform Z 1 F ðvÞðxÞ ¼ vðξÞeixξ dξ: ð2π Þ2 R2 Lemma 2.1. Let s be a positive real number such that 34 os o 1, then we have the following two inequalities. 1. Let g A L2 ð1Þ. Then ð  ΔÞ  s ∇g A L2 ðR2 Þ and there exists C 4 0 independent of g and s such that



ð  ΔÞ  s ∇g r

C ð1  sÞ3=2



g

L2 ð1Þ

:

ð2:1Þ

R 2. Let g A L2 ð2Þ such that R2 gðxÞ dx ¼ 0. Then ð  ΔÞ  s g A L2 ðR2 Þ and there exists C 4 0 independent of g and s such that



ð  ΔÞ  s g 2 r L

C ð1  sÞ

3=2



g

L2 ð2Þ

:

ð2:2Þ

Proof. We start by proving the inequality (2.1). For j A f1; 2g, using Fourier variables, one has



ð  ΔÞ  s ∂j g 22 r C L

Z

2 1

2

4s  2
gb
dξ þ g L2 jξj r 1
ξ
!ð2s  1Þ=s Z !ð1  sÞ=s Z

2

2s=ð1  sÞ 1
g b
rC dξ þ g L2

2s dξ jξj r 1
ξ
jξj r 1 r





C

g b 2L2s=ð1  sÞ þ g 2L2 : ð 1  sÞ

We now use the continuous injection of H 1 ðR2 Þ into L2s=ð1  sÞ ðR2 Þ. Looking at the computations of [8, p. 723–724], one can see that there exists a constant C 4 0 such that kukLp r CpkukH1

for all u A H 1 ðR2 Þ and 2 r p o þ 1:

ð2:3Þ

Notice that Cp is not the optimal constant in the previous inequality. Using the inequality (2.3), one has



C

g b 2H1 þ g 2L2 ð1  sÞ3

C

g 22 : r L ð1Þ 3 ð1  s Þ



ð  ΔÞ  s ∂j g 22 r L

We now prove the inequality (2.2). Since Fourier variables, we get Z




1


ð  ΔÞ  s g 22 ¼ ð2π Þ2 b ðξÞ
2 dξ


g L 2

4s R ξ

R

R2 f ðxÞ

dx ¼ 0, using

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O. Coulaud / International Journal of Non-Linear Mechanics 71 (2015) 132–151

Z






1
bðξÞ
2 dξ þ g 22

4s
g L

jξj r 1 ξ
Z
2 Z




1
1 r ð2π Þ2 ξ  ∇gbðσξÞdσ

dξ þ g 2L2

4s

0 jξj r 1
ξ
r ð2π Þ2

To obtain (2.6), we use Gagliardo–Niremberg's inequality as follows:

2



 

jxj ∇f 4 rC jxj2 ∇f 1=2 ∇ jxj2 ∇f 1=2 L

1=2 





jxj∇f 1=2 þ jxj2 ∇2 f 1=2 ; r C jxj2 ∇f


Z
2





1
1

bðσξÞ
dσ

dξ þ g 22 : ∇g

4s  2

L

0 jξj r 1 ξ

Z rC

and consequently inequality (2.5) implies (2.6).

Biot–Savart law: Let w be a real function defined on R2 . The Biot– Savart law is a way to build a divergence free vector field u such that curl u ¼ w. It is given by Z 1 ðx  yÞ ? uðxÞ ¼ ð2:10Þ

wðyÞ dy; 2π R2
x  y
2

Cauchy–Schwarz inequality and Fubini's theorem give Z 1Z








1


ð  ΔÞ  s g 22 r C bðσξÞ
2 dξ dσ þ g 2L2 :

4s  2
∇g L 0 jξj r 1
ξ
Using Hölder inequality, we get



ð  ΔÞ  s g 22 r C L

Z

1 0

Z

1

4  2=s dξ jξj r 1
ξ


 s s Z 1 rC 1s 0

Z j ξj r 1

!s Z j ξj r 1




∇g b ðσξÞ
2=ð1  sÞ dξ




∇ g b ðσξÞ
2=ð1  sÞ dξ

!1  s

!1  s

2 dσ þ g L2

2 dσ þ g L2 :

Z


2=ð1  sÞ 1

∇gbðζ Þ
dζ 2

!1  s

jζ j r σ σ

2 dσ þ g L2

 s s  1



∇ g b 22=ð1  sÞ þ g 22 : rC L L 1s 2s  1

1. Assume that 1 o p o 2 o q o 1 and 1=q ¼ 1=p  1=2. If w A Lp ðR2 Þ, then u A Lq ðR2 Þ2 and there exists C 40 such that kukLq r C kwkLp :

ð2:11Þ

2. Assume that 1 r p o 2 oq r1, and define α A ð0; 1Þ by the relation 1=2 ¼ α=p þ ð1  αÞ=q. If w A Lp ðR2 Þ \ Lq ðR2 Þ, then u A L1 ðR2 Þ2 and there exists C 4 0 such that

Finally, we use again the inequality (2.3) and obtain  s  1  2 2





g

ð  ΔÞ  s g 22 r C s b 2H2 þ g 2L2 L 1 s 2s  1 1 s

C

g 22 ; r L ð2Þ 3 ð1  sÞ which concludes the proof of this lemma.

where ðx1 ; x2 Þ ? ¼ ð x2 ; x1 Þ. The next two lemmas give estimates on the divergence free vector field u obtained from w via the Biot–Savart law. Lemma 2.3. Let u be the divergence free vector field given by (2.10).

The change of variables ζ ¼ σξ yields  s Z 1



ð  ΔÞ  s g 22 r C s L 1s 0

□

kukL1 r C kwkαLp kwk1Lq α :

ð2:12Þ

3. Assume that 1 o p o 1. If w A Lp ðR2 Þ, then ∇u A Lp ðR2 Þ4 and there exists C 40 such that

□

k∇ukLp rC kwkLp :

ð2:13Þ

Lemma 2.2. 1. Let 1 r p o þ 1 and f A Lp ðR2 Þ such that jxj2 f A Lp ðR2 Þ, then jxjf A Lp ðR2 Þ and the following inequality holds :



2 1=2

jxjf p r f 1=2

jxj f p : ð2:4Þ L L Lp 2

2. Let f A H ð2Þ, there exists C 4 0 such that

2 2



 

jxj ∇ f r C f þ jxj∇f þ jxj2 Δf :

ð2:5Þ

ð2:6Þ

Proof. The inequality (2.4) comes directly from Hölder's inequality. To prove the inequality (2.5), we show by a simple calculation that, for every j; k A f1; 2g, 

2







jxj ∂j ∂k f 2 r C f 2 þ jxj∇f 2 þ jxj2 Δf 2 : ð2:7Þ

ð2:8Þ

and furthermore 







  



∂j ∂k jxj2 f 2 r C Δ jxj2 f 2 r C f 2 þ jxj∇f 2 þ jxj2 Δf 2 : ð2:9Þ Combining (2.8) and (2.9) we get the inequality (2.7).

We refer to [18] for the proof of this lemma. Lemma 2.4. Let u be the divergence free vector field given by (2.10). 1. If w A L2 ð2Þ, then u A L4 ðR2 Þ2 and there exists C 4 0 such that

3. Let f A H 2 ð2Þ, then jxj2 ∇f A L4 ðR2 Þ and there exists C 4 0 such that

2











jxj ∇f 4 r C jxj2 ∇f 1=2 f 1=2 þ jxj∇f 1=2 þ jxj2 Δf 1=2 : L

Indeed, we notice that   jxj2 ∂j ∂k f ¼ ∂j ∂k jxj2 f  2δj;k f  2xj ∂k f  2xk ∂j f ;

In addition, div u ¼ 0 and curl u ¼ w.

kukL4 r C kwkL2 ð2Þ :

ð2:14Þ

2. If w A L2 ð2Þ \ H 1 ðR2 Þ, then u A L1 ðR2 Þ2 and there exists C 4 0 such that 1=2 1=2 kwk 2 : H1 L ð2Þ

kukL1 r C kwk

ð2:15Þ

for s A R, then 3. Let s A R. If ð  ΔÞðs  1Þ=2 w A L2 ðR2 Þ 2 s=2 2 2 ð  ΔÞ u A L ðR Þ and there exists C 40 such that











ð2:16Þ

ð  ΔÞs=2 u r C ð  ΔÞðs  1Þ=2 w : 4. Let s A R. If w A H s ðR2 Þ, then ∇u A H s ðR2 Þ4 and there exists C 4 0 such that k∇ukHs r C kwkHs :

ð2:17Þ

The proof of the two first inequalities is done in [24]. The two other inequalities are obvious when using Fourier variables. The next lemma is useful to get energy estimates in weighted Sobolev

O. Coulaud / International Journal of Non-Linear Mechanics 71 (2015) 132–151

spaces for solutions of (1.6). For a vector field u, we set
3
2
∇ u
¼

2 X



∂j ∂k ∂l ui

2

:

i;j;k;l ¼ 1

Lemma 2.5. Let w A L2 ðR2 Þ and u be the divergence free vector given by (2.10). 1 2 2 1. If

w2 A H ð1Þ, then  ∇ u A L ð1Þ and  there exists C 4 0 such that

∇ u 2 r C kwk 1 þ kjxj∇wk : ð2:18Þ H L ð1Þ

2. If w A H 2 ð1Þ, then jxj∇2 u A L4 ðR2 Þ and there exists C 4 0 such that





 

jxj∇2 u 4 r C ðkwk þ kjxj∇wkÞ1=2 k∇wk þ jxjΔw 1=2 : ð2:19Þ L 3. If w A H 2 ð2Þ, then u A jxj2 ∇3 u A L2 ðR2 Þ and there exists C 4 0 such that

2 3



 

jxj ∇ u r C kwk þ kjxj∇wk þ jxj2 Δw : ð2:20Þ R 4. If w A L2 ð1Þ and R2 wðxÞ dx ¼ 0, then u A H 1 ð1Þ and there exists a positive constant C such that kukþ kjxj∇ukr C kjxjwk: ð2:21Þ R 5. If w A H 1 ð2Þ and R2 wðxÞ dx ¼ 0, then jxj2 ∇2 u A L2 ðR2 Þ and there exists a positive constant C such that



jxj2 ∇2 u r C kwk 1 : ð2:22Þ H ð2Þ

137

which gives





1=2 

2

;

xi ∂j u 4 rC ðkwkþ kjxj∇wkÞ1=2 k∇wk þ jxjΔw

L

and the inequality (2.19) comes when summing for i A f1; 2g. In order to get the inequality (2.20), it suffices to obtain it for jxj2 ∂j ∂k2 u, where j; k A f1; 2g. One has Z
 
2

2 2 2
2

jxj ∂j ∂ u ¼ ð2π Þ2 b

dξ
Δ ξj ξk u k 2 Z R

2  




2 b

dξ rC
ξ ξj þ ξk Δu 2 R Z
Z


2




2 b
2
b

dξ þ þ
ξ ∇u
dξ
ξj þ ξk u R2 R2 ! 2

X



 2  2

Δðxi uÞ 2 r C ∇Δ jxj u þ k∇uk2 þ i¼1



2

2  r C jxj2 ∇Δu þ k∇uk2 þ jxjΔu





2 r C jxj2 ∇2 w þ kwk2 þ kjxj∇wk2 : Applying the inequality (2.5), we get (2.20). The proof of the inequality (2.21) is made in two steps. It is shown in [24] that kukr C kjxjwk:

ð2:25Þ

To finish the proof of the inequality (2.21), we notice that Z jxj2 ∂1 u2 ∂2 u1 dx: kjxjwk2 ¼ kjxj∂1 u2 k2 þ kjxj∂2 u1 k2  2 ð2:26Þ R2

Proof. Let us show the inequality (2.18). Let w belong to H 1 ð1Þ and u be the divergence free vector field obtained via the Biot–Savart law. From the inequality 2.13 of Lemma 2.3, we obtain

2

∇ u 2 r C k∇wk 2 : ð2:23Þ L L

Integrating by parts, one gets Z Z Z jxj2 ∂1 u2 ∂2 u1 dx ¼ jxj2 u2 ∂1 ∂2 u1 dxþ 2 x1 u2 ∂2 u1 dx 2 R2 R2 R2 Z Z jxj2 ∂1 ∂2 u2 u1 dxþ 2 x2 ∂1 u2 u1 dx: þ R2

R2

Since the divergence of u vanishes and since we are in dimension 2, it is enough to show the inequality



2

ð2:24Þ

xi ∂j uk rC ðkwkþ kjxj∇wkÞ;

Using the divergence free property of u and integrating by parts, we have

where i; j; k A f1; 2g. We omit k that does not appear in the following calculations. One has Z


2 
2




2 2 b

dξ
∂i ξj u

jxi j∂j2 u ¼ ð2π Þ 2 R Z Z

2

2
rC ξj ub
dξ þ C

ξ2j ∂i ub

dξ 2 2 R R Z
 

F Δðxi uÞ
2 dξ r C k∇uk2 þ C

Finally, integrating again by parts, we get Z jxj2 ∂1 u2 ∂2 u1 dx ¼ kjxj∂1 u1 k2 þ kjxj∂2 u2 k2  2kuk2 : 2

Z

Z

2 R

2

jxj2 ∂1 u2 ∂2 u1 dx ¼ kjxj∂1 u1 k2 þ kjxj∂2 u2 k2 þ 4

Z 2

R

x2 u2 ∂2 u2 dx þ 4

R2

x1 ∂1 u1 u1 dx:

R2

Thus, going back to (2.26), one has kjxj∇uk2 ¼ kjxjwk2 þ 2kuk2 : Combining this equality with (2.25), we get the inequality (2.21). The inequality (2.22) is obtained in the same way. □

R2



2 r C k∇uk2 þ C jxjΔu : Using the inequality (2.13) of Lemma 2.3 with p ¼ 2 and remarking that ∂1 w ¼ Δu2 and ∂2 w ¼ Δu1 , we obtain (2.24). Combining it with the inequality (2.23), we get (2.18). The inequality (2.19) is a direct consequence of (2.18) and Gagliardo–Niremberg inequality. Indeed, one has





1=2   1=2

2







xi ∂j u 4 r C xi ∂j2 u ∇ xi ∂j2 u

L

1=2 







2 2 1=2 r C xi ∂j2 u

:

∂j u þ xi ∂j ∇u

Furthermore, the inequalities (2.13) and (2.18) yield





1=2 

2

:

xi ∂j u 4 r C ðkwk þ kjxj∇wkÞ1=2 k∇wk þ xi ∇2 w

L

Making the same computations than the ones we made to establish (2.18), we obtain





 

xi ∇2 w r C k∇wkþ jxjΔw ;

3. Approximate solutions In this section, we introduce a “regularized” system of equations, whose solutions are more regular than the solutions of (1.2). Actually, this new system is very close to (1.2), and is obtained by 2 adding the small term εΔ w to (1.2). Here, the positive constant ε is supposed to be small and is devoted to tend to 0. Adding this term, we are able to prove the existence of solutions to the regularized system via a semi-group method. The presence of the term u:∇Δw would not let us obtain solutions to (1.2) by a semi-group method because of the too high degree of derivatives in this term compared to the linear term Δw. We introduce now the following regularized system of equations:     2 ∂t wε  α1 Δwε þ εΔ wε  Δwε þ uε ∇ wε  α1 Δwε 

  

  2 2  β div
Aε
∇wε  β div ∇
Aε
4 Aε ¼ 0; wεjt ¼ 0 ¼ w0 A H 2 ð2Þ;

ð3:1Þ t

where Aε ¼ ∇uε þ ð∇uε Þ .

138

O. Coulaud / International Journal of Non-Linear Mechanics 71 (2015) 132–151

on H 1 ðR2 Þ

The aim of this section is to prove the following theorem. Theorem 3.1. Let w0 A H 2 ð2Þ. For all ε 4 0, there exists t ε 4 0 and a unique solution wε of the system (3.1) such that       wε A C 1 ð0; t ε Þ; H 1 ð2Þ \ C 0 ½0; t ε Þ; H 2 ð2Þ \ C 0 ð0; t ε Þ; H 3 ð2Þ : Proof. First of all, we introduce the change of variable x~ ¼ γ x, where γ is a positive constant that is close to 0 and will be made more precise later. This is made in order to not have to consider restrictions on the size of α1. We note vε ðxÞ ¼ wε ðx=γ Þ. The system (3.1) provides a new system in vε that we will solve in H 2 ð2Þ     2 ∂t vε  α1 γ 2 Δvε þ εγ 4 Δ vε  γ 2 Δvε þ γ uε  ∇ vε  α1 γ 2 Δvε 

    

2

2 2  βγ ∇
Aε
 ∇vε  βγ 2
Aε
Δvε  β div ∇
Aε
4 Aε ¼ 0; vεjt ¼ 0 ¼ w0 ðx=γ Þ A H 2 ð2Þ:

ð3:2Þ

Although there are terms involving uε in this system, it is actually autonomous. In fact, one recover wε from vε and then recover uε via the Biot–Savart law (2.10) applied to wε . We set zε ðxÞ ¼ qðxÞvε ðxÞ;   where qðxÞ ¼ 1 þ jxj2 . To show the existence of a solution in H 2 ð2Þ to the system (3.2), we are reduced to show that there exists a solution in H 2 ðR2 Þ of the system   2 ∂t zε  γ 2 α1 Δzε  α1 γ 2 qΔq  1 zε  2γ 2 α1 q∇q  1  ∇zε þ εγ 4 Δ zε ¼ F ðzε Þ; zεjt ¼ 0 ¼ qw0 ðx=γ Þ A H 2 ðR2 Þ;

ð3:3Þ

where       2 Fðzε Þ ¼  εγ 4 qΔ q  1 zε þ γ 2 qΔ q  1 zε  γ quε ∇ q  1 zε  γ 2 α1 Δ q  1 zε 



2     2 þ βγ q∇
Aε
 ∇ q  1 zε þ βγ 2 q
Aε
Δ q  1 zε  

  2 ð3:4Þ þ βq div ∇
Aε
4 Aε :

A ¼ εγ 4 ðI  B  DÞ  1 Δ : 2

We rewrite the system (3.3) as follows: ∂t zε þ Aðzε Þ ¼ F~ ðzε Þ; zεjt ¼ 0 ¼ qw0 ðx=γ Þ A H 2 ðR2 Þ;

ð3:6Þ

where F~ ðzε Þ ¼ ðI  B DÞ  1 Fðzε Þ. To finish the proof of this theorem, we show that the operator A is sectorial on H 1 ðR2 Þ, which is equivalent to the fact that  A generates an analytic semigroup on H 1 ðR2 Þ. Then, we check that F~ is locally Lipschitz from bounded sets of a Sobolev space H s ðR2 Þ to H 1 ðR2 Þ, where 1 r s o 3. An easy computation leads to A ¼ εγ 4 ðI  BÞ  1 Δ  εγ 4 ðI  B  DÞ  1 DðI BÞ  1 Δ 2

2

¼ I þ εγ 4 ðI  BÞ  1 Δ  I  εγ 4 ðI  B  DÞ  1 DðI  BÞ  1 Δ ¼ J þ R; 2

2

where J ¼ I þ εγ 4 ðI  BÞ  1 Δ ; 2

R ¼  I  εγ ðI  B DÞ  1 DðI  BÞ  1 Δ : 4

2

Using the same method as the one used to invert ðI  B  DÞ, one can invert ðI  BÞ and define ðI  BÞ  1 from H  1 ðR2 Þ to H 1 ðR2 Þ. Consequently, J is well defined from H 3 ðR2 Þ to H 1 ðR2 Þ. In the remaining of this proof, we will show that  J generates an analytic semi-group on H 1 ðR2 Þ and then show that R satisfies the conditions of [28, Theorem 2.1, p. 81]. According to this result, it implies that  A generates an analytic semi-group on H 1 ðR2 Þ. In order to show that J is sectorial on H 1 ðR2 Þ, we associate it to a continuous and coercive bilinear form on H 2 ðR2 Þ  H 2 ðR2 Þ. To this end, we define a H1-scalar product which is suitable to J. Let us define, for u; v A H 1 ðR2 Þ, the bilinear form on H1 given by    hu; viH1 ¼ 1  α1 γ 2 qΔq  1 u; v L2 þ α1 γ 2 ð∇u; ∇vÞL2 :

We define the two linear operators B : DðBÞ ¼ H 1 ðR2 Þ-H  1 ðR2 Þ and D : DðDÞ ¼ L2 ðR2 Þ-H  1 ðR2 Þ as follows:

If γ is sufficiently small compared to α1, then h .,. iH1 is a scalar product on H 1 ðR2 Þ. Furthermore, for u A H 2 ðR2 Þ and v A H 1 ðR2 Þ, one has

BðzÞ ¼ α1 γ 2 Δz þ α1 γ 2 qΔq  1 z;

hu; viH1 ¼ ððI  BÞu; vÞL2 :

DðzÞ ¼ 2α1 γ 2 q∇q  1  ∇z:

Via Lax–Milgram theorem, it is easy to show that A ¼ ðI  B  DÞ is invertible. We define the bilinear form on H 1 ðR2 Þ     aðu; vÞ ¼ ðu; vÞL2 þ α1 γ 2 ð∇u; ∇vÞL2  α1 γ 2 qΔq  1 u; v L2  2α1 γ 2 q∇q  1  ∇u; v L2 : 1

2

1

2

We notice that a is obviously continuous on H ðR Þ  H ðR Þ. Using the fact that qΔq  1 and q∇q  1 are bounded on R2 , one has, for all u; v A H 1 ðR2 Þ,


aðu; vÞ
r Cðα1 ; γ Þkuk 1 kvk 1 ; H H where Cðα1 ; γ Þ is a positive constant depending on α1 and γ. We show now that a is coercive. Via an integration by parts, we get aðu; uÞ ¼ kuk þ α1 γ k∇uk  α1 γ 2

2

2

Z 2 R2

qΔ q

1

juj dx þ α1 γ 2

Z

  div q∇q  1 juj2 dx:

2 R2

  and div q∇q  1 , there exists C 4 0

Due to the boundedness of qΔq  1 such that   aðu; uÞ Z 1  α1 γ 2 C kuk2 þ α1 γ 2 k∇uk2 :

If we take γ sufficiently small, the bilinear form a is both continuous and coercive on H 1 ðR2 Þ. From the Lax–Milgram theorem, we conclude that for all, f A H  1 ðR2 Þ, there exists u A H 1 ðR2 Þ such that   ð3:5Þ aðu; vÞ ¼ f ; v H  1 H1 for all v A H 1 ðR2 Þ; and consequently ðI B  DÞ  1 is defined from H  1 ðR2 Þ to H 1 ðR2 Þ. We define A : DðAÞ ¼ H 3 ðR2 Þ-H 1 ðR2 Þ the linear differential operator

We define, using this scalar product, the bilinear form j on H 2 ðR2 Þ  H 2 ðR2 Þ associated to J by the formula   jðu; vÞ ¼ hu; viH1 þ εγ 4 Δu; Δv L2 : A short computation shows that, for u A H 3 ðR2 Þ and v A H 2 ðR2 Þ, one has   ð3:7Þ jðu; vÞ ¼ Ju; v H1 : Furthermore, if γ is small enough, using the definition of h .,. iH1 and j, we see that there exists Cðα1 ; ε; γ Þ 4 0 such that, for all u; v A H 2 ðR2 Þ, jðu; vÞ r Cðα1 ; ε; γ ÞkukH2 kvkH2 : Besides, it is simple to check that, if γ is mall enough, there exists Cðα1 ; γ ; εÞ 4 0 such that, for all u A H 2 ðR2 Þ, jðu; uÞ Z Cðα1 ; γ ; εÞkuk2H2 : The bilinear form j is thus continuous and coercive on H 2 ðR2 Þ and the operator J is consequently sectorial on H 1 ðR2 Þ. Additionally, the linear operator R is defined from H 2 ðR2 Þ to H 1 ðR2 Þ, and one can check that there exists Cðα1 ; γ ; εÞ 4 0 such that, for all u A H 3 ðR2 Þ, kRukH1 r Cðα1 ; γ ; εÞkukH2 :

ð3:8Þ 3

2

Applying the equality (3.7) to u A H ðR Þ, we get   jðu; uÞ ¼ Ju; u H1 for all u A H 3 ðR2 Þ:

O. Coulaud / International Journal of Non-Linear Mechanics 71 (2015) 132–151

Since j is coercive on H2, we obtain, via Cauchy–Schwartz inequality,



kuk2H2 r Cðα1 ; γ ; εÞ Ju H1 kukH1 for all u A H 3 ðR2 Þ: Going back to (3.8), the following property holds:



kRuk2H1 rCðα1 ; γ ; εÞ Ju H1 kukH1 for all u A H 3 ðR2 Þ:

139

  on M) such that, for all τ A τ0 ; τnε , the following inequality holds:

2







W ε ðτÞ 2 1 þ α1 e  τ ΔW ε ðτÞ 22 þ

jX j2 W ε ðτÞ 2 H L L

2

2 6 2  2τ

þ α1 e ð4:4Þ

jX j ΔW ε ðτÞ 2 rM γ ð1  θÞ : L

In particular, the Young inequality yields, for all δ 4 0,

2 kRuk2H1 r δ Ju H1 þ Cðα1 ; γ ; εÞkuk2H1 for all u A H 3 ðR2 Þ: By a classical result that we can find in [23],  A is thus the generator of an analytic semigroup on H 1 ðR2 Þ. Lastly, it is easy to check that F~ is Lipschitzian from the bounded sets of H 2 ðR2 Þ into H 1 ðR2 Þ. Combining several results from [23, Chapter 3] and [28, Section 6.3], there exists t ε 4  we conclude  that   0 and zε A C 1 ð0; t ε Þ; H 1 ðR2 Þ \ C 0 ½0; t ε Þ; H 2 ðR2 Þ \  a unique solution  C 0 ð0; t ε Þ; H 3 ðR2 Þ ofthe system (3.3). Thus, there exists a uniq  ue solution wε A C 1 ð0; t ε Þ; H 1 ð2Þ \ C 0 ½0; t ε Þ; H 2 ð2Þ \ C 0 ðð0; t ε Þ; H 3 ð2ÞÞ to the system (3.1). □

To simplify the notations in the following computations, we assume that 0 o γ r 1 and we take T sufficiently large so that α1 =T ¼ α1 e  τ0 r 1.   Since W ε A C 0 ½τ0 ; τε Þ; H 2 ð2Þ and the condition (1.9) holds, τnε is well defined. Furthermore, there exists constant C  a positive  independent of W0 such that, for all τ A τ0 ; τnε ,

2

2





η2 þ f ε 2H1 þ α1 e  τ Δf ε 2L2 þ

jX j2 f ε

2 þ α21 e  2τ

jX j2 Δf ε

2 L

ð4:5Þ

Indeed, using Cauchy–Schwartz inequality, we get Z η ¼ W 0 ðXÞ dX R2

Z

1 þ jX j2

¼

4. Energy estimates

L

r CM γ ð1  θÞ6 :

W 0 ðXÞ dX 2 R 1 þ jX j 0 11=2 Z  1=2 Z 2

1 B C 2
W 0 ðXÞ
2 dX dX 1 þ j X j r@ A   2 R2 R2 1 þ jX j2 2

In this section, we perform energy estimates on the regularized solutions of the third grade fluids equations in the weighted space H 2 ð2Þ. These estimates are independent of ε and allow us, in Section 5, to pass to the limit when ε tends to 0. Thus, we consider the solution wε ðt; xÞ of (3.1). Let T, T Z 1 be a fixed positive constant and τ0 ¼ log ðTÞ. We define W ε ðτ; XÞ, obtained from wε by the change of variables (1.4) and (1.5). A short computation shows that W ε satisfies the system   2 ∂τ W ε  α1 e  τ ΔW ε þ εe  τ Δ W ε  LðW ε Þ þ U ε   ∇ W ε  α1 e  τ ΔW ε þ α1 e  τ ΔW ε 

 X 2 þ α1 e  τ  ∇ΔW ε  βe  2τ div
Aε
∇W ε  β e  2τ 2  

  2 div ∇
Aε
4 Aε ¼ 0; div U ε ¼ 0; W εjτ ¼ τ0 ¼ W 0 ;

ð4:1Þ

where τ0 ¼ log ðTÞ, U ε is obtained from W ε via the Biot–Savart law (2.10), Aε ¼ ∇U ε þ ð∇U ε Þt and we recall that LðW ε Þ ¼ ΔW ε þW ε þ

X  ∇W ε : 2

By Theorem 3.1, it is clear that there exists τε 4 τ0 such that     W ε A C 1 ðτ0 ; τε Þ; H 1 ð2Þ \ C 0 ðτ0 ; τε Þ; H 3 ð2Þ :

ð4:2Þ

where G is the Oseen vortex sheet defined by (1.3) and V is the divergence free vector field obtained from G via the Biot–Savart law (2.10). Using the fact that LðGÞ ¼ 0, one has the equality     2 ∂τ f ε  α1 e  τ Δf ε þ εe  τ Δ f ε  Lðf ε Þ þ K ε  ∇ f ε  α1 e  τ Δf ε     þ ηV  ∇ f ε  α1 e  τ Δf ε þ ηK ε  ∇ G  α1 e  τ ΔG þ α1 e  τ Δf ε þ α1 e

 τX

τ

τX

τ

Considering the decomposition (4.2) and the smoothness of G, we obtain the inequality (4.5). To simplify the notations, in this section we write f instead of f ε , W instead of W ε , U instead of U ε and K instead of K ε . The aim of this section is to show that the inequality (1.10) of Theorem 1.1 holds for the regularized solutions of the system (4.1), provided that the condition (1.9) is satisfied by W0. To this end, we consider a fixed constant θ such that 0 o θ o 1 which is twice the rate of convergence of W to ηG in H 2 ð2Þ. In fact, we will show that, under the assumption (1.9), the decaying of f to 0 in H 2 ð2Þ is equivalent to e  θτ=2 . As it is explained in the introduction of this paper, the spectrum of L in L2 ðmÞ does not allow the rate of convergence to be better than e  τ=2 . In order to get the inequality (1.10), we construct in this section   an energy functional E ¼ EðτÞ such that, for every τ A τ0 ; τnε ,

2 EðτÞ  f ðτÞ H2 ð2Þ ; and thereexists a positive constant C ¼ Cðα1 ; β; θÞ such that, for all τ A τ0 ; τnε ,

We also assume that the initial data W 0 A H 2 ð2Þ satisfies the assumption (1.9) of Theorem 1.1, for some γ 40. Let R η ¼ R2 W 0 ðXÞ dX, we write the following decompositions: W ε ¼ ηG þf ε ; U ε ¼ ηV þ K ε ;

r C kW 0 kL2 ð2Þ :

 ∇Δf ε þ ηα1 e ΔG þ ηα1 e  ∇ΔG þ ηεe Δ G 2 2    





2  2 2  β e  2τ div
Aε
∇f ε þ η
Aε
∇G  βe  2τ div ∇
Aε
4 Aε ¼ 0: 2

ð4:3Þ Let M ¼ Mðα1 ; βÞ 4 2 be a positive constant which will be made more precise later. Let τnε A ðτ0 ; τε  be the largest time (depending

∂τ EðτÞ þ θEðτÞ r C γ e  τ :

ð4:6Þ

This inequality will enable us to show that τε ¼ þ1 and obtain, by the application of Gronwall Lemma, n

EðτÞ rC γ e  θτ

for all τ A ½τ0 ; þ 1Þ:

This functional is built as the sum of several intermediate energy functionals in various functions spaces, for which we perform convenient estimates.  ð1 þ θÞ=2 4.1. Estimates in H_

We start by performing an estimate of the solution of (4.3) in  ð1 þ θÞ=2 the homogeneous Sobolev space H_ ðR2 Þ. Combined with the other estimates, it will give us an estimate in the classical Sobolev space H  ð1 þ θÞ=2 ðR2 Þ. The motivation to do this comes from the fact that the H 1 -estimate that we will perform later (see Lemma 4.3) makes the term kuk2L2 appear on the right-hand side of our H 1 -energy inequality. In order to absorb this term, we look for an estimate in a Sobolev space of negative order. To this end, due

140

O. Coulaud / International Journal of Non-Linear Mechanics 71 (2015) 132–151

R to Lemma 2.1 and the fact that R2 f ðXÞ dX ¼ 0, for 34 rs o 1, one can apply the operator ð  ΔÞ  s to the equality (4.3) and take the inner L2 -product of it with ð  ΔÞ  s f . Through the computations that we will perform below, one can see that, in order to get the estimate (4.6), we have to choose at least s ¼ ð1 þ θÞ=2. Actually, since we have to absorb terms coming from the non-linear part of (4.3), it is more convenient to take ð1 þ θÞ=2 os o 1, for instance s ¼ ð3 þ θÞ=4. In [24], the considered operator was ð  ΔÞ  3=4 , which implied the restriction 0 o θ o 12. The next lemma summarizes the computations needed when applying ð  ΔÞ  s to (4.3) and taking the L2 -scalar product of it with ð  ΔÞ  s f . R Lemma 4.1. Let f A H 3 ð2Þ such that R2 f ðXÞ dX ¼ 0, then, for all 1 2 r s o 1 the three following equalities hold:   

2 X 1

 ∇f ; ð  ΔÞ  s f ð  ΔÞ  s ¼  s þ ð  ΔÞ  s f L2 ; 2 2 2 L

2  1

2  



ð  ΔÞ  s ðLðf ÞÞ; ð  ΔÞ  s f L2 ¼  ð  ΔÞ1=2  s f 2  s  ð  ΔÞ  s f L2 ; 2 L  

2 X



 ∇Δf ; ð  ΔÞ  s f ð  ΔÞ  s ¼ ðs þ 1Þ ð  ΔÞ1=2  s f 2 : ð4:7Þ 2 L L2

Taking the L2 -scalar product of (4.9) with ð  ΔÞ  ð3 þ θÞ=4 and taking into account the equalities    ð  ΔÞ  ð3 þ θÞ=4 ðLðf ÞÞ; ð  ΔÞ  ð3 þ θÞ=4 f 2 L

2 1 þ θ

2



 ð1 þ θÞ=4

 ð3 þ θ Þ=4

¼ ð  ΔÞ f þ f ;

ð  ΔÞ 4 and 

X  ∇Δf ; ð  ΔÞ  ð3 þ θÞ=4 f 2 L2 

2 7þθ



¼ α1 e  τ ð  ΔÞ  ð1 þ θÞ=4 f ; 4

α1 e  τ ð  ΔÞ  ð3 þ θÞ=4



given by Lemma 4.1, we obtain 

2

2 1





∂τ ð  ΔÞ  ð3 þ θÞ=4 f þ α1 e  τ ð  ΔÞ  ð1 þ θÞ=4 f

2

2 1 þ θ

2





þ εe  τ ð  ΔÞ  ð1  θÞ=4 f þ

ð  ΔÞ  ð3 þ θÞ=4 f

4  

2 1þθ þ 1þ α1 e  τ

ð  ΔÞ  ð1 þ θÞ=4 f

4   ¼ H ðτ; G; f Þ; ð  ΔÞ  ð3 þ θÞ=4 f 2 : L

Now, it remains to estimate the right-hand side of (4.10) that we write as   H ðτ; G; f Þ; ð  ΔÞ  ð3 þ θÞ=4 f 2 ¼ I 1 þI 2 þ I 3 þ I 4 þ I 5 ;

Proof. Using Fourier variables, it is easy to see that



2 ξ
ξ
2 b Xd ξ b X d b b

 ∇f ¼  f   ∇f and  ∇ Δf ¼ 2 ξ f þ ∇f : 2 2 2 2

L

The proof of this lemma is then obtained through the Plancherel formula and direct computations. □  ð1 þ θ Þ=2 In order to obtain a priori estimates of f in H_ ðR2 Þ, we define the functional 

2

2 1





E 1 ðτ Þ ¼

ð  ΔÞ  ð3 þ θÞ=4 f þ α1 e  τ ð  ΔÞ  ð1 þ θÞ=4 f : 2

The estimate in H_ the next lemma.

 ð1 þ θÞ=2

of f under the condition (4.4) is given in

    Lemma 4.2. Let W A C 1 ðτ0 ; τε Þ; H 1 ð2Þ \ C 0 ðτ0 ; τε Þ; H 3 ð2Þ be the solution of (4.1) satisfying the inequality (4.4) for some γ 4 0. There exist  γ 0 40 and T 0 Z 1 such that if T Z T 0 and γ r γ 0 , then, for all τ A τ0 ; τnε , E1 satisfies the inequality 

2 1θ α1 e  τ

ð  ΔÞ  ð1 þ θÞ=4 f

r CM3 γ ð1  θÞ2 e  2τ ∂τ E1 þ θE1 þ 1 þ 4 

2

2  2 þ CM γ ð1  θÞ2 f L2 ð2Þ þ ∇f þ α21 e  2τ Δf L2 ð1Þ ; ð4:8Þ where θ, 0 o θ o 1 is the fixed constant introduced at the beginning of Section 4. R Proof. Since according to Lemma 2.1, R2 f ðXÞ dX ¼ 0,  ð3 þ θÞ=4 ð  ΔÞ f is well defined. Thus, we apply ð  ΔÞ  ð3 þ θÞ=4 to the equality (4.3) and we get   ∂τ ð  ΔÞ  ð3 þ θÞ=4 f þ α1 e  τ ð  ΔÞð1  θÞ=4 f þ εe  τ ð  ΔÞð5  θÞ=4 f  ð  ΔÞ  ð3 þ θÞ=4 ðLðf ÞÞ  X  ∇ Δf  α1 e  τ ð  ΔÞð1  θÞ=4 f þ α1 e  τ ð  ΔÞ  ð3 þ θÞ=4 2 ¼ H ðτ; G; f ; W Þ; H ðτ; G; f ; W Þ ¼ ð  ΔÞ

     I1 ¼ ð  ΔÞ  ð3 þ θÞ=4  K  ∇ f  α1 e  τ Δf ; ð  ΔÞ  ð3 þ θÞ=4 f 2 ; L      I2 ¼ ð  ΔÞ  ð3 þ θÞ=4  ηV  ∇ f  α1 e  τ Δf ; ð  ΔÞ  ð3 þ θÞ=4 f 2 ; L      I3 ¼ ð  ΔÞ  ð3 þ θÞ=4  ηK  ∇ G  α1 e  τ ΔG ; ð  ΔÞ  ð3 þ θÞ=4 f 2 ; L    ð3 þ θ Þ=4 τ τX τ 2  ∇ΔG  ηεe Δ G ; ð ΔÞ  ð3 þ θÞ=4 f  ηα1 e ΔG  ηα1 e I 4 ¼ ð  ΔÞ 2 L2    ð3 þ θÞ=4  ð3 þ θ Þ=4 ¼ ð  ΔÞ J; ð ΔÞ f 2; L   

  2  ð3 þ θÞ=4  2τ I 5 ¼ ð  ΔÞ βe curl div
A
A ; ð  ΔÞ  ð3 þ θÞ=4 f 2 : L

The remaining of the proof of this lemma is devoted to the estimate of these terms. We recall that curl K ¼ f , curl V ¼ G and curl U ¼ W. Since the divergence of K vanishes, we obtain       ; ð  ΔÞ  ð3 þ θÞ=4 f 2 I 1 ¼ ð  ΔÞ  ð3 þ θÞ=4  div K f  α1 e  τ Δf L



  





r ð  ΔÞ  ð3 þ θÞ=4 ∇ K f  α1 e  τ Δf

ð  ΔÞ  ð3 þ θÞ=4 f : Using the inequalities (2.1) of Lemma 2.1 and (2.15) of Lemma 2.4, together with the Young and Hölder inequalities and the property (4.5), we get



 

C  ð3 þ θ Þ=4

K f  α 1 e  τ Δf 2

ð  Δ Þ f I1 r



L ð1Þ ð1  θÞ3=2





C



kK kL1 f  α1 e  τ Δf L2 ð1Þ ð  ΔÞ  ð3 þ θÞ=4 f

r 3=2 ð1  θÞ

2





C



f 2 f 1 f  α1 e  τ Δf 22 r μ ð  ΔÞ  ð3 þ θÞ=4 f þ L ð1Þ L ð2Þ H 3 μð1  θÞ

μ

ð4:9Þ

     K  ∇ f  α1 e  τ Δf  ηV  ∇ f  α1 e  τ Δf

  X  ηK  ∇ G α1 e  τ ΔG  ηα1 e  τ ΔG  ηα1 e  τ  ∇ΔG 2 

 2 2  ηεe  τ Δ Gþ βe  2τ curl div
A
A :

where



2 CM γ ð1  θÞ3 







f 22 þ α2 e  2τ Δf 22 r μ ð  ΔÞ  ð3 þ θÞ=4 f þ ; 1 L ð1Þ L ð1Þ

where   ð3 þ θ Þ=4

ð4:10Þ

ð4:11Þ where μ is a positive constant which is made more precise later. Similar computations and the inequality (4.5) give similar estimates for I2. One has

2 CM γ ð1  θÞ3 







f 22 þ α2 e  2τ Δf 22 : I 2 r μ ð  ΔÞ  ð3 þ θÞ=4 f þ 1 L ð1Þ L ð1Þ

μ

ð4:12Þ

O. Coulaud / International Journal of Non-Linear Mechanics 71 (2015) 132–151

Likewise, we estimate the term I3. Indeed, the same computations and the smoothness of G yield

2



C η2



f 2 f 1 G  α1 e  τ ΔG 22 I 3 r μ ð  ΔÞ  ð3 þ θÞ=4 f þ L ð1Þ μð1  θÞ3 L ð2Þ H

2 CMγ ð1  θÞ3 





f 22 þ f 2 1 ð1 þ α1 e  τ Þ: r μ ð  ΔÞ  ð3 þ θÞ=4 f þ L ð2Þ H

μ

Taking T0 sufficiently large so that α1 e  τ r 1, we get

2 CM γ ð1  θÞ3 





f 22 þ ∇f 2 : I 3 r μ ð  ΔÞ  ð3 þ θÞ=4 f þ L ð2Þ

μ

ð4:13Þ

Estimating I4 is simple, because of the smoothness of G. We R remark that R2 JðXÞ dX ¼ 0. Thus we can apply the inequality (2.2) of Lemma 2.1 to obtain



C
η
e  τ kGk 4

H ð3Þ  ð3 þ θÞ=4

J r :

ð  ΔÞ ð1  θÞ3=2 Using the above inequality and the smoothness of G, we can write

C ηe  τ



I4 r

ð  ΔÞ  ð3 þ θÞ=4 f

3=2 ð1  θÞ

2 CMγ ð1  θÞ3 e  2τ



: ð4:14Þ r μ ð  ΔÞ  ð3 þ θÞ=4 f þ

μ

It remains to estimate the term I5. The inequality (2.1) of Lemma 2.1 implies



 

C β e  2τ





2



I5 r

∇ A A 2 ð  ΔÞ  ð3 þ θÞ=4 f

3=2 L ð1Þ ð1  θÞ

2 C β 2 e  4 τ 

 2





2

r μ ð  ΔÞ  ð3 þ θÞ=4 f þ

∇ A A 2 : L ð1Þ μð1  θÞ3 A short computation leads to



 2

2



2



∇ A A 2 r C j∇U j2 ∇2 U 2 L ð1Þ

:

Using Hölder inequalities, the inequality (2.15) of Lemma 2.4 and the inequality (2.18) of Lemma 2.5, we get



 2

2



2

r C k∇U k41 ∇2 U 2

∇ A A

L ð1Þ

L

L2 ð1Þ

r kW k2L2 ð2Þ kW k2H1



kW k2L2

þ k∇W k2L2 ð1Þ



:

Finally, taking into account the inequality (4.4), we get

2 CM3 β2 γ 3 ð1  θÞ18 e  4τ  eτ



1þ I 5 r μ ð  ΔÞ  ð3 þ θÞ=4 f þ

μ



2 CM3 γ 3 ð1  θÞ18 e  3τ



: r μ ð  ΔÞ  ð3 þ θÞ=4 f þ

μ

α1

ð4:15Þ

The equality (4.10) and the inequalities (4.11)–(4.15) imply that 

2

2 1





∂τ ð  ΔÞ  ð3 þ θÞ=4 f þ α1 e  τ ð  ΔÞ  ð1 þ θÞ=4 f

2 

2 1  20μ þ θ



þ

ð  ΔÞ  ð3 þ θÞ=4 f

4  

2 1þθ þ 1þ α1 e  τ

ð  ΔÞ  ð1 þ θÞ=4 f

4 r

CM3 γ ð1  θÞ3 e  2τ

μ

þ



2

2  2  f L2 ð2Þ þ ∇f þ α21 e  2τ Δf L2 ð1Þ :







CMγ ð1  θÞ3 

f 22 þ ∇f 2 þ α2 e  2τ Δf 22 : 1 L ð2Þ L ð1Þ

μ

ð4:16Þ Setting μ ¼ ð1  θÞ=20, we finally get 

2 1θ α1 e  τ

ð  ΔÞ  ð1 þ θÞ=4 f

∂τ E 1 þ θ E 1 þ 1 þ 4 rCM3 γ ð1  θÞ2 e  2τ þ CMγ ð1  θÞ2

□

ð4:17Þ

4.2. Estimates in H 1 ðR2 Þ We next establish the H 1 -estimate of f. As explained earlier, we get it by performing the L2 -scalar product of (4.3) with f. In this section, we will see how useful Lemma 4.2 is for absorbing bad terms which appear in the computations made below. One defines the functional 

2  2 E2 ðτÞ ¼ 12 f þ α1 e  τ ∇f : The H 1 -estimate of f is given by the following lemma.     Lemma 4.3. Let W A C 1 ðτ0 ; τε Þ; H 1 ð2Þ \ C 0 ðτ0 ; τε Þ; H 3 ð2Þ be the solution of (4.1) satisfying the inequality (4.4) for some γ 4 0. There exist  γ 0 40 and T 0 Z 1 such that if T Z T 0 and γ r γ 0 , then, for all τ A τ0 ; τnε , E2 satisfies the inequality



2 1 2 β ∂τ E2 þ E2 þ ∇f þ e  2τ
A
∇f

2 2 

2

2

2



r f þ CM γ ð1  θÞ6 f þ jX j2 f þ CM 2 γ ð1  θÞ6 e  τ ; ð4:18Þ where θ, 0 o θ o 1 is the fixed constant introduced at the beginning of Section 4. Proof. Taking the L2 -inner product of (4.3) with f, performing several integrations by parts and taking into account the equalities

2

2 ð  Lðf Þ; f ÞL2 ¼ ∇f  12 f ; and

α1 e  τ

L ð1Þ

141



2 X  ∇ Δf ; f ¼ α1 e  τ ∇f ; 2 2 L

we obtain the equality

2

2



2 ∂τ E2 þ E2 þ εe  τ Δf þ ð1  α1 e  τ Þ ∇f þ β e  2τ
A
∇f

2 ¼ f þI 1 þ I 2 þ I 3 þ I 4 þ I 5 ;

ð4:19Þ

where     I 1 ¼  K  ∇ f  α 1 e  τ Δf ; f L 2 ;     I 2 ¼  η K  ∇ G  α 1 e  τ ΔG ; f L 2 ;     I 3 ¼  η V  ∇ f  α 1 e  τ Δf ; f L 2 ;  ε 2 X Δ G þ ΔG þ  ∇ΔG; f I 4 ¼  ηα1 e  τ 2 α1 L2  

  2 þ ηβe  2τ div
A
∇G ; f 2 ;   

 L  2 I 5 ¼ βe  2τ div ∇
A
4 A ; f 2 : L

We notice that, since K is divergence free, ðK  ∇f ; f ÞL2 ¼ 0. Integrating by parts and using the inequality (2.15) of Lemma 2.4 and the inequality (4.5), we obtain   I 1 ¼  α1 e  τ K Δf ; ∇f L2





r C α1 e  τ kK kL1 Δf

∇f

1=2 1=2



r C α1 e  τ f H1 f L2 ð2Þ Δf

∇f



pffiffiffiffiffiffi r C α1 M γ ð1  θÞ6 e  τ=2 ∇f

2 CM 2 γ 2 ð1  θÞ12  τ r μ ∇f þ e ;

μ

ð4:20Þ

where μ 4 0 will be made more precise later. By the same method, using the inequality (2.14) of Lemma 2.4 and the smoothness of G, one has     I 2 ¼ η K G  α1 e  τ ΔG ; ∇f L2

142

O. Coulaud / International Journal of Non-Linear Mechanics 71 (2015) 132–151







r
η
kK kL4 G  α1 e  τ ΔG L4 ∇f









r C ð1 þ α1 e  τ Þ
η
f L2 ð2Þ ∇f



2

2 CM γ ð1  θÞ6 2



f þ

r μ ∇f þ

jX j2 f :

μ



4=ð3 þ θÞ



∇f ð2 þ 2θÞ=ð3 þ θÞ r ð  ΔÞ  ð1 þ θÞ=4 f

   ð1 þ θÞ=2

2

1 þ θ 3

2 8



∇f 2 þ r

ð  ΔÞ  ð1 þ θÞ=4 f : 3 3þθ 3þθ 8

ð4:21Þ

Since, 0 o θ o 1, we obtain

2

2 1 2

f r ∇f þ 5

ð  ΔÞ  ð1 þ θÞ=4 f : 4

The same method gives   I 3 ¼  α1 e  τ η V Δf ; ∇f L2







 τ

r α1 e η kV kL1 Δf

∇f



pffiffiffiffiffiffi r C α1 M γ ð1  θÞ6 e  τ=2 ∇f

Thus, we have

2 CM 2 γ 2 ð1  θÞ12  τ r μ ∇f þ e :



2 1 2 β ∂τ E2 þ E2 þ ∇f þ e  2τ
A
∇f

4 2 

2

2

2





r 5 ð  ΔÞ  ð1 þ θÞ=4 f þ CM γ ð1  θÞ6 f þ jX j2 f

ð4:22Þ

μ

Because of the regularity of G, the estimate of I4 is simple. Indeed, an integration by parts and Hölder inequalities yield 







2 I 4 r C
η
ðε þ α1 Þe  τ f  ηβe  2τ
A
∇G; ∇f 2 L









r C
η
ðε þ α1 Þe  τ f þ C
η
β e  2τ k∇GkL1 k∇U k2L3 ∇f L3

þ CM 2 γ ð1  θÞ6 e  τ :

r CM 3 γ ð1  θÞ2 e  τ þ CM γ ð1  θÞ2 

2

2

2 2

2





 f þ ∇f þ α21 e  2τ Δf þ jX j2 f þ α21 e  2τ jX j2 Δf : ð4:29Þ

2

2

2

 ð1 þ θ Þ=4





Interpolating again f between ∇f and ð  ΔÞ f and taking γ sufficiently small, we obtain

2 1

1

2



∂τ E3 þ θE3 þ ð  ΔÞ  ð1 þ θÞ=4 f þ ∇f

2 8

2

2



r CM 3 γ ð1  θÞ2 e  τ þ CM γ ð1  θÞ2 α21 e  2τ Δf þ jX j2 f



2



þ α21 e  2τ jX j2 Δf :

H

r C ðε þ α1 ÞM γ ð1  θÞ6 e  τ þ CM 3=2 γ 3=2 ð1  θÞ9 e  3τ=2 r CM 3=2 γ ð1  θÞ6 e  τ :

ð4:23Þ

Finally, using the same arguments, due to the inequality (2.13) and the continuous injection of H 1 ðR2 Þ into L4 ðR2 Þ, one has  

  2 I 5 ¼  βe  2τ ∇
A
4 A; ∇f 2


L


r C β e  2τ k∇U kL4 ∇2 U L4
A
∇f





r C β e  2τ kW k 1 kW k 2
A
∇f

ð4:30Þ

H

eτ=2



r C β M γ ð1  θÞ6 e  2τ pffiffiffiffiffiffi
A
∇f

4.3. Estimates in H 2 ðR2 Þ

α1



2 β r e  2τ
A
∇f þ CM 2 γ 2 ð1  θÞ12 e  τ : 2

We now perform the H 2 -estimate of f. This is done with the same method as for the H 1 -estimate in the previous section. Indeed, we perform the L2 -product between (4.3) and  Δf and, after some computations, we see that the inequality (4.4) enables us to absorb all the terms involving the H 2 -norm of f. Combined with (4.30), we get an estimate in H2, where only terms with weighted norms remain. More precisely, we introduce the following functional: 

2  2 E4 ðτÞ ¼ 12 ∇f þ α1 e  τ Δf :

ð4:24Þ

Taking into account the inequalities (4.20)–(4.24) and assuming that γ r 1, we deduce from (4.19) that



2   2 β ∂τ E2 þE2 þ 1  3μ  α1 e  τ ∇f þ e  2τ
A
∇f

2 

2

2 CM γ ð1  θÞ6 2

2

f þ

r f þ

jX j2 f þCM γ ð1  θÞ6 e  τ :

μ

ð4:25Þ If we choose for instance μ ¼ α1 e  τ r 14, we get

1 12

and T0 large enough to have



2 1 2 β ∂τ E2 þE2 þ ∇f þ e  2τ
A
∇f

2 2 

2

2

2



r f þ CM γ ð1  θÞ6 f þ jX j2 f þCM 2 γ ð1  θÞ6 e  τ :

□

ð4:26Þ To achieve the H 1 -estimate of f, we have to combine the

2 inequalities (4.8) We can interpolate f between

2 and (4.18).



2



ð  ΔÞ  ð1 þ θÞ=4 f and ∇f . Indeed, via Hölder and Young inequalities, Zwe get

2

2ð1 þ θÞ=ð3 þ θÞ



2ð1 þ θÞ=ð3 þ θÞ



4=ð3 þ θÞ 1

f ¼ ð2π Þ2
ξ
f
f
dξ
b
b

2 1 þ θÞ=ð3 þ θ Þ R2

ð

ξ

0

12=ð3 þ θÞ Z ð1 þ θÞ=ð3 þ θÞ

2

b

2 1

b

2 A 2@
ξ

f
dξ r ð2π Þ d ξ f



1þθ R2
ξ
R2 Z

ð4:28Þ

We define E3 ¼ 6E1 þ E2 . The inequalities (4.8) and (4.28) give 

2 1

3 2



∂τ E3 þ θE3 þ 1 þ ð1  θÞα1 e  τ ð  ΔÞ  ð1 þ θÞ=4 f þ ∇f

2 4

Then, by the inequality (2.13), the continuous injection of H 1 ðR2 Þ into L3 ðR2 Þ and the inequalities (4.4) and (4.5), one obtains









I 4 r C
η
ðε þ α1 Þe  τ f þ C
η
β e  2τ kW k2L3 ∇f L3

  r C ðε þ α1 ÞM γ ð1  θÞ6 e  τ þ C
η
β e  2τ kW k2 1 ∇f þ Δf

H

ð4:27Þ

The H 2 -estimate of f is given by the lemma below.     Lemma 4.4. Let W A C 1 ðτ0 ; τε Þ; H 1 ð2Þ \ C 0 ðτ0 ; τε Þ; H 3 ð2Þ be the solution of (4.1) satisfying the inequality (4.4) for some γ 4 0. There exist  γ 0 40  and T 0 Z 1 such that if T Z T 0 and γ r γ 0 , then for all τ A τ0 ; τnε , E4 satisfies the inequality



2 1 2 β ∂τ E4 þ E4 þ Δf þ e  2τ
A
Δf

2 2 3 2 r ∇f þ CM 2 γ ð1  θÞ6 e  2τ þ CM 2 γ ð1  θÞ6 2 

2

2 2



 f þ ∇f þ jX j2 f ;

ð4:31Þ

where θ, 0 o θ o 1 is the fixed constant introduced at the beginning of Section 4. Proof. We take the L2 -product of (4.3) with  Δf . Doing several integrations by parts, it is easy to see that

2 2    Lðf Þ;  Δf L2 ¼ Δf  ∇f ;

O. Coulaud / International Journal of Non-Linear Mechanics 71 (2015) 132–151

where

and 

2 X 1 ¼ α 1 e  τ Δf :  α1 e  τ :∇Δf ; Δf 2 2 2 L

 ε 2 X I 13 ¼ ηα1 e  τ Δ G þ ΔG þ :∇ΔG; Δf ; 2 α L2  1 

  2 2  2τ

I 3 ¼ ηβe div A ∇G ; Δf 2 :

Furthermore, one also has 















βe  2τ div
A
2 ∇f ; Δf ¼ βe  2τ
A
Δf 2 þ βe  2τ

2 Z X j¼1

R2

L

A : ∂j A∂j f Δf dX:

Using Hölder inequalities, the inequality (2.13) of Lemma 2.3, the continuous injections of H 1 ðR2 Þ into L4 ðR2 Þ and the inequality (4.4), we get 2 Z X







βe  2τ A : ∂j A∂j f Δf dX r C β e  2τ
A
Δf

∇A∇f

j¼1

R2







r C βe  2τ
A
Δf

∇2 U L4 ∇f L4







r C βe  2τ
A
Δf k∇W kH1 ∇f H1



2 r μ1 β e  2τ
A
Δf



2  Cβ 2 þ e  2τ kW k2H2 ∇f þ Δf

μ1



2 r μ1 β e  2τ
A
Δf



 CMγ ð1  θÞ6  τ 

∇f 2 þ Δf 2 ; þ e μ1

where μ1 4 0 will be chosen later. Consequently, we get

2 



2  α1  2  ∂τ E4 þ εe  τ ∇Δf þ 1  e  τ Δf þ β 1  μ1 e  2τ
A
Δf

2

2 CM γ ð1  θÞ6  τ  2 2 

∇f þ Δf þ I 1 þ I 2 þI 3 þ I 4 þ I 5 ; r ∇f þ e

μ1

ð4:32Þ where     I 1 ¼ U  ∇ f  α 1 e  t Δf ; Δf L 2 ;     I 2 ¼ η K  ∇ G  α 1 e  t ΔG ; Δf L 2 ;  ε 2 X Δ G þ ΔG þ  ∇ΔG; Δf I 3 ¼ ηα1 e  t 2 α1 L2  

  2 þ ηβe  2t div
A
∇G ; Δf 2 ;   

 L  2 I 4 ¼ β e  2t div ∇
A
4 A ; Δf 2 :

Using the good regularity of G and the inequality (4.5), one can show that



I 13 r CM 1=2 γ 1=2 ð1  θÞ3 e  τ Δf

2 CM γ ð1  θÞ6  2τ r μ2 Δf þ e :

μ2

I 23

The estimate of is slightly more complicated. Actually, we can bound I 23 by two kinds of terms that we estimate separately. In fact, it is easy to see that Z Z











2
2



∇A

A
j∇Gj
Δf
dX þ C
η
βe  2τ
A

∇ G

Δf
dX: I 23 r C
η
βe  2τ R2

ð4:35Þ Each term on the right-hand side of (4.35) can be estimated in a convenient way. We use again the inequality (2.13) of Lemma 2.3, the inequality (4.4), the Hölder inequalities and the inequality (4.4). We get Z


















∇A

A
j∇Gj
Δf
dX r C
η
β e  2τ ∇2 U k∇GkL1
A
Δf

C
η
βe  2τ R2

r μ1 βe







2 C
η
2 β  2τ


k∇W k2 A Δf þ e

 2 τ


μ1

By the same way, we have Z 2 2 12



2
2









A

∇ G

Δf
dx r μ βe  2τ
A
Δf 2 þ CM γ ð1  θ Þ e  2τ ; C
η
βe  2τ 1

μ1



2 CM 2 γ 2 ð1  θÞ12  2τ e : I 23 r 2μ1 β e  2τ
A
Δf þ

μ1

Finally, assuming γ r 1, one has

2



2 CM 2 γ ð1  θÞ6  2τ e : I 3 r μ2 Δf þ 2μ1 β e  2τ
A
Δf þ minðμ1 ; μ2 Þ

Due to the Gagliardo–Niremberg inequality and the inequalities (2.13) and (4.4), it comes

2 I 1 r C k∇U k ∇f L4





r C k∇U k ∇f

Δf

ð4:33Þ

where μ2 4 0 will be chosen later. We now estimate I2 with the help of the inequality (2.15) of Lemma 2.4, the inequality (4.5) and the smoothness of G







I 2 r
η
kK kL1 G  α1 e  τ ΔG

Δf





1=2



1=2 r C
η
ð1 þ α1 e  τ Þ f H1 f L2 ð2Þ Δf



2

2 CM γ ð1  θÞ6 2 2



f þ ∇f þ

ð4:34Þ r μ 2 Δf þ

jX j2 f :

R2

þC βe  2τ

Z

R

2






2



∇A

A

∇ K

Δf
dX þC βe  2τ

Z R

R2

2



2
3



A

∇ K

Δf
dX: ð4:37Þ

We have to estimate each term on the right-hand side of the inequality (4.37). The first two ones can be estimated exactly like we did for I 23 . The inequality (2.13) of Lemma 2.3 and Gagliardo– Niremberg inequality yield Z




2









∇A

A

∇ K

Δf
dX r μ β e  2τ
A
Δf 2 C β e  2τ 1 R2

þ



C βe  2τ

∇2 U 24 ∇2 K 24 L L

μ1



2 r μ1 βe  2τ
A
Δf

þ

μ2

C β e  2τ

μ1

2 k∇W k2L4 ∇f L4



2 r μ1 βe  2τ
A
Δf

We rewrite I 3 ¼ I 13 þ I 23 ;

ð4:36Þ

It remains to estimate I4. Recalling that U ¼ ηV þ K, one has Z Z




2




2
3







∇A

A

∇ V

Δf
dX þC
η
βe  2τ
A

∇ V

Δf
dX I 4 r C
η
βe  2τ

R2

μ2

μ1



2 CM 2 γ 2 ð1  θÞ12  2τ r μ1 βe  2τ
A
Δf þ e :

and thus we have shown

Integrating by parts and using the divergence free property of K, one can show that 2 Z X ∂k U j ∂j f ∂k f dX: I1 ¼ 

2 CM γ ð1  θÞ6 2

∇f ; r μ2 Δf þ

R2

R2

L

j;k ¼ 1

143

þ

C β e  2τ

μ1







k∇W k ΔW

∇f

Δf :

144

O. Coulaud / International Journal of Non-Linear Mechanics 71 (2015) 132–151

inequality (4.41), it will give us an estimate inH 2 ð2Þ. To do this, we make the L2 -scalar product of (4.3) with jX j4 f  α1 e  τ Δf . We define the functional

 



2 E6 ðτÞ ¼ 12 jX j2 f  α1 e  τ Δf :

Due to the inequality (4.4), we get Z




2









∇A

A

∇ K

Δf
dX r μ βe  2τ
A
Δf 2 C β e  2τ 1 R2

þ

CM γ ð1  θÞ6 e  3τ=2 



μ1

2 2  ∇f þ Δf :

Before stating the lemma which contains the estimate of E6, we state a technical lemma, which gives the terms provided by the L2 -product of the linear terms of (4.3) with jX j4 f  α1 e  τ Δf .     Lemma 4.5. Let f A C 1 ðτ0 ; τε Þ; H 1 ð2Þ \ C 0 ðτ0 ; τε Þ; H 3 ð2Þ and HðX; τ; f Þ ¼ jX j4 ðf  α1 e  τ Δf Þ. For all τ A ðτ0 ; τε Þ, the next equalities hold.

2

2

2





1. ð  f ; HðX; τ; f ÞÞL2 ¼  jX j2 f þ 8α1 e  τ jX jf  α1 e  τ jX j2 ∇f :

By the same method, we obtain Z

2
3









A

∇ K

Δf
dX r μ βe  2τ
A
Δf 2 C β e  2τ 1 R2

þ

C β e  2τ

μ1



2 k∇U k2L1 ∇3 K



2 r μ1 βe  2τ
A
Δf

2 C β e  2τ þ k∇W k 1 k∇W k 2 Δf

μ1

L ð2Þ

H

6 τ



2 CM γ ð1  θÞ e r μ1 βe  2τ
A
Δf þ

μ1



2

2

2





2. ð  Δf ; HðX; τ; f ÞÞL2 ¼ α1 e  τ jX j2 Δf  8 jX jf þ jX j2 ∇f :



Δf 2 :

Finally, we have shown that

3.



2 CM 2 γ ð1  θÞ6 e  τ  2 2 

∇f þ Δf : I 4 r 4μ1 βe  2τ
A
Δf þ





2  



þ 3α1 e  τ jX j2 ∇f  ðα1 =2Þe  τ X  ∇Δf ; jX j4 f 2 :

μ1

ð4:38Þ Going back to (4.32) and taking into account the inequalities (4.33), (4.34), (4.36) and (4.38), we get 



2  α1  2  ∂τ E4 þ 1 3μ2  e  τ Δf þ 1  7μ1 β e  2τ
A
Δf

2 

2

2 CM 2 γ ð1  θÞ6 2 2 2



f þ ∇f þ Δf þ

r ∇f þ

jX j2 f

minðμ1 ; μ2 Þ CM γ ð1  θÞ  2τ e : minðμ1 ; μ2 Þ 2

þ



2

2 



X=2  ∇f ; HðX; τ; f Þ L2 ¼ 32 jX j2 f  24α1 e  τ jX jf

L







2 2

2 2 4. ð  Lðf Þ; HðX; τ; f ÞÞL2 ¼ 12 jX j f þ ð1 þ 2α1 e  τ Þ jX j ∇f



2  

2



þ α1 e  τ jX j2 Δf  ð8 þ 16α1 e  τ Þ jX jf  ðα1 =2Þe  τ X  ∇Δf ; jX j4 f 2 : L

    5. α1 e  τ X=2  ∇Δf ; HðX; τ; f Þ L2 ¼ ðα1 =2Þe  τ X  ∇Δf ; jX j4 f

1 1 Taking for instance μ1 ¼ 14 , μ2 ¼ 12 , large enough, we finally have

γ small enough and T ¼ eτ

0

L2



2



þ ð3α21 =2Þe  2τ jX j2 Δf :

6





2

2  



6. εe  τ Δ2 f ; HðX; τ; f Þ L2 ¼ εα1 e  2τ jX j2 ∇Δf  8 jX jΔf





2

2

2

2



þ εe  τ jX j2 Δf  8 jX j∇f þ 32 f  16 X  ∇f :



2 3 2 1 2 β ∂τ E4 þE4 þ Δf þ e  2τ
A
Δf r ∇f þ CM 2 γ ð1  θÞ6 e  2τ 2 2 2 

2





2 2



þ CM 2 γ ð1  θÞ6 f þ ∇f þ jX j2 f : □ ð4:39Þ In order to finish the H 2 -estimate of f we define a new functional E5 as a linear combination of E3 and E4 given by E5 ¼ 16E3 þ E4 : From the inequalities (4.30) and (4.31), it is clear that one has

2

2

2



∂τ E5 þ θE5 þ 8 ð  ΔÞ  ð1 þ θÞ=4 f þ 12 ∇f þ 12 Δf r CM 3 γ ð1  θÞ2 e  τ 

2

2 2 þ CM 2 γ ð1  θÞ2 f þ ∇f þ α21 e  2τ Δf



2

2





þ jX j2 f þ α21 e  2τ jX j2 Δf :

Proof. All these equalities are obtained via integrations by parts. We only show the first four ones, the others are obtained with the same method. Let us show the equality 1. Two integrations by parts imply

2

2  





 f ; jX j4 ðf  α1 e  τ Δf Þ 2 ¼  jX j2 f α1 e  τ jX j2 ∇f

L

 4α1 e  τ

2 Z X R2

j¼1



2

2





¼  jX j2 f α1 e  τ jX j2 ∇f

ð4:40Þ

Using the interpolation inequality (4.27) and taking γ small enough and τ0 ¼ log ðTÞ large enough, we finally obtain

2

2

2



∂τ E5 þ θE5 þ 7 ð  ΔÞ  ð1 þ θÞ=4 f þ 14 ∇f þ 14 Δf





2

2





r CM 3 γ ð1  θÞ2 e  τ þCM 2 γ ð1  θÞ2 jX j2 f þ α21 e  2τ jX j2 Δf : ð4:41Þ

 2α1 e  τ

4.4. Estimates in H (2) In order to achieve the estimate of f in H 2 ð2Þ, it remains to perform estimates in weighted spaces. Combined with the

2 Z X j¼1

2

R

  2 dX X j jX j2 ∂j f



2

2

2





¼  jX j2 f  α1 e  τ jX j2 ∇f þ 8α1 e  τ jX jf : The equality 2 is obtained through the same computations. We show now the third equality of this lemma. Integrating by parts, we obtain 

2

X j jX j2 f ∂j f dX



  X  ∇f ; jX j4 f  α1 e  τ Δf 2

¼ L2

2 Z X X j jX j4 



2  ∂j f dX 2 4 j¼1 R

þ α1 e  τ

2 Z X X j jX j4 ∂j f Δf dX 2 2 j¼1 R

O. Coulaud / International Journal of Non-Linear Mechanics 71 (2015) 132–151 2 Z

2 X X j jX j4 3



∂j f Δf dX: ¼ jX j2 f þ α1 e  τ 2 2 2 j¼1 R

Besides, integrating several times by parts, we get

α1 e

2 Z 2 Z X X X j jX j4 X j jX j4 ∂j f Δf dX ¼  α1 e  τ f ∂j Δf dX 2 2 2 2 j¼1 R j¼1 R Z jX j4 f Δf dX ¼  3α 1 e  τ R2  α1   e  τ X  ∇Δf ; jX j4 f 2 L 2

2

2



¼  24α1 e  τ jX jf þ 3α1 e  τ jX j2 ∇f



α1 2

  e  τ X  ∇Δf ; jX j4 f 2 ; L

and consequently  

  X  ∇f ; jX j4 f  α1 e  τ Δf 2

We now estimate J. One has J ¼ J1 þ J2 ;

ð4:44Þ

where !

τ

145



2

2 3



¼ jX j2 f  24α1 e  τ jX jf

2 L2

2 α  



1 þ 3α1 e  τ jX j2 ∇f  e  τ X  ∇Δf ; jX j4 f 2 : L 2

The fourth equality of this lemma is obtained by summing the first three ones. By the same method, we obtain easily the equalities 5 and 6 of this lemma. □ The H 2 ð2Þ-estimate of f is given in the following lemma.     Lemma 4.6. Let W A C 1 ðτ0 ; τε Þ; H 1 ð2Þ \ C 0 ðτ0 ; τε Þ; H 3 ð2Þ be the solution of (4.1) satisfying the inequality (4.4) for some γ 40. There exist  γ 0 40 and T 0 Z1 such that if T ZT 0 and γ r γ 0 , then for all τ A τ0 ; τnε , E6 satisfies the inequality







1  θ

2 2 1 2 2 α 1  τ 2 2 ∂τ E 6 þ θ E 6 þ

jX j f þ jX j ∇f þ e jX j Δf

4 8 4

2 1024

2 6 τ



rCM γ ð1  θÞ e þ f 1θ 

2  2 2 ð4:42Þ þ CM 2 γ 1=2 ð1  θÞ3 f þ ∇f þ Δf ; where θ, 0 o θ o 1 is the fixed constant introduced at the beginning of Section 4. Proof. To show this lemma, we perform the L2 -product of the   equality (4.3) with jX j4 f  α1 e  τ Δf . Applying Lemma 4.5, we obtain

2

2 

2

α2 1









∂τ E6 þ jX j2 f þ ð1 þ α1 e  τ Þ jX j2 ∇f þ α1 e  τ þ 1 e  2τ jX j2 Δf þ J 2 2



2

2

2 ¼ C εe  τ jX j∇f þ C εα1 e  2τ jX jΔf þ ð8 þ8α1 e  τ Þ jX jf

þ I1 þ I2 þ I3 þ I4 þ I5 ;

ð4:43Þ

where  

   2 ; J ¼  βe  2τ div
A
∇f ; jX j4 f  α1 e  τ Δf 2  L     I 1 ¼ K  ∇ f  α1 e  τ Δf ; jX j4 f  α1 e  τ Δf ; L2       4 I 2 ¼ η K  ∇ G  α1 e  τ ΔG ; jX j f  α1 e  τ Δf ; L2       I 3 ¼ η V  ∇ f  α1 e  τ Δf ; jX j4 f  α1 e  τ Δf ; L2     2 I 4 ¼  ηεe  τ Δ G; jX j2 f  α1 e  τ Δf L2    X þ ηα1 e  τ ΔG þ  ∇ΔG; jX j4 f  α1 e  τ Δf 2 L2  

   2 4  2τ τ

 ηβe div A ∇G ; jX j f  α1 e Δf ; L2   

     2 4 I 5 ¼  βe  2τ div ∇
A
4 A ; jX j f  α1 e  τ Δf : 2 L

 

  2 J 1 ¼  β e  2τ div
A
∇f ; jX j4 f 2 ; L  

  2 4  3τ

J 2 ¼ βα1 e div A ∇f ; jX j Δf Þ 2 : L

We estimate J1 and J2 separately. Integrating by parts, we obtain

2 Z X





2

2 X j jX j2
A
∂j ff dX: J 1 ¼ βe  2τ jX j2
A
∇f þ 4β e  2τ j¼1

R2

Using Hölder and Young inequalities, we obtain



2 Z

X



β



2

2
4βe  2τ X j jX j2
A
∂j ff dX

r e  2τ jX j2
A
∇f


2 2 R

j¼1

2 þ C β e  2τ jX jf k∇U k2L1 : Then, using the inequality (2.15) of Lemma 2.4, the inequality (2.4) of Lemma 2.2 and the conditions (4.4) and (4.5), we get



2 Z

X



β





2 2

2
4βe  2τ X j jX j A ∂j ff dX

r e  2τ jX j2
A
∇f


2 2 R

j¼1





2



 2τ

þ C βe

jX j f f k∇W kH1 k∇W kL2 ð2Þ





β

2 r e  2τ jX j2
A
∇f þ CM 2 γ 2 ð1  θÞ12 e  τ ; 2 and we conclude that





β

2 J 1 Z e  2τ jX j2
A
∇f  CM 2 γ 2 ð1  θÞ12 e  τ : 2

ð4:45Þ

By the same way, we estimate J2. A short computation shows that 2 Z

X





2 jX j4 ∂j A : A∂j f Δf dX: J 2 ¼ βα1 e  3τ jX j2
A
Δf þ 2βα1 e  3τ j¼1

R2

We define



2 Z X

4 jX j ∂j A : A∂j f Δf dX

I ¼

2βα1 e  3τ 2 R

j¼1 Applying Hölder inequalities and the continuous injection of H 1 ðR2 Þ into L4 ðR2 Þ, we obtain















I rC βα1 e  3τ jX j2
A
Δf

jX j2 ∇f 4 ∇2 U L4 L















rC βα1 e  3τ jX j2
A
Δf

jX j2 ∇f 4 ∇2 U H1 : L

Using the inequality (2.6) of Lemma 2.2 and the inequality (2.17) of Lemma 2.4, we get



1=2







I rC βα1 e  3τ jX j2
A
Δf

jX j2 ∇f



1=2

1=2

1=2



kW kH2 :  f þ jX j∇f þ jX j2 Δf

Due to the Young inequality and the condition (4.4), we obtain





β

2 I r α1 e  3τ jX j2
A
Δf

2 

2

2

2 2





þ C βα1 e  3τ kW k2H2 f þ ∇f þ jX j2 ∇f þ jX j2 Δf





β

2 r α1 e  3τ jX j2
A
Δf

2 

2

2

2 2





þ CM γ ð1  θÞ6 e  2τ f þ ∇f þ jX j2 ∇f þ jX j2 Δf :

146

O. Coulaud / International Journal of Non-Linear Mechanics 71 (2015) 132–151

Thus, we can conclude that





β

2 J 2 Z α1 e  3τ jX j2
A
Δf

2 

2

2

2 2





 CM γ ð1  θÞ6 e  2τ f þ ∇f þ jX j2 ∇f þ jX j2 Δf : ð4:46Þ Combining the inequalities (4.45) and (4.46) and going back to (4.44), we have shown that 











β

2

2 J Z e  2τ jX j2
A
∇f þ α1 e  τ jX j2
A
Δf CM 2 γ 2 ð1  θÞ12 e  τ 2 

2

2

2 2





 CM γ ð1  θÞ6 e  2τ f þ ∇f þ jX j2 ∇f þ jX j2 Δf : ð4:47Þ Taking into account the inequality (4.47), the equality (4.43) becomes

2

2 

2 α2 1









∂τ E6 þ jX j2 f þ ð1 þ α1 e  τ Þ jX j2 ∇f þ α1 e  τ þ 1 e  2τ jX j2 Δf

2 2 











β

2

2 þ e  2τ jX j2
A
∇f þ α1 e  τ jX j2
A
Δf

2

2

2

2 rC εe  τ jX j∇f þ C εα1 e  2τ jX jΔf þ ð8 þ 8α1 e  τ Þ jX jf



2

2

2 2





þ CM γ ð1  θÞ6 e  2τ f þ ∇f þ jX j2 ∇f þ jX j2 Δf

12  τ

þ CM γ ð1  θÞ e 2 2

þ I1 þ I2 þ I3 þ I4 þ I5 :

ð4:48Þ

It remains to estimate every Ii, i ¼ 1; …; 5. Using the divergence free property of K, integrating by parts and using Hölder inequalities, we get 2 Z X

2 I1 ¼  2 X j jX j2 K j
f  α1 e  τ Δf
dX j¼1

R2

 





 r C kK kL1 jX j2 f  α1 e  τ Δf

jX j f  α1 e  τ Δf :



2

2





r CM 1=2 γ 1=2 ð1  θÞ3 jX j2 f þ α21 e  2τ jX j2 Δf

2

2  þ f þ α21 e  2τ Δf : ð4:49Þ Using the inequality (2.14) of Lemma 2.4, one can bound I2 in a convenient way. Indeed, one has





 

 





I 2 r C
η
kK kL4 jX j2 ∇ G  α1 e  τ ΔG 4 jX j2 f  α1 e  t Δf

L



 









r C
η
f L2 ð2Þ jX j2 f þ α1 e  τ jX j2 Δf



2

2 2





r CM 1=2 γ 1=2 ð1  θÞ3 jX j2 f þ α21 e  2τ jX j2 Δf þ f : ð4:50Þ Via an integration by parts, due to the facts that VðXÞ  X ¼ 0 and div V ¼ 0, we show that I3 vanishes. Indeed 2 Z 

2  ηX I3 ¼ jX j4 V j ∂j
f  α1 e  τ Δf
dX 2 j ¼ 1 R2 ¼  2η

j¼1

R2



2 jX jX j V j
f  α1 e  τ Δf
dX ¼ 0:

L

It is easy, using the smoothness of G and the inequality (4.5), to see that





I 1 r C
η
e  τ kGk 3 f þ α1 e  τ Δf

H ð3Þ

4

6 τ

r CM γ ð1  θÞ e

:

The term I 24 is not really harder to estimate. Due to the inequality (2.13) of Lemma 2.3, the inequality (4.5), the continuous injection of H 1 ðR2 Þ into L4 ðR2 Þ and the inequality (4.4), we get



 











I 24 r
η
β e  2τ k∇U k2L4 jX j4 ΔG 1 þ k∇U kL4 ∇2 U L4 jX j4 ∇G 1 L

L

 

 f þ α 1 e  τ Δf

 



r C
η
βe  2τ kW k2L4 þ kW kL4 k∇W kL4 f þ α1 e  τ Δf

 



r C
η
e  2τ kW k2H1 þ kW kH1 kW kH2 f þ α1 e  τ Δf

r CM 2 γ 2 ð1  θÞ12 e  3τ=2 : Thus, assuming γ r 1, the following inequality holds: I 4 r CM 2 γ ð1  θÞ6 e  τ :

ð4:51Þ

It remains to estimate I5, which is the term that does not appear in the second grade fluids equations. We rewrite I 5 ¼ I 15 þ I 25 ; where

The inequalities (2.15) of Lemma 2.3 and (2.4) of Lemma 2.2, the 4 Young inequality ab r 34a4=3 þ 14b and the inequality (4.5) yield

1=2 1=2

1=2  

3=2

I 1 r C f H1 f L2 ð2Þ jX j2 f  α1 e  τ Δf f  α1 e  τ Δf





2  



2

rCM 1=2 γ 1=2 ð1  θÞ3 jX j2 f  α1 e  τ Δf þ f  α1 e  τ Δf

2 Z X

We rewrite I 4 ¼ I 14 þ I 24 , where    ε 2 X I 14 ¼  ηα1 e  τ Δ G þ ΔG þ  ∇ΔG; jX j2 f  α1 e  τ Δf ; 2 α1 L2  

   2 I 24 ¼  ηβe  2τ div
A
∇G ; jX j4 f  α1 e  τ Δf : 2

   

  2 I 15 ¼  β e  2τ div ∇
A
4 A ; jX j4 f 2 ; L

 

   2 I 25 ¼ βα1 e  3τ div ∇
A
4 A ; jX j4 Δf 2 : 

L

I 15 .

We begin by estimating After some computations, we notice that we have to estimate two kinds of terms. In fact, one has 1;2 I 15 r I 1;1 5 þ I5 ;

where  2τ I 1;1 5 ¼ C βe  2τ I 1;2 5 ¼ C βe

Z R2

Z R

2



2

jX j4
∇2 U
j∇U j
f
dX;



jX j4
∇3 U
j∇U j2
f
dX:

In order to simplify the notations, we set

δ ¼ Mγ ð1  θÞ6 : Applying the inequality (2.6) of Lemma 2.2 and the continuous injection of H 2 ðR2 Þ into L1 ðR2 Þ, we obtain





2



 2τ

jX j∇2 U L4 k∇U kL1 jX j2 f

I 1;1 5 r βe r C β e  2τ ðkW kþ kjX j∇W kÞðk∇W k









þ jX jΔW k∇U kH2 jX j2 f :

O. Coulaud / International Journal of Non-Linear Mechanics 71 (2015) 132–151

Then, using the inequalities (2.4) of Lemma 2.2 and (2.17) of Lemma 2.4 and the conditions (4.4) and (4.5), we get 

1=2



 2τ kW kþ k∇W k1=2 jX j2 ∇W

I 1;1 5 rC β e 

1=2





1=2





kW kH2 jX j2 f

 k∇W k þ ΔW jX j2 ΔW

where



1=2 and Then, we recall that W ¼ ηG þ f . Due to the fact that
η
r δ the smoothness of G, we obtain 

1=2

1=2 1=4

2  3τ =2 j j I 1;1 rC δ e δ þ δ X ∇f



5 

1=2



1=2 1=4  δ þ δ eτ=4 jX j2 Δf



1=2

jX j2 ∇f

7=4  3τ=2

e



1=2

jX j2 Δf



1=2

1=2



3=2 þ C δ e  5τ=4 jX j2 ∇f jX j2 Δf : þ Cδ

7=4  5τ=4

e

4=3

Using the Young inequalities ab r 14a4 þ 34b and ab r 13a3 þ 23b the inequality (4.5) and assuming γ r1, we finally obtain 

2



2 2 2  3τ =2 2

2 j j j j rC δ e þ C δ X ∇f þ X Δ f I 1;1





5

3=2

,



2=3 



4=3 e  2τ þ e  5τ=3 þC δ e  5τ=3 jX j2 ∇f





2

2



2 2

r C δ e  3τ=2 þ C δ jX j2 ∇f þ jX j2 Δf

þ Cδ

5=3





2 

2

e  2τ þ e  5τ=3 þC δe  5τ=2 þ C δ jX j2 ∇f



2

2





r CM 2 γ ð1  θÞ6 e  3τ=2 þ CM 2 γ 2 ð1  θÞ12 jX j2 ∇f þ jX j2 Δf : þ Cδ

5=3

ð4:54Þ

2;2 I 25 r I 2;1 5 þ I5 ;



1=2



1=2 1=4 :  δ þ δ eτ=4 jX j2 ΔW

2

Thus, combining the inequalities (4.52) and (4.53), we obtain 

2

2

2





I 15 rCM 2 γ ð1  θ Þ6 e  τ þ CM 2 γ ð1  θ Þ6 jX j2 f þ jX j2 ∇f þ jX j2 Δf :

It remains to estimate I 25 . Like in the case of I 15 , we have to consider two kinds of terms. Indeed, one can show that



1=2

1=2 1=4

r C δe  3τ=2 δ þ δ jX j2 ∇W

r C δ e  3τ=2 þ C δ

147



ð4:52Þ I 1;2 5 ,

In order to estimate we use the Hölder inequalities and obtain









1;2 2 I 5 rC βe  2τ jX j ∇3 U

jX j2 f 4 k∇U k2L8 : L

Then, using the Galiardo–Niremberg inequality, we notice that





1=2   1=2

2





jX j f 4 r jX j2 f ∇ jX j2 f

L

1=2 



2



1=2

jX jf þ

r C jX j2 f

:

jX j ∇f

The inequalities (2.13) of Lemma 2.3, (2.20) of Lemma 2.5 and the continuous injection of H 1 ðR2 Þ into L8 ðR2 Þ imply







 2τ

I 1;2

jX j2 ∇3 U

jX j2 f 4 k∇U k2L8 5 rC β e L

 



r C βe  2τ kW k þ kjX j∇W k þ jX j2 ΔW



1=2 



2



1=2

jX jf þ

kW k2H1 :  jX j2 f

jX j ∇f

Finally, using the conditions (4.4) and (4.5) and the Young 4=3 inequality ab r 14a4 þ 34b , we obtain

1=2

7=4  3τ =4

I 1;2 e

jX j2 f

5 rC δ

2



2 r C δ e  τ þC δ jX j2 f



2



r CM 2 γ 2 ð1  θÞ12 e  τ þ CM γ ð1  θÞ6 jX j2 f : ð4:53Þ

 3τ I 2;1 5 ¼ C α1 β e  3τ I 2;2 5 ¼ C α1 β e

Z 2 ZR

R2



2

jX j4
∇2 U
j∇U j
Δf
dX;



jX j4
∇3 U
j∇U j2
Δf
dX:

With the same tools than the ones used to estimate I 15 , one can bound I 2;1 5 . Due to Hölder inequalities and the continuous injection of H 2 ðR3 Þ into L1 ðR2 Þ, one has



2

 3τ

I 2;1

jX j2 Δf

jX j∇2 U L4 k∇U kL1 5 r C α1 β e



2



r C α1 βe  3τ jX j2 Δf

jX j∇2 U L4 k∇U kH2 : Then, using the inequality (2.6) of Lemma 2.2 and the inequality (2.17) of Lemma 2.4, we obtain





 

 3τ

I 2;1

jX j2 Δf ðkW k þ kjX j∇W kÞ k∇W kþ jX jΔW kW kH2 5 r C α1 β e Finally, the condition (4.4) and the Young inequality imply



3=2  τ

e jX j2 Δf

I 2;1 5 r Cδ

2



ð4:55Þ r CM 2 γ 2 ð1  θÞ12 e  2τ þ CM γ ð1  θÞ6 jX j2 Δf : Likewise, using the inequality (2.20) of Lemma 2.5 and the 3 continuous injection of H 2 ðR2 Þ into L1 ðR2 Þ, we get







 3τ

I 2;2

jX j2 Δf

jX j2 ∇3 U k∇U k2L1 5 r C βα1 e









r C βα1 e  3τ jX j2 Δf kW kþ kjX j∇W k þ jX j2 ΔW kW k2H3=2





1=2 r C δ e  2τ jX j2 Δf kW k2H3=2 : Using the well-known interpolation inequality 1=2 1=2 kvk 2 H1 H

kvkH3=2 r C kvk

for every v A H 2 ðR2 Þ;

we obtain, using again the condition (4.4) and the Young inequality,



1=2  2τ

I 2;2 e

jX j2 Δf kW kH1 kW kH2 5 r Cδ





3=2 r C δ e  3τ=2 jX j2 Δf



2



r CM 2 γ 2 ð1  θÞ12 e  3τ þ CM γ ð1  θÞ6 jX j2 Δf : ð4:56Þ Finally, the inequalities (4.55) and (4.56) imply

2



I 25 r CM 2 γ 2 ð1  θÞ12 e  3τ þ CM γ ð1  θÞ6 jX j2 Δf :

ð4:57Þ

Thus, combining the inequalities (4.54) and (4.57), we get 

2

2

2







I5 r CM 2 γ ð1  θÞ6 e  τ þ CM 2 γ ð1  θÞ6 jX j2 f þ jX j2 ∇f þ jX j2 Δf :

ð4:58Þ Taking into account the inequalities (4.49)–(4.51), and (4.58) and going back to (4.48), one has

2

2 

2 α2 1









∂τ E6 þ jX j2 f þ ð1 þ α1 e  τ Þ jX j2 ∇f þ α1 e  τ þ 1 e  2τ jX j2 Δf

2 2

2  ð8 þ 8α1 e  τ Þ jX jf





2 2 r C εe  τ jX j∇f þ C εα1 e  2τ jX jΔf þ CM 2 γ ð1  θÞ6 e  τ

148

O. Coulaud / International Journal of Non-Linear Mechanics 71 (2015) 132–151

  2 2 2 þ CM 2 γ 1=2 ð1  θÞ3 f þ ∇f þ Δf



2

2

2





þ CM 2 γ 1=2 ð1  θ Þ3 jX j2 f þ jX j2 ∇f þ jX j2 Δf :



2  2 2 þ CM 2 γ 1=2 ð1  θÞ3 f þ ∇f þ Δf :□

ð4:63Þ

ð4:59Þ 5. Proof of Theorem 1.1

Via the Young inequality and the condition (4.5), it is easy to check that

2

2

2

2



C εe  τ jX j∇f þ C εα1 e  2τ jX jΔf r ε2 jX j2 ∇f þ Ce  2τ ∇f



2

2



þ ε2 jX j2 Δf þC α21 e  4τ Δf



2

2





r ε2 jX j2 ∇f þ ε2 jX j2 Δf

þ CMð1  θÞ6 e  2τ :   We assume that ε2 r min 12; α1 e  τ0 =2 . The inequality (4.59) becomes

2 1

2 1





þ α1 e  τ jX j2 ∇f

∂τ E6 þ jX j2 f þ 2 2 

2 α1  τ α21  2τ

2

2 e þ e þ

jX j Δf  8α1 e  τ jX jf

2 2

2 r CM 2 γ ð1  θÞ6 e  τ þ 8 jX jf



2 2  2 þC 1 M 2 γ 1=2 ð1  θÞ3 f þ ∇f þ Δf



2

2

2







þ C 1 M 2 γ 1=2 ð1  θÞ3 jX j2 f þ jX j2 ∇f þ jX j2 Δf ; ð4:60Þ where C1 is a positive constant dependent on α1 and β. We take now γ sufficiently small so that C 1 M 2 γ 1=2 ð1  θÞ3 rð1  θÞ=4. We obtain  

2 θ 1  θ

2

2 1



þ α1 e  τ jX j2 ∇f

þ ∂τ E6 þ

jX j f þ 4 2 4 

2 α1  τ α21  2τ

2

2 e þ e þ

jX j Δf  8α1 e  τ jX jf

4 2

2 r CM 2 γ ð1  θÞ6 e  τ þ 8 jX jf



2  2 2 ð4:61Þ þ C 1 M 2 γ 1=2 ð1  θÞ3 f þ ∇f þ Δf : Using the inequality (2.4) of Lemma 2.2, one has

2 64



2 2



8 jX jf r h jX j2 f þ f

h

for all h 4 0:

Thus, we set h ¼ ð1  θÞ=8 and obtain  

2 θ 1  θ

2

2 1



þ α1 e  τ jX j2 ∇f

þ ∂τ E6 þ

jX j f þ 4 2 8 

2 α1  τ α21  2τ

2

2 e þ e þ

jX j Δf  8α1 e  τ jX jf

4 2

1024

f 2 r CM 2 γ ð1  θÞ6 e  τ þ 1θ 

2  2 2 þ C 1 M 2 γ 1=2 ð1  θÞ3 f þ ∇f þ Δf :

ð4:62Þ

2

2 α2

2

2 1









E6 ¼ jX j2 f þ α1 e  τ jX j2 ∇f þ 1 e  2τ jX j2 Δf  8α1 e  τ jX jf : 2 2







1  θ

2 2 1 2 2 α1  τ 2 2

jX j f þ jX j ∇f þ e jX j Δf

4 8 4

1024

f 2 r CM 2 γ ð1  θÞ6 e  τ þ 1θ

∂τ E6 þ θE6 þ

W ε ¼ ηG þ f ε ;

R where G is the Oseen vortex sheet given by (1.3), η ¼ R2 W 0 ðXÞ dX and f ε satisfies the equality (4.3). We define the functional E7 ¼

K E5 þ E 6 ; 1θ

where K is a large positive constant that will be made more precise later and E5 and E6 are the energy functionals defined in Section 4. If K is large enough, this energy is suitable to estimate the H 2 ð2Þ norm of f ε , as it is shown by the next lemma.     Lemma 5.1. Let f ε A C 1 ðτ0 ; τε Þ; H 1 ð2Þ \ C 0 ðτ0 ; τε Þ; H 3 ð2Þ . If K is large enough, there exist two positive constants C1 and C2 such that, for all τ A ðτ0 ; τε Þ, 

2

2



C1





f 2 1 þ α1 e  τ Δf 2 þ

E7 r

jX j2 f ε þ α21 e  2τ jX j2 Δf ε ; ε H ε 1θ 

2

2

2

2





C 2 f ε H1 þ α1 e  τ Δf ε þ jX j2 f ε þ α21 e  2τ jX j2 Δf ε rE7 : Proof. The first inequality of this lemma comes directly from the definition of E7. To prove the second one, we notice that





  

CK 

2

f 2 1 þ α1 e  τ Δf 2 þ 1

E7 Z

jX j2 f ε  α1 e  τ Δf ε : ε H ε 2 1θ Furthermore, we have already shown that

2

2







2

2

jX j f ε  α1 e  τ Δf ε ¼ jX j2 f ε þ 2α1 e  τ jX j2 ∇f ε



2

2



þ α21 e  2τ jX j2 Δf ε  16 jX jf ε : Via the Hölder and Young inequalities, we get

2

2







2

2

jX j f ε  α1 e  τ Δf ε Z jX j2 f ε þ2α1 e  τ jX j2 ∇f ε



2

2

2





þ α21 e  2τ jX j2 Δf ε  12 jX j2 f ε 128 f ε :

Integrating several times by parts, we notice that

Consequently, the inequality (4.62) can be written

In this section, we consider the solution W ε of (4.1) with initial data W0 satisfying the condition (1.9) for some γ 4 0 and we take advantage of the energy estimates obtained in Section 4 to show that W ε satisfies the inequality (1.10). Then, we pass to the limit when ε tends to 0 and show that W ε converges, up to a subsequence, to a weak solution of (1.6) which satisfies also the inequality (1.10). We recall that

Consequently, one has

2

2



 CK 





f 2 1 þ α1 e  τ Δf 2 þ 1

E7 Z

jX j2 f ε þ α1 e  τ jX j2 ∇f ε

ε H ε 4 1θ

2

2 α2



þ 1 e  2τ jX j2 Δf 64 f ε : 2 Thus, if K is big enough, we get the second inequality of this lemma. □   Lemma 5.2. Let W ε A C 0 ½τ0 ; τε Þ; H 3 ð2Þ be a solution of (4.1) satisfying the inequality (4.4) for some γ 4 0. There exist T 0 40 and   γ 0 4 0 such that if T ¼ eτ0 Z T 0 and γ r γ 0 , then, for all τ A τ0 ; τnε , E7 satisfies the inequality ∂τ E7 þ θE7 r CM 3 γ ð1  θÞe  τ :

ð5:1Þ

Proof. We take γ0 and T0 respectively as small and large as necessary to satisfy the conditions of Lemmas 4.2–4.6. According

O. Coulaud / International Journal of Non-Linear Mechanics 71 (2015) 132–151

Since f ε satisfies the inequality (4.5), one has η2 r C γ ð1  θÞ6 . Taking into account the inequality (5.6), it comes

2

2





2





W ε ðτÞ 2 1 þ

jX j2 W ε ðτÞ þ α1 e  τ ΔW ε ðτÞ þ α21 e  2τ jX j2 ΔW ε ðτÞ

H

to the inequalities (4.41) and (4.42), one has 



2 1

2 1

2 K



7 ð  ΔÞ  ð1 þ θÞ=4 f ε þ ∇f ε þ Δf ε

∂τ E 7 þ θ E 7 þ 4 4 1θ

2 α

2 1  θ

2



1 3 þ

jX j f ε þ e  τ jX j2 Δf ε rCM γ ð1  θÞe  τ 8 4

1024

f 2 þ CM 2 ð1  θÞγ 1=2 K þ 1θ ε 

2

2





 jX j2 f ε þ α21 e  2τ jX j2 Δf ε

r C γ ð1  θÞ6 þ E7 ðτ0 Þe  θðτ  τ0 Þ þ CM3 γ e  τ0 :





2

2  2 þ CM 2 ð1  θÞγ 1=2 f ε þ ∇f ε þ Δf ε :

2 Using the interpolation inequality (4.27) of f ε between

2 2 ‖ð  ΔÞ  ð1 þ θÞ=4 f k and ∇f and taking K large enough and ε

ε

γ small enough, we get

∂τ E7 þ θE7 r CM 3 γ ð1  θÞe  τ :

□

ð5:2Þ

Remark 5.1. We can see in the proofs of Lemmas 4.2–5.2 that does not depend on θ, but only on α1, β and M.

Before proving Theorem 1.1, we show an intermediate theorem. This one gives the same result than Theorem 1.1, but for the solutions of the regularized system (4.1). Theorem 5.1. Let θ be a constant such that 0 o θ r 1. There exist ε0 ¼ ε0 ðα1 ; βÞ 4 0, γ 0 ¼ γ 0 ðα1 ; βÞ 4 0 and T 0 ¼ T 0 ðα1 ; βÞ Z 0 such that, for all ε r ε0 , T ¼ eτ0 ZT 0 and W 0 A H 2 ð2Þ satisfying the condi1 tion (1.9) with γ  r γ 0 , there exist a unique    global solution W ε A C ðτ0 ; þ1Þ; H 1 ð2Þ \ C 0 ðτ0 ; þ 1Þ; H 3 ð2Þ of (4.1) and a positive constant C ¼ Cðα1 ; β ; θÞ 4 0 such that, for all τ Z τ0 , L ð2Þ

rC γ e

 θτ

;

R where η ¼ R2 W 0 ðxÞ dx and the parameters given in (1.1).

ð5:3Þ

α1 and β are fixed and

Proof of Theorem 5.1. Let W 0 A H 2 ð2Þ satisfying the condition (1.9) with 0 r γ r γ 0 and 0 r T 0 r T, where γ0 and T0 will be made more precise later. By Theorem 3.1, there exist τε 4 τ0 ¼ log ðTÞ and a solution Wε to the system (4.1) which belongs to C 1 ððτ0 ; τε Þ;  R H 1 ð2ÞÞ \ C 0 ðτ0 ; τε Þ; H 3 ð2Þ . Let η ¼ R2 W 0 ðXÞ dX, and f ε defined by the equality W ε ¼ ηG þf ε :

ð5:4Þ

Let M 4 2 be a positive constant that will be set later and τε A ½τ0 ; τε Þ be the highest positive time such that the inequality (4.4) holds. As shown at the beginning of Section 4, the inequality (4.5) holds on   τ0 ; τnε . We take T0 sufficiently large and γ0 and ε sufficiently small so that the results of Lemmas 4.2–5.2 occur. Consequently, there   exists C ¼ Cðα1 ; βÞ 4 0 such that, for all τ A τ0 ; τnε ,   ∂τ E7 eθτ r CM3 γ ð1  θÞe  ð1  θÞτ : ð5:5Þ n

  Integrating this inequality in time between τ0 and τ A τ0 ; τnε , we obtain   E7 ðτÞ r E7 ðτ0 Þe  θðτ  τ0 Þ þCM3 γ e  ð1  θÞτ0 e  θτ  e  τ : ð5:6Þ Due to the decomposition (5.4) and Lemma 5.1, for every

τ A τ0 ; τnε , one has



2

2





2





W ε ðτÞ 2 1 þ

jX j2 W ε ðτÞ þ α1 e  τ ΔW ε ðτÞ þ α21 e  2τ jX j2 ΔW ε ðτÞ

H r C η2 þ CE7 ðτÞ:

ð5:7Þ

Using again Lemma 5.1 and arguing like for the establishment of the inequality (4.5), we can show that



C 

f ðτ0 Þ 2 1 þ α1 e  τ0 Δf ðτ0 Þ 2 E 7 ðτ 0 Þ r ε ε H 1θ

2

2





þ jX j2 f ε ðτ0 Þ þ α21 e  2τ0 jX j2 Δf ε ðτ0 Þ

r C γ ð1  θÞ5 : Consequently, the inequality (5.7) becomes



2

2





2





W ε ðτÞ 2 1 þ

jX j2 W ε ðτÞ þ α1 e  τ ΔW ε ðτÞ þ α21 e  2τ jX j2 ΔW ε ðτÞ

H r C 1 γ ð1  θÞ5 þ C 2 M 3 γ e  τ0 ;

γ0

5.1. Regularized problem

 



1  α1 e  τ Δ W ε ðτÞ  ηG 22

149

ð5:8Þ

where C1 and C2 are two positive constants independent of W0 and θ. We set M ¼ 4C 1 =ð1  θÞ, and we get

2

2





2





W ε ðτÞ 2 1 þ

jX j2 W ε ðτÞ þ α1 e  τ ΔW ε ðτÞ þ α21 e  2τ jX j2 ΔW ε ðτÞ

H r

M γ ð1  θÞ6 þ C 2 M 3 γ e  τ0 : 4

ð5:9Þ

Finally, taking T0 sufficiently  large so that C 2 M 3 γ e  τ0 r M γ ð1  θÞ6 =4, we obtain, for all τ A τ0 ; τnε ,

2

2





2





W ε ðτÞ 2 1 þ

jX j2 W ε ðτÞ þ α1 e  τ ΔW ε ðτÞ þ α21 e  2τ jX j2 ΔW ε ðτÞ

H r

M γ ð1  θÞ6 : 2

ð5:10Þ

This inequality shows in particular that τnε ¼ τε and thus (5.10) holds for all τ A ½τ0 ; τε Þ. From the inequality (5.10), we deduce also that τε ¼ þ 1. Indeed, if τε o þ 1, the boundedness of W ε in H 2 ð2Þ on ½τ0 ; τε Þ given by (5.10) is a contradiction to the finiteness of τε . In particular, the inequality (5.6) occurs on ½τ0 ; þ1Þ. Applying Lemma 5.1 in the inequality (5.6), we finally obtain the inequality (5.3). □ 5.2. Existence of weak solutions in H2(2) Now, we show that under the hypotheses of Theorem 5.1, there exists a global weak solution W of (1.6) which belongs to   C 0 ½τ0 ; þ 1Þ; H 2 ð2Þ , and that this solution converges to the Oseen vortex sheet G when τ goes to infinity. To this end, we pass to the limit in the system (4.1) when ε tends to 0 and show that, up to a subsequence, W ε converges in some sense to a solution of the system (1.6) which satisfies the inequality (5.3). Let ðεn Þn A N be a sequence of positive numbers tending to 0. We consider the solution   W εn A C 1 ðτ0 ; þ 1Þ; H 1 ð2Þ \ C 0 ðτ0 ; þ 1Þ; H 3 ð2Þ of (4.1) which satisfies the conditions of Theorem 5.1. Due to technical reasons linked to the compactness properties of Sobolev spaces, it is more convenient to establish the convergence of W εn to W in every bounded regular domain of R2 . Let Ω be a bounded regular domain of R2 and τ1 be a fixed positive time such that τ0 o τ1 o þ 1. In what follows, H s ðΩÞ, s Z0, denotes the restrictions to Ω of the functions of the Sobolev space H s ðR2 Þ. From Theorem 5.1, we know  2 that W εn is bounded in L1 ½τ0 ; þ 1Þ; H ð2Þ uniformly  with respect to n. Consequently, there exists W A L1 ½τ0 ; τ1 ; H 2 ð2Þ such that   W εn, W weakly in Lp ½τ0 ; τ1 ; H 2 ðΩÞ for all p Z 2:

150

O. Coulaud / International Journal of Non-Linear Mechanics 71 (2015) 132–151

Looking at the system (4.1), we can see that ∂τ W εn is bounded in   L1 ½τ0 ; τ1 ; H 1 ðΩÞ uniformly with respect to n. This implies that W εn is equicontinuous in H 1 ðΩÞ. Indeed, for σ 1 ; σ 2 A ½τ0 ; τ1 , σ 2 Z σ 1 , we have

Z σ 2







W ε ðσ 2 Þ  W ε ðσ 1 Þ 1 ¼

∂τ W εn ðsÞ ds

n n



H ðΩÞ σ1

H ðΩÞ 1



r ðσ 2  σ 1 Þ ∂τ W εn ðsÞ L1 ð½τ ;τ ;H1 ðΩÞÞ : 0 1

Furthermore, for every τ A ½τ0 ; τ1 , the set ⋃n A N f ϵn ðτÞ is bounded in H 2 ðΩÞ and thus compact in H 1 ðΩÞ. Using the Arzela–Ascoli theorem, we get   W εn -W strongly in C 0 ½τ0 ; τ1 ; H 1 ðΩÞ : By interpolation, we can show that   W εn -W in C 0 ½τ0 ; τ1 ; H s ðΩÞ for all s o 2:

ð5:11Þ

This is enough to pass to the limit in the system (4.1) in the sense of the distributions on ½τ0 ; τ1   Ω and to show that W is a weak solution of the system (1.6). Since most of the terms of Eq. (4.1) have already been studied in [24], we 
will  show that the convergence
2 just holds for the term  div curl
Aεn
Aεn which does not appear in the second grade fluids equations.  We consider φ A C 1 0 ½τ 0 ; τ 1   Ω . For all τ A ½τ 0 ; τ 1 , we want to show that Z τZ


Aε ðτ; XÞ
2 Aε ðτ; XÞ♢∇2 φðτ; XÞ dX dτ n n Ω

τ0

Z τZ


Aðτ; XÞ
2 Aðτ; XÞ♢∇2 φðτ; XÞ dX dτ; ⟶ τ0

Ω

ð5:12Þ

when n tends to infinity, where, for A; B A M2 ðRÞ, we use the notation A♢B ¼

2  X  A1;j B2;j A2;j B1;j : j¼1

The term on the right-hand side of (5.12) appears naturally via two 2 integrations 
by
2 parts, when performing the L -scalar product of

div  curl A A with φ. The strong convergence of W εn to W in C 0 ½τ0 ; τ1 ; H 1 ðΩÞ implies directly the identity (5.12). Indeed, due to the continuous injection of H 1 ðΩÞ into L3 ðΩÞ, W εn converges to   0 3 W in C ½τ0 ; τ1 ; L ðΩÞ . Furthermore, the inequality (2.13) implies





Aε  A 3 r W ε W 3 ; n n L L   and consequently Aεn converges to A strongly in C 0 ½τ0 ; τ1 ; L3 ðΩÞ . This fact suffices to show that the identity (5.12) occurs. Thus W is a global weak solution of (1.6) which belongs to C 0 ð½τ 0 ; þ 1Þ; H 2 ð2ÞÞ. The fact that W satisfies the inequality (1.10) is a direct consequence of the weak convergence of W εn to W. Indeed, for all τ A ½τ0 ; þ 1Þ, W εn ðτÞ is bounded in H2 ð2Þ uniformly with respect to n and consequently we have ,

W εn ðτÞ W ðτÞ

weakly in H 2 ð2Þ

for all τ A ½τ0 ; þ 1Þ:

Since W εn satisfies the inequality (1.10), it implies that W also satisfies (1.10). 5.3. Uniqueness The aim of this part is to prove that the solution w of the system (1.2) obtained in Section 5.2 is unique in L2 ð2Þ. Let w1 and w2 be two solutions of (1.2) with the same initial data w0 A H 2 ð2Þ. Let u1 and u2 be the divergence free vector fields obtained via the Biot–Savart law respectively from w1 and w2. We also define Ai ¼ ∇ui þ ð∇ui Þt . Applying the Biot–Savart law to the system (1.1), we can see that, for i ¼1,2, the divergence free vector field ui

satisfies the system 

     2 ∂t ui  α1 Δui  Δui þcurl ui  α1 Δui 4 ui  β div
Ai
Ai þ ∇pi ¼ 0; div ui ¼ 0; uijt ¼ 0 ¼ u0 ;

ð5:13Þ

where u0 is obtained from w0 via the Biot–Savart  law. 1 2 þ Notice  that since  wi belongs to Lloc R ; H ð2Þ and ∂t wi belongs 1 þ 2 to L1 loc R ; H ðR Þ , the inequalities (2.11) and (2.13) imply in particular   p þ 2 2 ui A L1 for all p 4 2; loc R ; L ðR Þ   2 þ 2 4 ; ∇ui A L1 loc R ; H ðR Þ   p þ 2 2 for all p 4 2; ∂t ui A L1 loc R ; L ðR Þ   1 2 þ 2 2 ∂t Δui A Lloc R ; L ðR Þ : Consequently, the system (5.13) has a meaning in the sense of distributions. We note w ¼ w1  w2 , u ¼ u1  u2 , L ¼ L1  L2 and A ¼ A1  A2 . A short computation shows that u satisfies the system       ∂t u  α1 Δu  Δu þ curl u  α1 Δu 4 u1 þcurl u2  α1 Δu2 4 u 

 

 2 2 þ β div
A2
A2  β div
A1
A1 þ ∇q ¼ 0; div u ¼ 0; ujt ¼ 0 ¼ 0:

ð5:14Þ 2

2

Notice that, although u1 and u2 do not belong to L ðR Þ, the divergence free vector field u does. Indeed, since w1 and w2 have the same initial data, for all t Z 0, we have Z wðt; xÞ dx ¼ 0: R2

By application of Lemma 2.5, this fact implies that u belongs to L2 ðR2 Þ. Let t 0 40 be a fixed positive time.  We notice that both w1 and w2 are bounded in L1 ½0; t 0 ; H 2 ð2Þ . More precisely, one has 





sup w1 ðtÞ H2 ð2Þ þ w2 ðtÞ H2 ð2Þ r C: t A ½0;t 0 

Applying Lemma 2.3, it implies in particular 







 sup ui ðtÞ L4 þ ∇ui ðtÞ L1 þ Δui ðtÞ L4 r C

for i ¼ 1; 2:

t A ½0;t 0 

In order to show that u  0, we now perform estimates on the H 1 -norm of u. The uniqueness of the solutions of (5.13) has been shown in [4] for solutions with initial data in H 2 ðR2 Þ. In our case, the proof is slightly simpler, because the vector field u belongs to H 3 ðR2 Þ2 . We consider the L2 -inner product of (5.14) with u. First of all, integrating by parts, we notice that 











β div
A2
2 A2 
A1
2 A1 ; u

L2



2 A1 A1 
A2
A2 ; A 2 L β 



2



2 



2 ¼ A1 þ A2 A dx 4 R2 Z β 



2



2  þ A1  A2 ðA1 þ A2 Þ : A dx 4 R2 Z 

2

2 

2 β
A1
þ
A2

A
dx ¼ 4 R2 Z 

2

2 2 β
A1

A2
þ dx: 4 R2 ¼

β



2 2Z

Thus, using integrations by parts and the divergence free property of u, we have Z  1  β 



2



2 



∂t kuk2 þ αk∇uk2 þ k∇uk2 þ A dx A1 þ A2 2 4 R2 Z   2



β
A1
2 
A2
2 dx ¼ I 1 þI 2 ; þ 4 R2 ð5:15Þ

O. Coulaud / International Journal of Non-Linear Mechanics 71 (2015) 132–151

where     I 1 ¼ curl u2  α1 Δu2 4 u; u L2 ; I 2 ¼ ðcurl u 4 u1 ; uÞL2 ;   I 3 ¼  α1 curl Δu 4 u1 ; u L2 : A short computation shows that I1 vanishes. Indeed, we   set ω ¼ u2  α1 Δu2 and we recall the notation u ¼ u1 ; u2 ; 0 and curl ω ¼ ð0; 0; ∂1 ω2  ∂2 ω1 Þ. We have I 1 ¼ ðcurl ω 4 u; uÞL2     ¼  ð∂1 ω2  ∂2 ω1 Þu2 ; u1 L2 þ ð∂1 ω2  ∂2 ω1 Þu1 ; u2 L2 ¼ 0: Due to the boundedness of u1 in L4 ðR2 Þ, applying Hölder inequalities we obtain I 2 r ku1 kL4 k∇ukkuk   r Cðα1 Þ kuk2 þ α1 k∇uk2 : Using [27, Z I 3 r C α1

Lemma A.1], we check that Z


Δu1
j∇ujjuj dx þC α1 j∇u1 jj∇uj2 dx:

R2

R2

Using Hölder inequalities, the Gagliardo–Nirenberg inequality and 4=3 the Young inequality ab r 14a4 þ 34b , we obtain



I 3 r C α1 kukL4 Δu2 L4 k∇uk þ C α1 k∇u1 kL1 k∇uk2 r C α1 k∇uk3=2 kuk1=2 þ C α1 k∇uk2   r Cðα1 Þ kuk2 þ α1 k∇uk2 :

Going back to (5.15), we get     2 2 1 r CðαÞ kuk2 þ αk∇uk2 : 2 ∂t kuk þ αk∇uk

ð5:16Þ

Integrating in time this inequality between 0 and t A ½0; t 0  and applying the Gronwall lemma, we finally obtain





uðtÞ 2 þ α ∇uðtÞ 2 ¼ 0 for all t A ½0; t 0 : Since t0 is arbitrary, we conclude that u  0 on R þ . Consequently u is unique and so is w. Thus, the system (1.2) has a unique global   solution in the space C 0 R þ ; H 2 ð2Þ . Acknowledgement I would like to express my gratitude to Geneviève RAUGEL and Marius PAICU, who were very helpful for this work. References [1] C. Amrouche, D. Cioranescu, On a class of fluids of grade 3, Int. J. Non-Linear Mech. 32 (1) (1997) 73–88. [2] J.-M. Bernard, Weak and classical solutions of equations of motion for third grade fluids, M2AN 33 (6) (1999) 1091–1120. [3] D. Bresch, J. Lemoine, On the existence of solutions for non-stationary thirdgrade fluids, Int. J. Non-Linear Mech. 34 (1999) 485–498.

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