Asymptotic profiles for the third grade fluids׳ equations

International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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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

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a b s t r a c t

Article history: Received 6 October 2013 Received in revised form 20 February 2014 Accepted 19 March 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 Published by Elsevier Ltd.

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 not only 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 example, 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 þ u  ∇Ak  1 þ ð∇uÞt Ak  1 þAk  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 modelled through the stress tensor

s ¼  pI þ νA1 where ν 40 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

s ¼  pI þ νA1 þ α1 A2 þ α2 A21 þ βjA1 j2 A1 ;

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 s leads to the system ∂t ðu  α1 ΔuÞ  νΔu þ curlðu  α1 ΔuÞ 4 u ðα1 þ α2 Þ ðA  Δu þ 2 divðLLt ÞÞ  β divðjAj2 AÞ þ ∇p ¼ 0; div u ¼ 0; ujt ¼ 0 ¼ u0 ; ð1:1Þ t

where L ¼ ∇u, AðuÞ ¼ ∇u þ ð∇uÞ and 4 denotes the classical vectorial product of R3 . For matrices A; B A Md ðRÞ, we define A : B ¼ ∑di;j ¼ 1 Ai;j Bi;j and jAj2 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 nonNewtonian 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 equations (1.1) has been studied in various cases, on bounded domains of Rd , d ¼2,3 or in the whole space Rd (see [1–5,26]). 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 curlðI  α1 ΔÞ.

http://dx.doi.org/10.1016/j.ijnonlinmec.2014.03.013 0020-7462/& 2014 Published by Elsevier Ltd.

Please cite this article as: O. Coulaud, Asymptotic profiles for the third grade fluids' equations, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.03.013i

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O. Coulaud / International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

In 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 43. 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 , d ¼2,3, 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 us to show the existence of 2 þ d d solutions of (1.1) in the space L1 loc ðR ; H ðR Þ Þ. Besides, these solutions are unique if d ¼2. 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 ðw  α1 ΔwÞ  νΔw þ u  ∇ðw  α1 ΔwÞ  β divðjAj2 ∇wÞ  β divð∇ðjAj2 Þ 4 AÞ 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 curlðA  Δu þ 2 divðLLt ÞÞ ¼ 0, or equivalently there exists q such that A  Δu þ2 divðLLt Þ 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 us 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 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 finitedimensional manifold which is locally invariant by the semiflow associated with 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], JaffalMourtada 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 the additional term β divðjAj2 AÞ in the third grade fluids' equations. Sometimes, this term helps us 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.

Please cite this article as: O. Coulaud, Asymptotic profiles for the third grade fluids' equations, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.03.013i

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We consider the solution w of (1.2) and define W and U suchffi that pffiffiffiffiffiffiffiffiffi curl U ¼ W through the change of variables X ¼ x= t þ T and τ ¼ log ðt þ TÞ. We set   8 1 x > > ffi U log ðt þTÞ; pffiffiffiffiffiffiffiffiffiffi ; > < uðt; xÞ ¼ pffiffiffiffiffiffiffiffiffi t þT t þT   ð1:4Þ 1 x > > > wðt; xÞ ¼ W log ðt þ TÞ; pffiffiffiffiffiffiffiffiffiffi : : t þT t þT For τ Z log ðTÞ, we have ( Uðτ; XÞ ¼ eτ=2 uðeτ  T; eτ=2 XÞ;

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 self-similar solutions (see [11,12,14] or [25]), that is to say under pffiffiffiffiffiffiffiffiffi ffi the form ð1=ðt þ TÞÞFðx= t þ T Þ. 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 the sake of simplicity, we set Ai;j ¼ ∂j U i þ ∂i U j . Considering self-similar variables, one can see that W and its corresponding divergence free vector field U satisfy the system ∂τ ðW  α1 e  τ ΔWÞ LðWÞ þU  ∇ðW  α1 e  τ ΔWÞ þ α1 e  τ ΔW þ α1 e

2

 ∇ΔW  β e

 2τ

W jτ ¼ τ0 ¼ W 0 ;

ð1:6Þ

∂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 L2 ðmÞ ¼ fu A L2 ðR2 Þ : ð1 þ jxj2 Þm=2 u A L2 ðR2 Þg; equipped with the norm 1=2

R2 2

when τ- þ 1:

in L2 ð2Þ;

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  jXj =4  X 2 VðXÞ ¼ : ð1:8Þ X1 2π jXj2

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 H 1 ðmÞ ¼ fu A L2 ðmÞ : ∂j u A L2 ðmÞ; j A f1; 2gg; H 2 ðmÞ ¼ fu A H 1 ðmÞ : ∂j u A H 1 ðmÞ; j A f1; 2gg;

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 us 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

ð1 þ jxj2 Þm juðxÞj2 dx

f ðτÞ ¼ Oðe  τ=2 Þ

equipped with the norms

X  ∇W: 2

Z

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, so that sc ðLÞ ¼ λ A C : ReðλÞ r  12 . Since the second eigenvalue of L in L2 ð2Þ is  12 , the best result that we expect is

VðXÞ  X ¼ 0;

where τ0 ¼ log ðTÞ, W 0 ðXÞ ¼ eτ0 w0 ðeτ0 =2 XÞ and L is the linear differential operator defined by

:u:L2 ðmÞ ¼

In particular, the eigenvalue 0 is simple and the Oseen vortex G given by (1.3) is an eigenfunction of L associated with 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:

In particular, for every X A R2 , one has

 divðAj2 ∇WÞ

 βe  2τ divð∇ðjAj2 Þ 4 AÞ ¼ 0; div U ¼ 0;

LðWÞ ¼ ΔW þW þ

and the continuous spectrum   m 1 : sc ðLÞ ¼ λ A C : ReðλÞ r  2

WðτÞ ¼ ηG þ f ðτÞ;

Wðτ; XÞ ¼ eτ wðeτ  T; eτ=2 XÞ:

 τX

3

:

The spectrum of L in L ðmÞ is given in [18, Appendix A]. It is composed of the discrete spectrum   k sd ðLÞ ¼  : k A f0; 1; …; m  2g ; 2

2

:u:H1 ðmÞ ¼ ð:u:L2 ðmÞ þ :∇u:L2 ðmÞ Þ1=2 ¼

and

:u:H2 ðmÞ

2 2 ð:u:H1 ðmÞ þ :∇2 u:L2 ðmÞ Þ1=2

where j∇uj2 ¼ ∑2i ¼ 1 ð∂i uÞ2 and j∇2 uj2 ¼ ∑2i;j ¼ 1 ð∂i ∂j uÞ2 . 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 ; βÞ and T 0 ¼ T 0 ðα1 Þ Z 1 such that, for all W 0 A H 2 ð2Þ satisfying the condition 2

‖W 0 :H1 þ

α1 T

‖ΔW 0 ‖2L2 þ‖jXj2 W 0 ‖2L2 þ

α21 T2

‖jXj2 ΔW 0 ‖2L2 r γ ð1  θÞ6 ; 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 ð½τ 0 ; þ 1Þ; H 2 ð2ÞÞ of (1.6) and a positive constant C ¼ Cðα1 ; β ; θÞ such that, for all τ Z τ0     ð1:10Þ ð1  α1 e  τ ΔÞðWðτÞ  ηGÞ2L2 ð2Þ r C γ e  θτ ; R where η ¼ R2 W 0 ðXÞ dX, τ0 ¼ log ðTÞ and the parameters are fixed and given in (1.1).

α1 and β

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 in the right hand side of the inequality.

Please cite this article as: O. Coulaud, Asymptotic profiles for the third grade fluids' equations, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.03.013i

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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 also holds with β ¼ 0 and consequently describes the first order asymptotic profiles of the solutions of the second grade fluids' equation. 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 . 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). 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 Z 1 such that, for all w0 A H 2 ð2Þ satisfying the condition 1 2 2 T:w0 :L2 þT 2 :∇w0 :L2 þ ‖jxj2 w0 ‖2L2 T þ α1 T 3 :Δw0 :L2 þ 2

α21 T

:jxj2 Δw0 :L2 r γ ð1  θÞ6 ; 2

ð1:11Þ

for some T Z T 0 and 0 o γ r γ 0 , there exists a unique global solution w A C 0 ð½0; þ 1Þ; H 2 ð2ÞÞ of (1.2) such that, for all 1 rp r2, there exists a positive constant C ¼ Cðα1 ; β ; θÞ such that, for all t Z 0      x p ð1  α1 ΔÞ wðtÞ  η G pffiffiffiffiffiffiffiffiffi ffi L rC γ ðt þ TÞ  1  θ=2 þ 1=p ;  t þT t þT 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 q ð1  α1 ΔÞ uðtÞ  pffiffiffiffiffiffiffiffiffi ffiV pffiffiffiffiffiffiffiffiffiffi L rC γ ðt þ TÞ  1=2  θ=2 þ 1=q ;  t þT t þT 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 function 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 sc and sd, 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. Actually, we 2 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 also tends 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.

2. Biot–Savart law and auxiliary lemmas 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  kuk ¼ kuL2 ; 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 b Þ; ð  ΔÞs u ¼ F ðjξj2s u b (also denoted F ðuÞ) is the Fourier transform of u, given by where u Z b ðξÞ ¼ u uðxÞe  ixξ dx; R2

and F denotes the inverse Fourier transform Z 1 F ðvÞðxÞ ¼ vðξÞeixξ dξ: 2 ð2π Þ R2 Lemma 2.1. Let s be a positive real number such that 34 o s 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   C ð  ΔÞ  s ∇g  r kg:L2 ð1Þ : ð2:1Þ ð1  sÞ3=2 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 C :ð  ΔÞ  s g:L2 r :g:L2 ð2Þ : ð2:2Þ ð1 sÞ3=2

Proof. We start by proving the inequality (2.1). For j A f1; 2g, using Fourier variables, one has Z 1 2 bj2 dξ þ ‖g‖22 jg :ð  ΔÞ  s ∂j g:L2 r C 4s  2 L j ξ j j ξj r 1 Z ð2s  1Þ=s Z ð1  sÞ=s 1 bj2s=ð1  sÞ dξ rC d ξ j g þ ‖g‖2L2 2s jξj r 1 jξj jξj r 1  C   b 2 r g L2s=ð1  sÞ þ‖g‖2L2 : ð1  sÞ We now use the continuous injection of H 1 ðR2 Þ into L2s=ð1  sÞ ðR2 Þ. Looking at the computations of [8, pp. 723–724], one can see that there exists a constant C 4 0 such that :u:Lp r Cpku:H1

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 2

:ð  ΔÞ  s ∂j g:L2 r

C ð1  sÞ3

2

b:H1 þ‖g‖22 r :g L

C ð1  sÞ3

We now prove the inequality (2.2). Since Fourier variables, we get Z 1 bðξÞj2 dξ ‖ð  ΔÞ  s g‖2L2 ¼ ð2π Þ2 jg 2 jξj4s R

‖g‖2L2 ð1Þ : R

R2 f ðxÞ

dx ¼ 0, using

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r ð2π Þ2

Z

1

ξj4s

jξj r 1 j

and furthermore           ∂j ∂k ðjxj2 f Þ2 r C Δðjxj2 f Þ2 r Cðf 2 þ jxj∇f 2 þ jxj2 Δf 2 Þ:

bðξÞj2 dξ þ ‖g‖22 jg L

Z 2  1  1 ξ  ∇gbðsξÞds dξ þ ‖g‖2L2 4s  0 jξj r 1 jξj Z 2 Z  1  1 bðsξÞj ds dξ þ ‖g‖22 : rC j∇ g  4s  2 L j ξ j 0 j ξj r 1 r ð2π Þ2

Z

ð2:9Þ Combining (2.8) and (2.9) we get the inequality (2.7). To obtain (2.6), we use Gagliardo–Nirenberg's inequality as follows:    1=2  2  jxj ∇f  4 rC J jxj2 ∇f k :∇ðjxj2 ∇f Þ1=2 L

      r C jxj2 ∇f 1=2 ðjxj∇f 1=2 þ jxj2 ∇2 f 1=2 Þ;

Cauchy–Schwarz inequality and Fubini's theorem give Z 1Z 1 bðsξÞj2 dξ ds þ ‖g‖22 : j∇g ‖ð  ΔÞ  s g‖2L2 r C 4s  2 L 0 jξj r 1 jξj

and consequently inequality (2.5) implies (2.6).

j ξj r 1

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.

1  s  s s Z 1  Z bðsξÞj2=ð1  sÞ dξ rC j∇g 1s 0 jξj r 1 ds þ ‖g‖2L2 :

Lemma 2.3. Let u be the divergence free vector field given by (2.10).

The change of variables ζ ¼ sξ yields 1  s  s s Z 1  Z 1 bðζ Þj2=ð1  sÞ dζ j∇g ‖ð  ΔÞ  s g‖2L2 r C 2 1 s 0 jζ j r s s

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, q 2 2 then   u A LðR Þ and there exists C 4 0 such that u q r C w p : L L

ds þ ‖g‖2L2

C ð1  sÞ3

3. Assume that 1 o p o1. If w A Lp ðR2 Þ, then ∇u A Lp ðR2 Þ4 and there exists   C 4 0such  that ∇u p r C w p : ð2:13Þ L L

‖g‖2L2 ð2Þ ;

which concludes the proof of this lemma.

ð2:11Þ

2. Assume that 1 r p o2 oq r 1, 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 1 2 2 u  AL ðR Þ and  exists C 4 0 such that  there uL1 rC wαp w1q α : ð2:12Þ L L

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

Finally, we use again the inequality (2.3) and obtain  s s  1  2 2  b kg ‖ð  ΔÞ  s g‖2L2 r C 2H2 þ‖g‖2L2 1 s 2s  1 1 s r

□

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 ? wðyÞ dy; ð2:10Þ uðxÞ ¼ 2π R2 jx  yj2

Using Hölder inequality, we get s Z 1 Z 1 ‖ð  ΔÞ  s g‖2L2 r C d ξ 4  2=s 0 jξj r 1 jξj Z 1  s bðsξÞj2=ð1  sÞ dξ j∇g ds þ ‖g‖2L2

rC

5

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

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 jxj f A Lp ðR2 Þ and the following inequality holds: 1=2 1=2 ð2:4Þ :jxj f :Lp r‖f ‖Lp ‖jxj2 f ‖Lp : 2 that 2. Let  2f A2Hð2Þ, there exists C 4 0 such jxj ∇ f  rCðkf k þ kjxj∇f k þ kjxj2 Δf kÞ:

ð2:5Þ

3. Let f A H 2 ð2Þ, then jxj2 ∇f A L4 ðR2 Þ and there exists C 40 such that           jxj2 ∇f  4 rC jxj2 ∇f 1=2 f 1=2 þ jxj∇f 1=2 þ jxj2 Δf 1=2 Þ: L

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 Þ: ð2:7Þ Indeed, we notice that jxj2 ∂j ∂k f ¼ ∂j ∂k ðjxj2 f Þ  2δj;k f  2xj ∂k f  2xk ∂j f ;

ð2:8Þ

1. If w A L2 ð2Þ, then u A L4 ðR2 Þ2 and there exists C 4 0 such that ð2:14Þ :u:L4 r C:w:L2 ð2Þ : 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 :u:L1 rCkw:H1 :w:L2 ð2Þ : ð2:15Þ 3. Let s A R. If ð  ΔÞðs  1Þ=2 w A L2 ðR2 Þ for s A R, then 2 s=2 2 2 ð  ΔÞ u A L ðR Þ and there exists C 40 such that :ð  ΔÞs=2 u:r Ckð  ΔÞðs  1Þ=2 wk: ð2:16Þ 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 suchthat    ∇u s r C w s : ð2:17Þ H H The proof of the two first inequalities is shown in [24]. The two other inequalities are obvious when using Fourier variables. The next lemma is useful to get energy estimates in weighted Sobolev spaces for solutions of (1.6). For a vector field u, we set j∇3 uj2 ¼

2

∑

i;j;k;l ¼ 1

ð∂j ∂k ∂l ui Þ2 :

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Lemma 2.5. Let w A L2 ðR2 Þ and u be the divergence free vector given by (2.10). 1. If w A H 1 ð1Þ, then ∇2 u AL2 ð1Þ and there exists C 4 0 such that ð2:18Þ :∇2 u:L2 ð1Þ r CðkwH1 þ jxj∇wkÞ: 2 u A L4 ðR2 Þ and there exists C 4 0 such that 2. If w A H2 ð1Þ, then jxj∇ :jxj∇2 uL4 r Cð:w:þ jxj∇wkÞ1=2 ð:∇wk þkjxjΔwkÞ1=2 : ð2:19Þ 2

2

2

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 b Þj2 dξ :jxj2 ∂j ∂k2 u: ¼ ð2π Þ2 jΔðξj ξk u R2 Z b j 2 dξ rC jjξj2 ðξj þ ξk ÞΔu þ R2

2

3. If w A H ð2Þ, then u A jxj ∇ u A L ðR Þ and there exists C 4 0 such that   jxj2 ∇3 u r Cðkwk þ kjxj∇wk þ kjxj2 ΔwkÞ: ð2:20Þ 3

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 Ckjxjwk: 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 Ckw: 1 : ð2:22Þ H ð2Þ

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 ∇w 2 : ð2:23Þ L L 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  r Cðkwkþ kjxj∇wkÞ; where i; j; k A f1; 2g. We omit k that does not appear in the following calculations. One has Z 2 2 b Þj2 dξ :jxi j∂j2 u: ¼ ð2π Þ2 j∂i ðξj u Z rC

R2

R2

b j2 dξ þ C jξj u 2

r C:∇u: þ C

Z R

2

Z

b j2 dξ jξj ∂i u 2

R2

jF ðΔðxi uÞÞj2 dξ

   r C:∇u: þ C jxjΔu2 : 2

Using the inequality (2.13) of Lemma 2.3 with p ¼2 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–Nirenberg inequality. Indeed, one has    :xi ∂j2 u:L4 r Ckxi ∂j2 u1=2 ∇ðxi ∂j2 uÞ1=2

R2

Z

b j2 dξ þ jðξj þ ξk Þu

Z R2

b j2 dξ jjξj2 ∇u

 !

2

r C :∇Δðjxj2 uÞ: þ :∇u: þ ∑ :Δðxi uÞ: 2

2

2

i¼1

      r C jxj2 ∇Δu2 þ ∇u2 þ jxjΔu2       r C jxj2 ∇2 w2 þ w2 þ jxj∇w2 : 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 kuk r Ckjxjwk:

ð2:25Þ

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

Integrating by parts, one gets Z Z Z 2 jxj2 ∂1 u2 ∂2 u1 dx ¼ jxj2 u2 ∂1 ∂2 u1 dx þ 2 x1 u2 ∂2 u1 dx R2 R2 R2 Z Z þ jxj2 ∂1 ∂2 u2 u1 dx þ2 x2 ∂1 u2 u1 dx: R2

R2

Using the divergence free property of u and integrating by parts, we have Z Z   2 jxj2 ∂1 u2 ∂2 u1 dx ¼ kjxj∂1 u1 2 þ jxj∂2 u2 : þ 4 x2 u2 ∂2 u2 dx 2 R2 R2 Z þ4 x1 ∂1 u1 u1 dx: R2

Finally, integrating again by parts, we get Z      jxj2 ∂1 u2 ∂2 u1 dx ¼ kjxj∂1 u1 2 þ jxj∂2 u2 2  2u2 : 2 R2

Thus, going back to (2.26), one has       jxj∇u2 ¼ jxjw2 þ2u2 : Combining this equality with (2.25), we get the inequality (2.21). The inequality (2.22) is obtained in the same way. □ 3. Approximate solutions

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

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:

:xi ∇2 w: r Cðk∇wkþ kjxjΔwkÞ;

∂t ðwε  α1 Δwε Þ þ εΔ wε  Δwε þ uε ∇ðwε  α1 Δwε Þ

1=2

r C:xi ∂j2 u:

ð:∂j2 u:þ :xi ∂j2 ∇u:Þ1=2 :

Furthermore, the inequalities (2.13) and (2.18) yield :xi ∂j2 u:L4 r Cðkwk þkjxj∇wkÞ1=2 ðk∇wk þ kxi ∇2 wkÞ1=2 :

which gives :xi ∂j2 u:L4 r Cðkwk þkjxj∇wkÞ1=2 ðk∇wk þ kjxjΔwkÞ1=2 ; and the inequality (2.19) comes when summing for iA f1; 2g.

2

 β divðjAε j2 ∇wε Þ  β divð∇ðjAε j2 Þ 4 Aε Þ ¼ 0; wεjt ¼ 0 ¼ w0 A H 2 ð2Þ;

ð3:1Þ

t

where Aε ¼ ∇uε þ ð∇uε Þ . The aim of this section is to prove the following theorem.

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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 ðð0; t ε Þ; H 1 ð2ÞÞ \ C 0 ð½0; t ε Þ; H 2 ð2ÞÞ \ C 0 ðð0; t ε Þ; H 3 ð2ÞÞ:

∂t ðvε  α1 γ 2 Δvε Þ þ εγ 4 Δ vε  γ 2 Δvε þ γ uε  ∇ðvε  α1 γ 2 Δvε Þ 2

 βγ ∇ðjAε j2 Þ  ∇vε  βγ 2 jAε j2 Δvε  β divð∇ðjAε j2 Þ 4 Aε Þ ¼ 0; 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 ð1 þ jxj2 Þ. 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 ∂t ðzε  γ 2 α1 Δzε  α1 γ 2 qΔq  1 zε  2γ 2 α1 q∇q  1  ∇zε Þ þ εγ 4 Δ zε ¼ Fðzε Þ; 2

zεjt ¼ 0 ¼ qw0 ðx=γ Þ A H 2 ðR2 Þ;

ð3:3Þ

where Fðzε Þ ¼  εγ 4 qΔ ðq  1 zε Þ þ γ 2 qΔðq  1 zε Þ 2

 γ quε ∇ðq  1 zε  γ 2 α1 Δðq  1 zε ÞÞ

þ βγ q∇ðjAε j2 Þ  ∇ðq  1 zε Þ þ βγ 2 qjAε j2 Δðq  1 zε Þ þ β q divð∇ðjAε j2 Þ 4 Aε Þ:

ð3:4Þ

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: BðzÞ ¼ α1 γ 2 Δz þ α1 γ 2 qΔq  1 z; 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 ðqΔq  1 u; vÞL2  2α1 γ ðq∇q 2

1

 ∇u; vÞL2 :

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 Z 2 2 qΔq  1 juj2 dx aðu; uÞ ¼ :u: þ α1 γ 2 k∇u:  α1 γ 2 R2 Z divðq∇q  1 Þjuj2 dx: þ α1 γ 2 R2

Due to the boundedness of qΔq  1 and divðq∇q  1 Þ, there exists C 4 0 such that     aðu; uÞ Z ð1  α1 γ 2 CÞu2 þ α1 γ 2 ∇u2 : 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 for all v A H 1 ðR2 Þ;

A ¼ εγ 4 ðI  B  DÞ  1 Δ : 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

¼ I þ εγ ðI  BÞ 4

1

Δ  I  εγ ðI  B  DÞ 2

4

1

2

DðI  BÞ  1 Δ ¼ J þ R; 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 : 2

4

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 with 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 〈u; v〉H1 ¼ ðð1  α1 γ 2 qΔq  1 Þu; vÞL2 þ α1 γ 2 ð∇u; ∇vÞL2 : If γ is sufficiently small compared to α1, then 〈; 〉H1 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 〈u; v〉H1 ¼ ððI  BÞu; vÞL2 : We define, using this scalar product, the bilinear form j on H 2 ðR2 Þ  H 2 ðR2 Þ associated with J by the formula

We notice that a is obviously continuous on H 1 ðR2 Þ  H 1 ðR2 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    jaðu; vÞj r Cðα1 ; γ ÞkuH1 vH1 ;

aðu; vÞ ¼ 〈f ; v〉H  1 H1

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 on H 1 ðR2 Þ 2

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.

vεjt ¼ 0 ¼ w0 ðx=γ Þ A H 2 ð2Þ:

7

ð3:5Þ

jðu; vÞ ¼ 〈u; v〉H1 þ εγ 4 ðΔu; ΔvÞL2 : A short computation shows that, for u A H 3 ðR2 Þ and v A H 2 ðR2 Þ, one has jðu; vÞ ¼ 〈Ju; v〉H1 :

ð3:7Þ

Furthermore, if γ is small enough, using the definition of 〈; 〉H1 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 ÞuH2 vH2 : 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 ; εÞ:u:H2 : 2

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 40 such that, for all u A H 3 ðR2 Þ     Ru 1 r Cðα1 ; γ ; εÞu 2 : ð3:8Þ H H Applying the equality (3.7) to u A H 3 ðR2 Þ, we get jðu; uÞ ¼ 〈Ju; u〉H1

for all u A H 3 ðR2 Þ:

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Because j is coercive on H2, we obtain, via Cauchy–Schwartz inequality :u:H2 r Cðα1 ; γ ; εÞ:Ju:H1 :u:H1 2

for all u A H 3 ðR2 Þ:

Going back to (3.8), the following property holds: ‖Ru‖2H1

r Cðα1 ; γ ; εÞ:Ju:H1 :u:H1

3

2

for all u A H ðR Þ:

In particular, the Young inequality yields, for all δ 4 0 ‖Ru‖2H1 r δ‖Ju‖2H1 þ Cðα1 ; γεÞ‖u‖2H1

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], we conclude that there exists t ε 4 0 and a unique solution zε A C 1 ðð0; t ε Þ; H 1 ðR2 ÞÞ \ C 0 ð½0; t ε Þ; H 2 ðR2 ÞÞ \ C 0 ðð0; t ε Þ; H 3 ðR2 ÞÞ of the system (3.3). Thus, there exists a unique solution wε A C 1 ðð0; t ε Þ; H 1 ð2ÞÞ \ C 0 ð½0; t ε Þ; H 2 ð2ÞÞ \ C 0 ðð0; t ε Þ; H 3 ð2ÞÞ to the system (3.1). □ 4. Energy estimates 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 allows 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 ∂τ ðW ε  α1 e  τ ΔW ε Þ þ εe  τ Δ W ε  LðW ε Þ 2

τ

þ U ε  ∇ðW ε  α1 e ΔW ε Þ þ α1 e  τ ΔW ε X þ α1 e  τ  ∇ΔW ε  β e  2τ divðjAε j2 ∇W ε Þ 2  β e  2τ divð∇ðjAε j2 Þ 4 Aε Þ ¼ 0;

div U ε ¼ 0; W ε jτ ¼ τ 0 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 þð∇U ε Þt and we recall that LðW ε Þ ¼ ΔW ε þ W ε þ

X  ∇W ε : 2

‖W ε ðτ

Þ‖2H1

þ α1 e

τ

‖ΔW ε ðτ

Þ‖2L2

þ ‖jXj W ε ðτ 2

Þ‖2L2

þ α21 e  2τ ‖jXj2 ΔW ε ðτÞ‖2L2 r M γ ð1  θÞ6 :

ð4:4Þ

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 ð½τ0 ; τε Þ; H 2 ð2ÞÞ and the condition (1.9) holds, τnε is well defined. Furthermore, there exists a positive constant C independent of W0 such that, for all τ A ½τ0 ; τnε Þ       η2 þ f ε 2H1 þ α1 e  τ Δf ε 2L2 þ jXj2 f ε 2L2     ð4:5Þ þ α21 e  2τ jXj2 Δf ε 2L2 r CM γ ð1  θÞ6 : Indeed, using the Cauchy–Schwartz inequality, we get Z Z 1 þ jXj2 η ¼ W 0 ðXÞ dX ¼ W 0 ðXÞ dX 2 2 1 þ jXj2 R R ! 1=2 Z 1=2 Z 1 dX ð1 þ jXj2 Þ2 jW 0 ðXÞj2 dX r 2 2 R2 ð1 þ jXj Þ R2     r C W 0 L2 ð2Þ : 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 Section 1 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 ½τ0 ; τnε Þ EðτÞ  ‖f ðτÞ‖2H2 ð2Þ ; and there exists a positive constant C ¼ Cðα1 ; β; θÞ such that, for all τ A ½τ0 ; τnε Þ ∂τ EðτÞ þ θEðτÞ r C γ e  τ :

ð4:6Þ

This inequality will enable us to show that τε ¼ þ 1 and obtain, by the application of Gronwall lemma n

By Theorem 3.1, it is clear that there exists τε 4 τ0 such that W ε A C 1 ððτ0 ; τε Þ; H 1 ð2ÞÞ \ C 0 ððτ0 ; τε Þ; H 3 ð2ÞÞ: We also assume that the initial datum W 0 A H 2 ð2Þ satisfies the assumption (1.9) of Theorem 1.1, for some γ 4 0. Let R η ¼ R2 W 0 ðXÞ dX, we write the following decompositions: W ε ¼ ηG þ f ε ; U ε ¼ ηV þ K ε ;

such that, for all τ A ½τ0 ; τnε Þ, the following inequality holds:

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 ∂τ ðf ε  α1 e  τ Δf ε Þ þ εe  τ Δ f ε  Lðf ε Þ þ K ε  ∇ðf ε  α1 e  τ Δf ε Þ þ ηV  ∇ðf ε  α1 e  τ Δf ε Þ þ ηK ε  ∇ðG  α1 e  τ ΔGÞ þ α1 e  τ Δf ε X X 2 þ α1 e  τ  ∇Δf ε þ ηα1 e  τ ΔG þ ηα1 e  τ  ∇ΔG þ ηεe  τ Δ G 2 2  β e  2τ divðjAε j2 ∇f ε þ ηjAε j2 ∇GÞ  βe  2τ divð∇ðjAε j2 Þ 4 Aε Þ 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 on M)

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 θ Þ=2 4.1. Estimates in H_

We start by performing an estimate of the solution of (4.3) in the  ð1 þ θ Þ=2 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 H1-estimate that we will perform later (see Lemma 4.3) makes the term ‖u‖2L2 appear on the right hand side of our H1-energy inequality. In order to absorb this term, we look for an estimate in a Sobolev space of negative order. To this end, due to Lemma 2.1 and the fact R that R2 f ðXÞ dX ¼ 0, for 34 r s o1, 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

Please cite this article as: O. Coulaud, Asymptotic profiles for the third grade fluids' equations, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.03.013i

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O. Coulaud / International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

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 0o 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:       X 1 ð  ΔÞ  s  ∇f ; ð  ΔÞ  s f ¼  s þ ‖ð  ΔÞ  s f ‖2L2 ; 2 2 L2 ðð  ΔÞ  s ðLðf ÞÞ; ð  ΔÞ  s f ÞL2   1 ¼  ‖ð  ΔÞ1=2  s f ‖2L2  s  ‖ð  ΔÞ  s f ‖2L2 ; 2     X  ∇ Δf ; ð  ΔÞ  s f  ð  ΔÞ  s ¼ ðs þ 1Þ‖ð  ΔÞ1=2  s f ‖2L2 : 2 L2 ð4:7Þ

and

of f under the condition (4.4) is given in

Lemma 4.2. Let W A C 1 ððτ0 ; τε Þ; H 1 ð2ÞÞ \ C 0 ððτ0 ; τε Þ; H 3 ð2ÞÞ be the solution of (4.1) satisfying the inequality (4.4) for some γ 40. There exist γ 0 4 0 and T 0 Z1 such that if T Z T 0 and γ r γ 0 , then, for all τ A ½τ0 ; τnε Þ, E1 satisfies the inequality   1θ ∂τ E1 þ θ E1 þ 1 þ α1 e  τ :ð  ΔÞ  ð1 þ θÞ=4 f :2 r CM3 γ ð1  θÞ2 e  2τ 4 þ CM γ ð1  θÞ2 ð‖f ‖2L2 ð2Þ þ :∇f : þ α21 e  2τ ‖Δf ‖2L2 ð1Þ Þ; 2

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 θ Þ=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  α1 e  τ ð  ΔÞð1  θÞ=4 f   X  ∇Δf ¼ Hðτ; G; f ; WÞ; þ α1 e  τ ð  ΔÞ  ð3 þ θÞ=4 2

 ð  ΔÞ  ð3 þ θÞ=4 ðLðf ÞÞ

Hðτ; G; f ; WÞ ¼ ð  ΔÞ  ð3 þ θÞ=4 ð  K  ∇ðf  α1 e  τ Δf Þ  ηV  ∇ðf  α1 e  τ Δf Þ

X  ηK  ∇ðG  α1 e  τ ΔGÞ  ηα1 e  τ ΔG  ηα1 e  τ  ∇ΔG 2  ηεe

Δ G þ βe 2

 2τ

2

curl divðjAj AÞÞ:

Taking the L -scalar product of (4.9) with ð  ΔÞ into account the equalities 2

 ð3 þ θ Þ=4

ð  ð  ΔÞ  ð3 þ θÞ=4 ðLðf ÞÞ; ð  ΔÞ  ð3 þ θÞ=4 f ÞL2   1þθ 2 2 :ð  ΔÞ  ð3 þ θÞ=4 f : ; ¼ :ð  ΔÞ  ð1 þ θÞ=4 f : þ 4

 1 2 ∂τ ð:ð  ΔÞ  ð3 þ θÞ=4 f : þ α1 e  τ kð  ΔÞ  ð1 þ θÞ=4 f 2 Þ 2   1þθ 2 2 :ð  ΔÞ  ð3 þ θÞ=4 f : þ þ εe  τ :ð  ΔÞ  ð1  θÞ=4 f : 4     1þθ þ 1þ α1 e  τ :ð  ΔÞ  ð1 þ θÞ=4 f :2 4 ¼ ðHðτ; G; f Þ; ð  ΔÞ  ð3 þ θÞ=4 f ÞL2 :

ð4:10Þ

Now, it remains to estimate the right hand side of (4.10), that we write as ðHðτ; G; f Þ; ð  ΔÞ  ð3 þ θÞ=4 f ÞL2 ¼ I 1 þ I 2 þI 3 þ I 4 þ I 5 ;

I 3 ¼ ðð  ΔÞ  ð3 þ θÞ=4 ð  ηK  ∇ðG  α1 e  τ ΔGÞÞ; ð  ΔÞ  ð3 þ θÞ=4 f ÞL2 ;    X 2 I 4 ¼ ð  ΔÞ  ð3 þ θÞ=4  ηα1 e  τ ΔG  ηα1 e  τ  ∇ΔG  ηεe  τ Δ G ; 2  ð  ΔÞ  ð3 þ θÞ=4 f 2 ¼ ð  ΔÞ  ð3 þ θÞ=4 J; ð  ΔÞ  ð3 þ θÞ=4 f 2 ; L L  I 5 ¼ ð  ΔÞ  ð3 þ θÞ=4 ðβe  2τ curl divðjAj2 AÞÞ; 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 I 1 ¼ ðð  ΔÞ  ð3 þ θÞ=4 ð  divðKðf  α1 e  τ Δf ÞÞÞ; ð  ΔÞ  ð3 þ θÞ=4 f ÞL2 r :ð  ΔÞ  ð3 þ θÞ=4 ∇ðKðf  α1 e  τ Δf ÞÞ::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 :Kðf  α1 e  τ Δf Þ:L2 ð1Þ :ð  ΔÞ  ð3 þ θÞ=4 f : ð1  θÞ3=2 C r :K:L1 :f  α1 e  τ Δf :L2 ð1Þ :ð  ΔÞ  ð3 þ θÞ=4 f : ð1  θÞ3=2 C 2 :f : 2 :f : 1 ‖f  α1 e  τ Δf ‖2L2 ð1Þ r μ:ð  ΔÞ  ð3 þ θÞ=4 f : þ μð1  θÞ3 L ð2Þ H

I1 r

r μ:ð  ΔÞ  ð3 þ θÞ=4 f : þ 2

CM γ ð1  θÞ3

μ

ð‖f ‖2L2 ð1Þ þ α21 e  2τ ‖Δf ‖2L2 ð1Þ Þ; ð4:11Þ

ð4:9Þ

where

τ

given by Lemma 4.1, we obtain

I 2 ¼ ðð  ΔÞ  ð3 þ θÞ=4 ð  ηV  ∇ðf  α1 e  τ Δf ÞÞ; ð  ΔÞ  ð3 þ θÞ=4 f ÞL2 ;

 ð1 þ θÞ=2 In order to obtain a priori estimates of f in H_ ðR2 Þ, we define the functional  2 E1 ðτÞ ¼ 12 ð:ð  ΔÞ  ð3 þ θÞ=4 f : þ α1 e  τ kð  ΔÞ  ð1 þ θÞ=4 f 2 Þ:  ð1 þ θÞ=2

   X α1 e  τ ð  ΔÞ  ð3 þ θÞ=4  ∇Δf ; ð  ΔÞ  ð3 þ θÞ=4 f 2 L2   7þθ 2  ð1 þ θÞ=4 τ ¼ α1 e :ð  ΔÞ f: ; 4

I 1 ¼ ðð  ΔÞ  ð3 þ θÞ=4 ð  K  ∇ðf  α1 e  τ Δf ÞÞ; ð  ΔÞ  ð3 þ θÞ=4 f ÞL2 ;

X d ξjξj2 b  ∇Δf ¼ 2jξj2 b ∇f : fþ 2 2

The proof of this lemma is then obtained through the Plancherel formula and direct computations. □

The estimate in H_ the next lemma.

and 

where

Proof. Using Fourier variables, it is easy to see that Xd ξ  ∇f ¼  b f   ∇b f 2 2

9

and taking

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 I 2 r μ:ð  ΔÞ  ð3 þ θÞ=4 f : þ 2

CM γ ð1  θÞ3

μ

ð‖f ‖2L2 ð1Þ þ α21 e  2τ ‖Δf ‖2L2 ð1Þ Þ: ð4:12Þ

Likewise, we estimate the term I3. Indeed, the same computations and the smoothness of G yield I 3 r μ:ð  ΔÞ  ð3 þ θÞ=4 f : þ 2

C η2 :f : 2 :f : 1 ‖G  α1 e  τ ΔG‖2L2 ð1Þ μð1  θÞ3 L ð2Þ H

r μ:ð  ΔÞ  ð3 þ θÞ=4 f : þ 2

CM γ ð1  θÞ3

μ

ð‖f ‖2L2 ð2Þ þ ‖f ‖2H1 Þð1 þ α1 e  τ Þ:

Please cite this article as: O. Coulaud, Asymptotic profiles for the third grade fluids' equations, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.03.013i

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O. Coulaud / International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

10

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Taking T0 sufficiently large so that α1 e  τ r 1, we get I 3 r μ:ð  ΔÞ  ð3 þ θÞ=4 f : þ 2

CM γ ð1  θÞ

3

2

ð‖f ‖2L2 ð2Þ þ:∇f : Þ:

μ

ð4:13Þ

The H1-estimate of f is given by the following lemma.

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 :ð  ΔÞ  ð3 þ θÞ=4 J: r

Cjηje  τ :G:H4 ð3Þ ð1  θÞ3=2

:

Using the above inequality and the smoothness of G, we can write I4 r

ð1  θÞ3=2

:ð  ΔÞ

 ð3 þ θ Þ=4

2

ð1  θÞ3=2

r μ:ð  ΔÞ

‖∇ðjAj

CM γ ð1  θÞ e

μ

:

ð4:14Þ

:∇ðjAj2 AÞ:L2 ð1Þ :ð  ΔÞ  ð3 þ θÞ=4 f :

 ð3 þ θÞ=4

2

C β e  4τ

f: þ

μð1  θÞ

3

2

‖∇ðjAj

AÞ‖2L2 ð1Þ :

‖∇ðjAj2 AÞ‖2L2 ð1Þ r C‖∇U‖4L1 ‖∇2 U‖2L2 ð1Þ

μ

α1

18  3τ

CM γ ð1  θÞ e 3 3

μ

:

ð4:15Þ

CM γ ð1  θÞ3

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

2

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

ð4:16Þ

Setting μ ¼ ð1  θÞ=20, we finally get   1θ ∂τ E1 þ θE1 þ 1 þ α1 e  τ :ð  ΔÞ  ð1 þ θÞ=4 f :2 r CM3 γ ð1  θÞ2 e  2τ 4 þ CM γ ð1  θÞ ð‖f ‖2L2 ð2Þ þ :∇f : þ α21 e  2τ ‖Δf ‖2L2 ð1Þ Þ:

I 5 ¼ ðβe  2τ divð∇ðjAj2 Þ 4 AÞ; f ÞL2 :

r μ:∇f : þ

μ

2

where

I 1 ¼  α1 e  τ ðK Δf ; ∇f ÞL2      r C α1 e  τ K L1 Δf k∇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

2

ð4:19Þ

þ ηβe  2τ ðdivðjAj2 ∇GÞ; f ÞL2 ;

1 2 2 ∂τ ð:ð  ΔÞ  ð3 þ θÞ=4 f : þ α1 e  τ :ð  ΔÞ  ð1 þ θÞ=4 f : Þ 2   1  20μ þ θ 2 :ð  ΔÞ  ð3 þ θÞ=4 f : þ 4     1þθ þ 1þ α1 e  τ :ð  ΔÞ  ð1 þ θÞ=4 f :2 4

μ

we obtain the equality       ∂τ E2 þ E2 þ εe  τ Δf 2 þ ð1  α1 e  τ Þ∇f 2 þ β e  2τ jAj∇f 2  2 ¼ f  þ I 1 þ I 2 þI 3 þ I 4 þ I 5 ;

I 3 ¼  ηðV  ∇ðf  α1 e  τ Δf Þ; f ÞL2 ;   ε 2 X I 4 ¼  ηα1 e  τ Δ G þ ΔG þ  ∇ΔG; f 2 α1 L2

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

þ

    X  ∇ Δf ; f ¼ α1 e  τ ∇f 2 ; 2 2 L

I 2 ¼  ηðK  ∇ðG  α1 e  τ ΔGÞ; f ÞL2 ;

þ ‖∇W‖2L2 ð1Þ Þ:

Finally, taking into account the inequality (4.4), we get   2 CM 3 β γ 3 ð1  θÞ18 e  4τ eτ 2 I 5 r μ:ð  ΔÞ  ð3 þ θÞ=4 f : þ 1þ

CM 3 γ ð1  θÞ3 e  2τ

Proof. Taking the L2-inner product of (4.3) with f, performing several integrations by parts and taking into account the equalities     ð  Lðf Þ; f ÞL2 ¼ ∇f 2  12 f 2 ;

I 1 ¼ ðK  ∇ðf  α1 e  τ Δf Þ; f ÞL2 ;

r‖W‖2L2 ð2Þ ‖W‖2H1 ð‖W‖2L2

2

2

where θ, 0 o θ o 1, is the fixed constant introduced at the beginning of Section 4.

α1 e  τ

AÞ‖2L2 ð1Þ r C‖j∇Uj2 ∇2 U‖2L2 ð1Þ :

r μ:ð  ΔÞ  ð3 þ θÞ=4 f : þ

2

and

2

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

r

r :f : þ CM γ ð1  θÞ6 ð:f : þ :jXj2 f : Þ þ CM 2 γ ð1  θÞ6 e  τ ; ð4:18Þ

3  2τ

A short computation leads to 2

1 β 2 2 ∂τ E2 þ E2 þ :∇f : þ e  2τ :jAj∇f : 2 2

f:

It remains to estimate the term I5. The inequality (2.1) of Lemma 2.1 implies C β e  2τ

Lemma 4.3. Let W A C 1 ððτ0 ; τε Þ; H 1 ð2ÞÞ \ C 0 ððτ0 ; τε Þ; H 3 ð2ÞÞ be the solution of (4.1) satisfying the inequality (4.4) for some γ 4 0. There exist γ 0 4 0 and T 0 Z 1 such that if T ZT 0 and γ r γ 0 , then, for all τ A ½τ0 ; τnε Þ, E2 satisfies the inequality

2

C ηe  τ

r μ:ð  ΔÞ  ð3 þ θÞ=4 f : þ

I5 r

appear in the computations made below. One defines the functional     E2 ðτÞ ¼ 12 ðf 2 þ α1 e  τ ∇f 2 Þ:

CM 2 γ 2 ð1  θÞ12

μ

eτ;

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 ηðKðG  α1 e  τ ΔGÞ; ∇f ÞL2

r jηj:K:L4 :G  α1 e  τ ΔG:L4 :∇f :

r Cð1 þ α1 e  τ Þjηj:f :L2 ð2Þ :∇f :

□ ð4:17Þ

r μ:∇f : þ 2

CM γ ð1  θÞ6

μ

 2 ð:f : þ :jXj2 f 2 Þ:

4.2. Estimates in H 1 ðR2 Þ

The same method gives

We next establish the H1-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

I 3 ¼  α1 e  τ ηðV Δf ; ∇f ÞL2    r α1 e  τ jηjV L1 Δf kk∇f k   pffiffiffiffiffiffi r C α1 M γ ð1  θÞ6 e  τ=2 ∇f 

ð4:21Þ

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O. Coulaud / International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

r μ:∇f : þ 2

CM 2 γ 2 ð1  θÞ12

μ

11

r5:ð  ΔÞ  ð1 þ θÞ=4 f : þ CM γ ð1  θÞ6 2

e τ:

ð4:22Þ

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

L

2

2

2

:

ð4:28Þ

We define E3 ¼ 6E1 þ E2 . The inequalities (4.8) and (4.28) give ∂τ E3 þ θE3 þ ð1 þ 32 ð1  θÞα1 e  τ Þ:ð  ΔÞ  ð1 þ θÞ=4 f :

2

I 4 r Cjηjðε þ α1 Þe  τ kf k  ηβe  2τ ðjAj2 ∇G; ∇f ÞL2   r Cjηjðε þ α1 Þe  τ f k þ Cjηjβ e  2τ k∇G: 1 ‖∇U‖23 k∇f  L

6 τ

ð:f : þ :jXj f : Þ þ CM γ ð1  θÞ e 2

þ 14 :∇f : rCM 3 γ ð1  θÞ2 e  τ þ CMγ ð1  θÞ2 ð:f : þ :∇f : 2

L3

2

2

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

 ð1 þ θ Þ=4

þα

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 Cjηjðε þ α1 Þe  τ kf k þ Cjηjβe  2τ ‖W‖2L3 k∇f L3   r Cðε þ α1 ÞM γ ð1  θÞ6 e  τ þCjηjβ e  2τ ‖W‖2H1 ð:∇f  þ kΔf kÞ

2  2τ : 1e

Δf : þ :jXj f : þ α 2

2

2

2  2τ :jXj2 1e

2

Δf : Þ: 2

2

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 I 5 ¼  βe

 2τ

2

ð∇ðjAj Þ 4 A; ∇f ÞL2      r C β e  2τ ∇U L4 ∇2 U L4 jAj∇f k      r C β e  2τ W  1 W  2 jAj∇f k H

H

6  2τ

r C β M γ ð1  θÞ e

β

 eτ=2  pffiffiffiffiffiffijAj∇f 

α1

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

ð4:24Þ

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

β

∂τ E2 þ E2 þ ð1  3μ  α1 e  τ Þ:∇f : þ e  2τ :jAj∇f : 2 CM γ ð1  θÞ6 2 2 2 r:f : þ ð:f : þ:jXj2 f : Þ þ CM 2 γ ð1  θÞ6 e  τ : 2

2

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

2

þ CM γ ð1  θÞ2 ðα21 e  2τ :Δf : þ :jXj2 f : þ α21 e  2τ :jXj2 Δf : Þ: 2

2

□ ð4:26Þ

To achieve the H1-estimate of f, we have to combine the 2 inequalities (4.8) and (4.18). We can interpolate :f : between 2 2  ð1 þ θ Þ=4 :ð  ΔÞ f : and :∇f : . Indeed, via Hölder and Young inequalities, we get Z 1 2 :f : ¼ ð2π Þ2 jξj2ð1 þ θÞ=ð3 þ θÞ jb f j2ð1 þ θÞ=ð3 þ θÞ jb f j4=ð3 þ θÞ dξ 2ð1 þ θ Þ=ð3 þ θÞ R2 jξj !2=ð3 þ θÞ Z ð1 þ θÞ=ð3 þ θÞ Z 1 b2 2 rð2π Þ jf j dξ jξj2 jb f j2 dξ 1 þ θ R2 jξj R2 ð2 þ 2θÞ=ð3 þ θÞ

r:ð  ΔÞ  ð1 þ θÞ=4 f : :∇f :     ð1 þ θÞ=2 1þθ 3 2 8 2 2 :∇f : þ r :ð  ΔÞ  ð1 þ θÞ=4 f : : 3 3þθ 8 3þθ Since, 0 o θ o 1, we obtain       2 1  f r 4 ∇f 2 þ 5ð  ΔÞ  ð1 þ θÞ=4 f 2 : Thus, we have 1 β 2 2 ∂τ E2 þ E2 þ :∇f : þ e  2τ :jAj∇f : 4 2

2

ð4:30Þ 4.3. Estimates in H 2 ðR2 Þ We now perform the H2-estimate of f. This is done with the same method as for the H1-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 H2-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: E4 ðτÞ ¼ 12 ð:∇f : þ α1 e  τ :Δf : Þ: 2

2

The H2-estimate of f is given by the lemma below. Lemma 4.4. Let W A C 1 ððτ0 ; τε Þ; H 1 ð2ÞÞ \ C 0 ððτ0 ; τε Þ; H 3 ð2ÞÞ be the solution of (4.1) satisfying the inequality (4.4) for some γ 4 0. There exist γ 0 4 0 and T 0 Z 1 such that if T Z T 0 and γ r γ 0 , then for all τ A ½τ0 ; τnε Þ, E4 satisfies the inequality 1 β 3 2 2 2 ∂τ E4 þ E4 þ :Δf : þ e  2τ :jAjΔf : r :∇f : þ CM 2 γ ð1  θÞ6 e  2τ 2 2 2 2 2 2 þ CM 2 γ ð1  θÞ6 ð:f : þ :∇f : þ :jXj2 f : Þ; ð4:31Þ where θ, 0 o θ o 1 is the fixed constant introduced at the beginning of Section 4.

1 β 2 2 ∂τ E2 þ E2 þ :∇f : þ e  2τ :jAj∇f : 2 2 2 2 2 r:f : þ CM γ ð1  θÞ6 ð:f : þ:jXj2 f : Þ þ CM 2 γ ð1  θÞ6 e  τ :

4=ð3 þ θ Þ

2

f : and

∂τ E3 þ θE3 þ 12 :ð  ΔÞ  ð1 þ θÞ=4 f : þ 18 :∇f : r CM 3 γ ð1  θÞ2 e  τ

r Cðε þ α1 ÞM γ ð1  θÞ6 e  τ þCM 3=2 γ 3=2 ð1  θÞ9 e  3τ=2 r CM 3=2 γ ð1  θÞ6 e  τ :

ð4:29Þ

Proof. We take the L2-product of (4.3) with  Δf . Doing several integrations by parts, it is easy to see that ð  Lðf Þ;  Δf ÞL2 ¼ :Δf :  :∇f : ; 2

and   X 1 2 ¼ α1 e  τ :Δf : :  α 1 e  τ  ∇ Δf ; Δf 2 2 L2 Furthermore, one also has

βe  2τ ðdivðjAj2 ∇f Þ; Δf Þ 2

¼ β e  2τ :jAjΔf : þ β e  2τ ∑ 2

Z R2

j¼1

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 Z 2 β e  2τ ∑ A : ∂j A∂j f Δf dX rC βe  2τ :jAjΔf kk∇A∇f k j¼1

ð4:27Þ

2

R2

     r C βe  2τ jAjΔf ∇2 U L4 ∇f L4       r C βe  2τ jAjΔf ∇W H1 ∇f H1   r μ βe  2τ jAjΔf 2 1

þ

Cβ

μ1

    e  2τ ‖W‖2H2 ð∇f 2 þ Δf 2 Þ

Please cite this article as: O. Coulaud, Asymptotic profiles for the third grade fluids' equations, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.03.013i

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O. Coulaud / International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

12

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

  r μ1 βe  2τ jAjΔf 2    CM γ ð1  θÞ  τ  e ð∇f 2 þ Δf 2 Þ; þ 6

μ1

where μ1 4 0 will be chosen later. Consequently, we get  α1 2 2 2 ∂τ E4 þ εe  τ :∇Δf : þ 1  e  τ :Δf : þ β ð1  μ1 Þe  2τ :jAjΔf : 2  CM γ ð1  θÞ6  τ 2 2 r :∇f : þ e ð:∇f : þ :Δf 2 Þ þ I 1 þ I 2 þ I 3 þ I 4 þ I 5 ;

μ1

Lemma 2.3, the inequality (4.4), the Hölder inequalities and the inequality (4.4). We get Z   Cjηjβe  2τ j∇AjjAjj∇GjjΔf j dX r Cjηjβe  2τ k∇2 Ukk∇GL1 jAjΔf k R2

r μ1 β e  2τ :jAjΔf : þ 2

r μ1 β e  2τ :jAjΔf : þ 2

Z

2

I 1 ¼ ðU  ∇ðf  α1 e  t Δf Þ; Δf ÞL2 ;

and thus we have shown 2

2

Integrating by parts and using the divergence free property of K, one can show that Z 2 I1 ¼  ∑ ∂k U j ∂j f ∂k f dX: Due to the Gagliardo–Nirenberg 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 kk∇f kkΔf k 2

CM γ ð1  θÞ6

μ2

2

:∇f : ;

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 jηjkK L1 G  α1 e  τ ΔGkkΔf k r Cjηjð1 þ α1 e  τ Þ‖f ‖

μ1

r μ2 :Δf : þ 2

CM γ ð1  θÞ

2

2

2

ð:f : þ :∇f : þ :jXj2 f : Þ:

þ C β e  2τ

R2

j∇AjjAjj∇2 KjjΔf j dX þ C β e  2τ

ε 2 X Δ G þ ΔG þ  ∇ΔG; Δf 2 α1 I 23 ¼ ηβe  2τ ðdivðjAj2 ∇GÞ; Δf ÞL2 :

 L

2

R2

þ

CM γ ð1  θÞ6

μ2

þ

C β e  2τ

‖∇2 U‖2 :∇2 K:

2

4

C βe  2τ

Z

jAj2 j∇3 KjjΔf j dX r μ1 βe  2τ kjAjΔf : þ

R

Each term of the right hand side of (4.35) can be estimated in a convenient way. We use again the inequality (2.13) of the

C β e  2τ

μ1

 ‖∇U‖2L1 :∇3 K 2

  r μ1 βe  2τ jAjΔf 2 þ

C β e  2τ

þ

CM γ ð1  θÞ6 e  τ  

:∇W:H1 :∇W:L2 ð2Þ :Δf :    2τ  r μ1 β e jAjΔf 2

is slightly more complicated. Actually, we can The estimate of bound I 23 by two kinds of terms that we estimate separately. In fact, it is easy to see that Z Z I 23 r Cjηjβe  2τ j∇AjjAjj∇GjjΔf j dX þ Cjηjβe  2τ jAj2 j∇2 GjjΔf j dX: ð4:35Þ

μ1

2

2

e  2τ :

R2

   CM γ ð1  θÞ6 e  3τ=2  ð∇f 2 þ Δf 2 Þ:

By the same method, we obtain

I 23

R2

ð4:37Þ

R2

3

2

jAj2 j∇3 KjjΔf j dX:

L L4 μ1  2  2τ   r μ1 βe jAjΔf  C β e  2τ  þ ‖∇W‖2L4 :∇f 2L4 μ1   r μ1 βe  2τ jAjΔf 2 C β e  2τ þ :∇W::ΔW::∇f ::Δf :: μ1

;

Using the good regularity of G and the inequality (4.5), one can show that   I 1 r CM 1=2 γ 1=2 ð1  θÞ3 e  τ Δf  r μ2 :Δf : þ

R2

Due to the inequality (4.4), we get Z  j∇AjjAjj∇2 KjjΔf j dX r μ1 βe  2τ kjAjΔf 2 C β e  2τ

where 

Z

We have to estimate each term of 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– Nirenberg inequality yield Z  C β e  2τ j∇AjjAjj∇2 KjjΔf j dX r μ1 βe  2τ :jAjΔf 2

ð4:34Þ

I 3 ¼ I 13 þI 23 ;

ð4:36Þ

R2

We rewrite

I 13 ¼ ηα1 e  τ

e  2τ :

R2

Z

Δf :

μ2

e  2τ ;

It remains to estimate I4. Recalling that U ¼ ηV þ K, one has Z Z I 4 r Cjηjβe  2τ j∇AjjAjj∇2 V jjΔf j dX þ Cjηjβe  2τ jAj2 j∇3 VjjΔf j dX

1=2 1=2 ‖f ‖ 2 : H1 L ð2Þ 6

μ1

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

R2

r μ2 :Δf : þ

CM 2 γ 2 ð1  θÞ12

CM 2 γ 2 ð1  θÞ12

Finally, assuming γ r 1, one has

ðdivð∇ðjAj Þ 4 AÞ; Δf ÞL2 :

j;k ¼ 1

R2

I 23 r 2μ1 β e  2τ :jAjΔf : þ

þ ηβe  2t ðdivðjAj2 ∇GÞ; Δf ÞL2 ; I4 ¼ βe

 e  2τ :∇W 2

e  2τ :

jAj2 j∇2 GjjΔf j dx r μ1 β e  2τ kjAjΔf : þ

where

Cjηjβ e  2τ

 2t

μ1

μ1

By the same way, we have

ð4:32Þ

I 2 ¼ ηðK  ∇ðG  α1 e  t ΔGÞ; Δf ÞL2 ;   ε 2 X Δ G þ ΔG þ  ∇ΔG; Δf I 3 ¼ ηα1 e  t 2 α1 L2

CM 2 γ 2 ð1  θ Þ12

Cjηj2 β

μ1

μ1

2



Δf  2 :

Finally, we have shown that I 4 r 4μ1 β e  2τ :jAjΔf : þ 2

CM 2 γ ð1  θÞ6 e  τ

μ1

   ðk∇f 2 þ Δf 2 Þ:

ð4:38Þ

Please cite this article as: O. Coulaud, Asymptotic profiles for the third grade fluids' equations, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.03.013i

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O. Coulaud / International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

Going back to (4.32) and taking into account the inequalities (4.33), (4.34), (4.36) and (4.38), we get       α1 ∂τ E4 þ 1  3μ2  e  τ kΔf 2 þ ð1  7μ1 Þβ e  2τ jAjΔf 2 r ∇f 2 2

   ð8 þ 16α1 e  τ ÞjXjf 2

2

CM 2 γ ð1  θÞ6  2τ e : minðμ1 ; μ2 Þ

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

0

1 β 3 2 2 2 ∂τ E4 þ E4 þ :Δf : þ e  2τ :jAjΔf : r :∇f : þCM 2 γ ð1  θÞ6 e  2τ 2 2 2       þ CM 2 γ ð1  θÞ6 ðf 2 þ ∇f 2 þ jXj2 f 2 Þ: □ ð4:39Þ

In order to finish the H2-estimate of f we define a new functional E5 as a linear combination of E3 and E4 given by

From the inequalities (4.30) and (4.31), it is clear that one has

5:

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 ð  f ; jXj4 ðf  α1 e  τ Δf ÞÞL2 2

¼  :jXj2 f :  α1 e  τ :jXj2 ∇f :  4α1 e  τ ∑ 2

2

2 2 τ

þ 12 :∇f : þ 12 :Δf : r CM γ ð1  θÞ e 2

3

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

2

2

þ :jXj2 f : þ α21 e  2τ :jXj2 Δf : Þ: 2

2

ð4:40Þ

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

γ small

∂τ E5 þ θE5 þ 7:ð  ΔÞ  ð1 þ θÞ=4 f : þ 14 :∇f : 2

2

þ 14 :Δf : rCM 3 γ ð1  θÞ2 e  τ     þCM 2 γ ð1  θÞ2 ðjXj2 f 2 þ α21 e  2τ jXj2 Δf 2 Þ: 2

ð4:41Þ

4.4. Estimates in H 2 ð2Þ In order to achieve the estimate of f in H 2 ð2Þ, it remains to perform estimates in weighted spaces. Combined with the inequality (4.41), it will give us an estimate in H 2 ð2Þ. To do this, we make the L2-scalar product of (4.3) with jXj4 ðf  α1 e  τ Δf Þ. We define the functional   E6 ðτÞ ¼ 12 jXj2 ðf  α1 e  τ Δf Þ2 : Before stating the lemma which contains the estimate of E6, we state a technical lemma, which gives the terms provided by the L2product of the linear terms of (4.3) with jXj4 ðf  α1 e  τ Δf Þ. Lemma 4.5. Let f A C 1 ððτ0 ; τε Þ; H 1 ð2ÞÞ \ C 0 ððτ0 ; τε Þ; H 3 ð2ÞÞ and HðX; τ; f Þ ¼ jXj4 ðf  α1 e  τ Δf Þ. For all τ A ðτ0 ; τε Þ, the next equalities hold:   1: ð  f ; HðX; τ; f ÞÞL2 ¼  jXj2 f 2     þ 8α1 e  τ jXjf 2  α1 e  τ jXj2 ∇f 2 :   2: ð  Δf ; HðX; τ; f ÞÞL2 ¼ α1 e  τ jXj2 Δf 2  2  2 2  8jXjf  þ jXj ∇f  :      X 3 3:   ∇f ; HðX; τ; f Þ ¼ jXj2 f 2  24α1 e  τ jXjf 2 2 2 L2   þ 3α1 e  τ jXj2 ∇f 2

α1

e  τ ðX  ∇Δf ; jXj4 f ÞL2 : 2    1 ð  Lðf Þ; HðX; τ; f ÞÞL2 ¼ jXj2 f 2 þ ð1 þ 2α1 e  τ ÞjXj2 ∇f 2 2   þ α1 e  τ jXj2 Δf 2 

2

¼  :jXj2 f :  α1 e  τ :jXj2 ∇f :  2α1 e  τ ∑ 2

2

j¼1

Z R2

X j jXj2 f ∂j f dX

Z

R2

2

X j jXj2 ∂j ðf Þ dX

  2 2  ¼  :jXj2 f :  α1 e  τ :jXj2 ∇f : þ 8α1 e  τ jXjf 2 :

∂τ E5 þ θE5 þ 8:ð  ΔÞ  ð1 þ θÞ=4 f :

4:

4

j¼1

E5 ¼ 16E3 þ E4 :

2

τ

e ðX  ∇Δf ; jXj f ÞL2 : 2  X α α1 e  τ  ∇Δf ; HðX; τ; f Þ ¼ 1 e  τ ðX  ∇Δf ; jXj4 f ÞL2 2 2 L2   3α2 þ 1 e  2τ jXj2 Δf 2 : 2     εe  τ ðΔ2 f ; HðX; τ; f ÞÞL2 ¼ εα1 e  2τ ðjXj2 ∇Δf 2  8jXjΔf 2 Þ     þ εe  τ ðjXj2 Δf 2  8jXj∇f 2  2  2 þ 32f   16X  ∇f  Þ: 

6:

γ small enough and T ¼ eτ

α1



     CM 2 γ ð1  θÞ6  ðf 2 þ ∇f 2 þ Δf 2 þ minðμ1 ; μ2 Þ þ :jXj2 f : Þ þ

13

The equality (2) is obtained through the same computations. We show now the third equality of this lemma. Integrating by parts, we obtain   Z 2 X j jXj4 X  :∇f ; jXj4 ðf  α1 e  τ Δf Þ ∂j ðjf j2 Þ dX ¼ ∑ 2 2 2 4 j¼1 R L Z 2 X j jXj4 ∂j f Δf dX þ α1 e  τ ∑ 2 j ¼ 1 R2 Z 2 X j jXj4 3 2 ∂j f Δf dX: ¼ :jXj2 f : þ α1 e  τ ∑ 2 2 2 j¼1 R Besides, integrating several times by parts, we get ! Z Z 2 2 X j jXj4 X j jXj4 ∂j f Δf dX ¼  α1 e  τ ∑ α1 e  τ ∑ f ∂j Δf dX 2 2 j ¼ 1 R2 j ¼ 1 R2 Z ¼  3α1 e  τ jXj4 f Δf dX 

α1



α1

R2

τ

e ðX  ∇Δf ; jXj4 f ÞL2 2     ¼  24α1 e  τ jXjf 2 þ 3α1 e  τ jXj2 ∇f 2 2

e  τ ðX  ∇Δf ; jXj4 f ÞL2 ;

and consequently      X 3 ¼ jXj2 f 2  24α1 e  τ jXjf 2   ∇f ; jXj4 ðf  α1 e  τ Δf Þ 2 2 L2   α1   þ 3α1 e  τ jXj2 ∇f 2  e  τ ðX  ∇Δf ; jXj4 f ÞL2 : 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 ððτ0 ; τε Þ; H 1 ð2ÞÞ \ C 0 ððτ0 ; τε Þ; H 3 ð2ÞÞ be the solution of (4.1) satisfying the inequality (4.4) for some γ 4 0. There exist γ 0 4 0 and T 0 Z 1 such that if T Z T 0 and γ r γ 0 , then for all τ A ½τ0 ; τnε Þ, E6 satisfies the inequality 1θ 1 α1 2 2 2 :jXj2 f : þ :jXj2 ∇f : þ e  τ :jXj2 Δf : 4 8 4 1024 2 r CM 2 γ ð1  θÞ6 e  τ þ :f : þ CM 2 γ 1=2 ð1  θÞ3 1θ

∂τ E6 þ θE6 þ

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O. Coulaud / International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

      ðf 2 þ ∇f 2 þ Δf 2 Þ;

ð4:42Þ

where θ, 0 o θ o1, 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 jXj4 ðf  α1 e  τ Δf Þ. Applying Lemma 4.5, we obtain    1 ∂τ E6 þ jXj2 f 2 þ ð1 þ α1 e  τ ÞjXj2 ∇f 2 2   α2 2 þ α1 e  τ þ 1 e  2τ :jXj2 Δf : þJ 2      ¼ C εe  τ :jXj∇f 2 þ C εα1 e  2τ jXjΔf 2 þ ð8 þ 8α1 e  τ ÞjXjf 2 þI 1 þ I 2 þ I 3 þ I 4 þ I 5 ;

ð4:43Þ

where J ¼  βe  2τ ðdivðjAj2 ∇f Þ; jXj4 ðf  α1 e  τ Δf ÞÞL2 ;

I 1 ¼ ðK  ∇ðf  α1 e  τ Δf Þ; jXj4 ðf  α1 e  τ Δf ÞÞL2 ;

 :jXj jAjΔf ::jXj ∇f :L4 k∇ U L4 I rC βα1 e       rC βα1 e  3τ jXj2 jAjΔf jXj2 ∇f L4 ∇2 U H1 :  3τ

2

2

2

Using the inequality (2.6) of Lemma 2.2 and the inequality (2.17) of Lemma 2.4, we get    I rC βα1 e  3τ jXj2 jAjΔf kjXj2 ∇f 1=2  1=2  1=2  2 1=2   ðf  þ jXj∇f  þ jXj Δf  ÞW H2 : Due to the Young inequality and the condition (4.4), we obtain   β I r α1 e  3τ jXj2 jAjΔf 2 2         þC βα1 e  3τ ‖W‖2H2 ðf 2 þ ∇f 2 þ jXj2 ∇f 2 þ jXj2 Δf 2 Þ   β r α1 e  3τ jXj2 jAjΔf 2 2         þCM γ ð1  θÞ6 e  2τ ðf 2 þ ∇f 2 þ jXj2 ∇f 2 þ jXj2 Δf 2 Þ: Thus, we can conclude that

I 2 ¼ ηðK  ∇ðG  α1 e  τ ΔGÞ; jXj4 ðf  α1 e  τ Δf ÞÞL2 ;

  β J 2 Z α1 e  3τ jXj2 jAjΔf 2 2          CM γ ð1  θÞ6 e  2τ ðf 2 þ ∇f 2 þ jXj2 ∇f 2 þ jXj2 Δf 2 Þ:

I 3 ¼ ηðV  ∇ðf  α1 e  τ Δf Þ; jXj4 ðf  α1 e  τ Δf ÞÞL2 ;

I 4 ¼  ηεe  τ ðΔ G; jXj2 ðf  α1 e  τ Δf ÞÞL2   X þ ηα1 e  τ ΔG þ  ∇ΔG; jXj4 ðf  α1 e  τ Δf Þ 2 L2 2

ð4:46Þ Combining the inequalities (4.45) and (4.46) and going back to (4.44), we have shown that

 ηβe  2τ ðdivðjAj2 ∇GÞ; jXj4 ðf  α1 e  τ Δf ÞÞL2 ;

I 5 ¼  βe  2τ ðdivð∇ðjAj2 Þ 4 AÞ; jXj4 ðf  α1 e  τ Δf ÞÞL2 : We now estimate J. One has J ¼ J1 þ J2 ;

H 1 ðR2 Þ into L4 ðR2 Þ, we obtain

ð4:44Þ

where

  β 2 J Z e  2τ ð:jXj2 jAj∇f 2 þ α1 e  τ jXj2 jAjΔf : Þ  CM 2 γ 2 ð1  θÞ12 e  τ 2         CM γ ð1  θÞ6 e  2τ ðf 2 þ ∇f 2 þ :jXj2 ∇f 2 þ jXj2 Δf 2 Þ: ð4:47Þ

J 1 ¼  β e  2τ ðdivðjAj2 ∇f Þ; jXj4 f ÞL2 ;

Taking into account the inequality (4.47), the equality (4.43) becomes

We estimate J1 and J2 separately. Integrating by parts, we obtain Z 2 X j jXj2 jAj2 ∂j ff dX: J 1 ¼ βe  2τ :jXj2 jAj∇f : þ 4β e  2τ ∑2j ¼ 1

   1 ∂τ E6 þ jXj2 f 2 þ ð1 þ α1 e  τ ÞjXj2 ∇f 2 2   α2 2 þ α1 e  τ þ 1 e  2τ :jXj2 Δf : 2

J 2 ¼ βα1 e  3τ ðdivðjAj2 ∇f Þ; jXj4 Δf ÞÞL2 :

R2

2

and we conclude that 2

ð4:45Þ

By the same way, we estimate J2. A short computation shows that Z 2 2 jXj4 ∂j A : A∂j f Δf dX: J 2 ¼ βα1 e  3τ :jXj2 jAjΔf : þ 2βα1 e  3τ ∑ j¼1

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 Z 2 X j jXj2 K j jf  α1 e  τ Δf j2 dX I 1 ¼ 2 ∑ j¼1

R2

     r C K L1 jXj2 ðf  α1 e  τ Δf ÞkjXjðf  α1 e  τ Δf Þ:

r e  2τ :jXj2 jAj∇f : þ CM 2 γ 2 ð1  θÞ12 e  τ ; 2

β

2

þ CM 2 γ 2 ð1  θÞ12 e  τ þ I 1 þ I 2 þI 3 þ I 4 þ I 5 :

þC βe  2τ :jXj2 f ::f ::∇W:H1 :∇W:L2 ð2Þ

J 1 Z e  2τ :jXj2 jAj∇f :  CM 2 γ 2 ð1  θÞ12 e  τ : 2

2

 2 2 r C εe  τ :jXj∇f : þ C εα1 e  2τ :jXjΔf : þ ð8 þ 8α1 e  τ Þ:jXjf 2         þCM γ ð1  θÞ6 e  2τ ðf 2 þ ∇f 2 þ jXj2 ∇f 2 þ jXj2 Δf 2 Þ

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   Z   β 2 2   X j jXj2 jAj2 ∂j ff dX  r e  2τ :jXj2 jAj∇f : 4βe  2τ ∑ 2   2 j¼1 R

β

β

þ e  2τ ð:jXj2 jAj∇f : þ α1 e  τ :jXj2 jAjΔf : Þ 2

Using Hölder and Young inequalities, we obtain   Z   β 2     2τ 2 2 ∑ X j jXj jAj ∂j ff dX  r e  2τ :jXj2 jAj∇f 2 4βe 2   2 j¼1 R     þ C β e  2τ jXjf 2 ∇U 2L1 :

R2

We define   Z   2    3τ 4 ∑ jXj ∂j A : A∂j f Δf dX  I ¼ 2βα1 e 2   j¼1 R Applying Hölder inequalities and the continuous injection of

The inequalities (2.15) of Lemma 2.3 and (2.4) of Lemma 2.2, the 4 Young inequality ab r 34 a4=3 þ 14 b and the inequality (4.5) yield    1=2 1=2  I 1 r C‖f ‖H1 ‖f ‖L2 ð2Þ jXj2 ðf  α1 e  τ Δf Þ3=2 f  α1 e  τ Δf 1=2     r CM 1=2 γ 1=2 ð1  θÞ3 ðjXj2 ðf  α1 e  τ Δf Þ2 þ f  α1 e  τ Δf 2 Þ     r CM 1=2 γ 1=2 ð1  θÞ3 ðjXj2 f 2 þ α21 e  2τ jXj2 Δf 2     ð4:49Þ þ f 2 þ α2 e  2τ Δf 2 Þ: 1

Using the inequality (2.14) of Lemma 2.4, one can bound I2 in a convenient way. Indeed, one has     I 2 r CjηjkK L4 jXj2 ∇ðG  α1 e  τ ΔGÞL4 jXj2 ðf  α1 e  t Δf Þk r Cjηj:f :L2 ð2Þ ð:jXj2 f : þ α1 e  τ :jXj2 Δf :Þ

r CM 1=2 γ 1=2 ð1  θÞ3 ð:jXj2 f : þ α21 e  2τ :jXj2 Δf : þ :f : Þ: 2

2

2

ð4:50Þ

Please cite this article as: O. Coulaud, Asymptotic profiles for the third grade fluids' equations, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.03.013i

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O. Coulaud / International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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15

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 Z η 2 I3 ¼ ∑ jXj4 V j ∂j ðjf  α1 e  τ Δf j2 Þ dX 2 j ¼ 1 R2 Z 2 ¼  2η ∑ jXjX j V j jf  α1 e  τ Δf j2 dX ¼ 0:

Then, we recall that W ¼ ηG þ f . Due to the fact that jηj r δ the smoothness of G, we obtain

We rewrite I 4 ¼ I 14 þ I 24 , where   ε 2 X Δ G þ ΔG þ  ∇ΔG; jXj2 ðf  α1 e  τ Δf Þ ; I 14 ¼  ηα1 e  τ 2 α1 L2

Using the Young inequalities ab r 14 a4 þ 34 b and ab r 13 a3 þ 23 b , the inequality (4.5) and assuming γ r 1, we finally obtain  2 2  3τ =2 2 þ C δ ð:jXj2 ∇f : þ kjXj2 Δf 2 Þ I 1;1 5 r Cδ e   5=3 4=3 þ C δ ðe  2τ þe  5τ=3 Þ þ C δ e  5τ=3 jXj2 ∇f 2=3    2 2  r C δ e  3τ=2 þ C δ ðjXj2 ∇f 2 þ jXj2 Δf 2 Þ  5=3 2 þ C δ ðe  2τ þe  5τ=3 Þ þ C δe  5τ=2 þ C δ jXj2 ∇f 2     r CM 2 γ ð1  θÞ6 e  3τ=2 þ CM 2 γ 2 ð1  θÞ12 ðjXj2 ∇f 2 þ jXj2 Δf 2 Þ:

R2

j¼1

It is easy, using the smoothness of G and the inequality (4.5), to see that I 14 r Cjηje  τ kG:H3 ð3Þ ð:f k þ α1 e  τ kΔf kÞ r CM γ ð1  θÞ6 e  τ : The term I 34 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 jηjβe  2τ ð‖∇U‖2L4 kjXj4 ΔGL1 þ ∇U L4 ∇2 U L4 jXj4 ∇GL1 Þ    ðf  þ α1 e  τ Δf kÞ r Cjηjβe  2τ ð‖W‖2L4 þ :W:L4 :∇W:L4 Þð:f :þ α1 e  τ :Δf :Þ   r Cjηje  2τ ð‖W‖2H1 þ:W:H1 :W:H2 Þðf kþ α1 e  τ kΔf kÞ r CM 2 γ 2 ð1  θÞ12 e  3τ=2 : Thus, assuming γ r1, the following inequality holds: 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 I 15 ¼  βe  2τ ðdivð∇ðjAj2 Þ 4 AÞ; jXj4 f ÞL2 ; I 25 ¼ βα1 e  3τ ðdivð∇ðjAj2 Þ 4 AÞ; jXj4 Δf ÞL2 : We begin by estimating I 15 . 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 þI 5 ;

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

þ Cδ

þδ

þ Cδ

3=2  5τ=4

e

1=4

:jXj2 ∇f :

7=4  3τ=2

e

1=2

:jXj2 ∇f :

1=2

Þðδ

1=2

1=2

:jXj2 ∇f :

:jXj2 Δf :

1=2

þδ

þ Cδ

1=4 τ=4

e

:jXj2 Δf :

7=4  5τ=4

e

1=2

:jXj2 Δf :

Þ

1=2

: 3=2

ð4:52Þ I 1;2 5 ,

In order to estimate we use the Hölder inequalities and obtain  2 3   2τ  I 1;2 jXj ∇ U kjXj2 f :L4 ‖∇U‖2L8 : 5 r C βe Then, using the Gagliardo–Nirenberg inequality, we notice that   2     jXj f  4 r jXj2 f 1=2 ∇ðjXj2 f Þ1=2 L      r C jXj2 f 1=2 ðjXjf k þ jXj2 ∇f Þ1=2 : 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  2 3   2τ  I 1;2 jXj ∇ U kjXj2 f :L4 ‖∇U‖2L8 5 r C βe   2τ r C βe ðkW kþ jXj∇Wk þ kjXj2 ΔWkÞ      jXj2 f 1=2 ðjXjf k þ jXj2 ∇f Þ1=2 ‖W‖2H1 : Finally, using the conditions (4.4) and (4.5) and the Young 4=3 inequality ab r 14 a4 þ 34 b , we obtain   7=4  3τ =4  e jXj2 f 1=2 I 1;2 5 r Cδ  2 2 2 τ r C δ e þ C δjXj f    ð4:53Þ r CM 2 γ 2 ð1  θÞ12 e  τ þ CM γ ð1  θÞ6 jXj2 f 2 : Thus, combining the inequalities (4.52) and (4.53), we obtain   I 15 r CM 2 γ ð1  θÞ6 e  τ þ CM 2 γ ð1  θÞ6 ðjXj2 f 2  2 2  2 2 ð4:54Þ þ jXj ∇f  þ jXj Δf  Þ: 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

jXj4 j∇3 Ujj∇Uj2 jf j dX:

2;2 I 25 r I 2;1 5 þ I5 ;

R2

R2

where

In order to simplify the notations, we set

 3τ I 2;1 5 ¼ C α1 β e

δ ¼ Mγ ð1  θÞ : 6

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 I 1;1 :jXj∇2 U:L4 :∇U L1 jXj2 f k 5 r βe    r C βe  2τ ð:W  þ kjXj∇WkÞðk∇Wk þ kjXjΔWkÞk∇U H2 jXj2 f k: 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     2τ ðkW k þk∇W 1=2 jXj2 ∇W 1=2 Þ I 1;1 5 rC β e  1=2    2 1=2  2 ðk∇W k þ ΔW  jXj ΔW  ÞW H2 jXj f k r C δe  3τ=2 ðδ

r Cδ e

1=2

and

jXj4 j∇2 Uj2 j∇Ujjf j dX;

Z Z

2  3τ =2

4=3

I 24 ¼  ηβe  2τ ðdivðjAj2 ∇GÞ; jXj4 ðf  α1 e  τ Δf ÞÞL2 :

I 4 r CM 2 γ ð1  θÞ6 e  τ :

 3τ=2 ðδ I 1;1 5 r C δe

1=2

1=2

þδ

1=4

:jXj2 ∇W:

1=2

Þðδ

1=2

þδ

1=4 τ=4

e

:jXj2 ΔW:

1=2

Þ:

 3τ I 2;2 5 ¼ C α1 β e

Z 2

ZR

R2

jXj4 j∇2 Uj2 j∇UjjΔf j dX; jXj4 j∇3 Ujj∇Uj2 jΔf j dX:

With the same tools as 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  3τ :jXj2 Δf ::jXj∇2 U:L4 :∇U:L1 I 2;1 5 r C α1 β e 2

r C α1 βe  3τ :jXj2 Δf ::jXj∇2 U:L4 :∇U:H2 : 2

Then, using the inequality (2.6) of Lemma 2.2 and the inequality (2.17) of Lemma 2.4, we obtain    I 52;1 r C α1 βe  3τ jXj2 Δf ðkWk þkjXj∇WkÞðk∇Wk þkjXjΔWkÞkW H2

Please cite this article as: O. Coulaud, Asymptotic profiles for the third grade fluids' equations, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.03.013i

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O. Coulaud / International Journal of Non-Linear Mechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Finally, the condition (4.4) and the Young inequality imply  3=2  τ  e jXj2 Δf  I 2;1 5 r Cδ   r CM 2 γ 2 ð1  θÞ12 e  2τ þ CM γ ð1  θÞ6 jXj2 Δf 2 : ð4:55Þ 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  jXj2 Δf kjXj2 ∇3 Uk‖∇U‖2L1 I 2;2 5 r C βα1 e  2   3τ  r C βα1 e jXj Δf ðWkþ kjXj∇Wkþ kjXj2 ΔWkÞ‖W‖2H3=2   1=2 r C δ e  2τ jXj2 Δf ‖W‖2H3=2 : Using the well-known interpolation inequality       1=2  1=2   v H3=2 r C v 1 v 2 for every v A H 2 ðR2 Þ; H

H

we obtain, using again the condition (4.4) and the Young inequality     1=2  2τ  jXj2 Δf kW  1 W  2 e I 2;2 5 r Cδ H H   3=2 r C δ e  3τ=2 jXj2 Δf    r CM 2 γ 2 ð1  θÞ12 e  3τ þ CM γ ð1  θÞ6 jXj2 Δf 2 : ð4:56Þ Finally, the inequalities (4.55) and (4.56) imply   I 25 r CM 2 γ 2 ð1  θÞ12 e  3τ þ CM γ ð1  θÞ6 jXj2 Δf 2 :

ð4:57Þ

þ

α1 4

eτ þ

α21 2

  2 e  2τ :jXj2 Δf : 8α1 e  τ :jXjf 2

  r CM 2 γ ð1  θÞ6 e  τ þ 8jXjf 2       þC 1 M 2 γ 1=2 ð1  θÞ3 ðf 2 þ ∇f 2 þ Δf 2 Þ:

ð4:61Þ

Using the inequality (2.4) of Lemma 2.2, one has 2

2

8:jXjf : rh:jXj2 f : þ

64 2 :f : h

for all h 4 0:

Thus, we set h ¼ ð1  θÞ=8 and obtain     θ 1θ 1 2 2 þ α1 e  τ :jXj2 ∇f : þ :jXj2 f : þ ∂τ E6 þ 4 2 8   α1  τ α21  2τ 2 2 e þ e þ :jXj2 Δf : 8α1 e  τ :jXjf : 4 2  1024 f 2 r CM 2 γ ð1  θÞ6 e  τ þ 1θ       þC 1 M 2 γ 1=2 ð1  θÞ3 ðf 2 þ ∇f 2 þ Δf 2 Þ:

ð4:62Þ

Integrating several times by parts, we notice that

α2 1 2 2 2 E6 ¼ :jXj2 f : þ α1 e  τ :jXj2 ∇f : þ 1 e  2τ :jXj2 Δf : 2 2  8α1 e  τ :jXjf : : 2

Thus, combining the inequalities (4.54) and (4.57), we get   I 5 r CM 2 γ ð1  θÞ6 e  τ þ CM 2 γ ð1  θÞ6 ðjXj2 f 2  2 2  2 2 þ jXj ∇f  þ jXj Δf  Þ: ð4:58Þ

Consequently, the inequality (4.62) can be written 1θ 1 α1 2 2 2 :jXj2 f : þ :jXj2 ∇f : þ e  τ :jXj2 Δf : 4 8 4  1024  2 2 r CM 2 γ ð1  θÞ6 e  τ þ f  þ CM γ 1=2 ð1  θÞ3 1θ       □ ðf 2 þ ∇f 2 þ Δf 2 Þ:

∂τ E6 þ θE6 þ

Taking into account the inequalities (4.49)–(4.51) and (4.58) and going back to (4.48), one has    1 ∂τ E6 þ jXj2 f 2 þ ð1 þ α1 e  τ ÞjXj2 ∇f 2 2    α2 þ α1 e  τ þ 1 e  2τ :jXj2 Δf 2 2    ð8 þ 8α1 e  τ ÞjXjf 2   2 r C εe  τ :jXj∇f 2 þ C εα1 e  2τ jXjΔf : þ CM 2 γ ð1  θÞ6 e  τ    2  2 2 1=2 3  2 þ CM γ ð1  θÞ ð f þ ∇f  þ Δf  Þ       ð4:59Þ þ CM 2 γ 1=2 ð1  θÞ3 ðjXj2 f 2 þ jXj2 ∇f 2 þ jXj2 Δf 2 Þ: Via the Young inequality and the condition (4.5), it is easy to check that      2 C εe  τ :jXj∇f 2 þ C εα1 e  2τ jXjΔf 2 r ε2 jXj2 ∇f : þ Ce  2τ k∇f 2         þ ε2 jXj2 Δf 2 þ C α2 e  4τ Δf 2 r ε2 jXj2 ∇f 2 þ ε2 jXj2 Δf 2 1

þ CMð1  θÞ6 e  2τ : We assume that ε2 rminð12 ; α1 e  τ0 =2Þ. The inequality (4.59) becomes   1 1 2 2 þ α1 e  τ :jXj2 ∇f : ∂τ E6 þ :jXj2 f : þ 2 2    α1  τ α21  2τ 2 e þ e :jXj2 Δf :  8α1 e  τ :jXjf 2 þ 2 2   r CM 2 γ ð1  θÞ6 e  τ þ 8jXjf 2       þ C 1 M 2 γ 1=2 ð1  θÞ3 ðf 2 þ ∇f 2 þ Δf 2 Þ       þ C 1 M 2 γ 1=2 ð1  θÞ3 ðjXj2 f 2 þ jXj2 ∇f 2 þ jXj2 Δf 2 Þ; 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 r ð1  θÞ=4. We obtain      θ 1θ 1 2 þ α1 e  τ :jXj2 ∇f 2 þ :jXj2 f : þ ∂τ E6 þ 4 2 4

ð4:63Þ

5. Proof of Theorem 1.1 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 also satisfies the inequality (1.10). We recall that 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 ððτ0 ; τε Þ; H 1 ð2ÞÞ \ C 0 ððτ0 ; τε Þ; H 3 ð2ÞÞ. If K is large enough, there exist two positive constants C1 and C2 such that, for all τ A ðτ0 ; τε Þ     C1 2 2 ð:f : 1 þ α1 e  τ :Δf ε 2 þ jXj2 f ε 2 þ α21 e  2τ jXj2 Δf ε : Þ; 1θ ε H   2 2 2 C 2 ð:f ε :H1 þ α1 e  τ :Δf ε : þ :jXj2 f ε 2 þ α21 e  2τ jXj2 Δf ε : Þ rE7 : E7 r

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Proof. The first inequality of this lemma comes directly from the definition of E7. To prove the second one, we notice that E7 Z

CK 1 2 2 2 ð:f : 1 þ α1 e  τ :Δf ε : Þ þ :jXj2 ðf ε  α1 e  τ Δf ε Þ: : 2 1θ ε H

Furthermore, we have already shown that 2

2 2

Via the Hölder and Young inequalities, we get :jXj2 ðf ε  α1 e  τ Δf ε Þ: Z :jXj2 f ε : þ 2α1 e  τ :jXj2 ∇f ε : 2

2

2

þ α21 e  2τ :jXj2 Δf ε :  12 :jXj2 f ε :  128:f ε : : 2

2

2

Consequently, one has

∂τ ðE7 eθτ Þ r CM3 γ ð1  θÞe  ð1  θÞτ :

CK 1 2 2 2 2 E7 Z ð:f : 1 þ α1 e  τ :Δf ε : Þ þ :jXj2 f ε : þ α1 e  τ :jXj2 ∇f ε : 4 1θ ε H þ

α21 2

2

Lemma 5.2. Let W ε A C 0 ð½τ0 ; τε Þ; H 3 ð2ÞÞ be a solution of (4.1) satisfying the inequality (4.4) for some γ 40. There exist T 0 4 0 and γ 0 4 0 such that if T ¼ eτ0 Z T 0 and γ r γ 0 , then, for all τ A ½τ0 ; τnε Þ, E7 satisfies the inequality ∂τ E7 þ θE7 r CM γ ð1  θÞe

τ

τ0 and τ A ½τ0 ; τε Þ, we n

E7 ðτÞ r E7 ðτ0 Þe  θðτ  τ0 Þ þ CM3 γ ðe  ð1  θÞτ0 e  θτ e  τ Þ:

Thus, if K is big enough, we get the second inequality of this lemma. □

3

ð5:5Þ

Integrating this inequality in time between obtain

e  2τ :jXj2 Δf : 64:f ε : : 2

ð5:4Þ

Let M 4 2 be a positive constant that will be set later and τnε 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 ½τ0 ; τnε Þ. 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 ½τ0 ; τnε Þ

þ α21 e  2τ :jXj2 Δf ε :  16:jXjf ε : : 2

Proof. Let W 0 A H 2 ð2Þ satisfying the condition (1.9) with 0 r γ r γ 0 and 0 rT 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 W ε to the system (4.1) which belongs to C 1 ððτ0 ; τε Þ; H 1 ð2ÞÞ \ C 0 ðð R τ0 ; τε Þ; H 3 ð2ÞÞ. Let η ¼ R2 W 0 ðXÞ dX, and f ε defined by the equality W ε ¼ ηG þ f ε :

:jXj2 ðf ε  α1 e  τ Δf ε Þ: ¼ :jXj2 f ε : þ 2α1 e  τ :jXj2 ∇f ε : 2

17

:

ð5:1Þ

Due to the decomposition (5.4) and Lemma 5.1, for every τ A ½τ0 ; τnε Þ, one has :W ε ðτÞ:H1 þ:jXj2 W ε ðτÞ: þ α1 e  τ :ΔW ε ðτÞ: 2

2

2

þ α21 e  2τ :jXj2 ΔW ε ðτÞ:

2

r C η2 þCE7 ðτÞ:

Since f ε satisfies the inequality (4.5), one has η2 rC γ ð1  θÞ6 . Taking into account the inequality (5.6), it comes :W ε ðτÞ:H1 þ:jXj2 W ε ðτÞ: þ α1 e  τ :ΔW ε ðτÞ: 2

Proof. We take γ0 and T0 respectively as small and large as necessary to satisfy the conditions of the Lemmas 4.2–4.6. According to the inequalities (4.41) and (4.42), one has   K 1 1 2 2 2 ∂τ E 7 þ θ E 7 þ 7:ð  ΔÞ  ð1 þ θÞ=4 f ε : þ :∇f ε : þ :Δf ε : 4 4 1θ 1θ α 2 2 1 :jXj2 f ε : þ e  τ :jXj2 Δf ε : þ 8 4 1024 2 r CM 3 γ ð1  θÞe  τ þ :f : 1θ ε

ð5:6Þ

2

2

þ α21 e  2τ :jXj2 ΔW ε ðτÞ:

2

r C γ ð1  θÞ6 þ E7 ðτ 0 Þe  θðτ  τ0 Þ þ CM3 γ e  τ0 :

ð5:7Þ

Using again Lemma 5.1 and arguing like for the establishment of the inequality (4.5), we can show that C 2 2 ð:f ðτ0 Þ:H1 þ α1 e  τ0 :Δf ε ðτ0 Þ: 1θ ε 2 2 þ :jXj2 f ε ðτ0 Þ: þ α21 e  2τ0 :jXj2 Δf ε ðτ0 Þ: Þ r C γ ð1  θÞ5 :

E 7 ðτ 0 Þ r

þCM 2 ð1  θÞγ 1=2 Kð:jXj2 f ε : þ α21 e  2τ :jXj2 Δf ε : Þ    2 þCM 2 ð1  θÞγ 1=2 ð:f ε : þ :∇f ε 2 þ Δf ε 2 Þ:   Using the interpolation (4.27) of f ε 2 between    inequality 2   ð1 þ θÞ=4 2 and ∇f ε  and taking K large enough and γ fε  ð  ΔÞ small enough, we get

Consequently, the inequality (5.7) becomes

∂τ E7 þ θE7 r CM 3 γ ð1  θÞe  τ :

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

□

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.

γ0

:W ε ðτÞ:H1 þ:jXj2 W ε ðτÞ: þ α1 e  τ :ΔW ε ðτÞ: 2

2

þ α21 e  2τ :jXj2 ΔW ε ðτÞ:

r C 1 γ ð1  θÞ þ C 2 M γ e  τ0 ;

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

α1 and β are fixed and

ð5:8Þ

:W ε ðτÞ:H1 þ:jXj2 W ε ðτÞ: þ α1 e  τ :ΔW ε ðτÞ: 2

2

2

þ α21 e  2τ :jXj2 ΔW ε ðτÞ: r

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 condition (1.9) with γ r γ 0 , there exist a unique global solution W ε A C 1 ððτ0 ; þ 1Þ; H 1 ð2ÞÞ \ C 0 ððτ0 ; þ 1Þ; H 3 ð2ÞÞ of (4.1) and a positive constant C ¼ Cðα1 ; β ; θÞ 4 0 such that, for all τ Z τ0  2  ð5:3Þ ð1  α1 e  τ ΔÞðW ε ðτÞ  ηGÞ:L2 ð2Þ r C γ e  θτ ;

2

3

5

5.1. Regularized problem Before proving Theorem 1.1, we show an intermediate theorem. This one gives the same result as Theorem 1.1, but for the solutions of the regularized system (4.1).

2

2

M γ ð1  θÞ6 þ C 2 M 3 γ e  τ0 : 4

ð5:9Þ

Finally, taking T0 sufficiently large so M γ ð1  θÞ6 =4, we obtain, for all τ A ½τ0 ; τnε Þ

that

C 2 M 3 γ e  τ0 r

:W ε ðτÞ:H1 þ:jXj2 W ε ðτÞ: þ α1 e  τ :ΔW ε ðτÞ: 2

2

2

þ α21 e  2τ :jXj2 ΔW ε ðτÞ: r 2

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 also deduce 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). □

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5.2. Existence of weak solutions in H 2 ð2Þ 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 ð½τ0 ; þ 1Þ; H 2 ð2ÞÞ, 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 ððτ0 ; þ 1Þ; H 1 ð2ÞÞ \ C 0 ððτ0 ; þ 1Þ; H 3 ð2ÞÞ 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, Hs ðΩÞ, s Z 0, denotes the restrictions to Ω of the functions of the Sobolev space H s ðR2 Þ. From Theorem 5.1, we know that W εn is bounded in L1 ð½τ0 ; þ 1Þ; H 2 ð2ÞÞ uniformly with respect to n. Consequently, there exists W A L1 ð½τ0 ; τ1 ; H 2 ð2ÞÞ such that W εn

,W

weakly in

Lp ð½τ0 ; τ1 ; H 2 ðΩÞÞ

for all p Z2:

Looking at the system (4.1), we can see that ∂τ W εn is bounded in L1 ð½τ0 ; τ1 ; H 1 ðΩÞÞ uniformly with respect to n. This implies that W εn is equicontinuous in H 1 ðΩÞ. Indeed, for s1 ; s2 A ½τ0 ; τ1 , s2 Z s1 , we have Z s2 ∂τ W εn ðsÞds:H1 ðΩÞ :W εn ðs2 Þ  W εn ðs1 Þ:H1 ðΩÞ ¼ k s1

r ðs2  s1 Þ:∂τ W εn ðsÞ:L1 ð½τ0 ;τ1 ;H1 ðΩÞÞ : 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 ð½τ0 ; τ1 ; H s ðΩÞÞ

in

for all s o2:

Ω

⟶

Z τZ τ0

Ω

jAðτ; XÞj2 Aðτ; XÞ⋄∇2 φðτ; XÞ dX dτ;

weakly in H 2 ð2Þ for all τ A ½τ0 ; þ1Þ:

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 i¼ 1,2, the divergence free vector field ui satisfies the system ∂t ðui  α1 Δui Þ  Δui þ curlðui  α1 Δui Þ 4 ui  β divðjAi j2 Ai Þ þ ∇pi ¼ 0; div ui ¼ 0; uijt ¼ 0 ¼ u0 ;

ð5:11Þ

ð5:12Þ

when n tends to infinity, where, for A; B A M2 ðRÞ, we use the notation 2

A⋄B ¼ ∑ ðA1;j B2;j  A2;j B1;j Þ: j¼1

The term of the right hand side of (5.12) appears naturally via two integrations by parts, when performing the L2-scalar product of div curlðjAj2 AÞ with φ. The strong convergence of W εn to W in C 0 ð½τ0 ; τ1 ; H 1 ðΩÞÞ implies directly the identity (5.12). Indeed, due to the continuous injection of H 1 ðΩÞ into L3 ðΩÞ, W εn converges to W in C 0 ð½τ0 ; τ1 ; L3 ðΩÞÞ. Furthermore, the inequality (2.13) implies    :Aεn  AL3 r W εn  W L3 ;

ð5:13Þ

where u0 is obtained from w0 via the Biot–Savart law. 2 þ Notice that since wi belongs to L1 loc ðR ; H ð2ÞÞ and ∂t wi belongs 1 þ 2 to L1 ðR ; H ðR ÞÞ, the inequalities (2.11) and (2.13) imply in loc particular p þ 2 2 ui A L1 loc ðR ; L ðR Þ Þ

for all p 4 2;

2 þ 2 4 ∇ui A L1 loc ðR ; H ðR Þ Þ;

∂t Δ

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 just show that the convergence holds for the term  div curlðjAεn j2 Aεn Þ which does not appear in the second grade fluids' equations. We consider φ A C 1 0 ð½τ 0 ; τ 1   ΩÞ. For all τ A ½τ 0 ; τ 1 , we want to show that Z τZ jAεn ðτ; XÞj2 Aεn ðτ; XÞ⋄∇2 φðτ; XÞ dX dτ τ0

,

W εn ðτÞ W ðτÞ;

p þ 2 2 ∂t ui A L1 loc ðR ; L ðR Þ Þ

C 0 ð½τ0 ; τ1 ; H 1 ðΩÞÞ:

By interpolation, we can show that W εn -W

and consequently Aεn converges to A strongly in C 0 ð½τ0 ; τ1 ; L3 ðΩÞÞ. 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 ð½τ0 ; þ1Þ; 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 H 2 ð2Þ uniformly with respect to n and consequently we have

for all p 4 2;

2 þ 2 2 ui A L1 loc ðR ; L ðR Þ 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 ðu  α1 ΔuÞ  Δu þ curlðu  α1 ΔuÞ 4 u1 þcurlðu2  α1 Δu2 Þ 4 u þ β divðjA2 j2 A2 Þ  β divðjA1 j2 A1 Þ 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 the 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 ð½0; t 0 ; H 2 ð2ÞÞ. More precisely, one has sup ð:w1 ðtÞ:H2 ð2Þ þ :w2 ðtÞ:H2 ð2Þ Þ r C:

t A ½0;t 0 

Applying Lemma 2.3, it implies in particular     sup ðui ðtÞL4 þ ∇ui ðtÞkL1 þ kΔui ðtÞL4 Þ rC for i ¼ 1; 2: t A ½0;t 0 

In order to show that u  0, we now perform estimates on the H1norm 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

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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 βðdivðjA2 j2 A2  jA1 j2 A1 Þ; uÞL2 ¼ ðjA1 j2 A1  jA2 j2 A2 ; AÞL2 2Z

β

¼

4 þ

¼

R2

β

Z

4 Z

R2

β 4

þ

β 4

ðjA1 j2 þjA2 j2 ÞjAj2 dx

R2

ðjA1 j2  jA2 j2 ÞðA1 þ A2 Þ : A dx

ðjA1 j2 þjA2 j2 ÞjAj2 dx

Z

R2

ðjA1 j2  jA2 j2 Þ2 dx:

Thus, using integrations by parts and the divergence free property of u, we have Z 1 β 2 2 2 ∂t ð:u: þ α:∇u: Þ þ :∇u: þ ðjA1 j2 þ jA2 j2 ÞjAj dx 2 4 R2 Z β ðjA1 j2  jA2 j2 Þ2 dx ¼ I 1 þI 2 ; ð5:15Þ þ 4 R2 where I 1 ¼ ðcurlðu2  α1 Δu2 Þ 4 u; uÞL2 ; I 2 ¼ ðcurl u 4 u1 ; uÞL2 ;

I 3 ¼  α1 ðcurl Δu 4 u1 ; uÞL2 :

A short computation shows that I1 vanishes. Indeed, we set ω ¼ u2  α1 Δu2 and we recall the notation u ¼ ðu1 ; u2 ; 0Þ and curl ω ¼ ð0; 0; ∂1 ω2  ∂2 ω1 Þ. We have I 1 ¼ ðcurl ω 4 u; uÞL2

¼  ðð∂1 ω2  ∂2 ω1 Þu2 ; u1 ÞL2 þ ðð∂1 ω2  ∂2 ω1 Þu1 ; u2 ÞL2 ¼ 0:

Due to the boundedness of u1 in L4 ðR2 Þ, applying Hölder inequalities we obtain     I 2 r u1 L4 ∇ukuk  2   r Cðα1 Þðu þ α1 ∇u2 Þ: Using [27, Lemma A.1], we check that Z Z jΔu1 jj∇ujjuj dx þ C α1 j∇u1 jj∇uj2 dx: I 3 r C α1 R2

R2

Using Hölder inequalities, the Gagliardo–Nirenberg inequality and 4=3 the Young inequality ab r 14 a4 þ 34 b , we obtain          I 3 r C α1 uL4 Δu2 L4 ∇uk þC α1 ∇u1 L1 ∇u2       r C α1 ∇u3=2 u1=2 þ C α1 ∇u2  2  2 r Cðα1 Þðu þ α1 ∇u Þ: Going back to (5.15), we get         α∇u2 Þ r CðαÞðu2 þ α∇u2 Þ:

1  2 þ 2 ∂t ð u

ð5:16Þ

Integrating in time this inequality between 0 and t A ½0; t 0  and applying the Gronwall lemma, we finally obtain     uðtÞ2 þ α∇uðtÞ2 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 ðR þ ; H 2 ð2ÞÞ. Uncited reference [22].

19

64 65 66 I would like to express my gratitude to Genevieve Raugel and 67 Marius Paicu, whose advice was very helpful for this work. 68 69 References 70 71 [1] C. Amrouche, D. Cioranescu, On a class of fluids of grade 3, Int. J. Non-Linear 72 Mech. 32 (1) (1997) 73–88. 73 [2] J.-M. Bernard, Weak and classical solutions of equations of motion for third 74 grade fluids, Math. Model. Numer. Anal. 33 (6) (1999) 1091–1120. [3] D. Bresch, J. Lemoine, On the existence of solutions for non-stationary third75 grade fluids, Int. J. Non-Linear Mech. 34 (1999) 485–498. 76 [4] V. Busuioc, D. Iftimie, Global existence and uniqueness of solutions for the 77 equations of third grade fluids, Int. J. Non-Linear Mech. 39 (2004) 1–12. 78 [5] V. Busuioc, D. Iftimie, A non-Newtonian fluid with Navier boundary conditions, J. Dyn. Differ. Equ. 18 (2) (2006) 357–379. 79 [6] A. Carpio, Asymptotic behavior for the vorticity equations in dimensions two 80 and three, Commun. Partial Differ. Equ. 19 (1994) 827–872. 81 [7] A. Carpio, Large time behavior in incompressible Navier–Stokes equations, SIAM J. Math. Anal. 27 (2) (1996) 449–475. 82 [8] J.-Y. Chemin, C.-J. Xu, Inclusions de Sobolev en calcul de Weyl–Hörmander et 83 champs de vecteurs sous-elliptiques, Ann. Sci. Ecole Norm. Supér. (4) 30 6 84 (1997) 719–751. [9] D. Cioranescu, E.H. Ouazar, Existence and uniqueness for fluids of second 85 grade, Nonlinear Partial Differ. Equ. 109 (1984) 178–197 (Collège de France 86 Seminar Pitman). 87 [10] J.E. Dunn, R.L. Fosdick, Thermodynamics, stability and boundedness of fluids of 88 complexity 2 and fluids of second grade, Arch. Ration. Mech. Anal. 56 (1974) 191–252. 89 [11] M. Escobedo, O. Kavian, H. Matano, Large time behavior of solutions of a 90 dissipative semi-linear heat equation, Commun. Partial Differ. Equ. 20 (1995) 91 1427–1452. [12] M. Escobedo, E. Zuazua, Large-time behavior for convection diffusion equa92 tions in Rn , J. Funct. Anal. 100 (1991) 119–161. 93 [13] R.L. Fosdick, K.R. Rajagopal, Thermodynamics and stability of fluids of third 94 grade, Proc. R. Soc. Lond., Ser. A 369 (1738) (1980) 351–377. [14] V.A. Galaktionov, J.L. Vazquez, Asymptotic behaviour of nonlinear parabolic 95 equations with critical exponents, a dynamical system approach, J. Funct. Anal. 96 100 (1991) 435–462. 97 [15] G.P. Galdi, M. Grobbelaar-van Dalsen, N. Sauer, Existence and uniqueness of 98 classical solutions of the equations of motion for second-grade fluids, Arch. Ration. Mech. Anal. 124 (1993) 221–237. 99 [16] T. Gallay, G. Raugel, Scaling variables and asymptotics expansions in damped 100 wave equations, J. Differ. Equ. 150 (1998) 42–97. 101 [17] T. Gallay, G. Raugel, Scaling variables and stability of hyperbolic fronts, SIAM J. Math. Anal. 32 (1) (2000) 1–29. 102 [18] T. Gallay, E. Wayne, Invariant manifolds and the long time asymptotics of the 103 Navier–Stokes and vorticity equations on R2 , Arch. Ration. Mech. Anal. 163 104 (2002) 209–258. [19] T. Gallay, E. Wayne, Long time asymptotics of the Navier–Stokes and vorticity 105 equations on R3 , Philos. Trans. R. Soc. Lond., Ser. A: Math. Phys. Eng. Sci. 360 106 (1799) (2002) 2155–2188. 107 [20] T. Gallay, E. Wayne, Global stability of vortex solutions of the two-dimensional 108 Navier–Stokes equation, Commun. Math. Phys. 255 (1) (2005) 97–129. [21] T. Gallay, E. Wayne, Long time asymptotics of the Navier–Stokes equations in Q4109 R2 and R3 , ZAMM J. Appl. Math. Mech. 86 (4) (2006) 256–267. 110 [22] M. Hamza, M. Paicu, Global existence and uniqueness result of a class of third111 grade fluids equations, Nonlinearity 20 (2007) 1095–1114. [23] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes 112 in Mathematics, vol. 840, Springer-Verlag, 1981. Q5113 [24] B. Jaffal-Mourtada, Long time asymptotics of the second grade fluid equations 114 on R2 , Dyn. PDE 8 (2011) 185–223. [25] O. Kavian, Remarks on the large time behavior of a nonlinear diffusion 115 equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987) 423–452. 116 [26] M. Paicu, Global existence in the energy space of the solution of a non117 Newtonian fluid, Physica D 237 (2008) 1676–1686. 118 [27] M. Paicu, G. Raugel, A. Rekalo, Regularity of the global attractor and finitedimensional behavior for the second grade fluid equations, J. Differ. Equ. 252 119 (6) (2012) 3695–3751. 120 [28] A. Pazy, Semigroups of linear operators and applications to partial differential 121 equations, in: Applied Mathematical Sciences, vol. 44, Springer, 1983. [29] R.S. Rivlin, J.L. Ericksen, Stress-deformation relations for isotropic materials, J. 122 Ration. Mech. Anal. 4 (1955) 323–425. 123 [30] C. Truesdell, W. Noll, The Non-linear Field Theories of Mechanics, third 124 edition, Springer-Verlag, Berlin, 2004. 125 Acknowledgments

Please cite this article as: O. Coulaud, Asymptotic profiles for the third grade fluids' equations, International Journal of Non-Linear Mechanics (2014), http://dx.doi.org/10.1016/j.ijnonlinmec.2014.03.013i