Approximation of a Class of Incompressible Third Grade Fluids Equations

This paper discusses the approximation of weak solutions for a class of incompressible third grade fluids equations. We first introduce a family of perturbed slightly compressible third grade fluids equations (depending on a positive parameter ε) which approximate the incompressible equations as ε → 0. Then we prove the existence and uniqueness of weak solutions for the slightly compressible equations and establish that the solutions of the slightly compressible equations converge to the solutions of the incompressible equations.


Introduction
Fluids of differential type form an important class of non-Newtonian fluids.The fluids of grade , introduced by Rivlin and Ericksen [1], are the fluids for which the stress tensor is a polynomial of degree  in the first  Rivlin-Ericksen tensor defined recursively by where / = / +  × ∇ denotes the material derivative and (∇)  the transposition of the Jacobian matrix ∇.In [1], the constitutive relation of a particular of fluids of grade  is given by  = − +  ( 1 ,  2 , . . .,   ) , where  is the identity matrix of degree  and  is an isotropic polynomial of degree .
proved in [8] the regularity of the global attractor and finitedimensional behavior for the second grade fluids equations in the two-dimensional torus.
(i) Does the solution of the initial-boundary value problem of perturbed compressible third grade fluids equations ( 7)-( 9) uniquely exist?
The purpose of this paper is to give answers to the above two questions.When using the classical Faedo-Galerkin method to prove the existence of a weak solution for the compressible third grade fluid equations ( 7)-( 9), the main difficulty (compared with the incompressible case) comes from the presence of the term ∇  in the slightly compressible third grade fluids equations.Due to the presence of the term ∇  , the argument for the incompressible case (see, e.g., [7]) to obtain the bound of the derivative sequence {  /} ∞ =1 seems not applicable.This is caused essentially by the compressibility of the fluids.We know in the incompressible case div  = 0 and we can take  or  (see the notation in Section 2) as the phase space.Naturally, the term ∇ will disappear under the projection of the Helmhloz-Leray projector from L 2 (Ω) to .While the fluids are compressible, div  ̸ = 0 and we shall take L 2 (Ω) or H 1 0 (Ω) (or other Sobolev spaces) as the phase space.So the term ∇  will not disappear.To overcome this difficulty, we will use the Fourier transform (in time ) technique to obtain the boundedness of the fractional derivative in time variable  of the sequence The paper is organized as follows.In Section 2, we give some notations first and then the existence and uniqueness of weak solutions for (5).In Section 3, we describe the weak formulation of ( 7)-( 9) and then prove the existence and uniqueness of its weak solution.In Section 4, we show how the solutions of the slightly compressible third grade fluids equations converge to the solutions of the corresponding incompressible third grade fluids equations.

Preliminaries
In this paper, we denote by  the generic constant that can take different values in different places.
Then Problem 1 possesses a unique solution.
The proof of this lemma is similar with that of [7], and we omit it here.

The Existence and Uniqueness of Solutions for Slightly Compressible Third Grade Fluids Equations
In this section, we first give a description of the slightly compressible third grade fluids equations and then prove the existence and uniqueness of its weak solutions.
Since our ultimate purpose is to investigate the convergence of solutions of the slightly compressible third grade fluids equations to the solutions of the incompressible third grade fluids equations, we assume that  0 is given as in Problem 1.
Assume  0 , , and  1 are Hilbert spaces with the embedding being continuous and Let () be a function from R to  1 ; we denote by F[()] = ψ() its Fourier transform: The derivative in  of order  is the inverse Fourier transform of (2)  ψ(); that is, For given  > 0, define the space Then   is a Hilbert space with the norm For any set  ⊂ R, the subspace    of   is defined as the set of functions  ∈   with support contained in : Lemma 8 (see [13]).Assume  0 , , and  1 are Hilbert spaces satisfying ( 59) and (60).Then, for any bounded set  and ∀ > 0, the following compact embedding holds: Theorem 9. Let  ∈ (0, 1] be fixed.For any given  0 ∈  and  0 ∈  2 (Ω), Problem 7 possesses a unique solution {  ,   }.