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 ) which approximate the incompressible equations as . 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.
Fluids of differential type form an important class of non-Newtonian fluids. The fluids of grade , introduced by Rivlin and Ericksen , are the fluids for which the stress tensor is a polynomial of degree in the first Rivlin-Ericksen tensor defined recursively bywhere denotes the material derivative and the transposition of the Jacobian matrix . In , the constitutive relation of a particular of fluids of grade is given by where is the identity matrix of degree and is an isotropic polynomial of degree .
There are some references on the existence, uniqueness, and asymptotic behavior of solutions for second and third grade fluids equations; see, for example, [2–10] and the references therein. In [2–4], Azia et al. use the symmetry approach to obtain the analytical solutions of the third grade fluid equations. In , Busuioc and Iftime studied the existence of solutions to the following third grade fluids equations:in , with the coefficients , , , and satisfying the following hypotheses: Also, Busuioc and Iftimie in  proved that the third grade fluids equations (3) possess a global solution if the initial value is in and established the uniqueness of solutions for (3) when . Recently, Hamza and Paicu studied in  a particular case of the third grade fluids equations (3) in , where they assume . Then (3) becomewith the following assumptions on the coefficients , , , and :Hamza and Paicu in  proved the existence and uniqueness, as well as the stability (when ), of global solutions for (5) with natural regularity assumption on the initial data belonging to the energy space . They also proved that if the initial datum belongs to , then the solution belongs to for any positive time and they gave a control of the norm of the solution. Very recently, Paicu et al. proved in  the regularity of the global attractor and finite-dimensional behavior for the second grade fluids equations in the two-dimensional torus.
In the present paper, we first discuss the approximation of the incompressible third grade fluids equations (5) in three-dimensional bounded smooth domain, via the following slightly compressible third grade fluids equations depending on a parameter :with initial-boundary value conditions:
Compared with the incompressible third grade fluids equations (5), (7)-(8) overcome the computational difficulties connected with the constraint “.” It is easier to approximate than the original incompressible equations as the constraint “” has been replaced by the evolution equation (8). The third grade fluids equations (7)-(8) are called “slightly compressible” for the parameter will be taken to tend to zero. Obviously, the perturbed compressible third grade fluids equations (7)–(9) turn to be the incompressible third grade fluids equations (5) as . Following this clue, the questions are now the following.(i)Does the solution of the initial-boundary value problem of perturbed compressible third grade fluids equations (7)–(9) uniquely exist?(ii)Does the solution of (7)–(9) converge to the solution of the incompressible third grade fluids equations (5) as ?
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., ) to obtain the bound of the derivative sequence seems not applicable. This is caused essentially by the compressibility of the fluids. We know in the incompressible case 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 to . While the fluids are compressible, and we shall take or (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.
In this paper, we denote by the generic constant that can take different values in different places. is the 3D Lebesgue space with norm and is the 3D Sobolev space with norm and dual space , where and are the usual -Lebesgue space and Sobolev space on , respectively. When , we denote and also if there is no confusion. is the usual 3D Sobolev space and is its dual space (see ). Writeand denote by and the closure space of in norm and in norm, respectively. is the inner product of (or , ), and is the dual pairing between and (the dual space of ), or between and , or between and . Note
To put (5) and (7)–(9) into abstract forms, respectively, we now introduce some operators. Firstly, set (which is taken with zero boundary conditions). Then, using integration by parts, we haveSecondly, we define a continuous trilinear form (see, e.g., ) on (and in particular on ) byIf , one can checkFor (resp., ), we denote by the element of (resp., ) defined by , for any (resp.,), and setThe scalar product of two matrices and is denoted by . We set and for . We now define then the operator maps to viaAlso definethen maps to byUsing the notations and operators introduced above, we can express the weak formulation of (5) in the solenoidal field as
Problem 1. Let , for any given , to find satisfying
The proof of this lemma is similar with that of , and we omit it here.
Remark 3. Let , , and satisfy condition (6). Then for any given , for each solution obtained by Lemma 2, there exists a unique pressure corresponding to and, for each , . Moreover, and hence (one can refer to [13, Page 307] for a similar derivation)which satisfies in the sense of distribution thatTherefore, for any , there holdsand for any ,
Remark 4. If merely belongs to , condition (23) needs not make sense. But if and satisfies (26)-(27), then we will show thatBy the classical embedding theorem (see [13, 14]), we infer from (21) and (28) that . Therefore, (23) is meaningful.
3. 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 is given as in Problem 1.
For given , we consider the following initial-boundary value problem.
For any given , set . Find and such thatwhere the function (not appearing in (5)) is independent of and
Remark 5. Equation (29) contains the term which does not appear in (5). This is a stabilization term which corresponds to the substitution of the trilinear form for the form (see (14)), where the trilinear form is given byIn fact, if , then . But , . Equations (29)-(30) are the constructed slightly compressible third grade fluids equations. This artificial compressibility method has been used by Zhao and You  and Zhao  to approximate the incompressible convective Brinkman-Forchheimer equations and a class of incompressible non-Newtonian fluids equations.
We first consider the classical solution of (29)–(33). Assume that and . Then, for any and , multiplying (29) by and (30) by , we obtainThe above two equalities are still valid by a continuity argument for any and any .
Let be defined by (35) and set viaThen, like , the operator is continuous on .
Lemma 6. If , then
Proof. For almost all , , , and are elements of , and is an element of , and the measurability of the functions , , , and is easy to check. Now for any , we haveBy Hölder and Gagliardo-Nirenberg inequalities and the embedding , we haveSimilarly, for any , we haveFrom (42)–(44), we conclude thatTherefore,Equations (46)–(49) give (38)–(41), respectively. The proof is complete.
Problem 7. Let be fixed. For any given and , find and such that
Assume , , and are Hilbert spaces withthe embedding being continuous andLet be a function from to ; we denote by its Fourier transform:The derivative in of order is the inverse Fourier transform of ; that is,For given , define the spaceThen is a Hilbert space with the norm For any set , the subspace of is defined as the set of functions with support contained in :
Theorem 9. Let be fixed. For any given and , Problem 7 possesses a unique solution .
Proof. We will use the Faedo-Galerkin method to prove the existence of solutions for Problem 7. Consider an orthonormal basis of constituted of elements and an orthonormal basis of constituted of elements , .
For each positive integer , we define the approximate solution of Problem 7 bywhich satisfiesand the initial conditionswhere and are the projections of and from the spaces and to the spaces and , respectively.
Equations (68)–(70) form a Cauchy problem of first-order nonlinear ordinary differential equations (ODE) in for the functions and with the following initial conditions: By the standard theory of ODE, we have the existence of a solution defined on , . Moreover, if , then we have the following blow-up criteria and .
Multiplying (68) by () and (69) by () and then adding these resulting equalities, we obtainSet . Using integration by parts and Hölder inequality, we haveAlso integrating by parts, using the fact that is a symmetric matrix, we getSince and , we haveBy combining (72)–(75), we deduce thatIntegrating (76) in from to shows thatThus andWe now need an estimate of the fractional derivative in time of to pass to the limit in the nonlinear terms in (68). Set We write the relations (68) and (69) asSetThen both and have two discontinuities at and . Thus we have on the whole the followingwhere is defined similarly with that of and and are the Dirac distributions at and , respectively. Taking the Fourier transform on both sides of (84), we getSimilar with (83), we can define and and let and be the Fourier transforms of and , respectively. Multiplying (85) by () and (86) by () and then adding these resulting equalities, we obtainSince , integration by parts givesIt then follows from (87), (88), and then (78) thatWe next estimate the term . In fact, using the similar derivations as (73) and (74), we getInserting (90) into (89) givesFor some fixed , we have , . ThusEmploying Parseval equality, Poincaré inequality, and (79), we obtainBy (91) and the fact that , we obtainEquations (79), (80), and Parseval equality implyUsing Hölder inequality, (78), and the convergence of the infinite integral we obtainEquations (92)–(97) giveEquations (78)–(80), as well as Lemma 8 and (98), imply that we can extract a subsequence (still denoted by ) of such thatTaking and multiplying (68) (resp., (69)) by , integrating over , and then integrating the first term by parts, we obtain