GLOBAL STRONG SOLUTION TO THE DENSITY-DEPENDENT INCOMPRESSIBLE VISCOELASTIC FLUIDS

Transcription

1 GOBA STRONG SOUTION TO THE DENSITY-DEPENDENT INCOMPRESSIBE VISCOEASTIC FUIDS XIANPENG HU AND DEHUA WANG Abstract. The existence and uniueness of the global strong solution with small initial data to the three-dimensional density-dependent incompressible viscoelastic fluids is established. The local existence and uniueness of the global strong solution with small initial data to the three-dimensional compressible viscoelastic fluids is also obtained. A new method is developed to estimate the solution with weak regularity. Moreover, as a byproduct, we show the global existence and uniueness of strong solution to the densitydependent incompressible Navier-Stokes euations using a different techniue from [8]. All the results apply to the two-dimensional case. Contents 1. Introduction 2 2. Background of Mechanics for Viscoelastic Fluids 4 3. Main Results 6 4. ocal Existence Solvability of the density with a fixed velocity Solvability of the deformation gradient with a fixed velocity ocal existence via the fixed-point theorem Uniueness Global A Priori Estimates Dissipation of the deformation gradient Dissipation of the gradient of the density Global Existence Uniform estimates in time Refined estimates on ρ and E Proof of Theorem Acknowledgments 4 References 4 Date: April 29, Mathematics Subject Classification. 35A5, 76A1,76D3. Key words and phrases. Density-dependent incompressible viscoelastic fluids, strong solution, global existence, uniueness. 1

2 2 XIANPENG HU AND DEHUA WANG 1. Introduction Elastic solids and viscous fluids are two extremes of material behavior. Viscoelastic fluids show intermediate behavior with some remarkable phenomena due to their elastic nature. These fluids exhibit a combination of both fluid and solid characteristics, keep memory of their past deformations, and their behaviour is a function of these old deformations. Viscoelastic fluids have a wide range of applications and hence have received a great deal of interest. Examples and applications of viscoelastic fluids include from oil, liuid polymers, mucus, liuid soap, toothpaste, clay, ceramics, gels, some types of suspensions, to bioactive fluids, coatings and drug delivery systems for controlled drug release, scaffolds for tissue engineering, and viscoelastic blood flow flow past valves; see [1, 13, 35] for more applications. For the viscoelastic materials, the competition between the kinetic energy and the internal elastic energy through the special transport properties of their respective internal elastic variables makes the materials more untractable in understanding their behavior, since any distortion of microstructures, patterns or configurations in the dynamical flow will involve the deformation tensor. For classical simple fluids, the internal energy can be determined solely by the determinant of the deformation tensor; however, the internal energy of complex fluids carries all the information of the deformation tensor. The interaction between the microscopic elastic properties and the macroscopic fluid motions leads to the rich and complicated rheological phenomena in viscoelastic fluids, and also causes formidable analytic and numerical challenges in mathematical analysis. The euations of the density-dependent incompressible viscoelastic fluids of Oldroyd type [27, 28] in three spatial dimensions take the following form [12, 23, 29]: ρ t + divρu =, ρu t + div ρu u µ u + P ρ = divρff, F t + u F = u F, divu =, 1.1a 1.1b 1.1c 1.1d where ρ stands for the density, u R 3 the velocity, and F M 3 3 the set of 3 3 matrices the deformation gradient. The viscosity coefficient µ > is a constant. The increasing convex function P ρ = Aρ γ is the pressure, where γ > 1 and A > are constant. Without loss of generality, we set A = 1 in this paper. The symbol denotes the Kronecker tensor product and F means the transpose matrix of F. As usual we call euation 1.1a the continuity euation. For system 1.1, the corresponding elastic energy is chosen to be the special form of the Hookean linear elasticity: W F = 1 2 F 2, which, however, does not reduce the essential difficulties for analysis. The methods and results of this paper can be applied to more general cases. In this paper, we consider euations 1.1 subject to the initial condition: ρ, u, E t= = ρ x, u x, E x, x R 3, 1.2 and we are interested in the global existence and uniueness of strong solution to the initial-value problem near its euilibrium state in the three dimensional space R 3. Here the euilibrium state of the system 1.1 is defined as: ρ is a positive constant

3 STRONG SOUTIONS TO INCOMPRESSIBE VISCOEASTIC FUIDS 3 for simplicity, ρ = 1, u =, and F = I the identity matrix in M 3 3. We introduce a new unknown variable E by setting Then, 1.1c reads F = I + E. E t + u E = ue + u. 1.3 By a strong solution, we mean a triple ρ, u, F satisfying 1.1 almost everywhere with initial condition 1.2, in particular, ρ, u, F, t W 2,, 3, 6] for almost all t > in this paper. When density ρ is a constant, system 1.1 governs the homogeneous incompressible viscoelastic fluids, and there exist rich results in the literature for the global existence of classical solutions namely in H 3 or other functional spaces with much higher regularity; see [5, 6, 15, 16, 18, 19, 2, 23] and the references therein. When density ρ is not a constant, the uestion related to existence becomes much more complicated and not much has been done. In [17] the authors considered the global existence of classical solutions in H 3 of small perturbation near its euilibrium for the compressible viscoelastic fluids without the pressure term. One of the main difficulties in proving the global existence is the lacking of the dissipative estimate for the deformation gradient. To overcome this difficulty, the authors in [16] introduced an auxiliary function to obtain the dissipative estimate, while the authors in [18] directly deal with the uantities such as u + divf. Those methods can provide them with some good estimates, partly because of their high regularity of u, F. However, in this paper, we deal with the strong solution with much less regularity in W 2,, 3, 6], hence those methods do not apply. Thus, we need a new method to overcome this obstacle, and we find that a combination between the velocity and the convolution of the divergence of the deformation gradient with the fundamental solution of aplace operator will develop some good dissipative estimates reuired for the global existence. The global existence of strong solution to the initial-value problem is established based on the local existence and global uniform estimates. The local existence is obtained using a fixed point theorem without incompressible condition 1.1d, that is, the local existence holds for both the incompressible viscoelastic fluids 1.1a- 1.1d and the compressible viscoelastic fluids 1.1a-1.1c. The global existence and uniueness of strong solution also holds for the density-dependent incompressible Navier- Stokes euations when the deformation gradient does not appear, which, as a byproduct, gives a similar result to [8] but through a different techniue. The viscoelastic fluid system 1.1 can be regarded as a combination of the inhomogeneous incompressible Navier-Stokes euation with the source term divρff and the euation 1.1c. For the global existence of classical solutions with small perturbation near an euilibrium for the compressible Navier-Stokes euations, we refer the reader to [24, 25, 26, 3] and the references cited therein. We remark that, for the nonlinear inviscid elastic systems, the existence of solutions was established by Sideris-Thomases in [33] under the null condition; see also [31] for a related discussion. The existence of global weak solutions with large initial data of 1.1 is still an outstanding open uestion. In this direction for the homogeneous incompressible viscoelastic fluids, when the contribution of the strain rate symmetric part of u in the constitutive euation is neglected, ions-masmoudi in [22] proved the global existence of weak solutions with large initial data for the Oldroyd model. Also in-iu-zhang in [19] proved the

4 4 XIANPENG HU AND DEHUA WANG existence of global weak solutions with large initial data for the incompressible viscoelastic fluids when the velocity satisfies the ipschitz condition. When dealing with the global existence of weak solutions of the viscoelastic fluid system 1.1 with large data, the rapid oscillation of the density and the non-compatibility between the uadratic form and the weak convergence are two of the major difficulties. The rest of the paper is organized as follows. In Section 2, we recall briefly the densitydependent incompressible viscoelastic fluids from some basic mechanics and conservation laws. In Section 3, we state our main results, including the local and global existence and uniueness of the strong solution to the euations of the viscoelastic fluids, as well as to the incompressible Navier-Stokes euations with small data. In Section 4, we prove the local existence via a fixed-point theorem. In Section 5, we prove the uniueness of the solution obtained in Section 4. In Section 6, we establish some global a priori estimates, especially on the dissipation of the deformation gradient and gradient of the density. In Section 7, we first prove some energy estimates uniform in time and some refined estimates on the density and the deformation gradient, and then give the proof of the global existence. 2. Background of Mechanics for Viscoelastic Fluids To provide a better understanding of system 1.1, we recall briefly some background of viscoelastic fluids from mechanics in this section. First, we discuss the deformation gradient F. The dynamics of a velocity field ux, t in mechanics can be described by the flow map or particle trajectory xt, X, which is a time dependent family of orientation preserving diffeomorphisms defined by: { d dtxt, X = ut, xt, X, 2.1 x, X = X, where the material point X agrangian coordinate is deformed to the spatial position xt, X, the reference Eulerian coordinate at time t. The deformation gradient F is defined as Ft, X = x t, X, X which describes the change of configuration, amplification or pattern during the dynamical process, and satisfies the following euation by changing the order of differentiation: Ft, X ut, xt, X =. 2.2 t X In the Eulerian coordinate, the corresponding deformation gradient Ft, x is defined as Ft, xt, X = Ft, X. Euation 2.2, combined with the chain rule and 2.1, gives Ft, xt, X xt, X t Ft, xt, X + u Ft, xt, X = t Ft, xt, X + x t = Ft, X ut, xt, X ut, xt, X x = = t X x X ut, xt, X = Ft, X = u F, x

5 STRONG SOUTIONS TO INCOMPRESSIBE VISCOEASTIC FUIDS 5 which is exactly euation 1.1c. notations: Here, and in what follows, we use the conventional u ij = u i x j, u F i,j = u ik F kj, u F ij = u k F ij x k, and summation over repeated indices will always be well understood. In viscoelastic fluids, 1.1c can also be interpreted as the consistency of the flow maps generated by the velocity field u and the deformation gradient F. The difference between fluids and solids lies in the fact that, in fluids, such as Navier- Stokes euations [26], the internal energy can be determined solely by the determinant part of F euivalently the density ρ, and hence, 1.1c can be disregarded; while in elasticity, the energy depends on all information of F. In the continuum physics, if we assume the material is homogeneous, the conservation laws of mass and of momentum become [7, 16, 31] t ρ + divρu =, 2.3 and t ρu + divρu u µ u + P ρ = divdet F 1 SF, 2.4 with divu = where ρ det F = 1, 2.5 and S ij F = W. 2.6 F ij Here S, ρsf, W F denote Piola-Kirchhoff stress, Cauchy stress, and the elastic energy of the material, respectively. Recall that the condition 2.6 implies that the material is hyperelastic [23]. In the case of Hookean linear elasticity [15, 16, 2], W F = 1 2 F 2 = 1 2 trff, 2.7 where the notation tr stands for the trace operator of a matrix, and hence, SF = F. 2.8 Combining euations together, we obtain system 1.1. If the viscoelastic system 1.1 satisfies divρ F =, initially at t = with F = I + E, it was verified in [18] see Proposition 3.1 that this condition will insist in time, that is, divρtft =, for t. 2.9 Another hidden, but important, property of the viscoelastic fluids system 1.1 is concerned with the curl of the deformation gradient see [15, 16]. Formally, the fact that the agrangian derivatives commute and the definition of the deformation gradient imply Xk F ij = 2 x i X k X j = 2 x i X j X k = Xj F ik, which is euivalent to, in the Eulerian coordinates, F lk l F ij t, xt, X = F nj n F ik t, xt, X,

6 6 XIANPENG HU AND DEHUA WANG that is, F lk l F ij t, x = F nj n F ik t, x, which means that, using F = I + E, k E ij + E lk l E ij = j E ik + E nj n E ik. 2.1 According to 2.1, it is natural to assume that the initial condition of E in the viscoelastic fluids system 1.1 should satisfy the compatibility condition k E ij + E lk l E ij = j E ik + E nj n E ik Finally, if the density ρ is a constant, 1.1 becomes its corresponding homogeneous density-independent incompressible form see [6, 15, 16, 18, 19, 2] and references therein divu =, t u + u u µ u + P = divff, t F + u F = u F For more discussions on viscoelastic fluids and related models, see [4, 5, 7, 11, 12, 14, 19, 22, 23, 29, 34] and the references cited therein. 3. Main Results In this Section, we state our main results. As usual, the global existence is built on the local existence and global uniform estimates. In this paper, the standard notations for Sobolev spaces W s, and Besov spaces B s p [3] will be used. Throughout this paper, the real interpolation method [3] will be adopted and the following interpolation spaces will be needed and X 21 1 p p = R 3, W 2, R p,p = B21 1 p p, Y 1 1 p p = R 3, W 1, R p,p = B1 1 p p. Now we introduce the following functional spaces to which the solution and initial conditions of the system 1.1 will belong. Given 1 p, and T >, we set Q T = R 3, T, and W p,, T := { u W 1,p, T ; R 3 3 p, T ; W 2, R 3 3 : divu = } with the norm as well as with the norm We denote u W p,,t := u W 1,p,T ; R 3 + u p,t ;W 2, R 3, V p, := X 21 1 p p Y p p W 1, R 3 1 f, g V p, := f X 21 1 p p + f Y 1 1 p p W, T = W p,, T W 2,2, T, + g W 1, R 3.

7 STRONG SOUTIONS TO INCOMPRESSIBE VISCOEASTIC FUIDS 7 and V = V p, V 2,2. Our first result is the following local existence: Theorem 3.1 ocal existence for viscoelastic fluids. et T > be given and u, ρ, E V with p [2,, 3,. There exists a positive constant δ < 1, depending on T, such that if u, ρ 1, E V δ, then the initial-value problem 1.1a-1.1c with 1.2 as well as the initial-value problem have a uniue strong solution on R 3, T, satisfying u, ρ, F W, T W 1,p, T ; R 3 2 R 3 p, T ; W 1, R 3 W 1,2 R 3 1. Remark 3.1. The solution constructed in Section 4 later does not reuire the incompressible condition 1.1d, thus we have the local existence for both the compressible and incompressible cases. The solutions in Theorem 3.1 is local in time since δ = δ T implies that T is finite for a given δ 1. Remark 3.2. An interesting case is the case p. method and Theorem in [3], we have and W 21 1 p, B 21 1 p p = X 21 1 p p, W 1 1 p, B 1 1 p p Then, if we replace the functional space V p, V p, := Theorem 3.1 is still valid. = Z 1 1 p p. Indeed, by the real interpolation in Theorem 3.1 by W 21 1 p, R 3 3 W 1 1 p, R 3 3 W 1, R 3 1, The above local existence, with the aid of global estimates and the suitable choice of the smallness of the initial data, will result in the following global existence: Theorem 3.2 Global existence for viscoelastic fluids. Assume that p = 2 and 3, 6]; There exists a δ >, such that, for any δ with < δ δ 1, the initial data satisfies u, ρ 1, E V δ 2, 3.1 with compatibility condition 2.11, divu =, and divρ F = ; 3.2 In addition, the initial data satisfies 1 2 ρ u ρ E γ 1 ργ γρ + γ 1 dx δ 4, 3.3 and R 3 u 2 + u u 2 + u 2 + ρ 2 + E 2 δ4. 3.4

8 8 XIANPENG HU AND DEHUA WANG Then, there exists a µ > depending only on and determined by 6.18, such that if < µ µ, the initial-value problem has a uniue strong solution defined on R 3, with u, ρ, F W, T W 1,p, T ; R 3 2 R 3 p, T ; W 1, R 3 W 1,2 R 3 1, for each T >. Furthermore, the solution satisfies sup ut, ρt 1, Et V p, < δ. 3.5 t [, Remark 3.3. Notice that if > 3, then by Theorem 5.15 in [1], the imbedding W 1, R 3 C B R3 is continuous. Here, the notation C B R3 means the spaces of bounded, continuous functions in R 3. Hence the condition 3.1 implies that, if we choose δ sufficiently small, by Sobolev s imbedding theorem, there exists a positive constant C such that ρ C >, for a.e. x R Remark 3.4. Under assumption 3.2, the authors in [17, 18] showed that the property will insist in time, that is, for all t, divρf =. Remark 3.5. If the density ρ is a constant, for simplicity, ρ = 1, Theorems 3.1 and 3.2 become the analogous results of the homogeneous incompressible viscoelastic fluids In other words, following our argument in this paper, we can recover the global existence of strong solutions, or even classical solutions, of the homogeneous incompressible viscoelastic fluids near its euilibrium. An important conseuence of Theorem 3.2 is the case as E = and disregarding the euation 1.1c. In this case, one has the global existence of density-dependent incompressible Navier-Stokes euations, since the term on the right-hand side of 1.1b can be incorporated into the pressure. We state the result without proof as follows. Corollary 3.1 Global existence for Navier-Stokes euations. Assume that p = 2 and 3, 6]; There exists a δ >, such that, for any δ with < δ δ 1, the initial data satisfies u, ρ 1 V δ 2, with divu = ; In addition, the initial data satisfies 1 2 ρ u γ 1 ργ γρ + γ 1 dx δ 4. R 3 Then, the initial-value problem for the density-dependent incompressible Navier-Stokes euations has a uniue strong solution defined on R 3, such that u, ρ W, T W 1,p, T ; R 3 2 R 3 p, T ; W 1, R 3 W 1,2 R 3, for each T >. Furthermore, the solution satisfies sup ut, ρt 1 V p, < δ. t [,

9 STRONG SOUTIONS TO INCOMPRESSIBE VISCOEASTIC FUIDS 9 Remark 3.6. The similar result to Corollary 3.1 was shown in [8]. It is worthy of noticing that there is no assumption on the amplitude of the viscosity µ or condition 3.4 because those two conditions are only useful when dealing with the dissipation of the transformation gradient F. Actually, as seen from the argument later in this paper, Corollary 3.1 still holds if p 2 and > ocal Existence In this section, we prove the local existence of strong solution in Theorem 3.1. To this end, we introduce the following new variables by scaling s := ν 2 t, y := νx, vy, s := 1 ux, t, ry, s := ρx, t, Gy, s := Ex, t, ν where ν > will be determined later. Then, system 1.1, with 1.1c replaced by 1.3, becomes r t + divrv =, rv t + div rv v µ v + ν 2 P = ν 2 div G t + v G = vg + v, divv =. ri + GI + G, 4.1a 4.1b 4.1c 4.1d From 2.1, one has k G ij + G lk l G ij = j G ik + G nj n G ik. 4.2 Thus, if we denote by G i the i-th row of the matrix G or the i-th component of the vector G, then 4.2 becomes curl G i = G nj n G ik G lk l G ij. 4.3 The proof of local existence of strong solution with small initial data will be carried out through three steps by using a fixed point theorem. Instead of working on 1.1 directly, we will work on 4.1. We note that 4.1 is just a scaling version of 1.1. It can be seen from the argument below that we only need to verify the local existence in W p,, T, < T T, while initial data belongs to V p, Solvability of the density with a fixed velocity. et A j x, t, j = 1,..., n, be symmetric m m matrices in R n, T, fx, t and υ x be m-dimensional vector functions defined in R n, T and R n, respectively. For the following initial-value problem: n t υ + A j x, t j υ + Bx, tυ = fx, t, 4.4 i=1 υx, = υ x, we have

10 1 XIANPENG HU AND DEHUA WANG emma 4.1. Assume that A j [ C, T ; H s R n C 1, T ; H s 1 R n ] m m, j = 1,..., n, B C, T, H s 1 R n m m, f C, T, H s R n m, υ H s R n m, with s > n is an integer. Then there exists a uniue solution to 4.4, i.e, a function satisfying 4.4 pointwise. υ [ C[, T, H s R n C 1, T, H s 1 R n ] m Proof. This lemma is a direct conseuence of Theorem 2.16 in [26] with A x, t = I. To solve the density with respect to the fixed velocity, we have emma 4.2. Under the same conditions as Theorem 3.1, there is a uniue strictly positive function r := Sv W 1,p, T ; R 3, T ; W 1, R 3 which satisfies the continuity euation 4.1a and r 1, T ; R 3. Moreover, the density satisfies r,t ; R 3 CT, v W,T r R 3 + 1, 4.5 and the norm Sv 1 W 1, R 3 t is a continuous function in time. Here, and in what follows, C stands for a generic positive constant, and in some case, we will specify its dependence on parameters by the notation C. Proof. For the proof of the first part of this lemma, we refer the reader to Theorem 9.3 in [26], or the first part of the proof for emma 4.3 below. The positivity of density follows directly from the observations: by writing 4.1a along characteristics d dtxt = v, d rt, Xt = rt, Xtdivvt, Xt, X = x, dt and with the help of Gronwall s ineuality, inf x ρ exp t divvt R 3 dx rt, x sup ρ exp x t divvt R 3 dx. Now, we can assume that the continuity euation holds pointwise in the following form: t r + rdivv + v r =.

11 STRONG SOUTIONS TO INCOMPRESSIBE VISCOEASTIC FUIDS 11 Taking the gradient in both sides of the above identity, multiplying by r 2 r and then integrating over R 3, we get, by Young s ineuality 1 d dt r R 3 r divv dx + r r 1 divv dx R 3 R 3 + v r dx 1 v r dx R 3 R 3 r R 3 v R 3 + r 4.6 R 3 divv R divv r dx + r R 3 divv R 3 R 3 C r R 3 v W 2, R 3 + r R 3 divv R 3, since > 3. Then 4.5 follows from Gronwall s ineuality. Finally, noting from 4.6 and 4.5 that d dt r R 3 1, T, and hence d dt r 1 R 3 1, T, which together with 4.5 implies that r 1 t is continuous in time, and hence, R 3 r 1 R 3 t is continuous in time. Similarly, from the continuity euation, we know that t r 1 = divr 1v divv p, T ; R 3, which, together with the fact r 1, T ; R 3, yields r 1 C[, T ]; R 3. Hence, the uantity r 1 W 1, R 3 t is continuous in time. The proof of emma 4.2 is complete Solvability of the deformation gradient with a fixed velocity. Due to the hyperbolic structure of 4.1c, we can apply emma 4.1 again to solve the deformation gradient G in terms of the given velocity. For this purpose, we have emma 4.3. Under the same conditions as Theorem 3.1, there is a uniue function G := T v W 1,p, T ; R 3, T ; W 1, R 3 which satisfies the euation 4.1c. Moreover, the deformation gradient satisfies G,T ; R 3 CT, v W,T G R 3 + 1, 4.7 and, the norm G W 1, R 3 t is a continuous function in time. Proof. First, we assume that v C 1, T ; C R3, G C R3. Then, we can rewrite 4.1c in the component form as t G j + v G j = v G j + v j, for all 1 j 3. Applying emma 4.1 successively with A k x, t = v k x, ti for all 1 k 3, Bx, t = v, and fx, t = v j, we get a solution { } G l=1 C 1, T, H l 1 R 3, T ; H l R 3, which implies, by the Sobolev imbedding theorem, G k=1 C1, T ; C k R 3 = C 1, T ; C R

12 12 XIANPENG HU AND DEHUA WANG Next, for v W p,, T, there are two seuences: such that v n C 1, T ; C R 3, G n C R 3, v n v in W, T, G n G in W 1, R 3, thus v n v in CB, a, T for all a > where B, a denotes the ball with radius a and centered at the origin. According to the previous result 4.8, there are a seuence of functions {G n } n=1 C1, T ; C R 3 satisfying t G n + v n G n = v n G n + v n, 4.9 with G n = G n. Multiplying 4.9 by G n 2 G n, and integrating over R 3, using integration by parts and Young s ineuality, we obtain, 1 d G n dx dt R 3 = 1 v n G n dx + v n G n 2 G 2 ndx + v n G n 2 G n dx R 3 R 3 R G n v n + v n + v n. From Gronwall s ineuality, one obtains, G n dx R 3 t t G n dx + v n exp + 1 v n + v n dτ ds R 3 t exp + 1 v n + v n ds G n dx + R 3 Thus, t t v n ds exp + 1 v n + v n ds. G n,t ; R 3 CT, v p,t ;W 2, R 3 G R <. 4.1 Hence, up to a subseuence, we can assume that the seuence {v n } was chosen so that G n G weak-* in, T ; R 3.

13 STRONG SOUTIONS TO INCOMPRESSIBE VISCOEASTIC FUIDS 13 Taking the gradient in both sides of 4.9, multiplying by G n 2 G n and then integrating over R 3, we get, with the help of Hölder s ineuality and Young s ineuality, 1 d dt G n R 3 G n v n dx + G n G n 1 v n dx R 3 R 3 + v n G n dx 1 v n G n dx R 3 R 3 + v n G n 1 dx R 3 G n v n dx + G n G n 1 v n dx 4.11 R 3 R 3 + v n G n dx + 1 v n G n dx R 3 R 3 + v n G n 1 dx R 3 C G n R 3 v n W 2, R 3 + G n R v n W 2, R 3 C G n R 3 v n W 2, R 3 + G n R v n W 2, R 3, since > 3. Using Gronwall s ineuality and 4.1, we conclude from 4.11 that G n,t ; R 3 CT, v n W,T G n R 3 + 1, and hence, Therefore, G,T ; R 3 lim inf n G n,t ; R 3 CT, v W,T G R G n,t ;W 1, R 3 CT, v p,t ;W 2, R 3, G W 1, R 3 <. Furthermore, since > 3, we deduce G Q T and G Q T CT, v p,t ;W 2, R 3, G W 1, R 3 <. Passing to the limit as n in 4.9, we show that 4.1c holds at least in the sense of distributions. Therefore, t G p, T ; R 3, then G W 1,p, T ; R 3, and hence G C[, T ]; R 3. Finally, to show that the uantity G W 1, R 3 t is continuous in time, it suffices to show that G R 3 is continuous in time. Indeed, from 4.11, we know that d dt G R 3 t p, T, which, with 4.12, implies that G R 3 complete. C[, T ]. The proof of emma 4.3 is

14 14 XIANPENG HU AND DEHUA WANG 4.3. ocal existence via the fixed-point theorem. In order to solve locally system 4.1, we need to use the following fixed point theorem cf in [26]: Theorem 4.1 Tikhonov Theorem. et M be a nonempty bounded closed convex subset of a separable reflexive Banach space X and let F : M M be a weakly continuous mapping i.e. if x n M, x n x weakly in X, then F x n F x weakly in X as well. Then F has at least one fixed point in M. Now, let us consider the following operator ω := dω dt µ ω, ω Wp,, T. One has the following theorem by the maximal regularity of parabolic euations; see Theorem 9.2 in [26], or euivalently Theorem and Remark in [2] page 188. Theorem 4.2. Given 1 < p <, ω V p, and f p, T ; R 3 3, the Cauchy problem ω = f, t, T ; ω = ω, has a uniue solution ω := 1 ω, f W p,, T, and ω W p,,t C f p,t ; R 3 + ω V p, where C is independent of ω, f and T. Moreover, there exists a positive constant c independent of f and T such that ω W p,,t c sup t,t ωt V p,. Notice that Theorem 4.2 implies that the operator is invertible. Thus we define the operator Hv : W p,, T W p,, T by Hv := 1 v, t 1 Svv divsvv v + ν 2 P 1 P Sv + ν 2 divsvi + T vi + T v Then, solving system 4.1 is euivalent to solving To solve 4.14, we define Then, we prove first the following claim:, v = Hv B R := {v W p,, T : v W p,,t R}. emma 4.4. There are ν, T >, and < R < 1 such that HB R B R. Proof. et T >, < R < 1 and v B R. Since Sv solves 4.1a, we can rewrite operator H as Hv = 1 v, 1 Sv t v Svv v + ν 2 P 1 P Sv + ν 2 divsvi + T vi + T v. 4.15

15 STRONG SOUTIONS TO INCOMPRESSIBE VISCOEASTIC FUIDS 15 Thus, it suffices to prove that the terms in the above expression are small in the norm of p, T ; R 3 3. First of all, we begin to deal with the first term by letting r := Sv 1. Thus, r satisfies the euations { t r + divrv =, rx, = r 1. Repeating the argument in Section 4.2 again, we obtain r Q T C r,t ;W 1, R 3 C r 1 W 1, R 3 CT, v W,T r 1 W 1, R 3 CT, R CT R, where, by the formula of change of variables, we deduce that and r 1 R 3 ν 3 ρ 1 R 3 R, r R 3 ν 3 ρ R 3 R, if ρ 1 R 3 is small enough and ν > 1 is large enough. Hence, due to the assumption v B R, we obtain Secondly, by the Sobolev imbedding, 1 Sv t v p,t ; R 3 CT R v v dx v R 3 R 3 v R 3 C v R 3 v W 2, R 3, and thus, since W 1,p, T ; R 3 C[, T ]; R 3, we deduce T p v v T dx ds C v p R 3 R 3 v p W 2, R 3 ds C v p,t ; R 3 v p p,t ;W 2, R 3 Therefore, we get v 2p W,T CR2p. Svv v p,t ; R 3 CR Thirdly, for the term P Sv, we can estimate it as follows P Sv p,t ; R 3 CT sup { P η : CT 1 η CT } r R Fourthly, for the term divsvi + T vi + T v, we have divsvi + T vi + T v Sv I + T v 2 + 2Sv T v I + T v, and hence, 4.18 divsvi + T vi + T v p,t ; Sv I + T v 2 p,t ; + 2 Sv T v I + T v p,t ; CT M, 4.19

16 16 XIANPENG HU AND DEHUA WANG with M = max { G W 1, + 1, G R 3 + 1, r W 1, + 1, r R } 3 <. Combining together 4.16, 4.17, 4.18, 4.19, using the Theorem 4.2, and assuming parameter ν sufficiently large and R < 1 sufficiently small, we get The proof of emma 4.4 is complete. Thus, it is only left to show the following: Hv W,T CT R 2 + ν 2 R. emma 4.5. The operator H is weakly continuous from W p,, T into itself. Proof. Assume that v n v weakly in W p,, T, and set r n := Sv n, G n := T v n, then {r n } n=1 and {G n} n=1 are uniformly bounded in, T ; W 1, R 3 W 1,p, T ; R 3 by emmas 4.1 and 4.3. Hence, up to a subseuence, we can assume that r n r and G n G weakly in, T ; W 1, R 3 W 1,p, T ; R 3 and then strongly in C, T B, a for all a >. And at least the same convergence holds for v n. Thus, 4.1a and 4.1c follow easily from above convergence. Since r n r weakly* in, T ; W 1, R 3 W 1,p, T ; R 3, we can assume that P Sv n Sv n P Sv Sv weakly in p, T ; R 3 and hence, 1, P Sv n 1, P Sv weakly in W, T, since the strong continuity of 1 from p, T ; R 3 into W, T and the linearity of the operator imply also the weak continuity in these spaces. Similarly, since t v n t v weakly in p, T ; R 3 and r n r in C, T B, a for all a >, we have r e r n t v n r e r t v weakly in p, T ; R 3 and conseuently 1, r e r n t v n 1, r e r t v weakly in W, T. Since v n v weakly in W 1,p, T ; W 1, R 3 p, T ; W 1, R 3 which is compactly imbedded in to C[, T ]; B, a for all a >, we can assume that v n v strongly in, T ; B, a for all a >, and then weakly in p, T ; R 3. Hence Sv n v n v n Svv v 1, Sv n v n v n 1, Svv v weakly in W, T. Finally, due to the facts that r n r and G n G weakly in, T ; W 1, R 3 W 1,p, T ; R 3 and strongly in C, T B, a for all a >, we deduce that divsv n I + T v n I + T v n divsvi + T vi + T v weakly in p, T ; R 3. Therefore, 1, divsv n I + T v n I + T v n weakly in W, T. Thus, we can conclude that Hv n Hv weakly in W, T, 1, divsvi + T vi + T v due to the weak continuity of map 1 v,. The proof of emma 4.5 is complete.

17 STRONG SOUTIONS TO INCOMPRESSIBE VISCOEASTIC FUIDS 17 Therefore, by Theorem 4.1, there exists at least a fixed point v = Hv B R W, T, 4.2 of mapping H. The fixed point v provides a local in time solution ρ, u, F of system 1.1 near its euilibrium through the scaling with ν sufficiently large. The proof of the local existence in Theorem 3.1 is complete. The uniueness will be proved in the next section. 5. Uniueness In this section, we prove the uniueness of the local solution found in the previous section. Notice that, the argument in Section 4 yields that t v 2, T ; 2 R 3, r 2, T ; 2 R 3, G 2, T ; 2 R 3. Hence, using the interpolation, we deduce that t v p, T ; 3 R 3, r p, T ; 3 R 3, G p, T ; 3 R 3, where 1 = θ p θ p, 1 3 = θ θ, for some θ [, 1]. Now, assume that v 1, v 2 satisfying 4.2 for some T >. et r := Sv 1 Sv 2, v := v 1 v 2, G := T v 1 T v 2, with a little abuse of notations however, there should be no confusion in the rest of this section. Then, we have { t r + v 1 r + v Sv 2 + rdivv 1 + Sv 2 divv =, 5.1 r =. Multiplying 5.1 by r, and integrating over R 3, we get 1 d 2 dt r 2 1 r 2 divv 2 1 dx + v Sv2 r + r 2 divv 1 + rsv 2 divv dx =, 2 R 3 R 3 which yields d dt r 2 2 R 3 divv 1 r ε v Cε Sv 2 r ε v 2 2 R 3 + Cε Sv 2 2 r 2 2 divv 1 r ε v Cε Sv r ε v 2 2 R 3 + Cε Sv 2 2 r 2 2 η 1 ε r ε v 2 2 R 3, 5.2 where ε >, η 1 ε = divv 1 + Cε Sv Sv 2 2. Similarly, from 4.1c, we obtain { t G + v 1 G + v G 2 = v 1 G + vg 2 + v, G =. 5.3

18 18 XIANPENG HU AND DEHUA WANG Multiplying 5.3 by G, and integrating over R 3, we get 1 d 2 dt G 2 1 G 2 divv 2 1 dx + v T v 2 : Gdx 2 R 3 R 3 = G 2 v 1 dx + vt v 2 : Gdx + v : Gdx, R 3 R 3 R 3 which yields d dt G 2 2 R 3 divv 1 G ε v Cε T v 2 G ε v 2 2 R 3 + Cε T v 2 2 G ε v Cε G 2 2 divv 1 G ε v Cε T v G ε v 2 2 R 3 + Cε T v 2 2 G ε v Cε G 2 2 η 2 ε G ε v 2 2 R 3, where η 2 ε = divv 1 + Cε T v T v For each v j, j = 1, 2, we deduce from 4.1b that Sv j t v j µ v j = Sv j v j v j P Sv j + divsv j I + T v j I + T v j, v j = v. Subtracting these euations, we obtain, Sv 1 t v 1 Sv 2 t v 2 µ v = Sv 1 v 1 v 1 + Sv 2 v 2 v 2 P Sv 1 + P Sv 2 + divsv 1 I + T v 1 I + T v 1 divsv 2 I + T v 2 I + T v Since Sv 1 v 1 v 1 + Sv 2 v 2 v 2 = Sv 1 v v 1 Sv 1 Sv 2 v 2 v 1 Sv 2 v 2 v, and Sv 1 I + T v 1 I + T v 1 Sv 2 I + T v 2 I + T v 2 = Sv 1 GI + T v 1 + ri + T v 2 I + T v 1 + Sv 2 I + T v 2 G, we can rewrite 5.5 as Sv 1 t v µ v = r t v 2 Sv 1 v v 1 Sv 2 v 1 Sv 2 v 2 v P Sv 1 + P Sv 2 + divsv 1 GI + T v 1 + ri + T v 2 I + T v 1 + Sv 2 I + T v 2 G. 5.6

19 STRONG SOUTIONS TO INCOMPRESSIBE VISCOEASTIC FUIDS 19 Multiplying 5.6 by v, using the continuity euation 1.1a and integrating over R 3, we deduce that 1 d Sv 1 v 2 dx + µ v 2 dx 2 dt R 3 R 3 1 = R 3 2 Sv 1v 1 v v r t v 2 v Sv 1 v v 1 v Sv 2 v 1 v Sv 2 v 2 vv P Sv 1 v + P Sv 2 v with Sv 1 GI + T v 1 + ri + T v 2 I + T v 1 + Sv 2 I + T v 2 G vdx ε v Cε Sv 1 2 v 1 2 v ε v Cε t v r Sv 1 v 1 v v 2 v 1 r v ε v Cε Sv 2 2 v 2 2 v ε v Cεsup{P η : CT 1 η CT } 2 r ε v Cε Sv T v 1 2 G Sv T v 2 2 G r T v T v 2 2 5ε v η 3 ε r v G 2 2 η 3 ε = Cε Sv 1 2 v Cε tv Sv 1 v v 2 v 1 + Cε Sv 2 2 v Cεsup{P η : CT 1 η CT } 2 + Cε Sv T v Sv T v T v T v 2 2. Summing up 5.2, 5.4, and 5.7, by taking ε = µ 2, we obtain d Sv 1 v 2 + r 2 + G 2 dx + µ v 2 dx dt R 3 R 3 2η 3 ε + η 2 ε + η 1 ε v 2 + r G ηε, t Sv 1 v 2 + r 2 + G 2 dx, R 3 with η 3 ε + η 2 ε + η 1 ε ηε, t = min{min x R 3 Sv 1 x, t, 1} It is a routine matter to establish the integrability with respect to t of the function ηε, t on the interval, T. This is a conseuence of the regularity of v 1, v 2 W, T and the estimates in emmas 4.2 and 4.3 for Sv i, T v i with i = 1, 2. Therefore, 5.8, combining with Gronwall s ineuality, implies R 3 Sv1 v 2 + r 2 + G 2 dx =, for all t, T, 5.9

20 2 XIANPENG HU AND DEHUA WANG and conseuently v, r, G. Thus, the uniueness in Theorem 3.1 is established. 6. Global A Priori Estimates Up to now, we prove that for any given T, we can find a uniue solution to the scaling system 4.1. That is, we have proved the local existence of solution to the viscoelastic fluid system 1.1 and its uniueness. In order to establish the global existence for the uniue solution we constructed in the previous sections, we need to obtain some uniform a priori estimates which are independent of the time T. To simplify the presentation, we will focus on the case ν = 1, that is, system 1.1. We introduce the new variable: σ := ln ρ. Then, we have emma 6.1. Function σ satisfies t σ + u σ =, 6.1 in the sense of distributions. Moreover, the norm σt R 3 is continuous in time. Proof. We follow the argument in [26] Section 9.8 by denoting σ ε = S ε σ, where S ε is the standard mollifier in the spatial variables. Then, we have with t σ ε + u σ ε = R ε, R ε = u σ ε S ε u σ = u σ ε S ε u σ + σ ε u S ε σ u =: R 1 ε + R 2 ε. 6.2 Since σ, T ; R 3 and u p, T ; W 1, R 3, we deduce from emma 6.7 in [26] cf. emma 2.3 in [21] that R 1 ε as ε. Moreover, σ ε σ u 1,T ; R 3 σ σ ε p p 1,T ; R 3 u p,t ; R 3, and S ε σ u σ u in 1, T ; R 3 since σ u p, T ; R 3. Thus, we have R 2 ε in p, T ; R 3. Then, taking the limit as ε in 6.2, we get 6.1. Multiplying 6.1 by σ 2 σ, and integrating over R 3, we get 1 d dt σ R 3 R = j u k σ j σ k σ divu σ dx u σ + 1 divu σ C u W 2, σ. Dividing the above ineuality by σ 1, we obtain d dt σ C u W 2, σ.

21 STRONG SOUTIONS TO INCOMPRESSIBE VISCOEASTIC FUIDS 21 Since σ, T ; R 3 d, dt σ p, T. Thus, σ C, T. The proof of emma 6.1 is complete. For a given R = δ 1 as in Section 4, if the initial data satisfies u, ρ 1, E V δ 2 with < δ min{ 1 3, δ }, let T R be the maximal time T such that there is a solution of the euation u = Hu in B R. By virtue of emma 4.2, emma 4.3 and emma 6.1, we know that Su 1 W 1, R 3, σ and T u W 1, are continuous in the interval [, T R. On the other hand, under the assumptions on initial data and Remark 3.3, we know, if δ is sufficiently small, then σ R 3 1 C ρ R 3 δ Hence, there exists a maximum positive number T 1 such that max { Su 1 W 1,t, σ t, T u W 1,t } R 1 for all t [, T 1 ]. 6.3 Now, we denote T = min{t R, T 1 }. Without loss of generality, we assume that T <. Since > 3, we have ρ 1 R 3 C ρ 1 W 1, R 3 C R < 1 2, if R is sufficiently small. Hence, one obtains 1 2 ρ 3 2. On the other hand, for any given t, T, we can write and conseuently, ut p = u p + = u p + p t t t u p + p δ 2p + p t u p ds δ 2p + p Rp, u,t; Similarly, we have, for all t [, T ], d ds us p ds ut p us R 2 us s utdx 3 us p 1 s u ds p 1 p t 1 s u p p ds dt δ 2p + p Rp 1 p CR, t, T. 6.4 u,t; 2 CR.

22 22 XIANPENG HU AND DEHUA WANG 6.1. Dissipation of the deformation gradient. The main difficulty of the proof of Theorem 3.2 lies in the lack of estimates on the dissipation of the deformation gradient. This is partly because of the transport structure of euation 1.1c. It is worthy of pointing out that it is extremely difficult to directly deduce the dissipation of the deformation gradient. Fortunately, for the viscoelastic fluids system 1.1, as we can see in [6, 15, 16, 17, 18, 19, 2], some sort of combinations between the gradient of the velocity and the deformation gradient indeed induce some dissipation. To make this statement more precise, we rewritten the momentum euation 1.1b as, using 1.1a t u µ u dive = ρu u P ρ + divρi + E and prove the following estimate: + divρ 1E + divρee + 1 ρ t u, 6.5 emma 6.2. E p,t ; R 3 Cp,, µ R + R σ p,t ; R Proof. Now we introduce the function Z 1 x, t as Z 1 := E dive = Ex ydivedy, 6.7 R 3 where E is the fundamental solution of the aplacian in R 3. Then, 6.5 becomes t u µ u 1 µ Z 1 = F 1, 6.8 where, with the help of Remark 3.4, F 1 = ρu u P ρ + divρ 1E + divρee + 1 ρ t u. Also, from 1.3, we have Z 1 = Ex ydiv E t R 3 t dy = Ex ydiv u + ue u Edy. R 3 From 6.8 and 6.9, we deduce, denoting Z = u 1 µ Z 1, 6.9 t Z µ Z = F := F 1 F 2, 6.1 where F 2 = 1 µ u + 1 E div ue u E. µ Euation 6.1 with Theorem 4.2 implies that Z W,T Cp, Z X 21 p 1 + F p,t ; R 3 p Cp, 6.11 R + F p,t ; R 3.

23 1 µ R + CR STRONG SOUTIONS TO INCOMPRESSIBE VISCOEASTIC FUIDS 23 Next, we estimate F i p,t ; R 3, i = 1, 2, term by term. Indeed, for F 1, using 6.4, we have F 1 p,t ; R 3 ρ Q T u,t ; R 3 u p,t ; R 3 + α E p,t ; Ω + α σ p,t ; Ω E Q T + ρ 1 Q T E p,t ; R 3 + σ p,t ; R 3 E Q T + σ p,t ; R 3 E 2 Q T + E Q T E p,t ; R 3 + ρ 1 Q T t v p,t ; R 3 2R u p,t ;W 2, R 3 + α E p,t ; R α σ p,t ; R 3 E Q T + R σ p,t ; R 3 + R σ p,t ; R 3 + R E p,t ; R 3 + R 3 2 R α E p,t ; R 3 + R σ p,t ; R 3 + R E p,t ; R 3. Here, α = sup { xp x : 1 2 x 3 2} and in the first ineuality, we used the identity ρ = ρdive ρe due to Remark 3.4. And, for F 2, noting that E C 1, and from integrating by parts, x 2 we have F 2 1 µ u + C 1 ue u E, µ x 2 with ue u E p,t ; 3 +3 R 3 u p,t ; 3 R 3 E,T ; R 3 + u p,t ; 3 R 3 E,T ; R 3 R 3 2. Hence, one can estimate, by p estimate of Riesz potential, F 2 p,t ; R 3 1 µ u p,t ; R 3 + C 1 µ ue u E x 2 1 µ R + C µ ue u E p,t ; 3 3+ R 3 p,t ; R 3 Therefore, from 6.12 and 6.13, we obtain F p,t ; R 3 R µ R + α E p,t ; R 3 + R σ p,t ; R 3 + R E p,t ; R

24 24 XIANPENG HU AND DEHUA WANG Ineualities 6.11 and 6.14 imply that Z p,t ;W 2, R 3 Cp, 2R + 1 µ R + α E p,t ; R 3 + R σ p,t ; R 3 + R E p,t ; R Hence, we have, from 6.7 dive p,t ; R 3 µ Z p,t ;W 2, R 3 + u p,t ;W 2, R 3 Cp, µ 3R + 1 µ R + α E p,t ; R 3 + R σ p,t ; R 3 + R E p,t ; R On the other hand, from the identity 4.3, we deduce that curl E i p,t ; R 3 2 E Q T E p,t ; R 3 C E,T ;W 1, R 3 E p,t ; R 3 C R E p,t ; R Combining together 6.16 and 6.17, we obtain E p,t ; R 3 Cp, µ 3R + 1 µ R + α E p,t ; R 3 + R σ p,t ; R 3 + R E p,t ; R 3, and hence, by choosing R 1 2 and the assumption Cp, µα < 1, 6.18 one obtains 6.6. The proof of emma 6.2 is complete. Remark 6.1. Notice that, in view of the above argument, estimate 6.6 is actually valid for all t [, T ], that is, for all t [, T ], E p,t; R 3 Cp,, µ R + R σ p,t; R Dissipation of the gradient of the density. To make Theorem 3.2 valid, we need further the uniform estimate on the dissipation of the gradient of the density. emma 6.3. For any t, T, σ p,t; R 3 Cp,, µr. 6.19

25 STRONG SOUTIONS TO INCOMPRESSIBE VISCOEASTIC FUIDS 25 Proof. Multiplying 1.1b by σ σ 2 and integrating over R 3, we obtain µ + λ d dt σ + ρp ρ σ dx R 3 = µ u σ σ 2 dx ρ t u σ σ 2 dx R 3 R 3 ρu u σ σ 2 dx µ + λ u σ σ σ 2 dx R 3 R 3 + divρi + EI + E σ σ 2 dx. R 3 We estimate the right-hand side of 6.2 term by term, u σ σ 2 dx u σ 1 ; R 3 ρ t u σ σ 2 dx tu σ 1 ; R 3 ρu u σ σ 2 dx u u σ 1 R 3 u u W 2, σ 1 ; u σ σ σ 2 dx = j u k σ k σ j σ R R 2 dx + u k j k ln ρ j ln ρ σ 2 dx 3 3 R 3 = j u k σ k σ j σ 2 dx + 1 u k k σ 2 σ 2 dx R 3 2 R 3 = j u k σ k σ j σ 2 dx + u k k σ σ 1 dx R 3 R 3 = j u k σ k σ j σ 2 dx + 1 u k k σ dx R 3 R 3 = j u k σ k σ j σ 2 dx 1 σ divudx R 3 R 3 C u σ C u W 2, σ, and, due to 2.9, we can rewrite divρi + EI + E = ρe i + E i e j + E j i x j = e i + E i ρe j + E j + ρe j + E j e i + E i x j x j = ρe j + E j E i x j, then one has divρi + EI + E σ σ 2 dx = ρe j + E j E i σ i σ 2 dx R 3 R 3 x j E I + E σ 1 2 E R 3 σ

26 26 XIANPENG HU AND DEHUA WANG On the other hand, we have ρp ρ = P 1 + ρ 1 1 P ηρ ηρ 1 + 1P ηρ dη, for ρ close to 1 and conseuently { ρp ρ P 1 ρ 1 sup fx : 1 2 x 3 } 2 C { R sup fx : 1 2 x 3 } C R, 2 where fx = P x + xp x. Thus, from 6.2, we obtain µ + λ d dt σ + P 1 σ σ 1 u + t u + u W 2, u + C u W 2, σ + C E R 3 + R σ C σ 1 u + t u + u W 2, u + u W 2, σ + E R 3 + R σ, and hence, by assuming that R 1, one obtains µ + λ d dt σ P 1 σ 6.21 u + t u + u W 2, u + u W 2, σ + E. C σ 1 Multiplying 6.21 by σ p, we obtain µ + λ p d dt σ p P 1 σ p C σ p 1 u + t u + u W 2, u + u W 2, σ + E.

27 STRONG SOUTIONS TO INCOMPRESSIBE VISCOEASTIC FUIDS 27 Integrating the above ineuality over the interval, t, one obtains, by using 6.6, µ + λ σt p + 1 p 2 P 1 µ + λ σ p + C p t t σ p ds p 1 σ p p t 1 ds t u p p ds 1 u p p ds W 2, + u,t; R 3 + σ,t; R t + E p,t; R 3 µ + λ σ p + Cp,, µ R σ p p p,t ; R 3 t p 1 + Cp,, µr σ p p ds 1 + σ,t; R 3 + u,t; R 3. and hence, by letting R be so small such that Cp,, µ R < 1 4, one obtains µ + λ σt p + 1 p 4 P 1 t µ + λ σ p + Cp,, µr p σ p ds t Plugging 6.4 into 6.22, we obtain µ + λ σt p + 1 p 4 P 1 t µ + λ σ p + Cp,, µr p Then, Young s ineuality yields p 1 σ p p ds 1 + σ,t; + u,t;. σ p ds t µ + λ σt p + 1 p 8 P 1 for all t < T. Now, we let R be so small that p 1 σ p p ds 1 + σ,t; R 3. t σ p ds µ + λ δ 3 2 p + Cp,, µr p 1 + σ p,t; R 3 p, Cp,, µ 1 p R 1 + R < Due to the fact that σ R 3 δ 3 2, we can assume that σt < 1 2 R in some maximal interval, t max, T. If t max < T, then, σt max = 1 2 R and by 6.23, 1 R = σtmax Cp,, µ 2 1 p R1 + R < 1 R, 2

28 28 XIANPENG HU AND DEHUA WANG which is a contradiction. Hence, t max = T and σ 1 2 R, for all t [, T ] Thus, by 6.23, one obtains The proof of emma 6.3 is complete. We remark that, from 6.6 and 6.19, one has E p,t ; R 3 Cp,, µr Global Existence In this section, we prove the global existence in Theorem 3.2. Define { T max := sup T > : u W, T with u = Hu, such that, u W,T R, Su 1,T ;W 1, R, σ,t ; R, and T u,t ;W 1, } R, where R was constructed in the previous section. If T max =, we are done. From now on, we assume that T max < Uniform estimates in time. We now establish some estimates which are uniform in time T. First we prove the following energy estimates: emma 7.1. Under the same assumptions as Theorem 3.1, we have where C is a constant independent of T, T max. Proof. First we recall that and u 2,T ; 2 R 3 CR 2, 7.1 u,t ; 2 R 3 CR 2, 7.2 E,T ; 2 R 3 CR 2, 7.3 ρ 1,T ; 2 R 3 CR 2, 7.4 u W 1,2, T ; 2 R 3 2, T ; W 2,2 R 3 ρ, E W 1,2, T ; 2 R 3 2, T ; W 1,2 Ω. Multiplying euation 1.1b by u, and integrating over R 3, we obtain, using the conservation of mass 1.1a, d 1 dt R 3 2 ρ u γ 1 ργ + γ 1 dx + µ u 2 dx R 3 = ρff : udx. R 3

29 STRONG SOUTIONS TO INCOMPRESSIBE VISCOEASTIC FUIDS 29 Here, the notation A : B means the dot product between two matrices. Thus, we have 1 R 3 2 ρ u t γ 1 ργ + γ 1 dx + µ u 2 dxds R 3 1 = R 3 2 ρ u t γ 1 ργ + γ 1 dx ρff : udxds. R 3 From the conservation of mass 1.1a, one has 1 R 3 2 ρ u t γ 1 ργ γρ + γ 1 dx + µ u 2 dxds R 3 1 = R 3 2 ρ u t γ 1 ργ γρ + γ 1 dx ρff : udxds. R 3 On the other hand, due to euations 1.1c and 1.1a, we have ρ F 2 = ρ t t F 2 + 2ρF : F t = ρ t F 2 + 2ρF : u F u F = ρ t F 2 + 2ρF : u F ρu F 2 = ρ t F 2 + 2ρF : u F + divρu F 2 divρu F 2 = 2ρF : u F divρu F Integrating 7.6 over R 3, we arrive at 1 d ρ F 2 dx = ρf : u Fdx dt R 3 R 3 Since t R 3 ρf : u Fdxds = t R 3 ρff : udxds, we finally obtain, by summing 7.5 and 7.6, 1 R 3 2 ρ u ρ F t γ 1 ργ γρ + γ 1 dx + µ u 2 dxds R 3 1 = 2 ρ u ρ F γ 1 ργ γρ + γ 1 dx. R 3 Thanks to Remark 3.4, we have Hence, from 1.1c and 1.1a, we have ρi + E : u =. 7.8 t ρ tre =. 7.9

30 3 XIANPENG HU AND DEHUA WANG Therefore, from 7.8, 7.9 and the conservation of mass 1.1a, we finally arrive at 1 R 3 2 ρ u ρ E t γ 1 ργ γρ + γ 1 dx + µ u 2 dxds R 3 1 = 2 ρ u ρ E γ 1 ργ γρ + γ 1 dx R 4. R 3 Since µ > is a constant and ρ [ 1 2, 3 2 ], then ineualities follow from 7.1, and ineuality 7.4 follows from 7.1 and the following straightforward ineualities: for some η >, we have { x γ η x 1 2, if γ 2, 1 γx 1 η x 1 2, if x < 2 and γ < 2. The proof of emma 7.1 is complete. Based on the uniform estimates from Section 6, we have emma 7.2. Under the same assumptions as Theorem 3.2, for any T [, T max ]. Su 1,T ;W 1, < R, σ,t ; < R, 7.11 Proof. According to 6.24, it is obvious to see that Hence, we are only left to show max σ t < R. t [,T ] max t [,T ] Su 1 W 1,t < R. Indeed, for any t, T, we have, by using 1.1a and 6.19, Sut 1 α t = ρ 1 α + d ds Sus 1 α ds = ρ 1 α + α t Sus 1 α Sus 1 2 Sus 1 s Susdx ds R 3 where ρ 1 α + α δ 2α + α t t Sus 1 α 1 s Su ds 5 6p 3 6 Sus 1 α = p 1 p t 1 ds s Su p p ds, 5 6p

31 STRONG SOUTIONS TO INCOMPRESSIBE VISCOEASTIC FUIDS 31 From 1.1a and 1.1d, we obtain t ρ p,t ; R 3 = ρ u p,t ; R 3 2 σ,t ; R 3 u p,t ; R 3 CR 2. On the other hand, from the Gagliardo-Nirenberg ineuality, we have with θ = ρ 1 R 3 C ρ 1 θ 2 R 3 ρ 1 1 θ R 3 C ρ 1 θ 2 R 3 σ 1 θ R 3, Thus, by Hölder s ineuality, 7.4, and 6.19, one has ρ 1 5 6p C ρ 1 θ 3 6,T ; R 3,T ; 2 R 3 σ 1 θ p,t ; R 3 CR, which, together 7.12 and 7.13, yields Sut 1 CR Hence, according to 6.19, we obtain, by letting R be sufficiently small, max max { } Svt 1 W t [,T ] 1, R 3, σt R 3 < R The proof of emma 7.2 is complete. emma 7.3. For each 1 l 3, E satisfies t E + u E = u E + u in the sense of distributions, that is, for all ψ C Q T, we have T E T t ψdxdt + divuψ E R 3 R 3 T = u u E + E + u E + u for any T, T max. R 3 E + u E + u 7.15 ψdxdt, Proof. The proof is a direct application of the regularization. Indeed, one easily obtains, using 1.3, t S ε E + u S ε E = S ε t E + u E + u S ε E S ε u E Differentiate 7.16 with respect to x l, we get Sε E Sε E t + u = S ε ue + u = S ε ue + u + u S ε E S ε u E. + u S ε E S ε u E u S ε E. Notice that u S ε E S ε u E = u S ε E S ε + u S ε E u E S ε u E

32 32 XIANPENG HU AND DEHUA WANG According to emma 6.7 in [26] cf. emma 2.3 in [21], we know that u u S ε E S ε E, and E u S ε S ε u E, in 1, T ; R 3 as ε. Hence, u S ε E S ε u E in 1, T ; R 3. Thus, letting ε in 7.17, we deduce t E + u E = u E + u E + u E + u, in the sense of weak solutions. The proof of emma 7.3 is complete. Using 7.15, formally we have, E E t 2 E R 3 dx = u E u u E + E + u E + u E 2 E R 3 dx C u E R 3 + E Q T u W 2, E 1 R 3 + u W 2, E 1 C u u R 3 + R u W 2, E 1 R 3 + u W 2, E We remark that the rigorous argument for the above estimate involves a tedious regularization procedure as in DiPerna-ions [9], thus we omit the details and refer the reader to

33 STRONG SOUTIONS TO INCOMPRESSIBE VISCOEASTIC FUIDS 33 [9]. Using 7.18, one obtains E p t = E p t d + E p ds s ds = E p + p [ t E p E 2 E R 3 E p p t + C E p [ u E R 3 + R u W 2, E 1 R 3 + u W 2, E 1 ]ds E p p t + C p t p 1 δ 2p + C E p p t ds p δ 2p + C R p max E + R + 1. t [,T ] ] E s s dx ds E p 1 [ u E R R u W 2,]ds 1 u p p ds max W 2, E + R + 1 t [,T ] 7.19 Taking the summation over l in 7.19 and taking the maximum over the time t, one has, p max t [,T ] E p δ 2p + C R p max E + R + 1, t [,T ] and hence, by letting R, δ be sufficiently small and using 6.25, we obtain, max t [,T ] E p δ 2p + CR p < R p. 7.2 We are now left to deal with the uantity E R 3. To this end, from the Gagliardo- Nirenberg ineuality, we have with θ = E R 3 C E θ 2 R 3 E 1 θ R 3, Thus, by Hölder s ineuality, 7.3, and 6.25 E 5 6p C E θ 3 6,T ; R 3,T ; 2 R 3 E 1 θ p,t ; R 3 CR Hence, we have the following estimate: emma 7.4. Under the same assumptions as Theorem 3.2, it holds for any T [, T max ]. Proof. By 1.3, 6.25 and 7.2, and letting E,T ; R 3 < R, 7.22 α = 5 6p ,

34 34 XIANPENG HU AND DEHUA WANG one obtains, Et α t = E α + d ds Es α ds t Es α = E α + α = E α + α t t E α + α E α + α t E α + 2α Es R 2 Es s Esdx ds 3 [ ] Es α Es R 2 Es ue + u u E dx 3 ] Es α 1 [2 u E + u ds 5 6p 3 6 Es t R E α + CR E α 1 p 1 p dt 5 6p 3 6 Es 5 6p 3 6,T ; R 3 u p,t ;W 2, R 3 p 1 p dt Then, according to 7.21, one has, for all t [, T max ],. 2 sup Et R t,t max ds 7.23 Et α δ2α + CR α < R α, 7.24 if R is sufficiently small. Thus, 7.22 follows from The proof of emma 7.4 is complete. emma 7.4, together with 7.2 and emma 7.2, gives max max { Su 1 W 1,t, σ t, T u W 1,t } CR < R t [,T ] Similarly, we can obtain max max { Su 1 W 1,2t, σ 2t, T u W 1,2t } CR < R t [,T ] 7.2. Refined estimates on ρ and E. In order to prove Theorem 3.2, we need some refined estimates on ρ 2,T ; R 3 and E 2,T ; R 3. emma 7.5. for any T, T max. ρ 2,T ; R 3 R 2, 7.27 Proof. Taking the divergence in 1.1b, and using divu =, one obtains P ρ = divdivρee + divdivρe divρu u divρ 1 t u. 7.28