Density-dependent incompressible fluids with non-Newtonian viscosity

Czechoslovak Mathematical Journal F. Guillén-González Density-dependent incompressible fluids with non-Newtonian viscosity Czechoslovak Mathematical Journal, Vol. 54 (2004), No. 3, 637–656 Persistent URL: http://dml.cz/dmlcz/127917 Terms of use: © Institute of Mathematics AS CR, 2004 Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz Czechoslovak Mathematical Journal, 54 (129) (2004), 637–656 DENSITY-DEPENDENT INCOMPRESSIBLE FLUIDS WITH NON-NEWTONIAN VISCOSITY            , Sevilla (Received November 16, 2001) Abstract. We study the system of PDEs describing unsteady flows of incompressible fluids with variable density and non-constant viscosity. Indeed, one considers a stress tensor being a nonlinear function of the symmetric velocity gradient, verifying the properties of pcoercivity and (p−1)-growth, for a given parameter p > 1. The existence of Dirichlet weak solutions was obtained in [2], in the cases p⩾12/5 if d= 3 or p⩾2 if d= 2, dbeing the dimension of the domain. In this paper, with help of some new estimates (which lead to point-wise convergence of the velocity gradient), we obtain the existence of space-periodic weak solutions for all p⩾2. In addition, we obtain regularity properties of weak solutions whenever p⩾20/9 (if d= 3) or p⩾2 (if d= 2). Further, some extensions of these results to more general stress tensors or to Dirichlet boundary conditions (with a Newtonian tensor large enough) are obtained. Keywords: variable density, shear-dependent viscosity, power law, Carreau’s laws, weak solution, strong solution, periodic boundary conditions MSC 2000: 35B10, 35M10, 76A05 1. Introduction We consider that the domain Ωis a cube (of periodicity) in  d(d= 2 or 3) x= (xi)d i=1 ∈Ω = d Y i=1 (0, Li) (0 < Li<+∞,∀i= 1,...,d) and the time interval is t∈(0,+∞). Given f= (fi)d i=1 (external forces), %0and u0= (u0 i)d i=1 (initial density and velocity), the problem is to find %,u= (ui)d i=1 and π(density, velocity and pressure) This work has been partially supported by C.I.C.Y.T project MAR98-0486. 637 satisfying the following system of PDEs in (0,+∞)×Ω: (1) (∂t%+∂i(%ui) = 0, ∂iui= 0, ∂t(%ui) + ∂j(%ujui)−∂jτij (Du) + ∂iπ=%fi,∀i= 1,...,d, together with the initial conditions (2) %|t=0 =%0and (%u)|t=0 =%0u0in Ω, and the space-periodic boundary conditions: for a.e. t∈(0,+∞), (3) %(t, ·), uj(t, ·), π(t, ·)and ∂kuj(t, ·)are Li-periodic with respect to xi, for all i, j, k = 1,...,d. In (1), the summation is over repeated indices, ∂t,∂idenote partial derivatives with respect to tand xi,Du is the symmetric part of the velocity gradient (with the ij-component equal to Duij =1 2(∂iuj+∂jui)) and τ=τ(Du)denotes the (symmetric) extra-stress tensor, which is assumed to be a nonlinear function of Du. In such a case, (1) describes a class of non-Newtonian fluids, namely the fluids with shear-dependent viscosity. For simplicity, we consider τdefined by one of the following constitutive laws (see Section 5 for more general assumptions on τ): τ(Du) = {µ∞+µ0|Du|p−2}Du (power law),(4) τ(Du) = {µ∞+µ0(1 + |Du|)p−2}Du, or τ(Du) = {µ∞+µ0(1 + |Du|2)(p−2)/2}Du,)(Carreau’s laws)(5) where µ∞⩾0,µ0>0are constants, |Du|= (DuijDuij)1/2and p∈(1,+∞). In particular, one can write τ(Du) = µ(|Du|)Du =µ∞Du +µ0τp(Du). In the first equality, µ(|Du|)stands for the generalized viscosity function and, in the latter, we decompose the tensor into two parts: the Newtonian one (µ∞Du) and the purely non-Newtonian one (µ0τp(Du)). Properties of the tensor τpdiffer for p∈(2,+∞)and for p∈(1,2), which correspond respectively to dilatant fluids and pseudo-plastic fluids. For instance, the generalized viscosity function µ(|Du|) increases with respect to |Du|for dilatant fluids and decreases for pseudo-plastic ones. The case p= 2 corresponds to a Newtonian fluid. Constitutive laws (4) and (5) are frequently used in problems related to chemistry, biology, glaciology, geology, ... (see [8] and the references cited therein). 638 The main known results on this subject are the following. In the case %≡constant (i.e. for non-Newtonian variants of the Navier-Stokes equations), the existence of Dirichlet weak solutions was obtained by O. A. Ladyzhenskaya in [5] (for Carreau’s laws) and J. L. Lions in [6] (for the “p-laplacian” operator, i.e. τp=|∇u|p−2∇u), whenever p⩾11/5(if d= 3) or p⩾2(if d= 2). In the proofs, a combination of the monotone operator theory and the compactness method is used. On the other hand, the existence of space-periodic weak solutions was obtained by J. Málek, J. Nečas, M. Rokyta and M. R˚užička [9], whenever p > 9/5(if d= 3) or p > 3/2(if d= 2) The latter was shown without Newtonian viscosity (i.e. µ∞= 0). When µ∞>0, it is not difficult to extend these results to all p > 1 (d= 2 or 3). Recently, there has been another result on the existence of weak solutions valid for p > 2(d+ 1)/(d+ 2), see [3]. On the other hand, J. Málek, J. Nečas and M. R˚užička [10] have proved the existence of Dirichlet weak solutions when d= 3, for all p⩾2, in the cases µ∞>0or (µ∞= 0 and (5)). Moreover, if p⩾9/4, then the solution is strong and unique. For density-dependent fluids (%6=constant), first in the Newtonian case (p= 2), the existence of weak solutions was already proved by S. N. Antontsev and A. N. Kazhikhov in [1] (see also [4, 7]). Recently, when a power law is considered, the existence of Dirichlet weak solutions has been obtained in [2], whenever p⩾12/5(if d= 3) or p⩾2(if d= 2) (results that can be easily extended to Carreau’s laws). Therefore, there is a “gap” between p= 2 and p= 12/5in the 3Dcase. The main purpose of this paper is “filling” this gap, extending these last results to the cases p > 1(if µ∞>0) or p⩾2(if µ∞= 0 and (5)), when space-periodic boundary conditions are considered. Moreover, we will prove additional regularity properties of the weak solutions in the cases p⩾20/9 (d= 3) p > 1 (d= 2))(if µ∞>0) 639 or p⩾20/9 (d= 3) p⩾2 (d= 2))(if µ∞= 0 and (5)). For this purpose, the key is to obtain some new estimates for the velocity in an intermediate space between H1and H2, which lead to point-wise convergence of the velocity gradient. The derivation of these estimates is based on the extension of the technique presented in [8, 9] in the case of constant density, to non-constant density flows. 2. Definition of weak solutions and some consequences First of all, let us introduce some usual notation. For A⊂  klet X(A)be the space of scalar functions defined in A. Then X(A)d(or X(A)d×d) denotes the space of vector-valued (tensor-valued) functions whose components belong to X(A). On the other hand, let Xloc(A)represent the space of functions which belong to X(K), for all compact sets K⊂A. As usual, D(A)will denote the space of smooth functions with compact support in A. Furthermore, let q > 1(real) and k⩾1 (integer), then Lq(Wk,q) is used for the standard Lebesgue (Sobolev) spaces. Finally, by Lq(0, T;X(A)) we denote the Bochner spaces. Now, we will introduce the functional spaces related to the weak solutions. In order to consider the periodic conditions, we define Dper ={ψ∈C∞(  d): ψis Li-periodic with respect to xi,∀i= 1,...,d}, Lq per (or Wk,q per ) = the closure of Dper in Lq loc(  d)(in Wk,q loc (  d)). One has the following identities: Lq per ={g∈Lq loc(  d): gis Li-periodic with respect to xi}. (idem for Wk,q per , replacing Lq loc by Wk,q loc ). Finally, in order to consider the incompressibility equation (free divergence), we define Vper ={ϕ= (ϕi)d i=1 ∈(Dper)d:∂iϕi= 0 in  d}, Hper (or Vq per) = the closure of Vper in L2 loc(  d)d(in W1,q loc (  d)d). In the Hilbert case (q= 2), we will denote Wk,2 per ≡Hk per and V2 per ≡Vper. 640 Remark. In the definition of Lq per and Wk,q per , a restriction to the cube Ωmay be considered, extending the functions to all  dby periodicity. In this sense, if ∂Ω denotes the boundary of Ω, separated in parts Γi=∂Ω∩ {xi= 0},Γi+d=∂Ω∩ {xi=Li},∀i= 1,...,d, one also has Wk,q per ={g∈Wk,q(Ω): (∂αg)|Γi= (∂αg)|Γi+d∀|α|⩽k−1}, Hper ={v∈L2(Ω)d:∂ivi= 0 in Ω,(v·n)|Γi=−(v·n)|Γi+d}, Vq per ={v∈W1,q(Ω)d:∂ivi= 0 in Ω, v|Γi=v|Γi+d}. Here, nis the exterior normal vector to ∂Ω. Notice that, if v∈Hper, in particular v∈L2(Ω)dand ∂ivi∈L2(Ω), then v·nbelongs to H−1/2(∂Ω). In the sequel, all spatial norms will be taken in Ωand the space of functions related to the norm will be indicated by a subscript (for instance, kukL2denotes the L2(Ω)-norm of u). Definition 1. Under the following hypotheses for the data: (H1) %0∈L∞ per, %0⩾α > 0in Ω, u0∈Hper and f∈L2 loc([0,∞); (L2 per)d), a couple (%, u)is said to be a weak solution in (0,+∞)of the problem (1)–(3) if %∈L∞(0,∞;L∞ per),0< α ⩽%(t, x)⩽k%0kL∞a.e. (t, x),(6) u∈Lp loc([0,∞); Vp per)∩L∞ loc([0,∞); Hper),(7) Z∞ 0ZΩ {%∂tψ+%ui∂iψ}dxdt+ZΩ %0ψ(0) dx= 0 ∀ψ∈D([0,∞); Dper),(8) Z∞ 0ZΩ {%ui∂tϕi+ [%ujui−τij (Du)]Dϕij +%fiϕi}dxdt= 0(9) ∀ϕ∈D((0,∞); Vper), and, ∀v∈Vσ per (with σ= max{p, 2}), (10) ZΩ %uividx∈C[0,∞)with ZΩ %uividx(0) = ZΩ %0u0 ividx. Remarks. 1. By virtue of the regularity of (6)–(7), every term in (8)–(9) makes sense. 641 2. From (8), one has (cf. [7]) ∂t%+∂i(%ui) = 0 in D0((0,∞)×  d), %∈C([0,∞); Lq(Ω)) ∀q < +∞and %|t=0 =%0a.e. in Ω,(11) k%(t, ·)kLq=k%0kLq∀t∈[0,∞),∀q∈[1,∞).(12) 3. Taking in (9) ϕ=ϕ1(t)ϕ2(x)with ϕ1∈D(0,∞)and ϕ2∈Vper, one has h∂t(%ui) + ∂j(%ujui)−∂jτij (Du)−%fi,(ϕ2)ii= 0 in D0(0,∞) for all i= 1,...,d, where h·,·i denotes the duality in Ωin the distributional sense. Consequently, from De Rham’s lemma, we have that there exists a distribution π∈D0((0,∞)×Ω) such that (%, u, π)satisfies (1)2in D0((0,∞)×Ω), cf. [2]. 4. Incompressibility equation ∂iui= 0 and periodic conditions on uhave been imposed in the “essential” form; u(t, ·)∈Vp per for a.e. t∈(0,+∞). However, the initial conditions (%u)|t=0 =%0u0are verified in the sense of (10) (see also (13) below), because %u is only continuous in the weak topology of Vσ per. 3. Main results Theorem 1 (Existence of a weak solution. The case µ∞>0).If we assume µ∞>0and hypothesis (H1), then there exists a weak solution (%, u)in (0,+∞) of the problem (1)–(3) for all p > 1. Furthermore, u∈Lσ loc([0,∞); Vσ per)(with σ= max{p, 2}) and (13) k%1/2uk2 L2∈C[0,∞)with k%1/2uk2 L2(0) = k(%0)1/2u0k2 L2. Theorem 2 (Regularity of weak solutions. The case µ∞>0).Under hypotheses of Theorem 1, if moreover one assumes (H2) u0∈Vσ per, p ⩾20/9 (if d= 3) or p > 1 (if d= 2), then any weak solution (furnished by Theorem 1) verifies u∈L2 loc([0,∞); H2(Ω)d)∩L∞ loc([0,∞); Vσ per),(14) ∂tu∈L2 loc([0,∞); Hper) (recall that σ= max{p, 2}). 642 Theorem 3 (The case µ∞= 0 and Carreau’s laws).If we assume µ∞= 0,(H1) and the tensor τis defined as in (5), we have the statements of Theorems 1and 2whenever p⩾2, i.e. for all p⩾2there exists a weak solution (%, u)in (0,+∞) of (1)–(3), which has the additional regularity (14) if (H20)u0∈Vp per, p ⩾20/9 (if d= 3) or p⩾2 (if d= 2). Remarks. 1. When the Newtonian viscosity vanishes (µ∞= 0), we get no result for pseudoplastic fluids (i.e. for p < 2). 2. In particular, (14) implies that u∈C([0,∞); Hper). In addition, from (11), (10) and the bound form below of %0given in (H1), the following initial conditions for the velocity can be proved ([2]): u|t=0 =u0a.e. in Ω. 3. Under the regularity of (14), one can show that the pressure π∈L2 loc([0,∞); Lσ0 loc)(denoting by σ0the conjugate exponent of σ;1/σ + 1/σ0= 1) and (%, u, π) verifies (1)2in the L2 loc([0,∞); W−1,σ0 loc )sense. Moreover, we can directly deduce that ∂jτp ij(Du)∈L2 loc([0,∞); L6/(p+1)(Ω)), in the following cases: 2⩽p⩽5 (d= 3) 2⩽p < ∞(d= 2))(for the power law) or p⩽5 (d= 3) p < ∞(d= 2))(for Carreau’s laws). Then, in these cases, ∂iπ∈L2 loc([0,∞); L6/(p+1)(Ω)) and the system of PDEs (equivalent to (1)2), (15) %{∂tui+uj∂jui} − ∂jτij (Du) + ∂iπ=%fi,∀i= 1,...,d is satisfied point-wise a.e. in (0,∞)×Ω. On the other hand, the continuity equation (1)1is only satisfied in a variational sense, because a better regularity result for the density %is not known. It will be said that (%, u, π)(with the above properties) is a semi-strong solution in (0,+∞)of the problem (1)–(3). 4. In Theorem 2, if we consider (H2) but only with u0∈Hper, then using a standard “cut-off method” (in a neighborhood of t= 0), see [9] for instance, we 643 can show the following regularity results (similar to (14) but excluding the time t= 0): u∈L2 loc((0,∞); H2(Ω)d)∩L∞ loc((0,∞); Vσ per), ∂tu∈L2 loc((0,∞); Hper). 4. Outline of the proofs For simplicity, we outline the proofs of Theorems 1 and 2 simultaneously, i.e. for µ∞>0, and only some possible modifications to circumvent the case µ∞= 0 (Theorem 3) will be pointed out. We will start the proofs with u0∈Vσ per considering the case u0∈Hper at the end. The general method of these proofs is: construction of a sequence of approximated solutions (using a discretization in a space of Galerkin’s type) and a limit process using compactness in order to control the nonlinear terms. During all this section, we will denote by C, C1,... different positive constants. 4.1 Approximated solutions and a priori estimates. According to the method used in [2], we begin with a basis of Vσ per, furnished by regular functions (at least C1), {w1,...,wm,...}, and let Vmbe the subspace spanned by w1,...,wm. It will be said that (%m, um)is an m-th approximated solution to (1)–(3) if %m∈ C1([0,∞)×  d),%mis Li-periodic with respect to xi,um∈C1([0,∞); Vm)and (%m, µm)verifies ∂t%m+umi∂i%m= 0 in (0,∞)×  d, %m|t=0 =%0 min  d,(16)   ZΩ {%m(∂tumi+umj∂jumi)vi+τij (Dum)Dvij }dx=ZΩ %mfividx ∀v∈Vm∀t∈(0,∞), um|t=0 =u0 min Ω. (17) Here we choose %0 m∈C1 per and u0 m∈Vmsuch that %0 m→%0in Lq loc(  d)∀q < ∞with α⩽%0 m⩽k%0kL∞, u0 m→u0in Vσ per. For instance, we can define %0 m=%0?ξ1/m where (ξε)is a standard mollified sequence, and u0 m=Pmu0with Pmthe projection operator from Vσ per onto Vm. Notice that in the above definition of the m-th approximated solution, the PDEs system (15) has been discretized in space by a Galerkin method in (17), but the 644 one obtains I(um,∇um)⩽kumk2 Lp?k∇umk(5p−8)/2 Lpk∇umk(12−5p)/2 L3p ⩽Ckumk(5p−4)/2 VpIp(Dum)(12−5p)/2p ⩽εIp(Dum) + Cεkumk(5p−4)p/(7p−12) Vp. Since p < (5p−4)p/(7p−12) ⩽2p, if we use (P3) and (21), the second term on the right hand side can be bounded by Cεkumk{(5p−4)p/(7p−12)}−p VpC1 C1ZΩ Up(Dum) dx+|Ω|+kumkp L2. Finally, by (22), kumk{(5p−4)p/(7p−12)}−p Vpis bounded in L1 loc(0,∞). Consequently, inserting in (32) the above estimates for I(um,∇um)and taking into account (34), we obtain the following estimates by means of Gronwall’s lemma: ∀T > 0, (um)is bounded in L2(0, T ;H2(Ω)d)∩L∞(0, T;Vσ per), (∂tum)is bounded in L2(0, T ;Hper). Thanks to a classical compactness result of “Aubin-Lions” type, cf. [6], using the triplet of spaces (H2∩Vper),→,→Vper ,→Hper (we denote by ,→,→a compact embbeding and by ,→a continuous one), these estimates imply that (um)is relatively compact in L2(0, T ;Vper),∀T > 0. Hence, it is possible to complete the proof of Theorem 2 by a standard limit process. 4.6 Existence of a weak solution with u0∈Hper (Proof of Theorem 1). Now, we consider u0 m∈Vmwith u0 m→u0in Hper (instead of in Vσ per as before). Then, the bound for I(um,∇um)can not be used as in the previous case. Now, using interpolation inequalities, we have I(um,∇um)⩽kumk2 L6k∇umkL2k∇umkL6⩽εkumk2 H2+Cεkumk4 H1k∇umk2 L2. The second term on the right hand side can be bounded using (21) for q= 2 as follows: Cεkumk2 H1{kDumk2 L2+kumk2 L2}kDumk2 L2. 651 Now, we use (21) and the fact that (35)            kDumk2 L2⩽2 µ∞ J(um)(when µ∞>0) kDumk2 L2⩽2 pkDumkp Lp+p−2 p|Ω| ⩽C(Ω){J(um) + 1}(when µ∞= 0 and p⩾2). Inserting in (32) the above bound for I(um,∇um)one has (omitting constants) k∂tumk2 L2+kumk2 H2+Ip(Dum) + d dtJ(um) ⩽kfk2 L2+kumk2 H1{kumk2 L2+ 1}(1 + J(um))2. Dividing by (1 + J(um))2, one has (omitting constants) k∂tumk2 L2+kumk2 H2 (1 + J(um))2−d dt{(1 + J(um))−1}⩽kfk2 L2+kumk2 H1{kumk2 L2+ 1} and the right hand side is bounded in L1 loc([0,∞)). Therefore, integrating in (0, T), ZT 0 k∂tumk2 L2+kumk2 H2 (1 + J(um))2dt+1 1 + J(u0 m)⩽1 1 + J(um(T)) +C. In particular, although J(u0 m)could be not bounded, one has (36) ZT 0 kumk2 H2 (1 + J(um))2dt⩽C. Using (36), we can get the following estimate of um(of the “L2/3(0, T;H2)” type): ZT 0 kumk2/3 H2dt=ZT 0kumk2 H2 (1 + J(um))21/3 (1 + J(um))2/3dt ⩽C1/3ZT 0 (1 + J(um)) dt2/3 ⩽C. Finally, with help of the interpolation estimates between the Hkspaces kukH1+θ⩽Ckuk1−θ H1kukθ H2for 0⩽θ⩽1, one concludes (um)is bounded in L2/(1+2θ)(0, T;H1+θ),∀T > 0. 652 Since 2/(1+2θ)⩾1whenever 0⩽θ⩽1/2, the above estimate together with (24) and an adequate compactness result (considering the triplet of spaces (H1+θ∩Vper),→,→ Vper ,→Hper), cf. [11], imply that ∀T > 0, (37) (um)is relatively compact in L2/(1+2θ)(0, T ;Vper)for 0⩽θ⩽1/2. In particular, one can deduce point-wise convergence a.e. in (0,∞)×Ωfrom Dum to Du, hence (25) holds. Finally, by a standard limit process, one can complete the proof of Theorem 1. 5. Some extensions and open problems 5.1 More general non-Newtonian tensors. The same sort of results can be obtained for a non-Newtonian tensor τpdefined by a potential function Up, i.e. (38) Up:  d×d S→  +, Up∈C2and ∂Up ∂Dij (D) = τp ij(D)∀D∈  d×d S, which posesses the following properties: Up(0) = ∂Up ∂Dij (0) = 0,(39)  ∂2Up ∂Dij ∂Dkl (D) ⩽C1(|D|p−2 (1 + |D|)p−2∀D∈  d×d S,(40) ∂2Up ∂Dij ∂Dkl (D)Eij Ekl ⩾C2(|D|p−2|E|2 (1 + |D|)p−2|E|2∀D, E ∈  d×d S.(41) The above assertion is due to the fact that (38)–(41) lead to (P1), (P2) and (P3), cf. [9]. Indeed, (38)–(40) imply (P2) and the upper bound of (P3), whereas (38), (39) and (41) imply (P1) and the lower bound of (P3). We understand (40) and (41) in the sense that only one of the two conditions is considered: either (40)1and (41)1, which play the role of a power law, or (40)2and (41)2in the role of Carreau’s laws. On the other hand, a perturbation σpof τpcan be admitted (and then, the non-Newtonian tensor is τp+σp), where τpis determined by (38)–(41) and σpis 653 determined by a potential function Vp(σp ij ≡∂Vp/∂Dij) such that Vp(0) = ∂Vp ∂Dij (0) = 0 ∀ij,  ∂2Vp ∂Dij ∂Dkl (D) ⩽C0 1(|D|p−2 (1 + |D|)p−2∀D∈  d×d S, ∂2Vp ∂Dij ∂Dkl (D)Eij Ekl ⩾0∀D, E ∈  d×d S. Hence, it is easy to check that, in these cases, one will have the same results. 5.2 Dirichlet boundary conditions. According to [2], when one considers Dirichlet boundary conditions, the existence of weak solutions in the cases p < 12/5(if d= 3) and p < 2(if d= 2) are interesting open problems. The argument made in this paper for the periodic case now fails because of Aum6=−∆um(since ∆umdo not vanish on ∂Ω). In the case %≡constant and p⩾2, this difficulty has been circumvented in [10], multiplying −∆umby a cutoff function χεand considering the “mixed formulation” (with pressure), since the new test function −χε∆umis not solenoidal. On the other hand, when 1<p<2and Carreau’s laws are considered (i.e. assuming (40)2), results similar to Theorems 1 and 2 can be proved for the Dirichlet problem, assuming that the Newtonian viscosity “dominates” the non-Newtonian one in the following sense: µ∞> Cµ0with C > 0large enough. The main idea of this proof is to estimate the second term of (28) by −µ0ZΩ τp ij(Dum)D(Aum)ij dx=µ0ZΩ ∂2Up ∂Dij ∂Dkl (Dum)∂j(Dum)kl(Aum)idx ⩽µ0C1 d X i,j,k,l=1 ZΩ (1 + |Dum|)p−2|∂j(Dum)kl||(Aum)i|dx⩽µ0C1d2k∂2umk2 L2 (in the last estimate, we have used that 1< p < 2). Then, we could control this bound with the first term of (28), µ∞k∆umk2 L2, letting µ∞to be large enough (namely, whenever µ∞> Cµ0C1d2, with Cthe constant from inequality (31)). 5.3 Some open problems. Finally, we point out some related open problems: 654 1. Viscosity dependent on the density. If we assume µ∞=µ∞(%)and µ0= µ0(%), with µ∞, µ0:  +→  +continuous and strictly increasing functions, the existence of a weak solution is deduced in [2] when p⩾12/5(if d= 3) and p⩾2(if d= 2), using compactness and monotony arguments. The extension of this result to more general cases of this paper, is an interesting open problem. Arguing as in this paper, the main difficulty is to control the term ZΩ {µ∞(%m)(Dum)ij +µ0(%m)τp ij (Dum)}D(Aum)ij dx, which appears when Aumis taken as a test function. 2. The case µ∞= 0 and p < 2(for a power law or Carreau’s laws). From a physical point of view, this problem is more interesting for Carreau’s laws, cf. [8]. Now, the main difficulty is to control the term ZΩ %m∂tum·∆um which appears when −∆umis taken as a test function. 3. Uniqueness, even for semi-strong solutions and plarge. The more delicate question is that we do not know whether the additional regularity of ugiven in (14) implies additional regularity for the density gradient ∇%(for instance, of the Lqtype). With only %∈L∞, a standard uniqueness argument fails. On the other hand, a uniqueness result for a weak solution, whenever a sufficiently regular solution exists, could be developed, i.e. we obtain weak-regular uniqueness, cf. [7] for the density-dependent Newtonian case. 4. Existence of strong solutions (with regularity of the Lqtype for ∇%). At our knowledge, it is an open problem, even in the restrictive cases for small time (local in time solution) or for large time but assuming small data. References [1] S. A. Antonzev and A. V. Kazhikhov: Mathematical Study of Flows of Non-Homogeneous Fluids. Lectures at the University of Novosibirsk, U.S.S.R, 1973. (In Russian.) [2] E. Fernández-Cara, F. Guillén and R. R. Ortega: Some theoretical results for viscoplastic and dilatant fluids with variable density. Nonlinear Anal. 28 (1997), 1079–1100. [3] J. Frehse, J. Málek and M. Steinhauer: On existence results for fluids with shear dependent viscosity—unsteady flows. Partial Differential Equations, Praha 1998 Chapman & Hall/CRC, Res. Notes Math., 406, Boca Raton, FL, 2000, pp. 121–129. [4] A. V. Kazhikhov: Resolution of boundary value problems for nonhomogeneous viscous fluids. Dokl. Akad. Nauk 216 (1974), 1008–1010. [5] O. A. Ladyzhenskaya: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, 1969. 655 [6] J. L. Lions: Quelques méthodes de résolution des probl`emes aux limites non linéaires. Dunod, Gauthier-Villars, 1969. [7] P. L. Lions: Mathematical Topics in Fluid Mechanics. Volume 1, Incompressible models. Clarendon Press, 1996. [8] J. Málek, K. R. Rajagopal and M. R˚užička: Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity. Math. Models and Methods in Applied Sciences 5(1995), 789–812. [9] J. Málek, J. Nečas, M. Rokyta and M. R˚užička: Weak and Measure-Valued Solutions to Evolutionary PDEs. Chapman & Hall, 1996. [10] J. Málek, J. Nečas and M. R˚užička: On weak solutions of non-Newtonian incompressible fluids in bounded three-dimensional domains. The case p⩾2. Advances in Differential Equations 6(2001), 257–302. [11] J. Simon: Compact sets in Lp(0, T ;B). Ann. Mat. Pura Appl. 4(1987), 65–96. Author’s address: Depto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain, e-mail: [email protected]. 656