Semi-Galerkin approximation and strong solutions to the equations of the nonhomogeneous asymmetric fluids

Semi-Galerkin Approximation and Strong Solutions to the Equations of the Nonhomogeneous Asymmetric Fluids Jos´e L. Boldrini ∗ , Marko A. Rojas-Medar ∗ and Enrique Fern´ andez-Cara † Abstract This paper analyzes an initial/boundary value problem for a system of equations modelling the nonstationary flow of a nonhomogeneous incompressible asymmetric (polar) fluid. Under conditions similar to those usually imposed to the nonhomogeneous 3D Navier-Stokes equations, by using a spectral semi-Galerkin method, we prove the existence of a local in time strong solution. We also prove the uniqueness of the strong solution and some global existence results. Several estimates for the solutions and their approximations are given. These can be used to find useful error bounds of the Galerkin approximations. R´esum´e Dans ce papier, on analyse un probl`eme de valeurs initiales et valeurs aux limites pour un syste`eme d’´equations aux d´eriv´ees partielles qui mod´elise le flux instationnaire d’un fluide asymm´etrique incompressible non homog`ene. Sous des conditions similaires aux conditions usuellement impos´ees aux ´equations tridimensionelles de Navier-Stokes non homog`enes, `a l’aide d’une m´ethode de type semi-Galerkin, nous d´emontrons l’´existence d’une solution forte locale en temps. On ´etablit aussi l’unicit´e de solution forte et quelques r´esultats d’´existence globale. Tous ces ∗IMECC-UNICAMP, C.P. 6065, 13081-970, Campinas-SP, Brazil. Partially supported by CNPq-Brazil, Grants 300513/87-9 and 300116/93-4 (RN) and FAPESP-Brazil, Grant 01/07557-3. E-mails: [email protected], [email protected]. †Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain. Partially supported by D.G.E.S.-Spain, Grants PB98-1134 and BFM2000-1317. E-mail: [email protected]. 1 r´esultats reposent sur des estimations appropri´ees pour les solutions et leurs approximations qui permettent d’ailleurs d´eduire des estimations de l’erreur. 1 Introduction In this paper we will study the equations for the motion of a nonhomogeneous viscous incompressible asymmetric fluid. These equations will be considered in a set of the form Ω ×(0, T), where Ω ⊂R3is a bounded and regular domain with boundary ∂Ω and T > 0. Thus, let us denote by u,w,ρand pthe velocity field, the angular velocity of rotation of the fluid particles, the mass density and the pressure distribution, respectively. The governing equations are the following:              ρ(ut+ (u· ∇)u)−(µ+µr)∆u+∇p= 2µrcurl w+ρf, div u= 0, ρ(wt+ρ(u· ∇)w)−(ca+cd)∆w−(c0+cd−ca)∇div w + 4µrw= 2µrcurl u+ρg, ρt+u· ∇ρ= 0. (1) For simplicity, they will be completed with the following boundary and initial conditions (u(x, t) = 0, w(x, t) = 0 on ∂Ω×(0, T), u(x, 0) = u0(x), w(x, 0) = w0(x), ρ(x, 0) = ρ0(x) in Ω,(2) In (1), fand gare known density functions of external sources for the linear and the angular momentum of particles, respectively. The positive constants µ,µr,c0,caand cdcharacterize the physical properties of the fluid. Thus, µis the usual Newtonian viscosity; µr,c0,caand cdare additional viscosities related to the lack of symmetry of the stress tensor and, consequently, to the fact that the field of internal rotation wdoes not vanish. These constants must satisfy the inequality c0+cd> ca. The symbols ∇, ∆, div and curl denote the gradient,Laplacian,divergence and rotational operators, respectively; ut,wtand ρtstand for the time derivatives of u,wand ρ; the i-th components of (u·∇)uand (u·∇)w 2 in cartesian coordinates are given by [(u· ∇)u]i= n X j=1 uj ∂ui ∂xj and [(u· ∇)w]i= n X j=1 uj ∂wi ∂xj . We also have u· ∇ρ= n X j=1 uj ∂ρ ∂xj . For the derivation of equations (1) and a discussion on their physical meaning, see [5]. Observe that this system includes as a particular case the classical Navier-Stokes equations, which have been largely studied (see for instance the classical books by Ladyzhenskaya [10] and Temam [20] and the references therein). It also includes as a reduced model the nonhomogeneous Navier-Stokes system, that are less known (cf. [19],[9],[11],[18]). Concerning the model considered in this paper, let us recall that, under certain assumptions, by using linearization and an almost fixed point Theorem, Lukaszewicz established in [17] the existence of weak solutions for short time. In this same paper, there are considerations on the possible proof of the existence of strong solutions (assuming that the initial density is strictly separated from zero) by using the techniques of [15] and [16] (linearization and fixed point Theorems; recall that in [15] and [16] the density is a positive constant). In this paper we are also concerned with the existence of strong solutions of (1)–(2). However, since we are mainly motivated by techniques directly related to numerical applications, we have preferred an approach based on spectral semi-Galerkin methods. In this way, by assuming that the initial data are more regular than in [17] and the initial density is separated from zero, we will prove that more regular strong solutions exist. In this process, we will find appropriate estimates that become fundamental to derive error bounds for the Galerkin approximations, as was already explained in the previous paper [2]. Actually, these estimates can be viewed as one of the main objectives in this paper. The rest of the paper is organized as follows. In Section 2, we will explain what is a strong solution (u, w, ρ) of (1)–(2) and we will present our main results, Theorems 1 and 2: the existence and uniqueness of a local in time 3 strong solution. We will also introduce in this Section the main tool in this paper, namely a spectral semi-Galerkin approximation scheme for (1)–(2). In Section 3, we will deduce the a priori estimates needed to ensure the regularity of (u, w, ρ). To this end, we will combine arguments and techniques that have been used in other similar contexts by several authors, in particular by Heywood [6],[7], Kim [9] and Boldrini and Rojas-Medar [3]. In Section 4, we will use these a priori estimates to extract a convergent subsequence and, then, to pass to the limit in the equations. In this way, we will prove that a strong solution exists in a (possibly small) maximal time interval [0, T0), with T0≤T. Section 5 deals with uniqueness. There, we will prove that any strong solution must coincide with the solution furnished by Theorem 1. In Section 6, we prove the existence of a (regular) pressure. In particular, we see that the triplet (u, w, ρ) provided by Theorem 1 is such that, for some p, the equations (1) are satisfied a.e. in Ω ×(0, T0). Finally, Section 7 is concerned with the existence of global strong solutions for small regular data. 2 Preliminaries and Main Results In the sequel we will assume that Ω is a bounded domain in R3, with regular boundary ∂Ω. We will consider the usual Sobolev spaces Wm,q(Ω) = {f∈Lq(D) : k∂κfkLq(D)<+∞for |κ| ≤ m} for m≥1 and 1 ≤q≤ ∞ with the usual norms k·kWm,q . When q= 2, we will set Hm(Ω) = Wm,2(Ω). As usual, Hm 0(Ω) will stand for the closure of C∞ 0(Ω) in Hm(Ω). For simplicity, if Bis a Banach space with norm k · kB, the natural product norm in Bmwill be also denoted by k·kB. We will set V(Ω) = {v∈(C∞ 0(Ω))3: div v= 0 in Ω }, H= the closure of V(Ω) in (L2(Ω))3, V= the closure of V(Ω) in (H1 0(Ω))3. For any Banach-space Band any T > 0, we will denote by Lr(0, T;B) the Banach space of the B-valued (classes of) functions defined a.e. in [0, T] that are Lr-integrable in the sense of Bochner. Frequently, we will consider Banach spaces Lr(0, T ;B) with B=Wm,q(Ω). In such cases, for any v∈ Lr(0, T;Wm,q(Ω)), v(t) stands for the function v(·, t). 4 Let Pbe the orthogonal projection of (L2(Ω))3onto Hinduced by the usual Helmholtz decomposition of (L2(Ω))3. By definition, the Stokes operator is the unbounded linear mapping A:D(A)⊂H7→ H, with domain D(A) = V∩(H2(Ω))3, given by Av =P(−∆v)∀v∈D(A). It is well known that Ais a positive self-adjoint operator. It is characterized by the equalities (Aw, v) = (∇w, ∇v)∀w∈D(A),∀v∈V. Here and in the sequel, (·,·) stands for the usual scalar (L2(Ω))3-product. The associated norm will be denoted by k · k. We will also consider in the sequel the strongly uniformly elliptic operators L0and L, with D(L0) = D(L) = (H1 0(Ω))3∩(H2(Ω))3, L0z=−(ca+cd)∆z−(c0+cd−ca)∇div z∀z∈D(L) and Lz =L0z+ 4µrz∀z∈D(L). Due to the assumption c0+cd> ca,Lis indeed a positive operator. The following assumptions on the initial velocity, angular velocity and density will be imposed throughouth this paper: u0∈D(A),(3) w0∈D(L),(4) ρ0∈C1(Ω),0< α ≤ρ0(x)≤βa.e. in Ω. (5) We will also assume that f, g ∈L2(0, T; (H1(Ω))3), ft, gt∈L2(0, T; (L2(Ω))3).(6) Using the orthogonal projector Pand the operators Aand L, we can give a rigorous formulation of problem (1) −(2): Find a time T0∈(0, T ] and functions u∈C0([0, T0); D(A)) ∩C1([0, T0); H), w∈C0([0, T0); D(L)) ∩C1([0, T0); (L2(Ω))3) and ρ∈C1(Ω×[0, T0)), 5 such that                                (ρut, v) + (ρ(u· ∇)u, v) + (µ+µr)(Au, v) = 2µr(curl w, v) + (ρf, v) for 0 < t < T0,∀v∈V, (ρwt, ψ)+(ρ(u· ∇)w, ψ)+(Lw, ψ) = 2µr(curl u, ψ) + (ρg, ψ) for 0 < t < T0,∀ψ∈(H1 0(Ω))3, ρt+u· ∇ρ= 0 for (x, t)∈Ω×[0, T0), u(x, 0) = u0(x), w(x, 0) = w0(x), ρ(x, 0) = ρ0(x) for x∈Ω. (7) By definition, a triplet (u, w, ρ) with these properties is a strong solution of (1)–(2) in [0, T0). We will prove below that, under assumptions (3)– (6), the initial/boundary value problem (1)–(2) possesses exactly one strong solution in a maximal time interval. Actually, we will see that this strong solution is still more regular than stated above. To this end, let us first recall some properties of the Stokes operator A. If Ω is bounded and ∂Ω is of class C1,1, the mapping A:D(A)7→ H is one-to-one and onto (see for instance [1]). The inverse operator A−1is completely continuous as a mapping A−1:H7→ H. Also, Ais symmetric and, therefore, so is its inverse. Consequently, A−1possesses an orthogonal sequence of eigenfunctions {ϕk}which is complete in H,Vand D(A). We will denote by λkthe k-th associated eigenvalue (that is, Aϕk=λkϕkfor all k). It will be assumed that {ϕk}is orthonormal in H. Accordingly, the eigenfunctions {λ−1/2 kϕk}and {λkϕk}are complete and orthonormal respectively in V(endowed with the scalar product (∇u, ∇v) and D(A) (endowed with the scalar product (Au, Av)). Notice also that, if ∂Ω is a Cm+1,1manifold, then the eigenfunctions ϕk belong to (Hm+2(Ω))3. On the other hand, we will use the notation {ψk}and {γk}for the eigenfunctions and eigenvalues of L. Again, it is assumed that {ψk}is orthonormal for the L2-norm. The system {ψk}is complete in (L2(Ω))3, (H1 0(Ω))3 and D(L) and we have again regularity results for the eigenfunctions ψk when ∂Ω is a Cm+1,1manifold. Let Pkthe orthogonal projection of Honto the space Vkspanned by the kfirst eigenfunctions ϕ1, . . . , ϕkof A. Similarly, let Rkbe the orthogonal projection of L2(Ω) onto the space Wkspanned by the kfirst eigenfunctions ψ1, . . . , ψkof L. Then the solutions of (7) can be obtained by using a semiGalerkin method determined by the spaces Vkand Wkand the operators Pk and Rk. 6 More precisely, for each fixed k, we consider the following finite dimensional problem: Find Tk∈(0, T ], uk∈C1([0, Tk); Vk), wk∈C1([0, Tk); Wk) and ρk∈C1(Ω×[0, Tk)) such that                  Pk(ρkuk t+ρk(uk· ∇)uk−2µrcurl wk−ρkf)+(µ+µr)Auk= 0, Rk(ρkwk t+ρk(uk· ∇)wk−2µrcurl uk−ρkg+Lwk= 0 and ρk t+uk· ∇ρk= 0 for 0 < t < Tk, uk(0) = Pku0, wk(0) = Rkw0, ρk(0) = ρ0. (8) In (8), we have an initial value problem for a system of ordinary differential equations coupled to a transport equation. By using the characteristics method, it is not difficult to prove that (8) possesses exactly one solution (uk, wk, ρk) defined in a time interval [0, Tk). The a priori estimates established below prove that we can take in fact Tk=Tfor all k≥1. The k-th approximated problem (8) can also be written in the form                      (ρkuk t+ρk(uk· ∇)uk−2µrcurl wk−ρkf, v) + (µ+µr)(Auk, v) = 0, (ρkwk t+ρk(uk· ∇)wk−2µrcurl uk−ρkg, ψ) + (Lwk, ψ) = 0, for 0 < t < Tk,∀v∈Vk,∀ψ∈Wk, ρk t+uk· ∇ρk= 0 in Ω ×(0, Tk), uk(0) = Pku0, wk(0) = Rkw0, ρk(0) = ρ0. (9) The first main result in this paper is the following: Theorem 1 Assume that the initial data u0,w0and ρ0satisfy (3)–(5) and the external fields fand gsatisfy (6). Then (1)–(2) possesses exactly one strong solution (u, w, ρ)defined on a (possibly small) maximal time interval [0, T0), where T0≤T. The functions u,wand ρsatisfy P(ρut+ρ(u· ∇)u−2µrcurl w−ρf) + Au = 0,(10) ρwt+ρ(u· ∇)w−2µrcurl u+Lw =ρg (11) and ρt+u· ∇ρ= 0 a.e. in Ω×(0, T0). 7 Remark 1 As mentioned above, the functions u,wand ρsatisfy additional regularity properties. More precisely, the following will be proved for any small positive δand γ: u∈L2(0, T0;V)∩L∞(0, T0;H)∩C0([0, T0); D(A)) ∩L2(0, T0−γ; (H3(Ω))3)∩L∞(δ, T0−γ; (H3(Ω))3), ut∈L3/2(0, T0;V0)∩C0([0, T0); H)∩L2(0, T0−γ;V) ∩L2(δ, T0−γ;D(A)) ∩L∞(δ, T0−γ;V), utt ∈L2(δ, T0−γ;H). Similar regularity properties will also be established for the angular velocity w. Remark 2 As in the case of the classical Navier-Stokes equations, it can be proved that the strong solution furnished by Theorem 1 is global in time, i.e. it is defined for any t∈[0, T ) if the data u0,w0,ρ0,fand gare small enough. This situation will be analyzed in Section 7. The proof of Theorem 1 relies on appropriate estimates for the approximations (uk, wk, ρk). For future reference, let us gather them in the following: Proposition 1 Let (uk, wk, ρk)be the solution of (8). There exists T0>0 (independent of k) such that the following estimates hold for all t∈[0, T0): k∇uk(t)k2+k∇wk(t)k2≤F1(t), kuk t(t)k2+kwk t(t)k2+Zt 0 {k∇uk t(s)k2+k∇wk t(s)k2}ds ≤F2(t), kAuk(t)k2+k∆wk(t)k2≤F3(t), α0≤ρk(x, t)≤β0,(α0= inf Ωρ0, β0= sup Ω ρ0) k∇ρk(t)k2 L∞≤F4(t),kρk t(t)k2 L∞≤F5(t), Zt 0 {kuk(s)k2 H3+kwk(s)k2 H3}ds ≤F6(t), Zt 0 σ(s){kuk tt(s)k2+kwk tt(s)k2}ds ≤F7(t), σ(t){k∇uk t(t)k2+k∇wk t(t)k} ≤ F8(t), 8 σ(t){kuk(t)k2 H3+kwk(t)k2 H3} ≤ F9(t), Zt 0 σ(s){kAuk t(s)k2+k∆wk t(s)k2}ds ≤F11(t). Here, the Fiare nondecreasing continuous functions on [0, T0)and σ(t)≡ min{1, t}. Furthermore, the same estimates hold for the solution (u, w, ρ) furnished by Theorem 1. Remark 3 Proposition 1 implies the existence of a subsequence of approximations (uk, wk, ρk) that converges to the solution (u, w, ρ) in the following sense: (i) uk→u, wk→wstrongly in Lp(δ, T0−γ; (H3−ε(Ω))3), weakly in L2(0, T0−γ; (H3(Ω))3) and weakly-∗in L∞(δ, T0−γ; (H3(Ω))3), (ii) uk t→ut, wk t→wtstrongly in Lp(δ, T0−γ; (H1−ε(Ω))3), weakly in L2(0, T0−γ; (H1(Ω))3), weakly-∗in L∞(0, T0−γ; (L2(Ω))3) and weakly in L2(δ, T0−γ; (H2(Ω))2), (iii) uk tt →utt, wk tt →wtt weakly in L2(δ, T0−γ; (L2(Ω))3), (iv) ρk→ρstrongly in Lp(0, T0;C0,β(Ω)) (0 ≤β < 1) and uniformly in Ω×[0, T0−γ], (v) ∇ρk→ ∇ρweakly-∗in L∞(Ω ×(0, T0−γ))3, (vi) ρk t→ρtweakly-∗in L∞(Ω ×(0, T0−γ)). The above is true for all small δ,γand ε > 0 and any p∈(1,+∞). This justifies the regularity properties of uand wmentioned in Remark 1. Remark 4 Actually, more information can be obtained for ρkand ρ. Arguing as in [14], we deduce that the distribution function for the mass density, i.e. the function λ7→ meas {x∈Ω : ρ(x, t)≤λ} is independent of t. This leads in particular to the inequalities α0≤ρ(x, t)≤ β0. 9 Lemma 4 The approximations wksatisfy the following estimates for any t∈[0, T0): Zt 0 kwk(s)k2 W2,6ds ≤˜ F3(t),(34) Zt 0 k∇wk(s)k2 L∞ds ≤˜ F4(t).(35) Proof: For any ψ∈C∞ 0(Ω), we have (Lwk, ψ) =−(ρkwk t+ρk(uk· ∇)wk−2µrcurl uk−ρkg, ψ) ≡(ηk, ψ). (36) As before, ηkis uniformly bounded in L2(0, T0; (L6(Ω))3) for all T0< T0. Thus, wkis uniformly bounded in L2(0, T0; (W2,6(Ω))3). From the Sobolev embeddings, ∇wkis also uniformly bounded in L2(0, T0; (L∞(Ω))3). Lemma 5 The approximations ukand wksatisfy the following for all t∈ [0, T0): Zt 0 {kuk(s)k2 H3+kwk(s)k2 H3}ds ≤F6(t).(37) Proof: Again, we will take into account (33). From (30) and the estimates in Lemmas 1 and 2, it is not difficult to see that χkis uniformly bounded in L2(0, T0; (H1(Ω))3) for all T0< T0. Therefore, using Stokes regularity, we deduce that ukis also bounded in L2(0, T0; (H3(Ω))3). This proves the estimate for ukin (37). Arguing in a similar way for wk(starting from (36)), we finally deduce (37) for some F6. The following elementary remark will be useful for further estimates. Remark 5 Let h: (a, b)7→ Rbe a positive continuous function such that Zb a h(s)ds < +∞.(38) Then there exists a sequence {εn}with εn→a+such that εnh(εn)→0 as n→+∞. 16 Lemma 6 Under the assumptions in Theorem 1, we have the following for all t∈[0, T0): Zt 0 σ(s){kuk tt(s)k2+kwk tt(s)k2}ds ≤F7(t),(39) σ(t)(k∇uk t(t)k2+k∇wk t(t)k2)≤F8(t),(40) σ(t){kuk(t)k2 H3+kwk(t)k2 H3} ≤ F9(t).(41) Here, we have used the notation σ(t) = min{1, t}. Proof: Differentiating the first equation in (9) with respect to tand taking v=uk tt , in view of the estimates in the previous Lemmas, we get kuk ttk2+d dtk∇uk tk2≤G5(t)1 + k∇uk t(t)k2.(42) Multiplying (42) by σ(t) and integrating in (ε, t), we obtain Zt ε σ(s)kuk tt(s)k2ds +Zt ε σ(s)d dtk∇uk t(s)k2ds ≤Zt ε G5(s)σ(s)1 + k∇uk t(s)k2ds. Observe that Zt ε σ(s)d dtk∇uk t(s)k2ds =σ(t)k∇uk t(t)k2−σ(ε)k∇uk t(ε)k2−Zt ε σ0(s)k∇uk t(s)k2ds. (43) In view of Remark 5, we can choose ε=εnwith εn→0 and σ(εn)k∇uk t(εn)k2→0. Then σ(t)k∇uk t(t)k2+Zt 0 σ(s)k∇uk tt(s)k2ds ≤Zt 0 G5(s)σ(s)1 + k∇uk t(s)k2ds +Zt 0 k∇uk t(s)k2ds ≤G6(t)Zt 01 + k∇uk t(s)k2ds. (44) 17 In a similar way, we can deduce that σ(t)k∇wk t(t)k2+Zt 0 σ(s)k∇wk tt(s)k2ds ≤G7(t)Zt 01 + k∇uk t(s)k2+k∇wk t(s)k2ds (45) for all t∈[0, T0) for some G7. Hence, we have (39) and (40). The estimates (41) can be proved arguing as in the proof of Lemma 5 and using that (39) holds. This completes the proof. With similar arguments, the following Lemma also holds: Lemma 7 Under the hypotheses in Theorem 1, we have Zt 0 σ(s){kAuk t(s)k2+k∆wk t(s)k2}ds ≤F10(t) (46) for all t∈[0, T0). 4 Proof of Existence In view of the estimates given in Proposition 1, we can find a triplet (u, w, ρ) and a subsequence, again indexed by k, such that (uk, wk, ρk)→(u, w, ρ) in the sense indicated in Remark 3. We will now show that this suffices to pass to the limit in (9) and obtain (7). Thus, let us first prove that ZT0 0 (ρk(t)uk t(t), v)φ dt →ZT0 0 (ρ(t)ut(t), v)φ dt (47) and ZT0 0 (ρk(t)wk t(t), z)ζ dt →ZT0 0 (ρ(t)wt(t), z)ζ dt (48) as k→ ∞, for any v, z ∈(C∞ 0(Ω))3and any φ, ζ ∈ D(0, T). We have ZT0 0 (ρkuk t, v)φ dt ≤ZT0 0 ((ρk−ρ)uk t, v)φ dt +ZT0 0 (ρk(uk t−ut), v)φ dt . (49) 18 Observe that ZT0 0 ((ρk−ρ)uk t, v)φ dt ≤Cφ,v ZT0 0 kρk−ρk kuk tkdt for some T0< T0. Consequently, the first integral in the right hand side of (49) converges to zero. On the other hand, ZT0 0 (ρ(uk t−ut), v)φ dt =ZT0 0 (uk t−ut, ρv)φ dt for some T0< T0. Bearing in mind that uk t→utweakly-∗in L∞(Ω×(0, T0)) (for instance), we deduce that this integral also converges to zero. Thus, we have proved (47). The convergence of ρkwkin (48) can be proved similarly. Next, let us show that ZT0 0 (ρk(uk· ∇)uk, v)φ dt →ZT0 0 (ρ(u· ∇)u, v)φ dt (50) and ZT0 0 (ρk(uk· ∇)wk, z)ζ dt →ZT0 0 (ρ(u· ∇)w, z)ζ dt (51) as k→ ∞ for any v,z,φand ζas above. We will only prove (50), since the proof of (51) is similar. Notice that                              ZT0 0 (ρk(uk· ∇)uk, v)φ dt −ZT0 0 (ρ(u· ∇)u, v)φ dt =ZT0 0 ((ρk−ρ)(uk· ∇)uk, v)φ dt +ZT0 0 (ρ((uk−u)· ∇)uk, v)φ dt +ZT0 0 (ρ(u· ∇)(uk−u), v)φ dt. (52) Observe that the first integral in the right hand side of (52) converges to zero. Indeed, we have ZT0 0 ((ρk−ρ)(uk· ∇)uk, v)φ dt 19 ≤Cφ,vkρk−ρkL2(Ω×(0,T0)) ZT0 0ZΩ |(uk· ∇)uk|2dx dt!1/2 ≤Cφ,vkρk−ρkL2(Ω×(0,T0)) ZT0 0 kAukkk∇ukk3dt!1/2 and this converges to zero, in view of the estimates (14) and (22). The second integral in the right hand side of (52) is ZT0 0ZΩ ρ((uk−u)· ∇)uk·vφ dx dt ≤Cφ,vkρkL∞(Ω×(0,T)) ZT0 0 kuk−ukk∇ukkdt and also converges to zero. The third integral can be written in the form ZT0 0ZΩ ρ(u· ∇)(uk−u)·vφ dx dt. Since ∇uk→ ∇uweakly in L2(Ω ×(0, T ))3, it also converges to zero as k→+∞. By density, it is clear that (47), (48), (50) and (51) hold for any v∈D(A), any z∈D(L) and φ, ζ ∈C∞ 0(0, T). We can now pass to the limit in (9). Indeed, for any v∈Sj≥1Vjand any φ∈ D(0, T), we have ZT0 0 (ρkuk t+ρk(uk· ∇)uk−2µrcurl wk−ρkf−(µ+µr)∆uk, v)φ dt = 0 for all sufficiently large k. Letting k→+∞, we obtain hρut+ρ(u· ∇)u−2µrcurl w−ρf −(µ+µr)∆u, vi= 0 a.e. in (0, T0), for every v∈Sj≥1Vj. This gives P(ρut+ρ(u· ∇)u−2µrcurl w−ρf −(µ+µr)∆u) = 0 a.e. in Ω ×(0, T0). The passage to the limit in the equation for wkis analogous. In order to deal with the equation for the density, let us simply observe that, for instance, uk→ustrongly in L2(Ω×(0, T0)), ρk t→ρtweakly in L2(Ω×(0, T0)) and ∇ρk→ ∇ρweakly in L2(Ω ×(0, T0))3for all T0< T0. This gives ρt+u· ∇ρ= 0 20 in the distributional sense. Let us now check that the initial conditions are satisfied. We will first prove the following result: Proposition 2 Under the assumptions in Theorem 1, we have: lim t→0+ku(t)−u0kH1 0= 0 (53) and lim t→0+kw(t)−w0kH1 0= 0,(54) i.e. uand wassume the initial data continuously in the H1 0-norm. Proof: We will only prove (53), since (54) can be proved similarly. For all t, we have ∇uk(t)− ∇uk 0=Zt 0 ∇uk t(s)ds ∀k≥1. Therefore, in view of (21), k∇uk(t)− ∇uk 0k ≤ Zt 0 k∇uk t(s)kds ≤F2(T0/2)t(55) for any t∈[0, T0/2]. Now, notice that ukis bounded in L∞(0, T0/2; D(A)) and uk tis bounded in L∞(0, T0/2; V). From Aubin-Lions’ Lemma, we deduce that uk→ustrongly in C0([0, T0/2]; V). Thus, we can pass to the limit in (55), which gives k∇u(t)− ∇uk 0k ≤ F2(T0/2)t∀t∈[0, T0/2]. This proves (53). Proposition 3 Under the assumtions in Theorem 1, we have: lim t→0+kAu(t)−Au0kL2= 0 (56) and lim t→0+kut(t)−ut(0)kL2= 0.(57) 21 Proof: To prove (56), it is sufficient to show that lim sup t→0+ kAu(t)k ≤ kAu0k,(58) since we already know that kAu(t)kis bounded near t= 0 and u(t)→u0 strongly in V. From the first equation in (9) with v=Auk, after integration in time, we get kAuk(t)k2− kAuk 0k2=−Zt 0 (m(s), Auk t(s)) ds, where m(t) = 1 µ+µr (ρkuk t+ρk(uk· ∇)uk−2µrcurl wk−ρkf). We have Zt 0 (m(s, Auk t(s)) ds = (m(t), Auk(t)) −(m(0), Au0)−Zt 0 (m0(s), Auk(s)) ds. Therefore, in view of the estimates in Proposition 1, it is not difficult to prove that Zt 0 (m(s), Auk t(s)) ds ≤ |(ρk(uk· ∇)uk−2µrcurl wk−ρkf, Auk)(t) −(ρ0(u0· ∇)u0−2µrcurl w0−ρ0f(0), Au0)| +G8(T0/2) t+t1/4 for all t∈[0, T0/2], for some G8. We deduce that kAuk(t)k2≤ kAuk 0k2+G8(T0/2) t+t1/4 +|(ρk(uk· ∇)uk−2µrcurl wk−ρkf, Auk)(t) −(ρ0(u0· ∇)u0−2µrcurl w0−ρ0f(0), Au0)|. We know that, for each t∈[0, T0/2], Auk(t)→Au(t) weakly in L2(Ω) and uk(t)→u(t) strongly in V. Thus, lim inf k→+∞kAuk(t)k2≤ kAu(t)k2, 22 we have lim k→+∞(ρk(uk· ∇)uk−2µrcurl wk−ρkf, Auk)(t) = (ρ0(u0· ∇)u0−2µrcurl w0−ρ0f(0), Au0) and also kAu(t)k2≤ kAu0k2+G7(T0/2) t+t1/4. Obviously, this leads to (58). For the angular velocity, we have a similar result: Proposition 4 Under the assumtions in Theorem 1, we have: lim t→0+k∆w(t)−∆w0kL2= 0 (59) and lim t→0+kwt(t)−wt(0)kL2= 0.(60) Remark 6 The argument used in the proofs of these two Propositions can also be made at any t=t0∈(0, T0) instead of t= 0. This implies continuity from the right of u,ut,wand wtin the appropriate spaces. These arguments can also be adapted to prove continuity from the left at any t0∈(0, T0). In this way, we deduce the continuity properties indicated in Remark 1. 5 Proof of Uniqueness Let (v, ψ, σ) be a solution to (1)–(2) in [0, T0) and assume that (v, z, σ)∈ H0. Let us introduce (η, ξ, π), with η=u−v,ξ=w−ψand π=ρ−σ. Then these functions satisfy the following equations:              P(ρηt+σ(v· ∇)η)+(µ+µr)Aη =P(2µrcurl ξ+πf −πvt−π(u· ∇)u−σ(η· ∇)u), ρξt+σ(v· ∇)ξ+Lξ = 2µrcurl η+πg −πψt−π(u· ∇)w−σ(η· ∇)w, πt+u· ∇π=−η· ∇σ. (61) 23 Multiplying the first equation in (61) by ηand integrating over Ω, we obtain 1 2 d dtkρ1/2ηk2+ (µ+µr)k∇ηk2 = (2µrcurl ξ+πf −πvt−π(u· ∇)u−σ(η· ∇)u, η) +1 2(ρtη, η)−(σ(v· ∇)η, η). Now, estimating the above terms in the usual way, we obtain the following integral inequality: kη(t)k2+Zt 0 k∇η(s)k2ds ≤CZt 0kf(s)k2 L3+kvt(s)k2 L3+k∇u(s)k2kAu(s)k2kπ(s)k2ds +CZt 0 kξ(s)k2ds +CZt 0k∇v(s)k4+k∇u(s)k4+kρt(s)kL∞kη(s)k2ds. In a similar way, from the second equation in (61), we find kξ(t)k2+Zt 0 k∇ξ(s)k2ds ≤CZt 0kg(s)k2 L3+kψt(s)k2 L3+k∇u(s)k2k∆w(s)k2kπ(s)k2ds +CZt 0 kη(s)k2ds +1 2Zt 0 k∇η(s)k2ds +CZt 0k∇v(s)k2+k∆w(s)k2kρt(s)kL∞kξ(s)k2ds. On the other hand, multiplying the third equation in (61) by πand integrating with respect to xand tin Ω ×(0, t), we obtain: kπ(t)k2≤CZt 0 kη(s)kk∇σ(s)kL∞kπ(s)kds ≤Zt 0 kη(s)k2ds +CZt 0 k∇σ(s)k2 L∞kπ(s)k2ds. From these estimates, we deduce that kη(s)k2+kξ(s)k2+kπ(s)k2≤Zt 0 h(s)(kη(s)k2+kξ(s)k2+kπ(s)k2)ds 24 for all t∈[0, T0), where h(t) = C(1 + kfk2 H1+kgk2 H1+kvtk2 L3+kψtk2 L3 +k∇uk2(kAuk2+k∆wk2) +k∇uk4+k∆wk2+kρtkL∞+k∇vk4+k∇σk2 L∞). Observe that his an integrable function, in view of the regularity of (u, w, ρ) and (v, ψ, σ). Consequently, we can apply Gronwall’s Lemma, which gives kξ(t)k2+kη(t)k2+kπ(t)k2≡0, i.e. u=v,w=ψand σ=ρ. This ends the proof of Theorem 2. 6 Some Additional Results Concerning the Pressure We can now obtain some information on the pressure: Proposition 5 Under the assumptions of Theorem 1, there exists a function p∈C0([0, T0); H1(Ω)) such that ρut+ρ(u· ∇)u−(µ+µr)∆u+∇p= 2µrcurl w+ρf a.e. in Ω×(0, T0). Proof: Let (u, w, ρ) be the strong solution furnished by Theorem 1 and let us set j=ρ(f−ut−(u· ∇)u)+2µrcurl w+ (µ+µr)∆u. Then, from the regularity of (u, w, ρ) (see Remark 1), we easily deduce that j∈C0([0, T0); L2(Ω)) ∩L2(0, T0−γ;H1(Ω)) ∀γ > 0 (62) and jt∈L2(δ, T0−γ;L2(Ω)) ∀δ, γ > 0.(63) From (10), we have (j(t), v) = 0 ∀v∈ V(Ω), for ta.e. in [0, T0). Consequently, we deduce from De Rham’s Lemma that j=∇pfor some p∈ D0(Ω ×(0, T0)). Furthermore, since we have (62) and (62), we can choose psatisfying p∈C0([0, T0); H1(Ω)) ∩L2(0, T0−γ;H2(Ω)) ∀γ > 0 25