Some existence and uniqueness results for a time-dependent coupled problem of the Navier-Stokes kind

Some Existence and Uniqueness Results for a Time-Dependent Coupled Problem of the Navier-Stokes kind B. Climent and E. Fern´ andez-Cara∗ Abstract In this paper, we consider some systems which are close to the instationary Navier-Stokes equations. The structure of these systems is the following: An (N+ 1)-dimensional equation for motion (including the incompressibility condition) and a scalar equation involving an additional unknown, k=k(x, t). Among other things, they serve to model the behavior of certain turbulent flows. We are mainly concerned with existence and uniqueness results. The main difficulties are due to the scalar equation. In particular, the right side is typically in L1; furthermore, there are nonlinear terms of the kind ∇ · (µ(k)∇k) and ∇ · (B(k)), where µ and Bare general continuous functions (no growth condition at infinity is imposed). Following the previous work of other authors, it is crucial to introduce the notion of weak-renormalized solution. Our results provide existence in the two-dimensional case, as well as the uniqueness of regular solution in both the two and three-dimensional cases. ∗Department of Differential Equations and Numerical Analysis, University of Sevilla, Tarfia s/n, E-41012 Sevilla, Spain. Partially supported by D.G.I.C.Y.T. (Spain), Proyecto PB92– 0696. 0 Notation: •L1=L1(Ω), H1 0=H1 0(Ω), etc.; |·| (resp. k·k) denotes the usual norm in L2(resp. H1 0). •H−1=H−1(Ω) is the dual space of H1 0;k·k∗denotes the usual norm in H−1. •In general, if X=X(Ω) is a space of functions defined in the open set Ω and p≥1 , we denote by Lp(X) (resp. C0(X)) the Banach space Lp(0, T;X) (resp. C0([0, T]; X)). •z+= max(z, 0) for any real z. Contrarily, z+(resp. z−) denotes any real number z0> z (resp. z0< z) close enough to z. •TM(s) = sif s∈[−M, M] ; TM(s) = Msign sotherwise. •ξn,m =1 m(Tn+m−Tn) for all n, m ≥1 . •Sδ(s) = sif s∈[−δ, δ] ; Sδ(s) = sign sotherwise. •S:D=PN i,j=1 SijDij for any S={Sij}and D={Dij}. •For each p∈[1,+∞] , p∗is the associated Sobolev embedding exponent: p∗=Np N−pif p<N; 1 < p∗<+∞(arbitrary) if p=Nand p∗= +∞ otherwise. 1 1 Introduction. Motivation of the problem This paper is concerned with some nonlinear partial differential systems stemming from fluid mechanics. These are variants of the instationary Navier-Stokes equations and read as follows:          ∂tu−∇·νDu +|k|1/2Φ0(Du)+ (u· ∇)u+∇p=f , ∇ · u= 0 , ∂tk−∇·(µ(k)∇k+B(k)) + u· ∇k=ν0|Du|2+|k|1/2Φ0(Du):Du − |k|3/2ψ0(Du). (1) In (1), it is assumed that Du =∇u+t∇u. The functions D7→ Φ(D) , D7→ ψ0(D) , k7→ µ(k) and k7→ B(k) are prescribed. Given an open set Ω⊂IRN, a final time T > 0 , the function fand the coefficients ν > 0 and ν0∈[0, ν] , we will search for a triplet {u, p, k}satisfying (1) in Q= Ω ×(0, T) , together with appropriate initial conditions at t= 0 and boundary conditions on ∂Ω×(0, T) . Systems like (1) are motivated by turbulence modelling. More precisely, let U=U(x, t) and P=P(x, t) be respectively the velocity field and pressure distribution of the turbulent flow of a viscous incompressible fluid. Then, the couple {U, P}must satisfy the instationary Navier-Stokes equations: ∂tU−ν∆U+ (U· ∇)U+∇P=F , ∇ · U= 0 .(2) Let us denote by uand psome averaged variables (we write u=Uand p=P; uand pdescribe the mean flow). Let us put U=u+u0, P =p+p0. Then, instead of (2), it is appropriate to try to solve a system that should be satisfied by uand p. After some computations, one finds: ∂tu−∇·(νDu +R)+(u· ∇)u+∇p=f , ∇ · u= 0 ,(3) where f=f(x, t) is the averaged external forces field, i.e. f=Fand Ris the so called Reynolds tensor: R={Rij},with Rij =−u0 iu0 j. Since in (3) we still find the unknown variables u0 i, it is reasonable to introduce closing hypotheses relating Rto u. In the case of usual one-equation models, one imposes the following hypothesis of the Boussinesq kind: R=νTDu , where νT=νT(k) (an algebraic relation). (4) Here, k=1 2|u0|2is the mean turbulent kinetic energy. The problem is thus closed using (3), (4) and an additional PDE for k. 2 Unfortunately, when one tries to deduce an equation for k, one finds again terms in which the turbulent perturbations u0 i(and k0) appear. More precisely, one has: ∂tk−∇·ν∇k+ (−(p0+k0)u0)+u· ∇k=R:Du −ν 2|Du0|2.(5) Consequently, one has to replace (5) by an approximation. This is made by introducing new closing hypotheses: •Of course, (4) is used again in order to approximate the production term R:Du . •The approximation of the dissipation term ν 2|Du0|2is almost always the same: a constant times k3/2. •Contrarily, the approximation of −(p0+k0)u0has been achieved in several different ways in the litterature. In most papers, this term is replaced by c νT∇k, where cis an experimental constant (for instance, see [14], [15] and the references therein). In other papers, however, it is replaced by a vector function B(k) (see [6]). Hence, it is clear that equations like (1) can be used to describe the evolution of some turbulent flows. Another motivation for (1) can be found in non Newtonian mechanics. In this setting, uand pare the true velocity field and pressure, kis the temperature and it is assumed that the stress tensor τdepends on Du and kas follows: τ=ν0Du +kΦ0(Du). 2 The main results In this section, we present our main results. These are concerned with existence and uniqueness for systems of the kind (1) completed with appropriate initial and boundary conditions. Let us mention that the authors have considered a similar stationary problem in the previous paper [8]. At first sight, one could think that, for (1), the results can be obtained in the same way as in [8]. This is not true at least because of three reasons: •As shown below, there is a major difficulty for the existence proof when N= 3. This is related to the lack of regularity of ∂tu, typical of the evolution Navier-Stokes problem and its variants. •In order to prove that kis a solution to the third equation in (1), the arguments used in the stationary case, which have been taken in part from [12], do not work. This is maybe surprising but well known. Here, another argument, essentially due to D. Blanchard and H. Redwane [3], is used. 3 •The uniqueness results presented in [8] (and also their proofs) are different from those in this paper. Roughly speaking, in [8] we prove uniqueness for regular data and sufficiently small Reynolds number, i.e. large ν. Of course, it is not reasonable to expect results of this kind in the framework of the time-dependent problem (1). In this paper, we prove that regular solutions, in case they exist, must be unique. We will consider a simplified version of (1):      ∂tu−ν∆u−∇·(kΦ0(∇u)) + (u· ∇)u+∇p=f , ∇ · u= 0 , ∂tk−∇·(µ(k)∇k+B(k)) + u· ∇k=ν|∇u|2+kΦ0(∇u):∇u . (6) This is made for simplicity; in fact, the results in this section also hold for (1) with appropriate minor changes. The first, second and third equations in (6) will be known as the motion equation, the incompressibility condition and the energy equation, respectively. Our basic assumptions are the following: •Ω⊂IRNis a bounded, connected, open and regular set; T > 0 , ν > 0 and f∈L2(H−1) . •D7→ Φ(D) is C1, Φ0(0) = 0, |Φ0(D)| ≤ Const. and D7→ Φ0(D):Dis convex (consequently, it is also locally Lipschitz-continuous). In particular, D7→ Φ(D) is convex and one has (Φ0(D1)−Φ0(D2)) : (D1−D2)≥0 for all D1and D2. •k7→ µ(k) and k7→ B(k) are continuous functions; furthermore, µ(k)≥ µ0>0 for all k. We want to solve (6) in Q= Ω ×(0, T ) together with initial conditions u|t=0 =u0and k|t=0 =k0in Ω (7) and homogeneous Dirichlet conditions u= 0 and k= 0 on ∂Ω×(0, T).(8) Our main interest concerns general continuous functions µand B. This is motivated by the fact that, in turbulence modelling, an equation exactly satisfied by the true turbulent kinetic energy is unknown. Besides the usual spaces Lp(Lq) , L2(V) , etc., we will use the following: L={ψ∈L1(Q) ; TM(ψ)∈L2(H1 0)∀M > 0, lim n→+∞ 1 nZZn≤|ψ|≤2n |∇ψ|2dx dt = 0 } 4 (see the Notation). Moreover, e β,eµ, etc. will denote integrals of the corresponding β,µ,. . . For instance, e β(s) = Zs 0 β(σ)dσ for all s . Theorem 1 –Assume N= 2 ,u0∈Vand k0∈L1, with k0≥0. Under the previous assumptions, there exists {u, p, k}, with u∈L2(V)∩C0(H), p ∈L2(Q), k ∈ L , such that: 1. The couple {u, p}solves the first two equations in (6) together with the first initial condition in (7) in the usual weak sense. 2. k≥0and solves the energy equation in (6) and the second initial condition in (7) in the following sense: For all β∈W1,∞(IR) with compact support, one has      ∂te β(k)−∇·(β(k)(µ(k)∇k+B(k))) +β0(k)∇k·(µ(k)∇k+B(k)) + β(k) (u· ∇k) =β(k)ν|∇u|2+kΦ0(∇u):∇uin D0(Q). (9) In particular, ∂te β(k)∈L1(L1) + L2(H−1)and, for all q < 2, one has e β(k)∈C0(W−1,q). Furthermore, e β(k)|t=0 =e β(k0).(10) A triplet {u, p, k}as above will be called a weak-renormalized solution to (6), (7), (8). Renormalized solutions to PDE’s have been introduced by R. DiPerna and P.L. Lions in [9], in the framework of the Boltzmann equations. They have been used in connection with various nonlinear elliptic (resp. parabolic) equations by P. Benilan et al. [2], L. Boccardo et al. [5] and P.L. Lions and F. Murat [12],[13] (resp. by D. Blanchard and H. Redwane [3]). For the analysis of some problems similar to (1) and (6), weak-renormalized solutions were first considered by R. Lewandowski [11]. That we search for a renormalized solution kis motivated by the structure of the right side of the energy equation in (6) (typically in L1) and also by our interest in keeping µand Bas general as possible. 5 Theorem 2 –Under the assumptions in theorem 1, assume also that B≡0 and k0∈L∞. Then the solution {u, p, k}furnished by theorem 1satisfies eµ(k)∈\ q<2 Lq(W1,q 0),∇eµ(k) = µ(k)∇k , (11) ∂tk∈L1(L1) + Lq(W−1,q)for all q < 2 (12) and also the following:        h∂tk , φi+ZΩ µ(k)∇k· ∇φ+ZΩ (u· ∇k)φ =ZΩν|∇u|2+kΦ0(∇u):∇uφ∀φ∈ D(Ω) ,a.e. in (0, T ). If {u, p, k}is as in theorem 2, it will be said it is a weak solution to (6), (7), (8). As mentioned above, the situation considered in this theorem is the most frequently found in connection with one-equation turbulence models. Theorem 3 –Assume that N= 2 or N= 3 ,k7→ µ(k)is locally Lipschitzcontinuous, B≡0and D7→ Φ(D)is C2,with |Φ00(D)| ≤ Const. Let u0∈Vand k0∈W2,∞∩H1 0, with k0≥0. For i= 1,2, let {ui, pi, ki}be a (weak) solution to (6),(7),(8), with ui∈L∞(W1,∞)and suppose that (for instance) u2∈L2(W2,r), where r > N (r≥3if N= 3 ). Then {u1,∇p1, k1} and {u2,∇p2, k2}must coincide. Before giving the proofs of these results, let us make some remarks: 1. Theorems 1 and 2 provide existence for (6), together with suitable initial and boundary conditions when N= 2 . Other similar results can be found in [11]. It is not clear how to extend the proofs in order to cover the three-dimensional case (see the second and subsequent steps of the proof of theorem 1). 2. However, several more or less obvious generalizations of theorems 1 and 2 are possible. Thus, an existence result similar to theorem 1 can be deduced if we replace kΦ0(∇u) by a term of the form D2Ψ(k, ∇u) , where (s, D)7→ Ψ(s, D) satisfies appropriate assumptions. When N= 3, existence can also be obtained if one omits the nonlinear term (u· ∇)u. On 6 the other hand, theorem 2 holds as well when Bis not zero but Lipschitzcontinuous; for details, see [7]. An interesting situation arises when we simply assume µ(k)≥0 in (6). It seems also interesting to relax the assumption “D7→ Φ0(D) : Dis convex” in order to account for more general approximations of the Reynolds tensor. These cases are far from trivial and will be considered in a future work. 3. Some variants of theorem 3 can also be proved. Nevertheless, there is an important question that remains open: For N= 2 , is there uniqueness of weak-renormalized solution ? This seems to be complicate, but a positive answer would have a very interesting interpretation. 4. The existence of a weak-renormalized solution (and the existence and uniqueness of a weak solution) to the stationary analog of (6) have been established by the authors in [8] under reasonable assumptions (see also [1] and [11] for other similar results). 3 The proof of theorem 1 In the sequel, Cdenotes a constant which may depend on Ω and the other data in (6). The proof of theorem 1 consists of six steps: First step: The introduction of a family of approximations. For each ε > 0, we consider the following approximation to (6):      ∂tuε−∇·τε+ (uε· ∇)uε+∇pε=f , ∇ · uε= 0 , ∂tkε−∇·(µε(kε)∇kε+Bε(kε)) + uε· ∇kε=T1 ε(τε:∇uε). (13) Here, we have used the following notation: µε=T1 ε◦µ , Bε=B◦T1 ε, τε=ν∇uε+T1 ε(kε)+Φ0(∇uε). Again, these equations are required to be satisfied in Q= Ω ×(0, T ) together with the initial conditions uε|t=0 =u0and kε|t=0 =T1 ε(k0) in Ω (14) and the boundary conditions uε= 0 and kε= 0 on ∂Ω×(0, T).(15) The existence of a triplet {uε, pε, kε}satisfying (13), (14) and (15) can be established using (for instance) a Galerkin method. In fact, with this technique one 7 finds some nontrivial difficulties that can be solved arguing as in the following steps. One sees that uε∈L2(V)∩C0(H), kε∈L2(H1 0)∩C0(L2) and, also, that kε≥0 . Second step: A priori estimates and weak convergence. Using uεas a test function in the first equation in (13), one obtains: |uε(t)|2+Zt 0ZΩ τε:∇uε≤Cin (0, T). In particular, kuεkL∞(H)≤C , kuεkL2(V)≤C . (16) ¿From this, one also deduces by interpolation: uεis bounded in L(2a a−2)− (La),for all finite a > 2 (recall that z−is, for each z∈IR , an arbitrary real number z0< z , close enough to z). Since N= 2 , (16) suffices to deduce a bound for ∂tuεin L2(V0) : k∂tuεkL2(V0)≤C . Let us now consider the third equation in (13) (the energy equation). We will succesively use Sδ(kε) , TM(kε) and ξn,m(kε) as test functions (see the Notation). We easily find: kkεkL∞(L1)≤C , kTM(kε)kL2(H1 0)≤C·M(17) and also 1 mZZn≤kε≤n+m µε(kε)|∇kε|2≤ZZkε≥n T1 ε(τε:∇uε) + ZΩe ξn,m(T1 ε(k0)) (18) (recall that e ξn,m is the primitive of ξn,m which vanishes at zero). From (18), one sees that 1 mZZn≤kε≤n+m |∇kε|2≤C(n, m).(19) ¿From (17) and (19), arguing as in [4] and [13], one deduces the following: kkεkLq(W1,q 0)≤Cqfor all q < 2. By interpolation, it is also seen that kεis bounded in L(2b b−1)−Lb,for all finite b > 1. 8 Here, Yε M(z) = Zz 0 b0 ε(r)TM(r)dr and vε 0=vε(0) = eµ(k0) (observe that vε 0is bounded in L∞). The right side in (29) is ≤C˙ M, since Fε 1+Fε 2converges strongly in L1(Q) and 0 ≤Yε M(z)≤µ0Mz for all z≥0 . Hence, the first part of (28) holds. Now, let us use ξn,m(vε) as test function in (27). This gives: 1 mZZn≤vε≤n+m |∇vε|2≤ZZvε≥n |Fε 1+Fε 2|+ZΩb ξε n,m(vε 0). Here, we have introduced the function b ξn,m , given as follows: b ξε n,m(z) = Zz 0 b0 ε(r)ξn,m(r)dr ∀z . It is clear that 0 ≤b ξε n,m(z)≤µ0zfor all z≥0 , whence we easily deduce the second estimate in (28). 5 The proof of theorem 3 Let us set u=u1−u2and k=k1−k2. It is not difficult to see (in the usual way) that 1 2 d dt|u|2+νkuk2≤ −((u· ∇)u2, u)−(kΦ0(∇u2),∇u) (30) in (0, T) . The first term in the right side can be bounded as follows: |((u· ∇)u2, u)| ≤ k∇u2kL∞· |u|2. The second one satisfies: |(kΦ0(∇u2),∇u)|=|(Φ0(∇u2)∇k, u)+(k∇ · (Φ0(∇u2)), u)| ≤Ckkk·|u|+CkkkLbkD2u2kLr· |u|, where bis given by the identity 1 b+1 r+1 2= 1 . Since r > N , one has b < 2∗ and one obtains: |(kΦ0(∇u2),∇u)| ≤ C1 + kD2u2kLrkkk·|u|. Consequently, integrating (30) with respect to time in (0, t) , one is led to the following: 1 2|u(t)|2+νZt 0 ku(s)k2ds ≤Zt 0 k∇u2(s)kL∞· |u(s)|2ds +Zt 0 C1 + kD2u2(s)kLrkk(s)k·|u(s)|ds . (31) 15 In the sequel, we will derive an estimate of kkkin terms of kuk. In view of (31), it will then be possible to apply Gronwall’s lemma and to deduce that u≡0 , i.e. u1≡u2. First, notice that |µ(k1)−µ(k2)| ≤ l|k| for some l > 0 . This happens because µis locally Lipschitz-continuous and both k1and k2are bounded. In turn, this is a consequence of the fact that, in the PDE satisfied by ki, |∇ui|2∈L∞(Q) and Φ0(∇ui):∇ui∈L∞(Q). ¿From the results in [10], taking into account that k0∈W2,∞∩H1 0, one deduces: ki∈L2(W2,p) for all finite p(in particular, ∇ki∈L2(L∞)). ¿From the equations satisfied by k1and k2, we find: ∂tk+u1· ∇k−∇·(µ(k1)∇k) = −u· ∇k2 +∇ · ((µ(k1)−µ(k2))∇k2) + ν(∇u1+∇u2):∇u +k1(Φ0(∇u1):∇u1−Φ0(∇u2):∇u2) + kΦ0(∇u2):∇u2. Hence, 1 2 d dt|k|2+µ0kkk2≤ZΩ |∇k2|·|k|·|u| +lZΩ |∇k2|·|k| · |∇k|+νZΩ (|∇u1|+|∇u2|)|k| · |∇u| +CZΩ k1(1 + |∇u2|)|k| · |∇u|+CZΩ |∇u2|·|k|2 ≤ k∇k2kL∞· |k|·|u|+lk∇k2kL∞· |k|·kkk +ν(k∇u1kL∞+k∇u2kL∞) + Ckk1kL∞(1 + k∇u2kL∞)|k|·kuk +Ck∇u2kL∞· |k|2 in (0, T) . It is thus clear that, for some g∈L1(0, T) , one can write: d dt|k|2+µ0kkk2≤g(t)|k|2+kuk2.(32) After some elementary computations, we find Zt 0 kk(s)k2ds ≤GZt 0 ku(s)k2ds for all t , 16 where Gis a constant only depending on kgkL1. Consequently, going back to (31), one deduces: |u(t)|2+ 2νZt 0 ku(s)k2ds ≤2Zt 0 k∇u2(s)kL∞· |u(s)|2ds +CZt 0 ku(s)k2ds1 2Zt 01 + kD2u2(s)kLr2|u(s)|2ds1 2 ≤νZt 0 ku(s)k2ds +CZt 01 + k∇u2(s)kL∞+kD2u2(s)k2 Lr|u(s)|2ds . In other words, we have proved that |u(t)|2+νZt 0 ku(s)k2ds ≤Zt 0 h(s)|u(s)|2ds for all t for some nonnegative h∈L1(0, T) . This implies u≡0 ; from (32), one also has k≡0 . Therefore, the proof is completed. Acknowledgment: The authors are indebted to D. 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