Existence and uniqueness results for a coupled problem related to the stationary Navier-Stokes System

Existence and Uniqueness Results for a Coupled Problem Related to the Stationary Navier-Stokes System B. Climent and E. Fern´ andez-Cara ∗ June 13, 2007 Abstract In this paper, we consider some systems which are close to the stationary Navier-Stokes equations. The structure of these systems is the following: An N-dimensional equation for motion, the incompressibility condition and a scalar equation involving an additional unknown, k=k(x). Among other things, they serve to model the behavior of certain turbulent flows. Our main interest concerns existence and uniqueness. The main difficulties are due to the structure of the scalar equation; in particular, the right side is typically in L1and, furthermore, there are nonlinear terms of the kind ∇ · (µ(k)∇k) and ∇ · (B(k)), where µand Bare general continuous functions. ∗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. •z+= max(z, 0) for any real z. •TM(s) = sif s∈[−M, M] ; TM(s) = Msign sotherwise. •Lnis the piecewise linear even function satisfying Ln(s) = 1 if s∈[0, n] , Ln(s) = s n+ 2 if s∈[n, 2n] and Ln(s) = 0 if s > 2n. •S:D=PN i,j=1 SijDij for any S={Sij}and D={Dij}. •N0is the conjugate exponent of N, i.e. N0=N N−1. •For each p∈[1,∞] , p∗is the associated Sobolev embedding exponent: p∗=Np N−pif p<N; 1 < p∗<∞is arbitrary if p=Nand p∗=∞ otherwise. In particular, (N0)∗=N N−2if N≥3 . 1 1 Introduction. Description of the problem This paper is concerned with some nonlinear partial differential systems stemming from fluid mechanics. These are variants of the stationary Navier-Stokes equations and read as follows:          −∇ · (νDu +kΦ0(Du)) + (u· ∇)u+∇p=f , ∇ · u= 0 , −∇ · (µ(k)∇k+B(k)) + u· ∇k=ν0|Du|2+kΦ0(Du) : Du − |k|1/2kψ0(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 given. Once an open set Ω ⊂IRN and the data ν > 0 , ν0∈[0, ν] and fare fixed, we search for a solution {u, p, k} to (1), together with appropriate boundary value conditions. 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 a viscous incompressible fluid in turbulent regime. Then, the couple (U, P) must satisfy the instationary Navier-Stokes equations. Denoting by uand pthe corresponding time-averaged variables (that is to say, u=Uand p=P) and setting U=u+u0, P =p+p0, it is customary to replace the search of a solution to (1) by the analysis of a system that should be satisfied by uand p. After some computations, one finds: −∇ · (νDu +R)+(u· ∇)u+∇p=f , ∇ · u= 0 ,(2) where fis the time-averaged external forces field acting on the fluid particles and Ris the so called Reynolds tensor: R={Rij},with Rij =−u0 iu0 j. Since in (2) 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=F(k) (an algebraic relation). (3) Here, k=1 2|u0|2is the mean turbulent kinetic energy. The problem is thus closed using (2), (3) and an additional PDE for k. 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: −∇ · ν∇k+ (−(p0+k0)u0)+u· ∇k=R:Du −ν 2|Du0|2.(4) 2 Consequently, one has to replace (4) by an approximation. This is made by introducing new closing hypotheses: •There is general agreement in the approximation of the dissipation term ν 2|Du0|2. It is usually replaced by a constant times k3/2. •Of course, (3) is used again in order to approximate the production term R:Du. •The approximation of −(p0+k0)u0has been achieved by several authors in different ways. In most papers, this term is replaced by cνT∇k, where c is an experimental constant (for instance, see [13], [12] and the references therein). In others, it is replaced by a vector B(k) (see [7]). Hence, it is clear that equations like (1) can be used to describe the behavior of certain turbulent flows. Another motivation for (1) can be found in non Newtonian mechanics. In this setting, {u, p}are the true velocity field and pressure, kis the temperature and it is assumed that the stress tensor τdepends on Du and kas follows: τ=νDu +kΦ0(Du).(5) 2 The main results In the sequel, we will consider a simplified version of (1):      −ν∆u−∇·(kΦ0(∇u)) + (u· ∇)u+∇p=f , ∇ · u= 0 , −∇ · (µ(k)∇k+B(k)) + u· ∇k=ν|∇u|2+kΦ0(∇u) : ∇u . (6) This is made for convenience only; the results in this section also hold for (1) with appropriate changes. In (6), the first, second and third equations will be respectively known as the motion equation, the incompressibility condition and the energy equation. Our assumptions are the following: •Ω⊂IRNis a bounded, connected, open and regular set; ν > 0 and f∈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. 3 We want to solve (6) together with Dirichlet conditions for uand k: u= 0 and k= 0 on ∂Ω.(7) Our main interest concerns general continuous functions µand B. This is of course motivated by the fact that, in turbulence modelling, an equation exactly satisfied by kis unknown. Besides the usual spaces L2,H1 0,V, etc., we will use the following: L={ψ∈L1;TM(ψ)∈H1 0∀M > 0,lim n→∞ 1 nZn≤|ψ|≤2n |∇ψ|2dx = 0 } (see the Notation). Theorem 1–Under the previous assumptions, there exists {u, p, k}, with u∈V,p∈L2and k∈ L such that: 1. The couple {u, p}solves the first two equations in (6) in the usual weak or distributional sense. 2. k≥0and solves the third equation in (6) in the following sense: (−∇ · (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(8) in D0(Ω) for every β∈W1,∞(IR) with compact support. A triplet {u, p, k}as above will be called a weak-renormalized solution to (6). Renormalized solutions to PDE’s seem to have been introduced by R. DiPerna and P.L. Lions in [8], in the framework of the Boltzmann equation. They have been used in connection with various nonlinear elliptic equations by P. Benilan et al. [3], L. Boccardo et al. [6] and P.L. Lions and F. Murat [10] (see also [11]). In the analysis of existence results for problems similar to (1) and (6), weak-renormalized solutions were considered by R. Lewandowski [9] (see also [2]). That we look 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. Let us denote by ˜µthe following function: ˜µ(s) = Zs 0 µ(σ)dσ ∀s∈IR . Assume that, in theorem 1, one has B≡0 . Then it is not difficult to see that the solution {u, p, k}furnished by theorem 1 satisfies ˜µ(k)∈\ q<N0 W1,q 0,∇˜µ(k) = µ(k)∇k(9) 4 and also the following:   ZΩ µ(k)∇k· ∇φ+ZΩ (u· ∇k)φ=ZΩν|∇u|2+kΦ0(∇u) : ∇uφ ∀φ∈ D(Ω) . (10) In this case, it will be said that {u, p, k}is a weak solution to (6). Theorem 2–Assume that, in theorem 1,B≡0and D→Φ0(D) : Dis globally Lipschitz-continuous. Then, there exists ν0>0such that, when ν≥ν0, there exists at most one weak solution {u, p, k}to (6) with k≥0. Before giving the proofs of these results, let us make some remarks: 1. A very interesting question remains: When νis large and Bis not zero, is it still possible to prove the uniqueness of a renormalized solution ? 2. There are several other possible conditions for uniqueness, different from the assumption ν≥ν0. For instance, for fixed ν, one can also obtain at most one weak solution {u, p, k}to (6) with k≥0 if kfk∗is sufficiently small. 3. Results similar to those above can be proved for the instationary variant of (6). This will be analyzed in a forthcoming paper. 4. Several more or less obvious generalizations are possible. In particular, we find an interesting situation when we simply assume µ(k)≥0 in (6). This case is far from trivial and will also be the subject of future work. 3 The proof of theorem 1 In this section, Cdenotes a constant which may depend on N, Ω and the 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):        −ν∆uε− ∇ · (T1 ε(kε)+Φ0(∇uε)) + (uε· ∇)uε+∇pε=f , ∇ · uε= 0 , −∇ · T1 ε(µ(kε))∇kε+B(T1 ε(kε)+uε· ∇kε=T1 ε(τε:∇uε). (11) Here, we have used the following notation: τε=ν∇uε+T1 ε(kε)+Φ0(∇uε). 5 Of course, these equations are required to be satisfied in Ω , together with homogeneous Dirichlet conditions for uεand kεon ∂Ω . The existence of a triplet {uε, pε, kε}can be established using (for instance) a Galerkin method. In fact, some nontrivial difficulties are found with this technique that can be solved arguing as in the following steps. One finds that the solution belongs to the space V×L2×H1 0(Ω) and, also, that kε≥0 . Second step: A priori estimates and weak convergence. Using uεas a test function in the first equation in (11), one finds: ZΩ τε:∇uε≤C . (12) In particular, kuεk ≤ C . (13) In the energy equation in (11), let us use TM(kε) as test function. This gives: kTM(kε)k2≤C·M(14) On the other hand, if we choose ξn(kε) = T2n(kε)−Tn(kε) as test function in the same equation, it is not difficult to check that 1 nZn≤kε≤2n T1 ε(µ(kε))|∇kε|2≤Zkε≥n T1 ε(τε:∇uε),(15) whence 1 nZn≤kε≤2n |∇kε|2≤C . (16) ¿From (14) and (16), arguing as in [5] and [11], one deduces the following: kkεkW1,q 0≤Cq,∀q < N0.(17) Consequently, passing to a subsequence if necessary, it can be assumed that uε→uweakly in V, strongly in Lr∀r < 2∗and a.e., kε→kweakly in W1,q 0∀q < N0, strongly in Lp∀p < (N0)∗and a.e., TM(kε)→TM(k) weakly in H1 0∀M > 0. Obviously, one has k≥0 . Third step: uis, together with some p, a solution to the motion equation. For each ε > 0 , uεis a solution to the following variational inequality:        νZΩ ∇uε:∇v+ZΩ (uε· ∇)uε·v+ZΩ T1 ε(kε)Φ(∇v) ≥νZΩ |∇uε|2+ZΩ T1 ε(kε)Φ(∇uε) + hf, v −uεi ∀v∈V , uε∈V . 6 Taking limits as ε→0 , one obtains: νZΩ ∇u:∇v+ZΩ (u· ∇)u·v+ZΩ kΦ(∇v) ≥νlim inf ε→0ZΩ |∇uε|2+ lim inf ε→0ZΩ T1 ε(kε)Φ(∇uε) + hf, v −ui The first term in the right is bounded from below by νZΩ |∇u|2. In what concerns the second term, let us first notice that ZΩ T1 ε(kε)Φ(∇uε) = ZΩ (T1 ε(kε)−k)Φ(∇uε) + ZΩ kΦ(∇uε). Thus, taking into account that the function v7→ ZΩ kΦ(∇v) is lower semicontinous, we find: lim inf ε→0ZΩ T1 ε(kε)Φ(∇uε)≥lim ε→0ZΩ (T1 ε(kε)−k)Φ(∇uε) + lim inf ε→0ZΩ kΦ(∇uε) ≥ZΩ kΦ(∇u). Consequently, uis a solution to the variational inequality        νZΩ ∇u: (∇v− ∇u) + ZΩ (u· ∇)u·(v−u) + ZΩ kΦ(∇v) −ZΩ kΦ(∇u)≥ hf, v −ui ∀v∈V , u ∈V . (18) Now, taking in (18) the function vof the form u+tw , where w∈Vand t∈IR and letting t→0 , it is a standard matter to prove that usolves, together with some p∈L2, the first two equations in (6) in the usual weak sense. Fourth step: uεconverges strongly in V. ¿From the motion equation in (6), it is clear that ZΩν|∇u|2+kΦ0(∇u) : ∇u=hf, ui. On the other hand, choosing uεas test function in the first equation in (11), one has: ZΩν|∇uε|2+T1 ε(kε)Φ0(∇uε) : ∇uε=hf, uεi. 7 Consequently, lim ε→0ZΩν|∇uε|2+T1 ε(kε)Φ0(∇uε) : ∇uε=ZΩν|∇u|2+kΦ0(∇u) : ∇u, whence it is also clear that lim ε→0ZΩν|∇uε|2+kΦ0(∇uε) : ∇uε=ZΩν|∇u|2+kΦ0(∇u) : ∇u (recall that Φ0is uniformly bounded). Hence, 0 = lim ε→0ZΩν|∇uε|2+kΦ0(∇uε) : ∇uε−ZΩν|∇u|2+kΦ0(∇u) : ∇u ≥lim sup ε→0νZΩ |∇(uε−u)|2 + lim inf ε→0ZΩ kΦ0(∇uε) : ∇uε−ZΩ kΦ0(∇u) : ∇u). Here, the last term is ≥0 , in view of the lower semicontinuity of the function v7→ ZΩ kΦ0(∇v) : ∇v . Thus, lim ε→0ZΩ |∇(uε−u)|2= 0 . Fifth step: For all M > 0 , TM(kε) converges strongly in H1 0. We will use an argument due to P.L. Lions and F. Murat (see [10], [11]). Let us see that T1 ε(µ(kε))1 2∇TM(kε)→µ(k)1 2∇TM(k) strongly in L2∀M > 0 (19) (observe that µ(k)1 2∇TM(k) has a sense). Of course, (19) will suffice for our purposes. It has already been proved that T1 ε(τε:∇uε)→τ:∇ustrongly in L1and a.e. Here, we have introduced τ=ν∇u+kΦ0(∇u) . Choosing TM(kε) as test function in the energy equation in (11), one finds: ZΩ T1 ε(µ(kε))∇kε· ∇TM(kε) + ZΩ B(T1 ε(kε)) · ∇TM(kε) +ZΩ (uε· ∇kε)TM(kε) = ZΩ T1 ε(τε:∇uε)TM(kε). 8