EXISTENCE AND UNIQUENESS OF THE SOLUTION

COMMUNICATIONS ON PURE AND APPLIED ANALYSIS Volume 12, Number 1, January 2013

doi:10.3934/cpaa.2013.12. pp. –

EXISTENCE AND UNIQUENESS OF THE SOLUTION OF A BOUSSINESQ SYSTEM WITH NONLINEAR DISSIPATION

`re and Olivier Guibe ´ Dominique Blanchard† , Nicolas Bruye Laboratoire de Math´ ematiques Rapha¨ el Salem, UMR CNRS 6085, Universit´ e de Rouen ´ Avenue de l’universit´ e, BP12, 76801 Saint Etienne du Rouvray cedex, France

(Communicated by Alain Miranville) Abstract. We give existence and uniqueness results of the weak-renormalized solution for a class of nonlinear Boussinesq’s systems. The main tools rely on regularity results for the Navier-Stokes equations with precise estimates on the solution with respect to the data in dimension 2 and on the techniques involved for renormalized solutions of parabolic problems.

1. Introduction. Let Ω be an open bounded subset of RN and T > 0. We consider the following nonlinear Boussinesq’ type system ∂u − ∆u + u · Du + Dp = F (θ) ∂t div(u) = 0 ∂θ − ∆θ + u · Dθ = δ|Du|2 + f ∂t u(t = 0) = u0 , θ(t = 0) = θ0 u = 0,

θ=0

in (0, T ) × Ω,

(1.1a)

in (0, T ) × Ω,

(1.1b)

in (0, T ) × Ω,

(1.1c)

in Ω,

(1.1d)

on (0, T ) × ∂Ω.

(1.1e)

System (1.1a)–(1.1e) couples the Navier-Stokes equations (1.1a)-(1.1a) with the energy balance equation (1.1c). In this last equation we consider either δ = 1 and then it takes into account the mechanical dissipation |Du|2 , or δ = 0 which means that the dissipation is neglected. For the physical justification of Boussinesq’s systems we refer to [2], [9], [19]. Notice that, in general, the dissipation is neglected in such derivations although the Boussinesq’s approximation only concerns the introduction of the right hand side in the Navier-Stokes equations which is also often assumed to be linear with respect to θ. If the function F (θ) is replaced by a fixed function g in (1.1a), existence of solutions is proved in [15]. Similar systems describing a turbulent flow in a fluid are analyzed in [12]. In the case where |Du|2 is neglected in (1.1c) but for a nonlinear heat equation, existence and uniqueness are established in [13] using a maximum principle on the unknown θ. In [1] and [3], existence results are proved for more general systems than (1.1), including nonlinear internal energies b(θ) or nonlinear diffusion for the temperature. Indeed in these works and here, the function F is assumed to have an adequate growth at infinity (see (2.2)) 2000 Mathematics Subject Classification. Primary: 35Q35; Secondary: 35D05, 35D30, 76D03. Key words and phrases. Nonlinear systems of PDE, Navier-Stokes equations, parabolic equations.

1

2

` ´ DOMINIQUE BLANCHARD, NICOLAS BRUYERE AND OLIVIER GUIBE

so that F (θ) ∈ L2 ((0, T ); H −1 (Ω)). The main original results of the present paper concern the uniqueness of the solution for N = 2 in the two cases • f ∈ L1 ((0, T ) × Ω), θ0 ∈ L1 (Ω) and δ = 0 : system (1.1a)–(1.1e) without dissipation • f ∈ L2 ((0, T ) × Ω), θ0 ∈ L2 (Ω) and δ = 1 : system (1.1a)–(1.1e) with dissipation As far as we know, previous uniqueness results were obtained on simpler systems than (1.1a)–(1.1e) : either in the case δ = 0 and with bounded data for the heat equation (as in [13]) or for F = 0 in which case the Navier-Stokes equations are uncoupled from the heat equation (as in [15]). Indeed the difficulties come from the convection term u·Dθ which does not belong to L1 ((0, T ) × Ω) a priori (first case) and to the quadratic term |Du|2 (second case). In the case without dissipation, uniqueness of the solution is obtained in [10] and [11] through assuming that the data for the heat equation are more regular, namely that f ∈ L1 (0, T ; Lm (Ω)) and θ0 ∈ Lm (Ω) for 1 < m < 2. This allows to increase the regularity of θ and then of u · Dθ (the velocity u having the standard regularity of a weak solution of the Navier-Stokes equation, see Theorem 3.1 below). In the present paper we use a different technique which relies on regularity assumptions on Ω and u0 which allows us to get more regular solutions for the Navier-Stokes equations (using [Ger]). A key point in the proof of the uniqueness result is to obtain u · Dθ ∈ Lq ((0, T ) × Ω) for every q < 4/3. Since we deal in case one with L1 data we use throughout the paper the notion of renormalized solutions for the heat equation (1.1c). Indeed for such data, this framework is well known to provide uniqueness of the solution and stability results (see e.g. [4, 5]). Moreover it allows us to use the existence results for the system (1.1a)–(1.1e) which are established in [1] and [3], even if we slightly improve the admissible growth of F in Theorem 5.1 (but again with additional assumptions on Ω and u0 ). The paper is organized as follows. In Section 2, we set a few notations and assumptions on the data and we give the definition of a weak-renormalized solution of the system (1.1) for the case without dissipation. Section 3 is devoted to give some preliminaries recalls on the Navier-Stokes equations in the 2-dimensional case and on the renormalized solution of a heat equation. In Section 4 we prove the uniqueness of the solution for small data f and u0 . Section 5 is devoted to analyzed the case with dissipation. 2. Assumptions and definitions. Let Ω be an bounded open subset of RN (N = 2 or N = 3) with boundary ∂Ω, T > 0 and QT = (0, T ) × Ω. We recall the following classical functional spaces for the study of the Navier-Stokes equations N H10 (Ω) = H01 (Ω) , H10,σ (Ω) = {u ∈ H10 (Ω) ; div(u) = 0}, N

Lp (Ω) = (Lp (Ω)) ,

Lpσ (Ω) = H10,σ (Ω)

k·kLp (Ω)

.

As explained in the introduction, we use the same framework of solutions as in [1] or [3]. It consists in the standard weak formulation of the Navier-Stokes equations for the velocity and to a renormalized formulation for the heat equation. Although we will consider more regular data for the results of Sections 4 and 5, we recall in Definition 2.1 below this notion of solution under the same assumptions as in [1]. We assume that θ0 ∈ L1 (Ω), u0 ∈ L2σ (Ω) and f ∈ L1 (QT ),

(2.1)

EXISTENCE AND UNIQUENESS OF A BOUSSINESQ SYSTEM. . .

3

and that the function F : R 7→ RN is continuous and satisfies the growth assumption |F (r)| ≤ a + b|r|α , for every r ∈ R, (2.2) where 0 ≤ α < 2 and where a, b are nonnegative constants. For any real number k ≥ 0, the truncation at height k is the function Tk (s) = max(−k, min(k, s)). The definition of a weak-renormalized solution of the system (1.1) is recalled below. Definition 2.1. A couple of functions (u, θ) defined on QT is called a weakrenormalized solution of (1.1) if u ∈ L∞ (0, T ; L2σ (Ω)) ∩ L2 (0, T ; H10,σ (Ω));

(2.3)

θ ∈ L∞ (0, T ; L1 (Ω)); F (θ) ∈ L2 (0, T ; H −1 (Ω)); Tk (θ) ∈ L2 (0, T ; H01 (Ω)), ∀k ≥ 0 ; Z 1 |Dθ|2 dx dt = 0 ; lim n→+∞ n {n≤|θ|≤2n}

(2.4) (2.5) (2.6) (2.7)

and if for almost every t ∈ (0, T ), for every v ∈ H10,σ (Ω) Z Z Z ∂ u · v dx + Du · Dv dx + (u · D)u · v dx = hF (θ), viH−1 (Ω),H1 (Ω) ; (2.8) 0 ∂t Ω Ω Ω and for every S ∈ W 2,∞ (R) such that supp(S 0 ) is compact, for every ϕ ∈ Cc∞ ([0, T )× ¯ such that S 0 (θ)ϕ ∈ L2 (0, T ; H 1 (Ω)), Ω) 0 Z Z Z − ϕt S(θ) dx dt − ϕ(0)S(θ0 ) dx + S 00 (θ)|Dθ|2 ϕ dx dt QT Ω QT Z Z 0 + S (θ)Dθ · Dϕ dx dt + u · DS(θ)ϕ dx dt QT QT Z Z f S 0 (θ)ϕ dx dt. (2.9) |Du|2 S 0 (θ)ϕ dx dt + =δ QT

QT

Remark 1. In the above definition, we use the definition of renormalized solutions for parabolic problems introduced in [6] which is equivalent to the one developed in [4] or [5] and used in [1]. The following result is a straightforward consequence of Theorem 5.1 in [1] in which the authors prove existence of solutions to a Boussinesq’s type system in the case δ = 1 and f = 0. Indeed, since in this work the dissipation is merely in L1 (QT ), this existence result still holds true for f ∈ L1 (QT ) and δ = 0 or δ = 1. Theorem 2.2. Assume that N = 2, and that the assumptions (2.1)-(2.2) on the data hold true. Then: - if 0 ≤ 2α ≤ 1, there exists at least a weak-renormalized solution (u, θ) of problem (1.1) (in the sense of Definition 2.1). - if 1 < 2α < 3, there exists a real positive number η such that if (recall that the constant a is defined in (2.2)) a + ku0 k(L2 (Ω))2 + kθ0 kL1 (Ω) + ||f ||L1 (QT ) ≤ η, then there exists at least a weak-renormalized solution (u, θ) of problem (1.1) (in the sense of Definition 2.1). Moreover,

4


- if 0 ≤ 2α < 2 then F (θ) ∈ L2 (QT ) - if 2 ≤ 2α < 3 then F (θ) ∈ L2 ((0, T ); H −1 (Ω)). Let us just mention that the proof of Theorem 5.1 in [1] (or of Theorem 2.2 above) relies of a fixed point argument with respect to the unknown θ. The two main points are the uniqueness of the solution of the Navier-Stokes equations for N = 2 (together with the stability of the energy) and the uniqueness and stability of the renormalized solution of the heat equation for L1 data. In the case 1 < 2α < 3 the small character of the data insures the existence of a preserved ball (of a convenient Lebesgue space) in the fixed point method. Finally remark that in the case 2 ≤ 2α < 3, the regularity F (θ) ∈ L2 ((0, T ); H −1 (Ω)) is not a direct consequence of the standard estimates on renormalized solutions recalled in Theorem 3.4 in Subsection 3.2 below. It uses an estimate of θ in a convenient space of the type Lr ((0, T ); Lq (Ω)) (for details the reader is referred to Lemma 4.1 in [1]). As mentioned in the introduction, the main aim of the present paper is to establish uniqueness results of the solution provided by Theorem 2.2 under stronger assumptions on the data. In order to precise these assumptions and their impacts on the smoothness of the solutions we give a few recalls on the Navier-Stokes equations and on the renormalized solutions of parabolic equations in the following section. 3. Preliminary results and recalls. In this section we recall a few results on the Navier-Stokes equations in two dimensions and on the properties of the renormalized solutions for parabolic type equations. 3.1. Recalls on the Navier-Stokes equations. For g ∈ L2 (QT ) and u0 ∈ L2σ (Ω), let us consider the weak formulation of the Navier-Stokes equations (for the velocity u)  1   for almost every t and for every v ∈ H0,σ (Ω)   Z Z Z Z ∂ (3.1) u · v dx + Du · Dv dx + (u · D)u · v dx = gvdx,   ∂t Ω Ω Ω Ω   u(t = 0) = u . 0 We first recall the well known existence and uniqueness result in the 2-dimensional case (see e.g. [18]). Theorem 3.1. Assume N = 2. There exists a unique weak solution u ∈ L∞ (0, T ; L2σ (Ω)) ∩ L2 (0, T ; H10,σ (Ω)) of (3.1). The function u belongs to C([0, T ]; L2 (Ω)) and satisfies the energy equality Z Z tZ Z 1 1 2 2 |u| (t) dx + |Du| dx ds = |u0 |2 dx 2 Ω 2 Ω 0 Ω Z tZ + gu dx ds, for any t ∈ [0, T ]. (3.2) 0

Ω

Moreover if u1 , u2 are the weak solutions of (3.1) respectively for the data (g1 , u0,1 ), (g2 , u0,2 ) then the following estimate holds true Z Z tZ 2 |u1 − u2 | (t) dx + |Du1 − Du2 |2 dx ds Ω 0 Ω Z t Z Z 2 2 ≤C |g1 − g2 | dx ds + |u0,1 − u0,2 | , for a.e. t ∈ [0, T ] (3.3) 0

Ω

Ω


5

where C is a constant depending on ku0,2 kL2 (Ω) and kg2 kL2 (QT ) . Using Gagliardo-Nirenberg inequality (see [16]) and (3.3) , we deduce the standard estimate ku1 − u2 kL4 (QT ) ≤ C kg1 − g2 kL2 (QT ) + ku0,1 − u0,2 kL2 (Ω) (3.4) where the constant C depends on Ω, ku0,2 kL2 (Ω) and kg2 kL2 (QT ) . For more regular data, the following results are established in [14] (see Theorem 2 and Theorem 3). Theorem 3.2. Let Ω be an open subset of R2 with boundary ∂Ω ∈ C 2 . Suppose that u0 ∈ H10,σ (Ω) and g ∈ L2 (QT ). Then, we have the following estimates

∂u

1 2 + kuk + kgk ku k 2 (0,T ;H 2 (Ω)) ≤ C1 0 L L (Q ) H0 (Ω) T

∂t 2 L (QT ) 2 1 + ku0 kL2 (Ω) + kgkL2 (QT ) exp C2 ku0 k4L2 (Ω) + kgk4L2 (QT ) , (3.5) where C1 , C2 are constants depending only on Ω. Moreover, if u1 and u2 are solutions for the data (g1 , u0,1 ) and (g2 , u0,2 ), then

∂(u1 − u2 )

+ ku1 − u2 kL2 (0,T ;H 2 (Ω))

2 ∂t L (QT ) ≤ C3 kg1 − g2 kL2 (QT ) + ku0,1 − u0,2 kH10 (Ω) (3.6) 1/2 RT R R R where C3 depends on Ω, T, 0 |g |2 dx dt, Ω |u0,i |2 dx, Ω |Du0,i |2 dx (i = Ω i 1, 2) and is bounded w.r.t. these quantities. From (3.5) and using a standard interpolation argument we first deduce that kukL∞ (0,T ;H10 (Ω)) ≤ C(Ω) ku0 kH10 (Ω) + kgkL2 (QT ) 2 1 + ku0 kL2 (Ω) + kgkL2 (QT ) exp C2 ku0 k4L2 (Ω) + kgk4L2 (QT ) . (3.7) Then Sobolev’s embedding gives for any 1 ≤ r < +∞ kukL∞ (0,T ;Lr (Ω)) ≤ C(Ω) ku0 kH10 (Ω) + kgkL2 (QT ) 2 1 + ku0 kL2 (Ω) + kgkL2 (QT ) exp C2 ku0 k4L2 (Ω) + kgk4L2 (QT ) . (3.8) Secondly, the convenient Gagliardo-Nirenberg’s inequality which holds true for any v ∈ H 2 (Ω) ∩ H10 (Ω) 1

1

2 2 kDvkL4 (Ω) ≤ C(Ω)kvkH 2 (Ω) kv||H1 (Ω)) ,

(3.9)

0

gives here with (3.5) and (3.7) kDuk(L4 (QT ))2 ≤ C(Ω) ku0 kH10 (Ω) + kgkL2 (QT ) 2 1 + ku0 kL2 (Ω) + kgkL2 (QT ) exp C2 ku0 k4L2 (Ω) + kgk4L2 (QT ) . (3.10) Starting from (3.6) and proceeding as above we obtain for any 1 ≤ r < +∞ ku1 − u2 kL∞ (0,T ;Lr (Ω)) ≤ C3 kg1 − g2 kL2 (QT ) + ku0,1 − u0,2 kH10 (Ω) , (3.11)


6

where C3 depends on Ω, T , r, ku0,i kH10 (Ω) , kgi kL2 (QT ) (i = 1, 2) and is bounded w.r.t. these quantities. 3.2. Recalls on renormalized solutions for the heat equation. Let us consider the following heat equation  ∂θ     ∂t − ∆θ + u · Dθ = f in (0, T ) × Ω, (3.12) θ(t = 0) = θ0 in Ω,     θ=0 on (0, T ) × ∂Ω, with f ∈ L1 (QT ), θ0 ∈ L1 (Ω) and u ∈ L2 (0, T ; H10,σ (Ω)) ∩ L∞ (0, T ; L2 (Ω)). We recall below the definition of a renormalized solution of (3.12) (in the formulation given in [6]) and the main properties of this solution that we will use in the paper. Definition 3.3. A measurable function θ ∈ L∞ (0, T ; L1 (Ω)) is called a renormalized solution of (3.12) if Tk (θ) ∈ L2 (0, T ; H01 (Ω)), 1 n→+∞ n

Z

∀k ≥ 0 ;

(3.13)

|Dθ|2 dx dt = 0 ;

lim

(3.14)

{n≤|θ|≤2n}

¯ for every S ∈ W 2,∞ (R) such that supp(S 0 ) is compact, for every ϕ ∈ Cc∞ ([0, T )× Ω) 0 2 1 such that S (θ)ϕ ∈ L (0, T ; H0 (Ω)), Z

Z

−

Z

ϕ(0)S(θ0 ) dx + S 00 (θ)|Dθ|2 ϕ dx dt QT Z Z 0 f S 0 (θ)ϕ dx dt. u · DS(θ)ϕ dx dt = S (θ)Dθ · Dϕ dx dt +

ϕt S(θ) dx dt − QT

Z +

Ω

(3.15)

QT

QT

QT

Existence and uniqueness of a renormalized solution of (3.12) is e.g. a consequence of the results established in [1] (see Lemma 4.1) where a more general equation is considered. Moreover the regularity on θ given in the following theorem is well known (see [8] and [7]) since the convection term u · Dθ does not affect the underlying a priori estimates (due to div(u) = 0 and u = 0 on ∂Ω; see also [3]). Theorem 3.4. Suppose that f ∈ L1 (QT ) and θ0 ∈ L1 (Ω). The problem (3.12) admits a unique renormalized solution (in the sense of Definition 3.3). Moreover θ satisfies Dθ ∈ (Lq (QT ))N ,

∀q
σ > 1−β which is licit since β < 1/2. This implies βγ < 2 which allows us to apply again Theorem 3.4 and to obtain k1 + |θ1 |β + |θ2 |β kLγ (QT ) ≤ mes(QT )1/γ + C(β, γ)(kf kβL1 (QT ) + kθ0 kβL1 (Ω) ). (4.13) Indeed if β = 0, in the above inequality one has to replace the right hand side by 3 mes(QT )1/γ and the end of proof below is simpler. Finally, plugging (4.8), (4.13) in (4.11) yields kθ1 − θ2 kLσ (QT ) ≤ C4 mes(QT )1/γ + C(β, γ) kf kβL1 (QT ) + kθ0 kβL1 (Ω) kf kL1 (QT ) + kθ0 kL1 (Ω) kθ1 − θ2 kLσ (QT ) , (4.14)


9

where C4 depends on Ω, T, σ, α and ku0,i kH10 (Ω) . Choosing now kf kL1 (QT ) + kθ0 kL1 (Ω) small enough in such a way that C4 mes(QT )1/γ + C(β, γ)(kf kβL1 (QT ) + kθ0 kβL1 (Ω) ) kf kL1 (QT ) + kθ0 kL1 (Ω) < 1, (4.15) implies that θ1 = θ2 . It yields F (θ1 ) = F (θ2 ), then, by uniqueness of the weak solution of the Navier-Stokes equations (in the 2-dimensional case), we get u1 = u2 . The proof Theorem 4.1 is complete. 5. System with nonlinear dissipation. In this    ∂u − ∆u + u · Du + Dp = F (θ)     ∂t   div(u) = 0 ∂θ − ∆θ + u · Dθ = f + |Du|2   ∂t    u(t = 0) = u0 , θ(t = 0) = θ0    u = 0, θ = 0

section we consider the system in (0, T ) × Ω, in (0, T ) × Ω, in (0, T ) × Ω,

(5.1)

in Ω, on (0, T ) × ∂Ω,

with f ∈ L2 (QT ), θ0 ∈ L2 (Ω), u0 ∈ H10,σ (Ω), ∂Ω ∈ C 2 and the growth condition (2.2) on F with 0 ≤ α < 2. Under these assumptions, Theorem 2.2 gives the existence of at least a weak-renormalized solution to (5.1) such that F (θ) ∈ L2 (QT ) if 0 ≤ α < 1 (for small data in the case 1/2 < α < 1). Using the smooth character of f , θ0 , u0 and ∂Ω we show that this solution is actually a weak solution (which means that θ is a weak solution of the heat equation). Then we extend this existence result of a weak solution in the case 1 ≤ α < 2. As far as uniqueness of a small solution is concerned we assume that 0 ≤ α ≤ 1 together with a Lipschitz condition on F (see (5.4)). Theorem 5.1. Let Ω be an open subset of R2 with boundary ∂Ω ∈ C 2 . Assume that θ0 ∈ L2 (Ω) and u0 ∈ H10,σ (Ω). Let 0 ≤ α < 2. Suppose that F is continuous and satisfies the growth condition |F (r)| ≤ a + b|r|α ,

for every r ∈ R,

(5.2)

where a,b are nonnegative constant. Moreover, suppose that one of the following conditions is satisfied • 0 ≤ 2α ≤ 1 • 1 < 2α < 4 and a, ku0 kH10 (Ω) , kθ0 kL2 (Ω) , kf kL2 (Ω) are sufficiently small. Then, the problem (5.1) admits at least a weak solution. Theorem 5.2. Let Ω be an open subset of R2 with boundary ∂Ω ∈ C 2 . Assume that θ0 ∈ L2 (Ω) and u0 ∈ H10,σ (Ω). Suppose that F is continuous and satisfies the growth condition |F (r)| ≤ a + b|r|α , for every r ∈ R, (5.3) where 0 ≤ α ≤ 1 and the Lipschitz condition |F (r) − F (s)| ≤ L|r − s| 1 + |r|β + |s|β ,

for any r, s ∈ R,

(5.4)

where 0 ≤ β ≤ 1. Then, there is uniqueness of the weak solution of the problem (5.1) for sufficiently small solutions. More precisely, there exists R > 0 such that if (u1 , θ1 ) and (u2 , θ2 )

10


are two weak solutions of (5.1) satisfying θ1 , θ2 ∈ BL4 (QT ) (0, R),

(5.5)

Du1 , Du2 ∈ B(L4 (QT ))2 (0, R),

(5.6)

Dθ1 , Dθ2 ∈ BL2 (QT ) (0, R),

(5.7)

then θ1 = θ2 and u1 = u2 . Remark 3. Let us notice that deriving a priori estimates on the functions u and θ in the system (5.1) with respect to the data seems to be an arduous task because of the strong coupling of this system. This is the reason why the uniqueness result in Theorem 5.2, where we obtain uniqueness for small solutions, is different from the one in Theorem 4.1, or even the one established in [10] and [11], where uniqueness for small data is established. Proof of Theorem 5.1—The proof is divided into two cases, 0 ≤ α < 1 and 1 ≤ α < 2. As mentioned above, in the first case, we use Theorem 2.2 which insures the existence of the weak-renormalized solution such that F (θ) ∈ L2 (QT ), and we show that this solution is a weak solution because of the regularity assumption on the data. Whereas, in the second case, the proof relies on Schauder’s fixed point theorem, using the regularity results of the Navier-Stokes equations recalled in Section 3. 1st case: 0 ≤ α < 1. Theorem 2.2 implies that problem (5.1) admits at least a weakrenormalized solution with F (θ) ∈ L2 (QT ) (for small data if 1/2 < α < 1). Now, since ∂Ω ∈ C 2 and u0 ∈ H10 (Ω), estimate (3.10) of Section 3 gives Du ∈ L4 (QT ) so that |Du|2 ∈ L2 (QT ). It follows that θ is a solution of the heat equation with L2 data (recall that θ0 ∈ L2 (Ω)) and we conclude that (u, θ) is a weak solution of the problem (5.1). 2d case: 1 ≤ α < 2. The proof is based on Schauder’s fixed point theorem. Let θb b belongs to be in L2α (QT ) so that, thanks to the growth condition (5.2) on F , F (θ) 2 ∞ b L (QT ). Next, we associate to θ the unique weak solution u ∈ L (0, T ; L2 (Ω)) ∩ L2 (0, T ; H10 (Ω)) of the Navier-Stokes equations   b  ∂u − ∆u + u · Du + Dp = F (θ) in (0, T ) × Ω,     ∂t div(u) = 0 in (0, T ) × Ω, (5.8)   u(t = 0) = u in Ω,  0    u=0 on (0, T ) × ∂Ω. As above since ∂Ω ∈ C 2 and u0 ∈ H10 (Ω), estimate (3.10) gives |Du|2 ∈ L2 (QT ). Thus due to f ∈ L2 (QT ) and θ0 ∈ L2 (Ω), the problem  ∂θ 2   in (0, T ) × Ω,  ∂t − ∆θ + u · Dθ = f + |Du| (5.9) θ(t = 0) = θ0 in Ω,    θ=0 on (0, T ) × ∂Ω, admits a unique weak solution θ ∈ L2 (0; T ; H01 (Ω)) ∩ L∞ (0, T ; L2 (Ω)). Then we have θ ∈ L4 (QT ) so that θ ∈ L2α (QT ) (since α < 2). As a consequence, we can define the following map Ψ : L2α (QT ) −→ L2α (QT ) θb 7−→ θ.


11

In what follows we show that Ψ has a fixed point θ which gives a weak solution of (5.1). We use Schauder’s fixed point theorem, through proving that Ψ is continuous, compact and preserves a ball of L2α (QT ). Ψ is continuous: let (θbn ) be a sequence of functions belonging to L2α (QT ) which b Indeed we denote by un and u the solutions of the Navierconverges strongly to θ. b Subtracting the equations Stokes equations associated respectively to θbn and θ. b and taking θn − θ as a test satisfied respectively by θn = Ψ(θbn ) and θ = Ψ(θ) function in the resulting equation gives Z Z tZ 2 |θn − θ| (t) dx + |Dθn − Dθ|2 dx ds Ω 0 Ω Z tZ ≤ |Dun − Du||Dun + Du||θn − θ| dx ds 0 Ω Z tZ + |un − u||Dθn ||θn − θ| dx ds, a.e. t ∈ (0, T ). (5.10) 0

Ω

Using H¨ older’s inequality and Gagliardo-Nirenberg’s inequality, we get kθn − θkL4 (QT ) ≤ C(Ω) kDun − Duk(L2 (QT ))2 k|Dun | + |Du|kL4 (QT ) + kun − ukL4 (QT ) kDθn kL2 (QT ) . (5.11) Next applying estimates (3.3) and (3.4) to un − u leads to b L2 (Q ) , ku0 kL2 (Ω) kF (θbn ) − F (θ)k b L2 (Q ) kθn − θkL4 (QT ) ≤ C Ω, kF (θ)k T T × kDun k(L4 (QT ))2 + kDuk(L4 (QT ))2 + kDθn kL2 (QT ) . (5.12) Now θbn converges to θb strongly in L2α (QT ), thus the growth condition (5.2) on F b strongly in L2 (QT ). Thus, estimate (3.10) implies that F (θbn ) converges to F (θ) shows that kDun k(L4 (QT ))2 is bounded independently of n. Since indeed kDθn kL2 (QT ) ≤ C k|Dun |2 kL2 (QT ) + ||f ||L2 (QT ) + kθ0 kL2 (Ω) (5.13) = C kDun k2(L4 (QT ))2 + ||f ||L2 (QT ) + kθ0 kL2 (Ω) , we deduce that kDθn kL2 (QT ) is also bounded independently of n. Finally estimate (5.12) leads to b L2 (Q ) , kθn − θkL4 (QT ) ≤ CkF (θbn ) − F (θ)k T where C is a constant independent of n. This gives the strong convergence of θn to θ in L4 (QT ) and since 2α < 4 θn −→ θ in L2α (QT ), which shows the continuity of Ψ. Ψ is compact: let (θbn ) be a sequence of functions bounded in L2α (QT ). Then, by the growth condition (5.2) on F , F (θbn ) is bounded in L2 (QT ). With the same argument to that used in the proof of the continuity of Ψ, it follows that un is bounded in L4 (QT ) and θn is bounded in L4 (QT ). Writing un · Dθn = − div(un θn ) since un is divergence free, we obtain that un · Dθn is bounded in L2 (0, T ; H −1 (Ω)). Therefore, from the standard estimates on the weak solution of the heat equation, we get • θn is bounded in L2 (0, T ; H01 (Ω))


12

∂θn = |Dun |2 + ∆θn − un · Dθn is bounded in L2 (QT ) + L2 (0, T ; H −1 (Ω)). ∂t A classical Aubin’s type lemma (see e.g. [17]), implies that for a subsequence still indexed by n θn −→ ϑ in L2 (QT ) and a.e. in QT , where ϑ belongs to L2 (QT ). Thanks to the Vitali theorem, the above pointwise convergence and the fact that (θn ) is bounded in L4 (QT ) together with 2α < 4, imply that, up to a subsequence, •

θn −→ ϑ in L2α (QT ). This proves the compactness of the map Ψ. Ψ preserves a ball: let us consider the function ψ(s) = (1 + s2 ) exp(C2 s4 ),

(5.14)

where the constant C2 which depends on Ω is defined in Theorem 3.2. For any b we claim that θb ∈ L2α (QT ) and θ = Φ(θ), h 2 b 2α2α kθkL2α (QT ) ≤ C(α, Ω) a2 mes(QT ) + b2 kθk L (QT ) + ku0 kH10 (Ω) i 1 b α2α 2 2 2 , (5.15) + ku k + kf k + kθ k ×ψ 2 a mes(QT ) 2 + bkθk 0 0 L (Ω) L (Ω) L (Ω) L (QT ) Indeed, on the one hand, from the definition (5.9) of θ, we have kθkL4 (QT )

≤ C(Ω) k|Du|2 kL2 (QT ) + ||f ||L2 (QT ) + kθ0 kL2 (Ω) = C(Ω) kDuk2(L4 (QT ))2 + ||f ||L2 (QT ) + kθ0 kL2 (Ω) .

(5.16)

On the other hand, the estimate (3.10) applied to the solution u of (5.8) yields b L2 (Q ) + ku0 kH1 (Ω) ψ kF (θ)k b L2 (Q ) + ku0 kL2 (Ω) . kDuk(L4 (QT ))2 ≤ C(Ω) kF (θ)k T T 0 Thus, the growth condition (5.2) on F implies that 1 b α2α 1 + ku k kDuk(L4 (QT ))2 ≤ C(Ω) a mes(QT ) 2 + bkθk 0 H0 (Ω) L (QT ) 1 b α2α × ψ a mes(QT ) 2 + bkθk L (QT ) + ku0 kL2 (Ω) . (5.17) Gathering (5.16) and (5.17) leads to h 2 b 2α2α kθkL4 (QT ) ≤ C(Ω) a2 mes(QT ) + b2 kθk + ku k 1 0 L (QT ) H0 (Ω) i 1 2 α b 2α ×ψ a mes(QT ) 2 + bkθk L (QT ) + ku0 kL2 (Ω) + kf kL2 (Ω) + kθ0 kL2 (Ω) . Finally, the fact that 2α < 4 and H¨older’s inequality give (5.15). Since ψ is a continuous and increasing function on R+ with ψ(0) = 1 and since 2α > 1 there exists R > 0 and η > 0 such that C(α, Ω) (η + b2 R2α )ψ(η + bRα ) + η ≤ R. At the expense of choosing the quantities a, ku0 kH10 (Ω) , kθ0 kL2 (Ω) , kf kL2 (Ω) sufficiently small, it follows that h C(α, Ω) a2 mes(QT ) + ku0 k2H1 (Ω) + b2 R2α 0 i 1 2 α 2 × ψ a mes(QT ) + bR + ku0 kL2 (Ω) + kf kL2 (Ω) + kθ0 kL2 (Ω) ≤ R. (5.18)


13

Finally (5.15) and (5.18) implies that b L2α (Q ) ≤ R =⇒ kθkL2α (Q ) ≤ R, kθk T T which means that Ψ preserves the ball of center 0 and radius R of L2α (QT ). This achieves the proof of Theorem 5.1. Proof of Theorem 5.2—Let (u1 , θ1 ), (u2 , θ2 ) be two weak solutions of the problem (5.1). Subtracting the equations satisfied respectively by θ1 and θ2 and taking θ1 −θ2 as a test function imply that Z Z tZ 2 |Dθ1 − Dθ2 |2 dx ds |θ1 − θ2 | (t) dx + 0 Ω Ω Z tZ |Du1 − Du2 ||Du1 + Du2 ||θ1 − θ2 | dx ds ≤ Ω 0 Z tZ + |u1 − u2 ||Dθ1 ||θ1 − θ2 | dx ds, a.e. t ∈ (0, T ). (5.19) 0

Ω

By the same arguments as the ones used in order to obtain (5.11), we get kθ1 − θ2 kL4 (QT ) ≤ C Ω, kF (θ1 )kL2 (QT ) , ku0 kL2 (Ω) kF (θ1 ) − F (θ2 )kL2 (QT ) × kDu1 k(L4 (QT ))2 + kDu2 k(L4 (QT ))2 + kDθ1 kL2 (QT ) . (5.20) Then, the Lipschitz condition (5.4) satisfied by F , H¨older’s inequality and the fact that β ≤ 1 imply that kF (θ1 ) − F (θ2 )kL2 (QT ) ≤Lk(θ1 − θ2 )(1 + |θ1 |β + |θ2 |β )kL2 (QT ) ≤Lkθ1 − θ2 kL4 (QT ) k1 + |θ1 |β + |θ2 |β kL4 (QT ) ≤C(L, Ω)kθ1 − θ2 kL4 (QT ) 1 + kθ1 kβL4 (QT ) + kθ2 kβL4 (QT ) . Plugging this last inequality in (5.20) yields kθ1−θ2 kL4 (QT ) ≤ C(L, Ω, kF (θ1 )kL2 (QT ) , ku0 kL2 (Ω) ) 1+kθ1 kβL4 (QT ) +kθ2 kβL4 (QT ) × kDu1 k(L4 (QT ))2 + kDu2 k(L4 (QT ))2 + kDθ1 kL2 (QT ) kθ1 − θ2 kL4 (QT ) . (5.21) Now, as in the proof of Theorem 4.1, if kθi kL4 (QT ) , kDui k(L4 (QT ))2 and kDθi kL2 (QT ) are sufficiently small, we have C(L, Ω, kF (θ1 )kL2 (QT ) , ku0 kL2 (Ω) ) 1 + kθ1 kβL4 (QT ) + kθ2 kβL4 (QT ) × kDu1 k(L4 (QT ))2 + kDu2 k(L4 (QT ))2 + kDθ1 kL2 (QT ) < 1. (5.22) Thus (5.21) implies that θ1 = θ2 which in turn leads to u1 = u2 as in Theorem 4.1. 6. Concluding comments. The uniqueness results given in the present paper are restricted to the two dimensional case N = 2. Indeed the extension to higher dimension is confronted to the lack of uniqueness results for the Navier-Stokes equations for N ≥ 3 and for an applied force of the type F (θ) (together with the stability of the energy). Actually the regularity of θ, even in the case without dissipation where θ is a weak solution of the heat equation, precludes the use of the partial uniqueness results on the Navier-Stokes system (see e.g. [18] and [15]). Moreover a key point in our proves is the regularity result of [14] which is limited to N = 2. Up to now we are not aware of similar results for higher dimension, even for specific geometry

14


as radial domain. Similarly the results of [14] hardly seems to be reachable with a viscosity which depends upon the temperature field in the Navier-Stokes equations. By contrast, the consideration of temperature dependent diffusion (possibly nonlinear) should be possible using the techniques developed here and the results of [5] and [6]. Finally let us notice that uniqueness of the solution in the case f ∈ L1 (QT ) and δ = 1 remains an open problem. Indeed the proof of Theorem 5.2 relies on estimates (5.19) and (5.20) which are false if θ1 and θ1 are solutions of the heat equation with L1 data. Acknowledgements. The authors thank the referee of this manuscript for his comments and remarks. REFERENCES [1] A. Attaoui, D. Blanchard and O. Guib´ e, Weak-renormalized solution for a nonlinear Boussinesq system, Differential Integral Equations, 22 (2009), 465–494. [2] C. Bernardi, B. M´ etivet and B. Pernaud-Thomas, Couplage des ´ equations de Navier-Stokes et de la chaleur: le mod` ele et son approximation par ´ el´ ement finis, RAIRO Mod´ el. Math. Anal. Num´ er, 29 (1995), 871–921. [3] D. Blanchard, A few result on coupled systems of thermomechanics, In “On the Notions of Solution to Nonlinear Elliptic Problems: Results and Developments,” Quaderni di Matematica, 23 (2009), 145–182. [4] D. Blanchard and F. Murat, Renormalized solutions of nonlinear parabolic problems with L1 data: existence and uniqueness, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1137–1152. [5] D. Blanchard, F. Murat and H. Redwane, Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems, J. Differential Equations, 177 (2001), 331–374. [6] D. Blanchard and A. Porretta, Stefan problems with nonlinear diffusion and convection, J. Differential Equations, 210 (2005), 383–428. [7] L. Boccardo, A. Dall’Aglio, T. Gallou¨ et and L. Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal., 147 (1997), 237–258. [8] L. Boccardo and T. Gallou¨ et, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149–169. [9] J. Boussinesq, “Th´ eorie analytique de la chaleur,” volume 2. Gauthier-Villars, Paris, 1903. [10] N. Bruy` ere, “Comportement asymptotique de probl` emes pos´ es dans des cylindres. Probl` emes d’unicit´ e pour les syst` emes de Boussinesq,” PhD thesis, Universit´ e de Rouen, 2007. [11] N. Bruy` ere, Existence et unicit´ e de la solutions faible-renormalis´ ee pour un syst` eme non lin´ eaire de Boussinesq, C. R. Math. Acad. Sci. Paris, 346 (2008), 521–526. [12] B. Climent and E. Fern´ andez-Cara, Some existence and uniqueness results for a timedependent coupled problem of the Navier-Stokes kind, Math. Models Methods Appl. Sci., 8 (1998), 603–622. [13] J. I. D´ıaz and G. Galiano, Existence and uniqueness of solutions of the Boussinesq system with nonlinear thermal diffusion, Topol. Methods Nonlinear Anal., 11 (1998), 59–82. [14] C. Gerhardt, Lp -estimates for solutions to the instationary Navier-Stokes equations in dimension two, Pacific J. Math., 79 (1978), 375–398. [15] P-L Lions, “Mathematical Topics in Fluid Mechanics,” Vol. 1, volume 3 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press Oxford University Press, New York, 1996. Incompressible models, Oxford Science Publications. [16] L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 733–737. [17] J. Simon, Compact sets in the space Lp (0, T ; B), Ann. Mat. Pura Appl., 146 (1987), 65–96. [18] R. Temam, “Navier-Stokes Equations,” AMS Chelsea Publishing, Providence, RI, 2001. Theory and Numerical Analysis. [19] R. Temam and A. Miranville, “Mathematical Modeling in Continuum Mechanics,” Cambridge University Press, New York, second edition, 2005.

Received August 2011; revised December 2012.


E-mail address: [email protected] E-mail address: [email protected]

15