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On the uniqueness for the Boussinesq system with non linear diffusion. (Sur l’unicité pour le système de Boussinesq avec diffusion non linéaire.) (French) Zbl 1156.35074
The author considers uniqueness problem for the 2D Boussinesq system (BS) with nonlinear diffusion, in critical spaces \[ \left\{ \begin{matrix} \partial_t \vec v+(\vec v \cdot \nabla) \vec v -\text{div}(2\mu(\theta){\mathcal M})+\nabla p=0,\\ \partial_t \theta+(\vec v \cdot \nabla )\theta =0,\\ \text{div}\;\vec v=0,\\ \left. \left(\vec v,\theta\right)\right| _{t=0}=\left(\vec v^0,\theta^0\right), \end{matrix} \right. \tag{B} \] where \({\mathcal M}\) is the strain tensor, \(\vec v=(v_1,v_2)\) is the velocity, \(p\) is the pressure and the kinematic viscosity \(\mu\) is a positive \(C^{\infty}\) function satisfying the uniform lower bound \[ 0<\underline{\mu}\leq \mu(s)\;\;\text{for any}\;s>0. \tag{visc} \] The main result is the following: provided that \( v^0_j\in \overset{.}{B}^{-1}_{\infty,1}({\mathbb R}^2) \cap L^2({\mathbb R}^2)\) for \(j=1,2\), with \(\vec v^{\;0}\) divergence-free and \(\theta^0\in \overset{.}{B}^{1}_{2,1}({\mathbb R}^2)\) and if \(\mu\) satisfies (visc), there exists \(\epsilon>0\) small enough such that if \[ \| \mu(\theta_0)-1\| _{L^{\infty}}\leq \epsilon\text{ and } \| \theta^0\| _{ \overset{.}{B}^{1}_{2,1}({\mathbb R}^2)}\leq \epsilon, \] then there exists a \(T(\theta^0,\vec{v}^{\;0})\) such that (B) has a unique solution \((\vec{v},\theta)\) satisfying \[ v_j\in C_b([0,T); \overset{.}{B}^{-1}_{\infty,1}({\mathbb R}^2)) \cap L^1([0,T);\overset{.}{B}^{-1}_{\infty,1}({\mathbb R}^2)) \cap L^{\infty}([0,T);L^2({\mathbb R}^2)) \cap L^2([0,T);\overset{.}{H}^{1}({\mathbb R}^2)), \] for \(j=1,2\) and \(\theta \in C_b([0,T);\overset{.}{B}^{1}_{2,1}({\mathbb R}^2)).\)

MSC:
35Q35 PDEs in connection with fluid mechanics
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