an:05500788 Zbl 1156.35074 Abidi, Hammadi On the uniqueness for the Boussinesq system with non linear diffusion FR J. Math. Pures Appl. (9) 91, No. 1, 80-99 (2009). 00244574 2009
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35Q35 Boussinesq system; Uniqueness; Paradifferential calculus 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 $$\theta \in C_b([0,T);\overset{.}{B}^{1}_{2,1}({\mathbb R}^2)).$$ Bernard Ducomet (Bruy??res le Ch??tel)