On the uniqueness for the Boussinesq system with non linear diffusion (Q998972)

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 \(\theta \in C_b([0,T);\overset{.}{B}^{1}_{2,1}({\mathbb R}^2)).\)