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Global well-posedness for the 2-D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity. (English) Zbl 1231.35180
The authors consider the Cauchy problem for the 2D Boussinesq system $\begin{cases} \partial_t \vec v+\vec v \cdot \nabla \vec v -\nabla \cdot\left(\nu(\theta) \nabla \vec v\right)+\nabla p=\theta \vec e,\\ \partial_t \theta+\vec v \cdot \nabla \theta -\nabla \cdot\left(\kappa (\theta)\nabla \theta\right)=0,\\ \nabla\cdot \vec v=0,\\ \left. \left(\vec v,\theta\right)\right|_{t=0}=\left(\vec v_0,\theta_0\right), \end{cases} \tag{1}$ where $$\vec e=(0,1)$$, $$\vec v=(v_1,v_2)$$ is the velocity, $$p$$ is the pressure, the kinematic viscosity $$\nu$$ is a positive function satisfying $C_0^{-1}\leq \nu(\theta)\leq C_0,$ and the diffusivity coefficient $$\kappa$$ is is a positive function satisfying $C_0^{-1}\leq \kappa(\theta)\leq C_0.$ They prove that, provided that $$\theta_0\in H^s$$ and $$\vec v_0$$ is a divergence-free $$(H^s)^2$$-vector-field with $$s>2$$, system (1) admits a unique global solution such that $\vec v\in C({\mathbb R}_+;(H^s)^2)\cap L^2({\mathbb R}_+;(H^{s+1})^2),$ and $\theta \in C({\mathbb R}_+;H^s)\cap L^2({\mathbb R}_+;H^{s+1}),$ for any $$T>0$$.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
##### Keywords:
2D Boussinesq system; global well-posedness
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##### References:
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