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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 $$\cases \partial_t \vec v+\vec v \cdot \nabla \vec v -\nabla \cdot\left(\nu(\theta) \nabla \vec v\right)+\nabla p=\theta \vec e,\\ \partial_t \theta+\vec v \cdot \nabla \theta -\nabla \cdot\left(\kappa (\theta)\nabla \theta\right)=0,\\ \nabla\cdot \vec v=0,\\ \left. \left(\vec v,\theta\right)\right|_{t=0}=\left(\vec v_0,\theta_0\right), \endcases \tag1$$ 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 $T>0$.

35Q35PDEs in connection with fluid mechanics
76D03Existence, uniqueness, and regularity theory
35B30Dependence of solutions of PDE on initial and boundary data, parameters
Full Text: DOI
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