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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 \(T>0\).

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
Full Text: DOI
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