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The initial value problem for a generalized Boussinesq model: Regularity and global existence of strong solutions. (English) Zbl 0861.35080
Summary: We study the global existence in time, as well as the regularity, of strong solutions of the partial differential equations of evolution type corresponding to a generalized Boussinesq model for thermically driven flows $\partial_tu- \text{div}(\nu(\varphi)\nabla u)+ u \nabla u-\alpha\varphi g+\nabla p=h,\quad \text{div }u=0,$ $\partial_t\varphi-\text{div}(k(\varphi)\nabla\varphi)+ u \nabla\varphi= f\quad\text{in} \quad (0,T]\times\Omega,$ where $$\Omega$$ is a bounded domain in $$\mathbb{R}^N$$, $$N=2$$ or 3. The model includes the case in which the fluid viscosity and thermal conductivity depend on the temperature.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35B65 Smoothness and regularity of solutions to PDEs