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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.

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