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On the Boussinesq system with nonlinear thermal diffusion. (English) Zbl 0897.35065
The authors deal with the Boussinesq system of hydrodynamics \[ u_t+ (u\cdot \nabla)u -\text{div} \bigl(\mu (\theta) D(u)\bigr) +\nabla p=F (\theta), \quad \text{div} u=0, \] \[ \theta_t +u\cdot \nabla \theta- \Delta\varphi (\theta)=0. \] which is appropriately transformed to allow the use of projection technique used for Navier-Stokes equations. Under suitable conditions they show the existence of solutions in a domain \(\Omega \subset \mathbb{R}^N\), and some time interval \([0,T]\). The solution is searched in the space \(L^\infty (0,T;L^2 (\Omega)) \cap L^2(0,t; W_\sigma^{1,2} (\Omega))\). An interative procedure is proposed to decouple \(u\) and \(\theta\).

MSC:
35Q35 PDEs in connection with fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
80A20 Heat and mass transfer, heat flow (MSC2010)
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