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Existence and uniqueness of solutions of the Boussinesq system with nonlinear thermal diffusion. (English) Zbl 0916.35087
The authors consider a quasilinear parabolic system which arises from the conventional Boussinesq approximation when the fluid viscosity, specific heat and thermal conductivity are temperature-dependent. The main purpose is to prove the existence and uniqueness of weak solution for the above system. To this end, the authors give a variational formulation of the problem in appropriate function spaces and construct an iterative scheme to decouple the system. Some results for Navier-Stokes equations and the application of regularization and duality techniques allow to prove the existence of weak solutions for the decoupled systems, which, after the passage to limit in the iterative scheme, provides the unique solvability of the original problem. The two-dimensional cases corresponding to singular and degenerate diffusion are considered separately.
Reviewer: O.Titow (Berlin)

35Q35 PDEs in connection with fluid mechanics
35D05 Existence of generalized solutions of PDE (MSC2000)
80A20 Heat and mass transfer, heat flow (MSC2010)
35K55 Nonlinear parabolic equations
76R10 Free convection
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