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Weak solutions for a class of non-Newtonian fluids with energy transfer. (English) Zbl 0974.35090
The author studies a nonlinear convection problem with temperature-dependent coefficients for a rather general class of non-Newtonian fluids with energy dissipation. After introducing a variational formulation of the problem, the author proves the existence of a non-unique weak solution in Sobolev spaces by the fixed point technique, employing Galerkin method, Yosida approximation, and Lagrange multipliers for the associated generalized Navier-Stokes system. The uniqueness of the solution is guaranteed after prescribing the diffusion coefficient and convective as well as dissipative terms.

35Q35 PDEs in connection with fluid mechanics
76A05 Non-Newtonian fluids
76M30 Variational methods applied to problems in fluid mechanics
80A20 Heat and mass transfer, heat flow (MSC2010)
76R10 Free convection
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