We establish long‐time and large‐data existence of a weak solution to the problem describing three‐dimensional unsteady flows of an incompressible fluid, where the viscosity and heat‐conductivity coefficients vary with the temperature. The approach reposes on considering the equation for the total energy rather than the equation for the temperature. We consider the spatially periodic problem.
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Feireisl et al. (2006) studied this question.
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