We consider the abstract initial value problem for the system of evolution equations which describe motion of incompressible viscous and heat-conductive fluids in a bounded domain. It is difficulty of our problem that we do not neglect the viscous dissipation function in contrast to the Boussinesq approximation. This problem has uniquely a mild solution locally in time for general initial data, and globally in time for small initial data. Moreover, a mild solution of this problem can be a strong or classical solution under appropriate assumptions for initial data. We prove the above properties by the theory of analytic semigroups on Banach spaces.
"The initial value problem for motion of incompressible viscous and heat-conductive fluids in Banach spaces." Hiroshima Math. J. 40 (3) 371 - 402, November 2010. https://doi.org/10.32917/hmj/1291818851