  We consider two hydrodynamic model problems (one incompressible and one
compressible) with three dimensional fluid flow on the torus and
temperature-dependent viscosity and conductivity. The ambient heat for the
fluid is transported by the flow and fed by the local energy dissipation,
modeling the transfer of kinetic energy into thermal energy through fluid
friction. Both the viscosity and conductivity grow with the local temperature.
We prove a strong a priori bound on the enstrophy of the velocity weighed
against the temperature for initial data of arbitrary size, requiring only that
the conductivity be proportionately larger than the viscosity (and, in the
incompressible case, a bound on the temperature as a Muckenhoupt weight).
