Conditional regularity of very weak solutions to the Navier-Stokes-Fourier system

摘要

We consider a class of (very) weak solutions to the Navier-Stokes-Fourier system describing the time evolution of the density ρ, the absolute temperature?, and the macroscopic velocity u. It is shown that a weak solution emanating from smooth initial data is regular as long as u and ? are bounded and ||divχu||L∞ integrable in the existence interval (0, T). Using the method of relative energy we first show that any weak solution enjoying the above mentioned regularity coincides with a strong one as long as the latter exists. In such a way, the proof reduces to showing that the life span of the strong solution can be extended to the desired existence interval (0, T).