Local existence and conditional regularity for the Navier-Stokes-Fourier system driven by inhomogeneous boundary conditions

2024/09/20 by Abbatiello, Anna, Basaric, Danica, Chaudhuri, Nilasis +1
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2409.13459

Abstract

We consider the Navier-Stokes-Fourier system with general inhomogeneous Dirichlet-Neumann boundary conditions. We propose a new approach to the local well-posedness problem based on conditional regularity estimates. By conditional regularity we mean that any strong solution belonging to a suitable class remains regular as long as its amplitude remains bounded. The result holds for general Dirichlet-Neumann boundary conditions provided the material derivative of the velocity field vanishes on the boundary of the physical domain. As a corollary of this result we obtain: Blow up criteria for strong solutions, Local existence of strong solutions in the optimal L^p-L^q framework, Alternative proof of the existing results on local well posedness.

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