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

TL;DR

This paper establishes local existence and regularity criteria for the Navier-Stokes-Fourier system with inhomogeneous boundary conditions, using a new approach based on conditional regularity estimates.

Contribution

It introduces a novel method for proving local well-posedness and regularity of solutions under inhomogeneous boundary conditions, extending previous results.

Findings

Blow-up criteria for strong solutions

Local existence in optimal L^p-L^q spaces

Alternative proof of existing well-posedness results

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.