Well-Posedness for 2D Combustion Model in Bounded Domains and Serrin-Type Blowup Criterion

TL;DR

This paper establishes the global existence, uniqueness, and blowup criteria for 2D combustion models in bounded domains, extending Serrin's blowup criterion by removing velocity conditions under certain boundary conditions.

Contribution

It provides new well-posedness results and extends Serrin's blowup criterion specifically for the 2D combustion model with boundary conditions.

Findings

Global existence of weak and strong solutions under small initial density.

Extension of Serrin's blowup criterion to the 2D combustion model.

Removal of velocity boundary condition in the blowup criterion.

Abstract

We investigate the 2D combustion model with Dirichlet boundary conditions and slip boundary conditions in bounded domains. The global existence of weak and strong solutions and the uniqueness of strong solutions are obtained provided the initial density is small in some precise sense. Using the energy method and the estimates of boundary integrals, we obtain the a priori bounds if the density and velocity field. In addition, we prove the local existence of the strong solutions via iterative method and the contraction mapping theorem. Finally, we extend the well known Serrion's blowup criterion to the 2D combustion model. Under the suitable boundary conditions, the Serrin's condition on the velocity can be removed in this criteria.