The Compressible Navier-Stokes Equations on the Multi-Connected Domains

TL;DR

This paper studies the mathematical properties of the compressible Navier-Stokes equations on multi-connected domains, focusing on solvability, Green's functions, and the behavior of solutions with large initial data including vacuum.

Contribution

It introduces new methods for analyzing the equations on complex domains using conformal mappings and Green's functions, advancing understanding of global solutions and large-time behavior.

Findings

Established multi-solvability of stationary systems on general domains.

Derived commutator estimates using Green's functions and conformal mappings.

Proved global well-posedness and analyzed large-time behavior with vacuum and large initial data.

Abstract

This paper investigates the isentropic compressible Navier-Stokes equations on k-connected domains under Navier-slip boundary conditions. We study the multi-solvability of the stationary systems on general domains, which is closely related with the Cauchy-Riemann systems and critical points of harmonic functions on the domain. Then based on the structure of Green's functions, the commutator estimates are obtained on the circular domains and extended to general domains with the help of conformal mappings. Moreover, we will utilize these assertions to discuss the global well-posedness and large time behaviours of the non-stationary systems on general domains with large initial values containing vacuum.