Global Existence of Strong and Weak Solutions to 2D Compressible Navier-Stokes System in Bounded Domains with Large Data and Vacuum

TL;DR

This paper proves the global existence of strong and weak solutions for the 2D compressible Navier-Stokes system with vacuum and large initial data in bounded domains, using advanced boundary and mapping techniques.

Contribution

It establishes the first global existence results for this system with large data and vacuum, employing conformal mapping and Green's function methods.

Findings

Global existence of solutions without size restrictions on initial data

Effective handling of boundary effects via conformal mapping

Representation of viscous flux enabling control of singularities

Abstract

We study the barotropic compressible Navier-Stokes system where the shear viscosity is a positive constant and the bulk one proportional to a power of the density with the power bigger than one and a third. The system is subject to the Navier-slip boundary conditions in a general two-dimensional bounded simply connected domain. For initial density allowed to vanish, we establish the global existence of strong and weak solutions without any restrictions on the size of initial value. To get over the difficulties brought by boundary, on the one hand, we apply Riemann mapping theorem and the pull back Green's function method to get a pointwise representation of the effective viscous flux. On the other hand, observing that the orthogonality is preserved under conformal mapping due to its preservation on the angle, we use the slip boundary conditions to reduce the integral representation to the desired commutator form whose singularities can be cancelled out by using the estimates on the spatial gradient of the velocity.