Global weak solutions to the isentropic compressible Navier-Stokes equations with vacuum and unbounded density in a half-plane under Dirichlet boundary conditions

TL;DR

This paper proves the existence of global weak solutions to the isentropic compressible Navier-Stokes equations in a half-plane, allowing vacuum and unbounded density, using new estimates and extending Hoff's framework to unbounded domains.

Contribution

It introduces a novel intermediate regularity class for weak solutions that handles vacuum, unbounded density, and boundary conditions in an unbounded domain, extending Hoff's theory.

Findings

First global weak solutions with vacuum in a half-plane under Dirichlet conditions.

Solutions admit unbounded densities and have intermediate regularity.

The analysis employs Green function methods and new estimates.

Abstract

We establish the global existence of a class of weak solutions to the isentropic compressible Navier-Stokes equations in a half-plane with Dirichlet boundary conditions, allowing for vacuum both in the interior and at infinity, under a suitably small initial total energy. The solutions constructed here admit unbounded densities and lie in an intermediate regularity regime between the finite-energy weak solutions of Lions-Feireisl and the framework of Hoff. This result generalizes previous works of Hoff (Comm. Pure Appl. Math. 55 (2002), pp. 1365-1407) and Perepelitsa (Arch. Ration. Mech. Anal. 212 (2014), pp. 709-726) concerning discontinuous solutions by allowing vacuum states and unbounded density. Our analysis relies on the Green function method and new estimates involving the specific structure of the equations and the geometry of the half-plane. To the best of our knowledge, this is the first result concerning global weak solutions within Hoff's framework on an unbounded domain that simultaneously accommodates Dirichlet boundary conditions and far-field vacuum. The intermediate-regularity class developed here may be viewed as a natural extension of Hoff's theory, precisely tailored to overcome the two corresponding obstructions: the lack of global space-time control of the effective viscous flux arising from far-field vacuum and the absence of boundary-induced regularity gains in the no-slip setting.