Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Shuai Wang, Xin Zhong

TL;DR

The paper addresses the global existence of weak solutions to the isentropic compressible Navier–Stokes equations in the half-plane with no-slip Dirichlet boundary, allowing vacuum and unbounded density under a small initial energy $C_0$. It advances Hoff’s intermediate regularity framework to a boundary-value, unbounded-domain setting by introducing an $L^\theta$ density regime and a Green function decomposition of the effective viscous flux $F=(2\mu+\lambda)\div\mathbf{u}-P(\rho)$, exploiting the boundary identity $x_2F=0$ to control regularity near the boundary and far-field vacuum. A two-step approximation plus Zlotnik-type bounds yield a global weak solution in the Hoff framework, despite the lack of an $L^\infty$ density bound. This work extends Hoff’s and Perepelitsa’s results to an unbounded half-plane with Dirichlet boundary conditions and demonstrates a boundary-compatible mechanism for handling vacuum and boundary effects in tandem, providing new a priori estimates and a pathway for further studies in more complex geometries.

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.

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

The paper addresses the global existence of weak solutions to the isentropic compressible Navier–Stokes equations in the half-plane with no-slip Dirichlet boundary, allowing vacuum and unbounded density under a small initial energy . It advances Hoff’s intermediate regularity framework to a boundary-value, unbounded-domain setting by introducing an density regime and a Green function decomposition of the effective viscous flux , exploiting the boundary identity to control regularity near the boundary and far-field vacuum. A two-step approximation plus Zlotnik-type bounds yield a global weak solution in the Hoff framework, despite the lack of an density bound. This work extends Hoff’s and Perepelitsa’s results to an unbounded half-plane with Dirichlet boundary conditions and demonstrates a boundary-compatible mechanism for handling vacuum and boundary effects in tandem, providing new a priori estimates and a pathway for further studies in more complex geometries.

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.
Paper Structure (9 sections, 16 theorems, 178 equations)

This paper contains 9 sections, 16 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main result
  4. Strategy of the proof
  5. Preliminaries
  6. Auxiliary results and inequalities
  7. An exposition on the boundary
  8. A priori estimates
  9. Proof of Theorem \ref{['t1.1']}

Key Result

Theorem 1.1

Let the assumptions 1.12, 1.7, 1.9, and 1.11 be satisfied. Then there exists a positive constant $\varepsilon$ depending only on $\alpha, \hat{\rho}, M, a, \gamma, \mu, \lambda$, and $\eta_0$ such that if the problem a1--a4 admits a global weak solution $(\rho,\mathbf{u})$ in the sense of Definition d1.1 satisfying, for any $0<T<\infty$, with $\sigma=\sigma(t)\triangleq\min\{1,t\}$, and for som

Theorems & Definitions (34)

  • Definition 1.1
  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 24 more