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Global weak solutions and incompressible limit to the isentropic compressible Navier-Stokes equations in 2D bounded domains with ripped density and large initial data

Shuai Wang, Guochun Wu, Xin Zhong

TL;DR

This work proves global existence of weak solutions to the 2D isentropic compressible Navier–Stokes equations in bounded convex domains with rippled density and large initial data under Navier–slip boundaries, provided the bulk viscosity is sufficiently large. It introduces and leverages the effective viscous flux and a Desjardins-type logarithmic interpolation inequality to overcome curvature-related boundary difficulties and to obtain uniform a priori bounds independent of the lower density bound and of $\lambda$. The authors also justify a rigorous incompressible limit as $\lambda\to\infty$, establishing convergence to an inhomogeneous incompressible Navier–Stokes system with density transported by the limiting velocity. Overall, the paper extends prior half‑plane results to curved boundaries, clarifies the role of boundary geometry via a Hodge decomposition, and broadens the regime of large-data, vacuum-allowing global weak solutions for compressible flows.

Abstract

This paper is a continuation of our previous work (arXiv:2507.03505), where the global existence and incompressible limit of weak solutions to the isentropic compressible Navier-Stokes equations in the half-plane with ripped density and large initial data were established. We extend such results to the case of two-dimensional bounded convex domains under a Navier-slip boundary condition. To overcome difficulties in the presence of a curved boundary, some new estimates based on the effective viscous flux and a Desjardins-type logarithmic interpolation inequality play decisive roles.

Global weak solutions and incompressible limit to the isentropic compressible Navier-Stokes equations in 2D bounded domains with ripped density and large initial data

TL;DR

This work proves global existence of weak solutions to the 2D isentropic compressible Navier–Stokes equations in bounded convex domains with rippled density and large initial data under Navier–slip boundaries, provided the bulk viscosity is sufficiently large. It introduces and leverages the effective viscous flux and a Desjardins-type logarithmic interpolation inequality to overcome curvature-related boundary difficulties and to obtain uniform a priori bounds independent of the lower density bound and of . The authors also justify a rigorous incompressible limit as , establishing convergence to an inhomogeneous incompressible Navier–Stokes system with density transported by the limiting velocity. Overall, the paper extends prior half‑plane results to curved boundaries, clarifies the role of boundary geometry via a Hodge decomposition, and broadens the regime of large-data, vacuum-allowing global weak solutions for compressible flows.

Abstract

This paper is a continuation of our previous work (arXiv:2507.03505), where the global existence and incompressible limit of weak solutions to the isentropic compressible Navier-Stokes equations in the half-plane with ripped density and large initial data were established. We extend such results to the case of two-dimensional bounded convex domains under a Navier-slip boundary condition. To overcome difficulties in the presence of a curved boundary, some new estimates based on the effective viscous flux and a Desjardins-type logarithmic interpolation inequality play decisive roles.

Paper Structure

This paper contains 10 sections, 16 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main results
  4. Strategy of the proof
  5. Preliminaries
  6. Auxiliary results and inequalities
  7. Uniform estimates for $F$, $\mathop{\mathrm{curl}}\nolimits\mathbf{u}$, $\nabla\mathbf{u}$, and $\dot{\mathbf{u}}$
  8. A priori estimates
  9. Proof of Theorem \ref{['t1.1']}
  10. Proof of Theorem \ref{['t1.2']}

Key Result

Theorem 1.1

Let z1.7 and z1.8 be satisfied, there exists a positive number $D$ depending only on $\hat{\rho}$, $a$, $\gamma$, $\mu$, and $\Omega$ such that if then the problem a1--a3 admits a global weak solution $(\rho,\mathbf{u})$ in the sense of Definition d1.1 satisfying where $\sigma\triangleq\min\{1,t\}$.

Theorems & Definitions (33)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.2
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • ...and 23 more