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Nishida-Smoller type large solutions for the compressible Navier-Stokes equations with slip boundary conditions in 3D exterior domains

Minghong Xie, Saiguo Xu, Yinghui Zhang

TL;DR

This work proves the global existence of classical solutions to the isentropic compressible Navier–Stokes equations in a 3D exterior domain under Navier-slip boundary conditions, allowing vacuum and large initial energy when the adiabatic exponent $\gamma$ is near $1$. The authors develop a Nishida–Smoller type approach by building a time-dependent a priori energy framework and proving a density upper bound that is independent of time, carefully handling boundary contributions and exterior-domain geometry via boundary-adapted localization and Hodge decomposition techniques. A key scaling $(\gamma-1)E_0^{59/3} \le C$ (along with smallness of the slip-boundary matrix $A$) enables global classical existence and yields long-time behavior analogous to bounded-domain results, while the assumption $\rho_\infty=0$ is used to achieve uniform pressure control. These results extend Lions–Feireisl weak-solution theory into a regime of large energy with near-isothermal behavior in exterior domains, opening paths toward understanding large-data classical solutions in more complex geometries.

Abstract

This paper investigates the global existence of classical solutions to the isentropic compressible Navier-Stokes equations with slip boundary condition in a three-dimensional (3D) exterior domain. It is shown that the classical solutions with large initial energy and vacuum exist globally in time when the adiabatic exponent $γ>1$ is sufficiently close to 1 (near-isothermal regime). This constitutes an extension of the celebrated result for the one-dimensional Cauchy problem of the isentropic Euler equations that has been established in 1973 by Nishida and Smoller (Comm. Pure Appl. Math. 26 (1973), 183-200). To the best of our knowledge, we establish the first result on the global existence of large-energy solutions with vacuum to the compressible Navier-Stokes equations with slip boundary condition in a 3D exterior domain, which improves previous related works where either small initial energy is required or boundary effects are ignored.

Nishida-Smoller type large solutions for the compressible Navier-Stokes equations with slip boundary conditions in 3D exterior domains

TL;DR

This work proves the global existence of classical solutions to the isentropic compressible Navier–Stokes equations in a 3D exterior domain under Navier-slip boundary conditions, allowing vacuum and large initial energy when the adiabatic exponent is near . The authors develop a Nishida–Smoller type approach by building a time-dependent a priori energy framework and proving a density upper bound that is independent of time, carefully handling boundary contributions and exterior-domain geometry via boundary-adapted localization and Hodge decomposition techniques. A key scaling (along with smallness of the slip-boundary matrix ) enables global classical existence and yields long-time behavior analogous to bounded-domain results, while the assumption is used to achieve uniform pressure control. These results extend Lions–Feireisl weak-solution theory into a regime of large energy with near-isothermal behavior in exterior domains, opening paths toward understanding large-data classical solutions in more complex geometries.

Abstract

This paper investigates the global existence of classical solutions to the isentropic compressible Navier-Stokes equations with slip boundary condition in a three-dimensional (3D) exterior domain. It is shown that the classical solutions with large initial energy and vacuum exist globally in time when the adiabatic exponent is sufficiently close to 1 (near-isothermal regime). This constitutes an extension of the celebrated result for the one-dimensional Cauchy problem of the isentropic Euler equations that has been established in 1973 by Nishida and Smoller (Comm. Pure Appl. Math. 26 (1973), 183-200). To the best of our knowledge, we establish the first result on the global existence of large-energy solutions with vacuum to the compressible Navier-Stokes equations with slip boundary condition in a 3D exterior domain, which improves previous related works where either small initial energy is required or boundary effects are ignored.

Paper Structure

This paper contains 5 sections, 20 theorems, 218 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Proof of Theorem \ref{['thm-global-CNS-exterior']}
  4. Lower-order a priori estimates
  5. The mathematical analysis on three terms about density.

Key Result

Theorem 1.1

Let $\Omega$ be the exterior of a simply connected bounded domain $D$ in $\mathbb{R}^3$ and its boundary $\partial\Omega$ is smooth. For given positive constants $M$ and $\tilde{\rho}\geq\rho_{\infty}+1$, suppose that the $3\times3$ symmetric matrix $A$ in Navier-slip-condition is smooth and positiv and the compatibility condition for some $g\in L^2$. If $\rho_{\infty}=0$, then the initial-bounda

Theorems & Definitions (42)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1
  • Lemma 2.2: see Theorem 2.1 in Crispo2004
  • Lemma 2.3: see Zlotnik2000
  • Lemma 2.4
  • ...and 32 more