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Nishida-Smoller type large solutions and exponential growth for the compressible Navier-Stokes equations with slip boundary conditions in 3D bounded domain

Saiguo Xu, Yinghui Zhang

Abstract

This paper concerns the isentropic compressible Navier-Stokes equations in a three-dimensional (3D) bounded domain with slip boundary conditions and vacuum. It is shown that the classical solutions to the initial-boundary-value problem of this system with large initial energy and vacuum exist globally in time and have an exponential decay rate which is decreasing with respect to the adiabatic exponent $γ>1$ provided that the fluid is nearly isothermal (namely, the adiabatic exponent is close enough to 1). 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). In addition, it is also shown that the gradient of the density will grow unboundedly with an exponential rate when the initial vacuum appears (even at a point). In contrast to previous related works, where either small initial energy are required or boundary effects are ignored, this establishes the first result on the global existence and exponential growth of large-energy solutions with vacuum to the 3D isentropic compressible Navier-Stokes equations with slip boundary conditions.

Nishida-Smoller type large solutions and exponential growth for the compressible Navier-Stokes equations with slip boundary conditions in 3D bounded domain

Abstract

This paper concerns the isentropic compressible Navier-Stokes equations in a three-dimensional (3D) bounded domain with slip boundary conditions and vacuum. It is shown that the classical solutions to the initial-boundary-value problem of this system with large initial energy and vacuum exist globally in time and have an exponential decay rate which is decreasing with respect to the adiabatic exponent provided that the fluid is nearly isothermal (namely, the adiabatic exponent is close enough to 1). 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). In addition, it is also shown that the gradient of the density will grow unboundedly with an exponential rate when the initial vacuum appears (even at a point). In contrast to previous related works, where either small initial energy are required or boundary effects are ignored, this establishes the first result on the global existence and exponential growth of large-energy solutions with vacuum to the 3D isentropic compressible Navier-Stokes equations with slip boundary conditions.
Paper Structure (5 sections, 21 theorems, 223 equations)

This paper contains 5 sections, 21 theorems, 223 equations.

Table of Contents

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

Key Result

Theorem 1.1

Let $\Omega$ be a simply connected bounded domain in $\mathbb{R}^3$ and its smooth boundary $\partial\Omega$ has a finite number of 2-dimensional connected components. For given positive constants $M$ and $\tilde{\rho}$, suppose that the $3\times3$ symmetric matrix $A$ in Navier-slip-condition is sm and the compatibility condition for some $g\in L^2$. Then the initial-boundary value problem Large

Theorems & Definitions (41)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3: see Zlotnik2000
  • ...and 31 more