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Global solutions of the one-dimensional compressible Euler equations with nonlocal interactions via the inviscid limit

Jose A. Carrillo, Gui-Qiang G. Chen, Difan Yuan, Ewelina Zatorska

Abstract

We are concerned with the global existence of finite-energy entropy solutions of the one-dimensional compressible Euler equations with (possibly) damping, alignment forces, and nonlocal interactions: Newtonian repulsion and quadratic confinement. Both the polytropic gas law and the general gas law are analyzed. This is achieved by constructing a sequence of solutions of the one-dimensional compressible Navier-Stokes-type equations with density-dependent viscosity under the stress-free boundary condition and then taking the vanishing viscosity limit. The main difficulties in this paper arise from the appearance of the nonlocal terms. In particular, some uniform higher moment estimates for the compressible Navier-Stokes equations on expanding intervals with stress-free boundary conditions are obtained by careful design of the approximate initial data.

Global solutions of the one-dimensional compressible Euler equations with nonlocal interactions via the inviscid limit

Abstract

We are concerned with the global existence of finite-energy entropy solutions of the one-dimensional compressible Euler equations with (possibly) damping, alignment forces, and nonlocal interactions: Newtonian repulsion and quadratic confinement. Both the polytropic gas law and the general gas law are analyzed. This is achieved by constructing a sequence of solutions of the one-dimensional compressible Navier-Stokes-type equations with density-dependent viscosity under the stress-free boundary condition and then taking the vanishing viscosity limit. The main difficulties in this paper arise from the appearance of the nonlocal terms. In particular, some uniform higher moment estimates for the compressible Navier-Stokes equations on expanding intervals with stress-free boundary conditions are obtained by careful design of the approximate initial data.
Paper Structure (24 sections, 29 theorems, 298 equations)

This paper contains 24 sections, 29 theorems, 298 equations.

Table of Contents

  1. Introduction
  2. Definitions and Main Theorems
  3. Entropy and entropy flux pairs for CEEs
  4. Existence of solutions to CEEs
  5. Existence of solutions to CNSEs and their inviscid limit
  6. Uniform Estimates of Approximate Solutions
  7. Conservation of mass and Lagrangian coordinates
  8. Basic energy estimate
  9. Second moment estimate
  10. Higher-order estimates of the density and the pressure
  11. Inviscid Limit for Polytropic Gases
  12. Choice of a special entropy and entropy flux pair
  13. Higher integrability of the velocity
  14. $H_{\rm loc}^{-1}(\mathbb{R}_+^2)$--Compactness
  15. Proof of Theorem \ref{['thm3.3']} for the polytropic equation of state
  16. ...and 9 more sections

Key Result

Theorem 2.2

Consider problem 1.1 with initial data 1.6 satisfying 1.7--2.1. Let $P(\rho)$ satisfy hypotheses ($\mathcal{H}$ ). Then there exists a global-in-time finite-energy entropy solution $(\rho, m )(t,x)$ of problem 1.1 and 1.6, in the sense of Definition definition-Euler. In particular, there exists a gl

Theorems & Definitions (36)

  • Definition 2.1
  • Theorem 2.2: Existence of solutions of CEEs with nonlocal interactions
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6: Inviscid limit for CNSEs with nonlocal interactions
  • Remark 2.7
  • Remark 2.8
  • Lemma 3.1
  • Lemma 3.2
  • ...and 26 more