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Global well-posedness and large-time behavior of classical solutions to the Euler-Navier-Stokes system in R^3

Feimin Huang, Houzhi Tang, Guochun Wu, Weiyuan Zou

Abstract

In this paper, we study the Cauchy problem of a two-phase flow system consisting of the compressible isothermal Euler equations and the incompressible Navier-Stokes equations coupled through the drag force, which can be formally derived from the Vlasov-Fokker-Planck/incompressible Navier-Stokes equations. When the initial data is a small perturbation around an equilibrium state, we prove the global well-posedness of the classical solutions to this system and show the solutions tends to the equilibrium state as time goes to infinity. In order to resolve the main difficulty arising from the pressure term of the incompressible Navier-Stokes equations, we properly use the Hodge decomposition, spectral analysis, and energy method to obtain the $L^2$ time decay rates of the solution when the initial perturbation belongs to $L^1$ space. Furthermore, we show that the above time decay rates are optimal.

Global well-posedness and large-time behavior of classical solutions to the Euler-Navier-Stokes system in R^3

Abstract

In this paper, we study the Cauchy problem of a two-phase flow system consisting of the compressible isothermal Euler equations and the incompressible Navier-Stokes equations coupled through the drag force, which can be formally derived from the Vlasov-Fokker-Planck/incompressible Navier-Stokes equations. When the initial data is a small perturbation around an equilibrium state, we prove the global well-posedness of the classical solutions to this system and show the solutions tends to the equilibrium state as time goes to infinity. In order to resolve the main difficulty arising from the pressure term of the incompressible Navier-Stokes equations, we properly use the Hodge decomposition, spectral analysis, and energy method to obtain the time decay rates of the solution when the initial perturbation belongs to space. Furthermore, we show that the above time decay rates are optimal.
Paper Structure (11 sections, 15 theorems, 215 equations)

This paper contains 11 sections, 15 theorems, 215 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Auxiliary lemmas
  5. The a priori estimates
  6. Reformulated system and local existence
  7. Time-independent energy estimates
  8. Time decay estimates of the E-NS system
  9. Time decay estimates of the linear equations
  10. Time decay estimates of the nonlinear equations
  11. The proof of Theorems \ref{['thm1']}-\ref{['thm3']}

Key Result

Theorem 1.1

Assume that for some integer $s\geq 3$, the initial data $(a_0-a_*,u_0,v_0)\in H^s(\mathbb{R}^3)$ satisfies where $\varepsilon_0$ is a small positive constant and ${\rm{div}}v_0=0$, then the Cauchy problem Main2-in-data admits a unique global classical solution $(a,u,v)$ such that for any $t\in\mathbb{R}_+$. Additionally, when $(a_0-a_*,u_0,v_0)\in L^1(\mathbb{R}^3)$, then it holds that where t

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 15 more