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Exponential Stability of the Inhomogeneous Navier-Stokes-Vlasov System in Vacuum

Hai-Liang Li, Ling-Yun Shou, Yue Zhang

TL;DR

The paper addresses the exponential stability of the inhomogeneous Navier-Stokes-Vlasov system in vacuum, proving global existence and uniqueness of strong solutions when the initial energy is small or viscosity is large. It introduces a novel energy framework with higher-order, time-weighted inequalities to overcome degeneracy near vacuum and to obtain uniform bounds on the macroscopic density. Under a conditional bound on the particle density, it establishes exponential decay of both the fluid energy and the distribution function, including convergence in the Wasserstein sense to an equilibrium profile with velocity concentration at zero. The work also provides comprehensive higher-order estimates and rigorously derives the large-time behavior of the coupled fluid-particle system, yielding precise decay rates and velocity-space localization, along with a weak-solution theory via regularization and compactness. These results advance understanding of kinetic-fluid interactions in unbounded domains with vacuum and offer techniques applicable to related fluid-particle models.

Abstract

In this paper, we study the asymptotic behaviors of solutions to the inhomogeneous Navier-Stokes-Vlasov system in $\mathbb{R}^{3}\times\mathbb{R}^{3}$, where the initial fluid density is allowed to vanish. We establish the uniform bound of the macroscopic density associated with the distribution function and prove the global existence and uniqueness of strong solutions to the Cauchy problem with vacuum for either small initial energy or large viscosity coefficient. The uniform boundedness and the presence of vacuum enable us to show that as the time evolves, the fluid velocity decays, while the distribution function concentrates towards a Dirac measure in velocity centred at $0$, with an exponential rate. In order to overcome the degeneracy in the momentum equations, we develop an energy argument based on higher order functional inequalities designed for fluid-particle coupled structures.

Exponential Stability of the Inhomogeneous Navier-Stokes-Vlasov System in Vacuum

TL;DR

The paper addresses the exponential stability of the inhomogeneous Navier-Stokes-Vlasov system in vacuum, proving global existence and uniqueness of strong solutions when the initial energy is small or viscosity is large. It introduces a novel energy framework with higher-order, time-weighted inequalities to overcome degeneracy near vacuum and to obtain uniform bounds on the macroscopic density. Under a conditional bound on the particle density, it establishes exponential decay of both the fluid energy and the distribution function, including convergence in the Wasserstein sense to an equilibrium profile with velocity concentration at zero. The work also provides comprehensive higher-order estimates and rigorously derives the large-time behavior of the coupled fluid-particle system, yielding precise decay rates and velocity-space localization, along with a weak-solution theory via regularization and compactness. These results advance understanding of kinetic-fluid interactions in unbounded domains with vacuum and offer techniques applicable to related fluid-particle models.

Abstract

In this paper, we study the asymptotic behaviors of solutions to the inhomogeneous Navier-Stokes-Vlasov system in , where the initial fluid density is allowed to vanish. We establish the uniform bound of the macroscopic density associated with the distribution function and prove the global existence and uniqueness of strong solutions to the Cauchy problem with vacuum for either small initial energy or large viscosity coefficient. The uniform boundedness and the presence of vacuum enable us to show that as the time evolves, the fluid velocity decays, while the distribution function concentrates towards a Dirac measure in velocity centred at , with an exponential rate. In order to overcome the degeneracy in the momentum equations, we develop an energy argument based on higher order functional inequalities designed for fluid-particle coupled structures.
Paper Structure (7 sections, 18 theorems, 187 equations)

This paper contains 7 sections, 18 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Conditional large-time behavior
  3. Uniform a-priori estimates
  4. The bootstrap
  5. Large-time behavior of the distribution function
  6. Higher order estimates
  7. Proofs of main results

Key Result

Theorem 1.1

Let $\mu>0$, $\kappa\geq1$ and $R_{0}>0$ be given constants. Assume that the initial data $(\rho_{0},u_{0},f_{0})$ satisfies Then, there exists a constant $\varepsilon_{0}>0$ depending on $M$, $\|\rho_{0}\|_{L^{\frac{3}{2}}\cap L^{\infty}}$, $\|f_{0}\|_{L^{1}_{x,v}}$ and $\|(1+|v|^{2})f_{0}\|_{ L^{1}_{v}(L^{\infty}_{x})}$ such that if the Cauchy problem NSV--inda admits a unique global solution

Theorems & Definitions (34)

  • Theorem 1.1
  • Remark 1.1
  • Definition 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 3.1
  • Lemma 3.2
  • ...and 24 more