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Large-time asymptotics of periodic two-dimensional Vlasov-Navier-Stokes flows

Raphaël Danchin, Ling-Yun Shou

TL;DR

This work analyzes the long-time behavior of the 2D Vlasov-Navier-Stokes system on the torus for both homogeneous and inhomogeneous fluids. Using a modulated energy framework and a hierarchy of Lyapunov functionals, it proves convergence to monokinetic equilibria with algebraic decay for general data and exponential decay under small initial particle distributions, extended to piecewise-constant density in the inhomogeneous setting. The authors derive time-dependent regularity bounds, construct global weak solutions via regularization and compactness, and provide detailed asymptotics showing traveling-wave decay profiles for the particle distribution alongside convergence of the fluid density and velocity. Collectively, the results yield a sharp, quantitative picture of how kinetic particles coupled to a viscous fluid relax to traveling-wave-type states in periodic two-dimensional domains.

Abstract

We study the large-time behavior of finite-energy weak solutions for the Vlasov-Navier-Stokes equations in a two-dimensional torus. We focus first on the homogeneous case where the ambient (incompressible and viscous) fluid carrying the particles has a constant density, and then on the variable-density case. In both cases, large-time convergence to a monokinetic final state is demonstrated. For any finite energy initial data, we exhibit an algebraic convergence rate that deteriorates as the initial particle distribution increases. When the initial particle distribution is suitably small, then the convergence rate becomes exponential, a result consistent with the work of Han-Kwan et al. [17] dedicated to the homogeneous, three-dimensional case, where an additional smallness condition on the velocity was required. In the non-homogeneous case, we establish similar stability results, allowing a piecewise constant fluid density with jumps.

Large-time asymptotics of periodic two-dimensional Vlasov-Navier-Stokes flows

TL;DR

This work analyzes the long-time behavior of the 2D Vlasov-Navier-Stokes system on the torus for both homogeneous and inhomogeneous fluids. Using a modulated energy framework and a hierarchy of Lyapunov functionals, it proves convergence to monokinetic equilibria with algebraic decay for general data and exponential decay under small initial particle distributions, extended to piecewise-constant density in the inhomogeneous setting. The authors derive time-dependent regularity bounds, construct global weak solutions via regularization and compactness, and provide detailed asymptotics showing traveling-wave decay profiles for the particle distribution alongside convergence of the fluid density and velocity. Collectively, the results yield a sharp, quantitative picture of how kinetic particles coupled to a viscous fluid relax to traveling-wave-type states in periodic two-dimensional domains.

Abstract

We study the large-time behavior of finite-energy weak solutions for the Vlasov-Navier-Stokes equations in a two-dimensional torus. We focus first on the homogeneous case where the ambient (incompressible and viscous) fluid carrying the particles has a constant density, and then on the variable-density case. In both cases, large-time convergence to a monokinetic final state is demonstrated. For any finite energy initial data, we exhibit an algebraic convergence rate that deteriorates as the initial particle distribution increases. When the initial particle distribution is suitably small, then the convergence rate becomes exponential, a result consistent with the work of Han-Kwan et al. [17] dedicated to the homogeneous, three-dimensional case, where an additional smallness condition on the velocity was required. In the non-homogeneous case, we establish similar stability results, allowing a piecewise constant fluid density with jumps.

Paper Structure

This paper contains 14 sections, 18 theorems, 242 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. The homogeneous case
  4. The inhomogeneous case
  5. The homogeneous case
  6. Algebraic convergence rate for large data
  7. Higher order Lyapunov functionals
  8. Time-dependent regularity estimates
  9. Exponential decay rate in the case of small distribution functions
  10. Large-time exponential asymptotics
  11. Construction of the solutions
  12. The inhomogeneous case
  13. Regularity estimates of the Vlasov equation
  14. Technical lemmas

Key Result

Theorem 1.1

Let the fluid domain $\Omega$ be the torus $\mathbb{T}^d$ or the whole space $\mathbb{R}^d$ with $d=2,3.$ Assume that Then, homoVNS supplemented with homod admits a global-in-time distributional solution verifying If $d=2$ and, in addition, $|v|^qf_0\in L^\infty(\Omega\times\mathbb{R}^2)$ for some $q>4,$ then uniqueness is true.

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.1
  • Remark 2.2
  • Proposition 3.1
  • proof
  • Lemma 3.1
  • ...and 20 more