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Landau damping in mixed hyperbolic-kinetic systems and thick sprays

D. Bian, B. Després, V. Fournet, E. Grenier

TL;DR

This work analyzes a thick-spray model that couples a Vlasov equation for particles with barotropic compressible Euler fluid dynamics through the fluid pressure gradient, revealing a Landau-damping–type mechanism for acoustic waves whose fate (damping or amplification) is determined by the sign of \\partial_v f_0 at the spray’s phase speed. By deriving and studying the dispersion relation expressed in terms of the scaled frequency \\sigma = \\omega/|k|, the authors show that amplification leads to linear ill-posedness in Sobolev spaces, while damping corresponds to a stable, well-posed regime locally in time. They further demonstrate that Landau-damping–type phenomena are universal for hyperbolic-kinetic couplings, illustrating this first in a scalar case and then for general systems via perturbation theory. The results provide a concrete criterion linking stability to the kinetic profile near the phase velocity and offer insight into the dynamics of thin versus thick sprays, with potential implications for multi-physics models where hyperbolic and kinetic descriptions interact.

Abstract

This article is devoted to the study of a model of thick sprays which combines the Vlasov equation for the particles and the barotropic compressible Euler equations to describe the fluid, coupled through the gradient of the pressure of the fluid. We prove that sound waves interact with particles of nearby velocities, which results in a damping or an amplification of these sound waves, depending on the sign of the derivative of the distribution function at the sound speed. This mechanism is very similar to the classical Landau damping which occurs in the Vlasov-Poisson system. If the sound waves are amplified then the thick spray model is linearly ill-posed in Sobolev spaces, even locally in time. We also show that such Landau damping type phenomena naturally arise when we couple an hyperbolic system of conservation laws with the Vlasov equation.

Landau damping in mixed hyperbolic-kinetic systems and thick sprays

TL;DR

This work analyzes a thick-spray model that couples a Vlasov equation for particles with barotropic compressible Euler fluid dynamics through the fluid pressure gradient, revealing a Landau-damping–type mechanism for acoustic waves whose fate (damping or amplification) is determined by the sign of \\partial_v f_0 at the spray’s phase speed. By deriving and studying the dispersion relation expressed in terms of the scaled frequency \\sigma = \\omega/|k|, the authors show that amplification leads to linear ill-posedness in Sobolev spaces, while damping corresponds to a stable, well-posed regime locally in time. They further demonstrate that Landau-damping–type phenomena are universal for hyperbolic-kinetic couplings, illustrating this first in a scalar case and then for general systems via perturbation theory. The results provide a concrete criterion linking stability to the kinetic profile near the phase velocity and offer insight into the dynamics of thin versus thick sprays, with potential implications for multi-physics models where hyperbolic and kinetic descriptions interact.

Abstract

This article is devoted to the study of a model of thick sprays which combines the Vlasov equation for the particles and the barotropic compressible Euler equations to describe the fluid, coupled through the gradient of the pressure of the fluid. We prove that sound waves interact with particles of nearby velocities, which results in a damping or an amplification of these sound waves, depending on the sign of the derivative of the distribution function at the sound speed. This mechanism is very similar to the classical Landau damping which occurs in the Vlasov-Poisson system. If the sound waves are amplified then the thick spray model is linearly ill-posed in Sobolev spaces, even locally in time. We also show that such Landau damping type phenomena naturally arise when we couple an hyperbolic system of conservation laws with the Vlasov equation.
Paper Structure (9 sections, 3 theorems, 78 equations)

This paper contains 9 sections, 3 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Study of the dispersion relation
  3. The dispersion relation
  4. Comparison with the genuine Landau damping
  5. The "thin spray" regime
  6. The spectrally unstable case
  7. "Universality" of the Landau damping
  8. The scalar case
  9. The case of a system

Key Result

Theorem 1.1

Let $\alpha_0 \in (0,1)$, $\rho_0 > 0$, $u_0 \in {\Bbb R}$ and $f_0(v)$ be given. Let us assume that $f_0(v)$ is analytic and can be extended to the complex strip $| \Im v | \le \delta$ for some positive $\delta$. Let us moreover assume that, on this strip, $| f(v) | \le C_0 \exp(- C_1 |v|^2)$ for s

Theorems & Definitions (4)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Proposition 4.1