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Fujita-Kato Solutions and Optimal Time Decay for the Vlasov-Navier-Stokes System in the Whole Space

Raphaël Danchin

TL;DR

The article addresses global well-posedness and long-time behavior for the Vlasov-Navier-Stokes system in $\,\mathbb{R}^3$, extending the Fujita–Kato framework from the Navier–Stokes equation to a kinetic-fluid coupling. It introduces a hierarchy of energy functionals, including the primary $E_0$ and the higher-order $E_1$, and proves optimal decay rates $E_0(t)\sim t^{-3/2}$ and $E_1(t)\sim t^{-5/2}$ under small-data assumptions, aided by a Nash-type Lyapunov approach and Besov-space decay controls. A detailed Fujita–Kato-type theory is developed: for small initial velocity in $\dot H^{1/2}$ and localized initial kinetic data, there exists a unique global solution, with a local-in-time construction on $[0,1]$ extended globally using a Log-Lipschitz stability framework. The paper also analyzes the large-time behavior of the kinetic distribution, establishing Wasserstein-based convergence towards a monokinetic state and providing precise asymptotics for the density $\rho$ and current $j$, thereby refining the understanding of the spray-fluid interaction. The results rely on Friedrichs regularization, maximal regularity in Besov spaces, and Nash-type decay arguments that yield optimal rates and stability for the coupled system.

Abstract

We are concerned with the construction of global-in-time strong solutions for the incompressible Vlasov-Navier-Stokes system in the whole three-dimensional space. One of our goals is to establish that small initial velocities with critical Sobolev regularity and sufficiently well localized initial kinetic distribution functions give rise to global and unique solutions. This constitutes an extension of the celebrated result for the incompressible Navier-Stokes equations (NS) that has been established in 1964 by Fujita and Kato. If in addition the initial velocity is integrable, we establish that the total energy of the system decays to 0 with the optimal rate t^{-3/2}, like for the weak solutions of (NS). Our results partly rely on the use of a higher order energy functional that controls the regularity $H^1$ of the velocity and seems to have been first introduced by Li, Shou and Zhang in the context of nonhomogeneous Vlasov-Navier-Stokes system. In the small data case, we show that this energy functional decays with the rate t^{-5/2}.

Fujita-Kato Solutions and Optimal Time Decay for the Vlasov-Navier-Stokes System in the Whole Space

TL;DR

The article addresses global well-posedness and long-time behavior for the Vlasov-Navier-Stokes system in , extending the Fujita–Kato framework from the Navier–Stokes equation to a kinetic-fluid coupling. It introduces a hierarchy of energy functionals, including the primary and the higher-order , and proves optimal decay rates and under small-data assumptions, aided by a Nash-type Lyapunov approach and Besov-space decay controls. A detailed Fujita–Kato-type theory is developed: for small initial velocity in and localized initial kinetic data, there exists a unique global solution, with a local-in-time construction on extended globally using a Log-Lipschitz stability framework. The paper also analyzes the large-time behavior of the kinetic distribution, establishing Wasserstein-based convergence towards a monokinetic state and providing precise asymptotics for the density and current , thereby refining the understanding of the spray-fluid interaction. The results rely on Friedrichs regularization, maximal regularity in Besov spaces, and Nash-type decay arguments that yield optimal rates and stability for the coupled system.

Abstract

We are concerned with the construction of global-in-time strong solutions for the incompressible Vlasov-Navier-Stokes system in the whole three-dimensional space. One of our goals is to establish that small initial velocities with critical Sobolev regularity and sufficiently well localized initial kinetic distribution functions give rise to global and unique solutions. This constitutes an extension of the celebrated result for the incompressible Navier-Stokes equations (NS) that has been established in 1964 by Fujita and Kato. If in addition the initial velocity is integrable, we establish that the total energy of the system decays to 0 with the optimal rate t^{-3/2}, like for the weak solutions of (NS). Our results partly rely on the use of a higher order energy functional that controls the regularity of the velocity and seems to have been first introduced by Li, Shou and Zhang in the context of nonhomogeneous Vlasov-Navier-Stokes system. In the small data case, we show that this energy functional decays with the rate t^{-5/2}.
Paper Structure (20 sections, 10 theorems, 274 equations)

This paper contains 20 sections, 10 theorems, 274 equations.

Table of Contents

  1. Main results
  2. Energy estimates
  3. Decay estimates and control of the Lipschitz norm of the velocity
  4. Decay estimates for the energy functionals
  5. Getting the Lipschitz bound
  6. Proving theorem \ref{['thm:main1']}
  7. A stability result
  8. The proof of existence
  9. Convergence of the distribution of particles
  10. The Fujita-Kato theorem for the Vlasov-Navier-Stokes system
  11. The incompressible Navier-Stokes equations with source term
  12. Construction of a solution on an interval of size $1$
  13. Uniqueness
  14. Continuation of the solution
  15. The case of an initial velocity in $\dot B^{1/2}_{2,1}$
  16. ...and 5 more sections

Key Result

Theorem 1.1

Assume that the initial distribution function $f_0$ satisfies and that $u_0$ is a divergence free vector-field in $L^1(\mathbb R^3)\cap H^1(\mathbb R^3).$ There exists a small constant $c_0$ depending only on such that if In all the paper, $\int$ means $\int_{\mathbb R^3}.$ then $(VNS)$ admits a unique global solution $(f,u,P)$ satisfying the energy balance: and such that: Furthermore, the fo

Theorems & Definitions (21)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Corollary 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Proposition 2.1
  • ...and 11 more