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An elementary approach to the pressureless Euler-Navier-Stokes system

Raphaël Danchin

TL;DR

The paper studies the global well-posedness and long-time behavior of the pressureless Euler-Navier-Stokes system in $\mathbb{R}^3$, arising from a monokinetic limit of the Vlasov-Navier-Stokes equations. It develops an elementary energy-method framework, introducing the energies $E_0$, $E_1$ (and dissipations $D_0$, $D_1$) and a time-weighted pair to obtain decay estimates for the kinetic and fluid components, as well as a description of the density’s asymptotics. Global existence of strong solutions is established for bounded initial density with finite mass, without smallness on $\rho_0$, under smallness assumptions on the velocity fields, and the authors derive optimal decay rates and a long-time convergence for $\rho$. The work also addresses stability/uniqueness, an mollification-based existence scheme, and a treatment of critical Besov regularity, thereby connecting the VNS and ENS frameworks without relying on Fourier analysis. The methodology provides a self-contained, energetically grounded approach with Besov tools to handle low-frequency decay, offering a solid foundation for future rigorous links between kinetic and hydrodynamic descriptions.

Abstract

The pressureless Euler-Navier-Stokes system can be obtained formally from the Vlasov-Navier-Stokes system, under the assumption that the distribution function describing the density of particles is monokinetic. Its study has been the subject of several recent papers, which have established the global existence of solutions with high enough regularity, for small initial data. In this work, we demonstrate the global existence of strong solutions in the whole space case, without assuming the initial density to be small and regular: it suffices for it to be bounded and for the total mass to be finite. In passing, we obtain optimal decay estimates for the energy and dissipation functionals. As a corollary, we get a long-time description of the density. All these results are based on an elementary energy method, with no need of sophisticated Fourier analysis tools.

An elementary approach to the pressureless Euler-Navier-Stokes system

TL;DR

The paper studies the global well-posedness and long-time behavior of the pressureless Euler-Navier-Stokes system in , arising from a monokinetic limit of the Vlasov-Navier-Stokes equations. It develops an elementary energy-method framework, introducing the energies , (and dissipations , ) and a time-weighted pair to obtain decay estimates for the kinetic and fluid components, as well as a description of the density’s asymptotics. Global existence of strong solutions is established for bounded initial density with finite mass, without smallness on , under smallness assumptions on the velocity fields, and the authors derive optimal decay rates and a long-time convergence for . The work also addresses stability/uniqueness, an mollification-based existence scheme, and a treatment of critical Besov regularity, thereby connecting the VNS and ENS frameworks without relying on Fourier analysis. The methodology provides a self-contained, energetically grounded approach with Besov tools to handle low-frequency decay, offering a solid foundation for future rigorous links between kinetic and hydrodynamic descriptions.

Abstract

The pressureless Euler-Navier-Stokes system can be obtained formally from the Vlasov-Navier-Stokes system, under the assumption that the distribution function describing the density of particles is monokinetic. Its study has been the subject of several recent papers, which have established the global existence of solutions with high enough regularity, for small initial data. In this work, we demonstrate the global existence of strong solutions in the whole space case, without assuming the initial density to be small and regular: it suffices for it to be bounded and for the total mass to be finite. In passing, we obtain optimal decay estimates for the energy and dissipation functionals. As a corollary, we get a long-time description of the density. All these results are based on an elementary energy method, with no need of sophisticated Fourier analysis tools.
Paper Structure (5 sections, 3 theorems, 170 equations)

This paper contains 5 sections, 3 theorems, 170 equations.

Table of Contents

  1. A priori estimates for the global existence
  2. The proof of Theorems \ref{['thm:main1']} and \ref{['thm:main1b']}
  3. Uniqueness and stability
  4. The existence scheme
  5. The case with critical regularity

Key Result

Theorem 1

Assume that $\rho_0\in L^1\cap L^\infty$ is nonnegative, that $u_0\in L^{3/2}\cap H^1$ with ${\rm div}\, u_0=0,$ that $w_0\in C^{0,1}$ and that $\sqrt{\rho_0}\, w_0\in L^2.$ There exists a constant $c_0>0$ depending only on $M_0,$$\|\rho_0\|_{L^\infty}$ and $\|u_0\|_{L^{3/2}}$ such that if then System (ENS) supplemented with initial data $(\rho_0,w_0,u_0)$ has a unique global solution $(\rho,w,u,

Theorems & Definitions (6)

  • Theorem 1
  • Remark 2
  • Theorem 3
  • Remark 4
  • Theorem 5
  • Remark 6