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The Navier-Stokes limit of kinetic equations for low regularity data

Kleber Carrapatoso, Isabelle Gallagher, Isabelle Tristani

TL;DR

The paper proves a rigorous Navier–Stokes limit for kinetic equations (Boltzmann with/without cutoff and Landau) under low-regularity data by embedding the kinetic–fluid transition in a unified, sharp functional setting. It uses a fixed-point strategy on the Duhamel formulation, decomposing the kinetic semigroup into low- and high-frequency parts and carefully bounding data, source, linear, and nonlinear contributions. A macro-micro decomposition together with hypocoercivity-based spectral analysis yields precise convergence of the kinetic fluctuation to the NSF solution while the microscopic part vanishes in the same low-regularity norm, with the initial data designed via a Fourier-cutoff and smallness-balancing parameters α,β. The approach extends strong-solution convergence results to data with regularity as low as H^{1/2}_x, and covers not-too-soft collision regimes for Boltzmann and Landau, offering a unified, quantitative route to the fluid limit. This has significance for rigorous derivations in kinetic theory from mesoscopic to macroscopic scales, with potential applicability to broader kinetic models and ill-prepared data scenarios in future work.

Abstract

In this paper, we investigate the link between kinetic equations (including Boltzmann with or without cutoff assumption and Landau equations) and the incompressible Navier-Stokes equation. We work with strong solutions and we treat all the cases in a unified framework. The main purpose of this work is to be as accurate as possible in terms of functional spaces. More precisely, it is well-known that the Navier-Stokes equation can be solved in a lower regularity setting (in the space variable) than kinetic equations. Our main result allows to get a rigorous link between solutions to the Navier-Stokes equation with such low regularity data and kinetic equations.

The Navier-Stokes limit of kinetic equations for low regularity data

TL;DR

The paper proves a rigorous Navier–Stokes limit for kinetic equations (Boltzmann with/without cutoff and Landau) under low-regularity data by embedding the kinetic–fluid transition in a unified, sharp functional setting. It uses a fixed-point strategy on the Duhamel formulation, decomposing the kinetic semigroup into low- and high-frequency parts and carefully bounding data, source, linear, and nonlinear contributions. A macro-micro decomposition together with hypocoercivity-based spectral analysis yields precise convergence of the kinetic fluctuation to the NSF solution while the microscopic part vanishes in the same low-regularity norm, with the initial data designed via a Fourier-cutoff and smallness-balancing parameters α,β. The approach extends strong-solution convergence results to data with regularity as low as H^{1/2}_x, and covers not-too-soft collision regimes for Boltzmann and Landau, offering a unified, quantitative route to the fluid limit. This has significance for rigorous derivations in kinetic theory from mesoscopic to macroscopic scales, with potential applicability to broader kinetic models and ill-prepared data scenarios in future work.

Abstract

In this paper, we investigate the link between kinetic equations (including Boltzmann with or without cutoff assumption and Landau equations) and the incompressible Navier-Stokes equation. We work with strong solutions and we treat all the cases in a unified framework. The main purpose of this work is to be as accurate as possible in terms of functional spaces. More precisely, it is well-known that the Navier-Stokes equation can be solved in a lower regularity setting (in the space variable) than kinetic equations. Our main result allows to get a rigorous link between solutions to the Navier-Stokes equation with such low regularity data and kinetic equations.

Paper Structure

This paper contains 26 sections, 16 theorems, 291 equations.

Table of Contents

  1. Introduction
  2. Kinetic equations
  3. Hydrodynamic limit
  4. Functional framework and notation
  5. State of the art
  6. Main result
  7. Sketch of the proof and plan of the paper
  8. Preliminaries
  9. Proof of the theorem
  10. Some results on the operators $U^\varepsilon$ and $\Psi^\varepsilon$
  11. Estimates on $U^\varepsilon$ and $\Psi^\varepsilon$
  12. Refined estimates on $\Psi^\varepsilon$
  13. Nonlinear estimates
  14. The equation on $\delta^{\varepsilon}$: proof of Proposition \ref{['estimates on delta eps']}
  15. Continuity estimates for $U^\varepsilon$
  16. ...and 11 more sections

Key Result

Theorem 1

Let $3/2 < \ell \leqslant 2$ be given. Consider $(\rho_{\rm in}, u_{\rm in}, \theta_{\rm in}) \in H^{\frac{1}{2}}(\mathbb T^3)$ that are mean-free, such that $u_{\rm in}$ is divergence free and $\rho_{\rm in}+\theta_{\rm in} = 0$. Let $(\rho,u,\theta)$ be the unique solution to eq:NSF associated wit and $\mathbf P_0^\perp f_{\rm in}^\varepsilon$ is arbitrary, going to zero in the sense that for s

Theorems & Definitions (31)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 3.1
  • Proposition 3.2
  • proof : Proof of Theorem \ref{['theo:main']}
  • Proposition 4.1
  • Proposition 4.2
  • ...and 21 more