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Hydrodynamic limit for the non-cutoff Boltzmann equation

Chuqi Cao, Kleber Carrapatoso

Abstract

This work deals with the non-cutoff Boltzmann equation for all type of potentials, in both the torus $\mathbf{T}^3$ and in the whole space $\mathbf{R}^3$, under the incompressible Navier-Stokes scaling. We first establish the well-posedness and decay of global mild solutions to this rescaled Boltzmann equation in a perturbative framework, that is for solutions close to the Maxwellian, obtaining in particular integrated-in-time regularization estimates. We then combine these estimates with spectral-type estimates in order to obtain the strong convergence of solutions to the non-cutoff Boltzmannn equation towards the incompressible Navier-Stokes-Fourier system.

Hydrodynamic limit for the non-cutoff Boltzmann equation

Abstract

This work deals with the non-cutoff Boltzmann equation for all type of potentials, in both the torus and in the whole space , under the incompressible Navier-Stokes scaling. We first establish the well-posedness and decay of global mild solutions to this rescaled Boltzmann equation in a perturbative framework, that is for solutions close to the Maxwellian, obtaining in particular integrated-in-time regularization estimates. We then combine these estimates with spectral-type estimates in order to obtain the strong convergence of solutions to the non-cutoff Boltzmannn equation towards the incompressible Navier-Stokes-Fourier system.
Paper Structure (33 sections, 24 theorems, 351 equations)

This paper contains 33 sections, 24 theorems, 351 equations.

Table of Contents

  1. Introduction
  2. The Boltzmann equation
  3. The Navier-Stokes-Fourier system
  4. Main results
  5. Well-posedness for the rescaled Boltzmann equation
  6. Well-posedness for the Navier-Stokes-Fourier system
  7. Hydrodynamic limit
  8. Organization of the paper
  9. Linearized Boltzmann operator
  10. Boundedness and regularization estimates
  11. Decay estimates: Hard potentials in the torus
  12. Decay estimates: Soft potentials in the torus
  13. Decay estimates: Hard potentials in the whole space
  14. Decay estimates: Soft potentials in the whole space
  15. Well-posedness and regularization for the rescaled Boltzmann equation
  16. ...and 18 more sections

Key Result

Theorem 2.1

Let $\ell=0$ in the hard potentials case $\gamma+2s \ge 0$, and $\ell \ge 0$ in the soft potentials case $\gamma+2s<0$. There is $\eta_0>0$ small enough such that for all $\varepsilon \in (0,1]$ the following holds:

Theorems & Definitions (47)

  • Remark 1.1
  • Theorem 2.1: Global well-posedness and decay for the Boltzmann equation
  • Remark 2.1
  • Theorem 2.2: Global well-posedness for the Navier-Stokes-Fourier system
  • Theorem 2.3: Hydrodynamic limit
  • Remark 2.2
  • Proposition 3.1
  • Proposition 3.2
  • Remark 3.1
  • proof
  • ...and 37 more