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Strong diffusive limit of the Boltzmann equation with Maxwell boundary condition

Yan Guo, Junhwa Jung, Fujun Zhou

TL;DR

This work establishes a global-in-time strong diffusive limit for the Boltzmann equation with Maxwell boundary conditions across the full range $\alpha\in[0,1]$ in the diffusive scaling $\varepsilon\partial_t F + v\cdot\nabla_x F = \varepsilon^{-1}Q(F,F)$. It introduces a twofold strategy: (i) an $\varepsilon$-stretching method to obtain uniform $L^{\infty}$ control via a single-bounce mechanism on stretched domains, and (ii) a dissipative decomposition around a carefully crafted rotating Maxwellian $\tilde{\mu}$ to handle the near-specular regime $0\le\alpha\ll\varepsilon$. The main results show strong convergence to the incompressible Navier–Stokes–Fourier system with boundary behavior dictated by the limiting ratio $\lambda=\frac{1}{\sqrt{2\pi}}\lim_{\varepsilon\to0}\frac{\alpha}{\varepsilon}$, yielding Dirichlet when $\lambda=\infty$ and Navier-slip when $0<\lambda<\infty$ (and the perfect Navier slip in the near-specular limit). These findings extend prior weak convergence results to strong convergence in general domains and provide a robust framework for the coupling of kinetic boundary interactions with hydrodynamic limits.

Abstract

While weak diffusive limit from the Boltzmann equation to the incompressible Navier-Stokes-Fourier system was established for the Maxwell boundary condition within renormalized solutions framework [Saint.Raymond2009][Jiang-Masmoudi2017], the corresponding strong diffusive limit has remained outstanding except when the accommodation coefficient $α\sim \varepsilon^{1/2}$ [Jiang-Masmoudi2017]. We establish global in time strong diffusive limit for all accommodation coefficients $α\in [0, 1]$ within strong solutions framework. The main novelties of our proof include: (1) a $\varepsilon$-stretching method for reduction to a single-bounce $L^\infty$ estimate; (2) a dissipation estimate for a carefully constructed rotating Maxwellian in the near-specular regime $α\ll \varepsilon$.

Strong diffusive limit of the Boltzmann equation with Maxwell boundary condition

TL;DR

This work establishes a global-in-time strong diffusive limit for the Boltzmann equation with Maxwell boundary conditions across the full range in the diffusive scaling . It introduces a twofold strategy: (i) an -stretching method to obtain uniform control via a single-bounce mechanism on stretched domains, and (ii) a dissipative decomposition around a carefully crafted rotating Maxwellian to handle the near-specular regime . The main results show strong convergence to the incompressible Navier–Stokes–Fourier system with boundary behavior dictated by the limiting ratio , yielding Dirichlet when and Navier-slip when (and the perfect Navier slip in the near-specular limit). These findings extend prior weak convergence results to strong convergence in general domains and provide a robust framework for the coupling of kinetic boundary interactions with hydrodynamic limits.

Abstract

While weak diffusive limit from the Boltzmann equation to the incompressible Navier-Stokes-Fourier system was established for the Maxwell boundary condition within renormalized solutions framework [Saint.Raymond2009][Jiang-Masmoudi2017], the corresponding strong diffusive limit has remained outstanding except when the accommodation coefficient [Jiang-Masmoudi2017]. We establish global in time strong diffusive limit for all accommodation coefficients within strong solutions framework. The main novelties of our proof include: (1) a -stretching method for reduction to a single-bounce estimate; (2) a dissipation estimate for a carefully constructed rotating Maxwellian in the near-specular regime .

Paper Structure

This paper contains 25 sections, 43 theorems, 700 equations.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Strong Limit Result for the Case $\varepsilon\lesssim \alpha\leq 1$
  4. Methodology 1: Streaching Method for $L^\infty$ Estimate
  5. Strong Limit Result for the Case $0\leq \alpha \ll \varepsilon$
  6. Methodology 2: Dissipative Decomposition Mechanism
  7. Background and Progress
  8. Notations
  9. $L^\infty$ Estimate
  10. $L^\infty$ Estimate for the Semigroup
  11. $L^\infty$ Estimate for the Linear Equation
  12. Strong Limit for the Case $\varepsilon \lesssim \alpha \leq 1$
  13. Energy Estimate
  14. Macroscopic $L^2$ and $L^6$ Estimates
  15. Nonlinear Estimates
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $F_0 = \mu + \varepsilon \sqrt{\mu} f_0 \geq 0$. Then there exists $\varepsilon_0>0$ such that for all $0<\varepsilon<\varepsilon_0$, if the initial fluctuation satisfies for some small constant $\delta_0>0$ independent of $\varepsilon$, then the Boltzmann equation with Maxwell boundary condition F - Boltzmann equation admits a unique global strong solution $F = \mu + \varepsilon \sqrt{\mu}

Theorems & Definitions (83)

  • Theorem 1.1: Case $\varepsilon\lesssim \alpha\leq 1$
  • Proposition 1.2
  • Proposition 1.3
  • Theorem 1.4: Case $0\leq \alpha \ll \varepsilon$
  • Proposition 1.5
  • Lemma 2.1
  • proof : Proof.
  • Lemma 2.2: Single-bounce for specular trajectory
  • proof : Proof.
  • Lemma 2.3: No further bounce for diffuse trajectory
  • ...and 73 more