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Knudsen boundary layer equations with incoming boundary condition: full range of cutoff collision kernels and Mach numbers of the far field

Ning Jiang, Yi-Long Luo, Yulong Wu, Tong Yang

TL;DR

The paper analyzes the nonlinear Knudsen boundary layer for the Boltzmann equation in a half-space with incoming boundary data, encompassing the full range of angular cutoff kernels ($-3<\gamma\leq 1$) and all Mach numbers $\mathcal{M}^\infty$ of the far field. It introduces a novel $(x,v)$-mixed weight $\sigma$ and an artificial damping mechanism to overcome limitations of the linearized operator's null space, yielding uniform $L^\infty_{x,v}$ and $L^2_{x,v}$ bounds and exponential decay in $x$ and $v$ for the Knudsen layer. A linear theory is developed with solvability conditions that depend on $\mathcal{M}^\infty$ (codimension matching a table of cases), and this extends to a nonlinear theory proving existence and uniqueness for small data in a consistent $\mathbb{VSS}$ framework. The results advance the understanding of hydrodynamic limits with incoming boundary conditions and provide a robust toolkit for analyzing kinetic boundary layers across the full parameter range.

Abstract

This paper establishes tahe existence and uniqueness of the nonlinear Knudsen layer equation with incoming boundary conditions. It is well-known that the solvability conditions of the problem vary with the Mach number of the far Maxwellian $\mathcal{M}^\infty$. We consider full ranges of cutoff collision kernels (i.e., $- 3 < γ\leq 1$) and all the Mach numbers of the far field in the $L^\infty_{x,v}$ framework. Additionally, the solution exhibits exponential decay $\exp \{- c x^\frac{2}{3 - γ} - c |v|^2 \}$ for some $c > 0$. To address the general angular cutoff collision kernel, we introduce a $(x,v)$-mixed weight $σ$. The proof is essentially bsed on adding an artificial damping term.

Knudsen boundary layer equations with incoming boundary condition: full range of cutoff collision kernels and Mach numbers of the far field

TL;DR

The paper analyzes the nonlinear Knudsen boundary layer for the Boltzmann equation in a half-space with incoming boundary data, encompassing the full range of angular cutoff kernels () and all Mach numbers of the far field. It introduces a novel -mixed weight and an artificial damping mechanism to overcome limitations of the linearized operator's null space, yielding uniform and bounds and exponential decay in and for the Knudsen layer. A linear theory is developed with solvability conditions that depend on (codimension matching a table of cases), and this extends to a nonlinear theory proving existence and uniqueness for small data in a consistent framework. The results advance the understanding of hydrodynamic limits with incoming boundary conditions and provide a robust toolkit for analyzing kinetic boundary layers across the full parameter range.

Abstract

This paper establishes tahe existence and uniqueness of the nonlinear Knudsen layer equation with incoming boundary conditions. It is well-known that the solvability conditions of the problem vary with the Mach number of the far Maxwellian . We consider full ranges of cutoff collision kernels (i.e., ) and all the Mach numbers of the far field in the framework. Additionally, the solution exhibits exponential decay for some . To address the general angular cutoff collision kernel, we introduce a -mixed weight . The proof is essentially bsed on adding an artificial damping term.
Paper Structure (33 sections, 19 theorems, 296 equations, 1 figure, 3 tables)

This paper contains 33 sections, 19 theorems, 296 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction and main results
  2. The description of the problem
  3. Overdetermined far-field condition $\lim_{x \to + \infty} f (x, v) = 0$
  4. Toolbox
  5. Notations
  6. Functions spaces
  7. Energy functionals
  8. Main results
  9. Linear problem \ref{['KL']}
  10. Application to nonlinear problem
  11. Outline of existence of the solutions to the system $\ref{['KL']}$
  12. Methodology and novelties
  13. Historical remarks
  14. Incoming and Maxwell reflection boundary conditions
  15. Organization of current paper
  16. ...and 18 more sections

Key Result

Lemma 1.1

(Lemma 2.3 of Golse-2008-BIMA) The projections $\mathbb{P}$ and $\mathcal{P}^0$ satisfy the relations Moreover, $(\mathcal{P}^0 f, f ) \geq 0.$

Figures (1)

  • Figure 1: Derivation of uniform bounds for the approximate equation \ref{['A1']}. Here we denote by $g_\sigma = e^{\hbar \sigma} g$.

Theorems & Definitions (37)

  • Lemma 1.1
  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2: Exponential decay
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • ...and 27 more