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Global Diffusive Expansion of Boltzmann Equation in exterior Domain

Junhwa Jung

TL;DR

The paper addresses the hydrodynamic limit of the Boltzmann equation in exterior domains with diffusive boundary, proving a global diffusive expansion to the incompressible Navier–Stokes–Fourier system. It introduces a novel $L^2-L^3-L^6$ splitting of the macroscopic part to extend the $L^2-L^\infty$ framework to unbounded domains, circumventing the failure of Poincaré’s inequality. The main result constructs a global solution $F=\mu + \varepsilon\sqrt{\mu}(f_1 + \varepsilon f_2+ \varepsilon^{1/2}R)$ with detailed decay and regularity bounds on $f_1,f_2,R$, and establishes energy and $L^\infty$ estimates that enable a contraction mapping argument. This work extends the Guo framework to exterior domains and provides a rigorous justification of the diffusive limit in unbounded settings, with potential implications for kinetic-fluid coupling in exterior geometries. $

Abstract

The study of flows over an obstacle is one of the fundamental problems in fluids. In this paper we establish the global validity of the diffusive limit for the Boltzmann equations to the Navier-Stokes-Fourier system in an exterior domain. To overcome the well-known difficulty of the lack of Poincare's inequality in unbounded domain, we develop a new $L^2-L^3-L^6$ splitting to extend $L^2-L^\infty$ framework into the unbounded domain.

Global Diffusive Expansion of Boltzmann Equation in exterior Domain

TL;DR

The paper addresses the hydrodynamic limit of the Boltzmann equation in exterior domains with diffusive boundary, proving a global diffusive expansion to the incompressible Navier–Stokes–Fourier system. It introduces a novel splitting of the macroscopic part to extend the framework to unbounded domains, circumventing the failure of Poincaré’s inequality. The main result constructs a global solution with detailed decay and regularity bounds on , and establishes energy and estimates that enable a contraction mapping argument. This work extends the Guo framework to exterior domains and provides a rigorous justification of the diffusive limit in unbounded settings, with potential implications for kinetic-fluid coupling in exterior geometries. $

Abstract

The study of flows over an obstacle is one of the fundamental problems in fluids. In this paper we establish the global validity of the diffusive limit for the Boltzmann equations to the Navier-Stokes-Fourier system in an exterior domain. To overcome the well-known difficulty of the lack of Poincare's inequality in unbounded domain, we develop a new splitting to extend framework into the unbounded domain.
Paper Structure (15 sections, 22 theorems, 302 equations)

This paper contains 15 sections, 22 theorems, 302 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Basic Notation
  4. Boundary condition
  5. Main result
  6. Mathematical difficulty
  7. Historical background
  8. Estimation of the Fluid Equations
  9. Estimation of the fluid terms in the remainder of the Boltzmann equation
  10. Time decay of the fluid equations
  11. Energy estimate
  12. $L^{\infty}$ Estimate
  13. $L^2$- $L^3$ - $L^6$ Splitting
  14. Nonlinear Estimate
  15. Results

Key Result

Theorem 1.2

\newlabelmainthm0 Let $\Omega$ be a $C^2$ bounded open set of $\mathbb{R}^3$ and $\Omega^c = \mathbb{R}^3 \backslash \bar{\Omega}$. For any $0< \varepsilon \ll 1$ consider the boundary value problem Suppose the initial datum takes the form of $F_0 = \mu + \varepsilon \sqrt{\mu} f_0 \ge 0$ such that where satisfies where $\omega(v) = e^{\beta \left\lvert v\right\rvert^2}$ with $0 < \beta \ll 1

Theorems & Definitions (55)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Definition 1.8
  • Remark 1.9
  • Proposition 2.1
  • ...and 45 more