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Asymptotic behavior of large-amplitude solutions to the Boltzmann equation with soft interactions in $L^p_v L^\infty_x$ spaces

Jong-in Kim, Gyounghun Ko

Abstract

In this paper, we study the global well-posedness of the Boltzmann equation within the $L_{v}^{p}L_{x}^{\infty}$ framework for soft potential models with angular cutoff in a periodic box $\mathbb{T}^3$. By using a time-involved weight function, inspired by the works of [Liu-Yang,2017], [Duan-Yang-Zhao,2013], [Ko-Lee-Park,2022], we overcome the absence of a spectral gap. An analytical difficulty in the $L_v^p L_x^\infty$ setting is that the standard arguments used in [Ko-Lee-Park,2022], [Li,2022] for the nonlinear loss term are no longer applicable when dealing with time integration involving the collision frequency. To resolve this, we introduce a modified solution operator. Furthermore, we control the nonlinear gain term by deriving pointwise estimates bounded by $L_v^p$ and $L_v^\ell$ (for some $\ell <p$) norms. Thanks to the smallness of the initial relative entropy and Grönwall's inequality, we prove the global existence of unique solutions for large-amplitude initial data and obtain a sub-exponential convergence rate toward equilibrium.

Asymptotic behavior of large-amplitude solutions to the Boltzmann equation with soft interactions in $L^p_v L^\infty_x$ spaces

Abstract

In this paper, we study the global well-posedness of the Boltzmann equation within the framework for soft potential models with angular cutoff in a periodic box . By using a time-involved weight function, inspired by the works of [Liu-Yang,2017], [Duan-Yang-Zhao,2013], [Ko-Lee-Park,2022], we overcome the absence of a spectral gap. An analytical difficulty in the setting is that the standard arguments used in [Ko-Lee-Park,2022], [Li,2022] for the nonlinear loss term are no longer applicable when dealing with time integration involving the collision frequency. To resolve this, we introduce a modified solution operator. Furthermore, we control the nonlinear gain term by deriving pointwise estimates bounded by and (for some ) norms. Thanks to the smallness of the initial relative entropy and Grönwall's inequality, we prove the global existence of unique solutions for large-amplitude initial data and obtain a sub-exponential convergence rate toward equilibrium.
Paper Structure (20 sections, 25 theorems, 338 equations)

This paper contains 20 sections, 25 theorems, 338 equations.

Table of Contents

  1. Introduction
  2. Boltzmann equation
  3. Notations
  4. Perturbation framework near Maxwellian
  5. Main results
  6. Arrangement of the paper
  7. Preliminaries
  8. Estimates on linear operators
  9. Estimates on nonlinear operators
  10. Relative entropy
  11. Small perturbation problem in $L^p_vL^\infty_x$
  12. $L^2$ estimate
  13. $R(f)$ estimate
  14. $L^p_vL^\infty_x$ estimate
  15. Global Existence and Sub-exponential decay
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $p$ and $\beta$ satisfy the condition Assume that $F_0(x,v) = \mu(v) + \mu^{1/2}(v) f_0(x,v) \ge 0$ satisfying Then there is $\eta_0 \ll 1$ so that there exists a constant $\varepsilon_1 = \varepsilon_1(\eta_0) >0$, depending only on $\eta_0$, such that if then there exists a unique solution $F(t,x,v) =\mu(v) + \mu^{1/2}(v) f(t,x,v) \ge 0$ to the Boltzmann equation Boltzmanneq with initia

Theorems & Definitions (44)

  • Theorem 1.1: Small perturbation problem
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.2: Large amplitude problem
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 34 more