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Global Stability of the Boltzmann Equation for a Polyatomic Gas with Initial Data Allowing Large Oscillations

Gyounghun Ko, Sung-jun Son

TL;DR

$F(v,I)=M(v,I)+\sqrt{M(v,I)}\,f(t,x,v,I)$ is the perturbation framework for the polyatomic Boltzmann equation on $\mathbb{T}^3$; the paper proves global well-posedness of a mild solution with an $L^{\infty}$ velocity-weight bound and small initial relative entropy, and establishes exponential convergence to the Maxwellian. A key technical advance is a pointwise bound on the gain term $\Gamma_+(f,f)$, which, together with a new operator $\mathcal{R}(f)$ that absorbs the nonlinear loss, enables a Grönwall-type argument via double Duhamel. Relative entropy $\mathcal{E}(F)$ provides a quantitative measure of proximity to equilibrium and is nonincreasing, supplying a priori control that closes the estimates for large-amplitude data (in $L^{\infty}$ after weight). Overall, the work extends global well-posedness and exponential relaxation results to polyatomic gases with large oscillations in the initial data, under a small relative-entropy constraint, using a refined kinetic-entropy framework and precise nonlinear gain-term bounds.

Abstract

In this paper, we consider the Boltzmann equation for a polyatomic gas. We establish that the mild solution to the Boltzmann equation on the torus is globally well-posed, provided the initial data that satisfy bounded velocity-weighted $L^{\infty}$ norm and the smallness condition on the initial relative entropy. Furthermore, we also study the asymptotic behavior of solutions, converging to the global Maxwellian with an exponential rate. A key point in the proof is to develop the pointwise estimate on the gain term of non-linear collision operator for Grönwall's argument.

Global Stability of the Boltzmann Equation for a Polyatomic Gas with Initial Data Allowing Large Oscillations

TL;DR

is the perturbation framework for the polyatomic Boltzmann equation on ; the paper proves global well-posedness of a mild solution with an velocity-weight bound and small initial relative entropy, and establishes exponential convergence to the Maxwellian. A key technical advance is a pointwise bound on the gain term , which, together with a new operator that absorbs the nonlinear loss, enables a Grönwall-type argument via double Duhamel. Relative entropy provides a quantitative measure of proximity to equilibrium and is nonincreasing, supplying a priori control that closes the estimates for large-amplitude data (in after weight). Overall, the work extends global well-posedness and exponential relaxation results to polyatomic gases with large oscillations in the initial data, under a small relative-entropy constraint, using a refined kinetic-entropy framework and precise nonlinear gain-term bounds.

Abstract

In this paper, we consider the Boltzmann equation for a polyatomic gas. We establish that the mild solution to the Boltzmann equation on the torus is globally well-posed, provided the initial data that satisfy bounded velocity-weighted norm and the smallness condition on the initial relative entropy. Furthermore, we also study the asymptotic behavior of solutions, converging to the global Maxwellian with an exponential rate. A key point in the proof is to develop the pointwise estimate on the gain term of non-linear collision operator for Grönwall's argument.
Paper Structure (8 sections, 6 theorems, 101 equations, 2 figures)

This paper contains 8 sections, 6 theorems, 101 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results and scheme of the proof
  3. Preliminaries
  4. The operator $K$
  5. Pointwise estimate on non-linear terms
  6. Relative entropy
  7. A priori estimate
  8. $\mathcal{R}(f)$ estimate

Key Result

Theorem 1.1

Define the velocity weight function Then, for any $M_0>0$, there exists $\varepsilon = \varepsilon(M_0)>0$ such that if initial data satisfy that $F_0 (x,v,I) = M(v,I)+ \sqrt{M(v,I)} f_0 (x,v,I) \geq 0$, the conservation laws laws with $(M_I,J_I,E_I)=(0,0,0)$, and then the initial value problem poly be on the polyatmoic Boltzmann equation admits a unique global-in-time mild solution $F(t,x,v,I)=

Figures (2)

  • Figure 1: This figure describes the changing ratio of kinetic energy to internal energy for two polyatomic molecules after a collision. To aid readers' understanding, the phenomenon of increasing internal energy and decreasing kinetic energy ($R_1 < R_2 < R_3$) is illustrated purely from the perspective of kinetic energy.
  • Figure 2: Representation of variables

Theorems & Definitions (11)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • ...and 1 more