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

TL;DR

This paper proves the global stability and exponential convergence to equilibrium of solutions to the Boltzmann equation for polyatomic gases with large initial oscillations, under certain boundedness and entropy conditions.

Contribution

It establishes the global well-posedness and asymptotic stability of solutions for the Boltzmann equation for polyatomic gases with large initial oscillations.

Findings

Global solutions exist under bounded velocity-weighted norms.

Solutions converge exponentially to the Maxwellian equilibrium.

Developed pointwise estimates for the collision operator's gain term.

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 Gr\"{o}nwall's argument.