Hypocoercivity for the linear Boltzmann equation with confining forces

TL;DR

This paper establishes exponential convergence to equilibrium for solutions of the linear Boltzmann equation with confining forces, under certain initial data and potential conditions, extending previous results to more general forces.

Contribution

It proves hypocoercivity and exponential decay rates for the linear Boltzmann equation with a broad class of confining potentials, including non-parabolic cases.

Findings

Exponential convergence to steady state under specified conditions.

Extension of hypocoercivity results to non-parabolic confining potentials.

Applicable to initial data satisfying conservation laws.

Abstract

This paper is concerned with the hypercoercivity property of solutions to the Cauchy problem on the linear Boltzmann equation with a confining potential force. We obtain the exponential time rate of solutions converging to the steady state under some conditions on both initial data and the potential function. Specifically, initial data is properly chosen such that the conservation laws of mass, total energy and possible partial angular momentums are satisfied for all nonnegative time, and a large class of potentials including some polynomials are allowed. The result also extends the case of parabolic forces considered in [4] to the non-parabolic general case here.