Hypocoercivity for the Linear Semiconductor Boltzmann Equation with Boundaries and Uncertainties

TL;DR

This paper proves exponential convergence to equilibrium for the semiconductor Boltzmann equation with boundaries and uncertainties by constructing a modified entropy functional, extending hypocoercivity techniques to uncertain models.

Contribution

It introduces a hypocoercivity framework for the semiconductor Boltzmann equation with boundary conditions and uncertainties, including a new entropy functional and regularity analysis.

Findings

Exponential decay to equilibrium established

Entropy functional is equivalent to a weighted norm

Method extended to models with uncertainties

Abstract

In this paper, we establish hypocoercivity for the semiconductor Boltzmann equation with the presence of an external electrical potential under the Maxwell boundary condition. We will construct a modified entropy Lyapunov functional, which is proved to be equivalent to some weighted norm of the corresponding function space. We then show that the entropy functional dissipates along the solutions, and the exponential decay to the equilibrium state of the system follows by a Gronwall type inequality. We also generalize our arguments to situations where uncertainties in our model arise,and the hypocoercivity method we have established is adopted to analyze the regularity of the solutions along the random space.