Asymptotic expansion of solutions to the drift-diffusion equation with fractional dissipation

TL;DR

This paper investigates the long-term behavior of solutions to a drift-diffusion equation with fractional dissipation, deriving high-order asymptotic expansions especially for cases with small fractional Laplacian exponents.

Contribution

It provides the first detailed asymptotic expansion of solutions for small fractional Laplacian exponents, extending understanding beyond large exponent cases.

Findings

Derived high-order asymptotic expansions for solutions.

Analyzed the behavior of solutions with small fractional Laplacian exponents.

Extended the theoretical framework for drift-diffusion equations with fractional dissipation.

Abstract

The initial-value problem for the drift-diffusion equation arising from the model of semiconductor device simulations is studied. The dissipation on this equation is given by the fractional Laplacian. When the exponent of the fractional Laplacian is large, large-time behavior of solutions is known. However, when the exponent is small, the perturbation methods used in the preceding works would not work. Large-time behavior of solutions to the drift-diffusion equation with small exponent is discussed. Particularly, the asymptotic expansion of solutions with high-order is derived.