The volume-filling property of semiconductor model with cross-diffusion phenomenon

TL;DR

This paper proves that semiconductor models with cross-diffusion are volume-filling when initial conditions are bounded, using a regularized entropy method to establish global existence and boundedness of solutions.

Contribution

It introduces a regularized entropy approach to demonstrate the volume-filling property of semiconductor models with cross-diffusion.

Findings

Bounded initial data leads to bounded weak solutions.

The regularized entropy ensures positive semi-definiteness of the product matrix.

Semiconductor models are classified as volume-filling cross-diffusion systems.

Abstract

Semiconductor model is a system of parabolic partial differential equations with cross-diffusion phenomenon. Previous results showed that a weak solution exists and is not bounded in general. So semiconductor model was categorized as a cross-diffusion system without volume-filling. In this work, we show that once the initial value is bounded, there exists a weak solution that is also bounded. This result indicates that semiconductor model is also a volume-filling cross-diffusion system. The entropy method is a major tool in global existence analysis of cross-diffusion systems. We notice that traditional entropies in volume-filling cases may not provide required positive semi-definiteness result for the existence proof. In this situation, we regularize the entropy and diffusion matrix simultaneously. The product between Hessian matrix of the regularized entropy and regularized diffusion matrix is positive semi-definite, then we apply the entropy method to show volume-filling property of semiconductor model.