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arXiv · 2203.14311

Bounded weak solutions to cross-diffusion semiconductor model with electron-hole scattering

Xi Lin
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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 bounded weak solutions. In this work, we show that once the initial value is bounded, there exists a weak solution that is also bounded. 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, a transformation of variables technique has been applied. The product between Hessian matrix of the entropy and replacement diffusion matrix is positive semi-definite, then we apply the entropy method to show semiconductor model has a bounded weak solution.

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Xi Lin. 2026-09-14. Bounded weak solutions to cross-diffusion semiconductor model with electron-hole scattering. https://arxiv.org/abs/2203.14311

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