On steady states for the Vlasov-Schr\"odinger-Poisson system

TL;DR

This paper constructs and analyzes a class of steady states for the Vlasov-Schr"odinger-Poisson system, revealing their structure, stability, and uniqueness using advanced mathematical techniques.

Contribution

It introduces a method to construct and analyze steady states for a hybrid kinetic-quantum model, addressing challenges from lack of compactness.

Findings

Existence of 2D kinetic/1D quantum steady states

Finite subband structure and monotonicity established

Conditional dynamical stability demonstrated

Abstract

The Vlasov-Schr\"odinger-Poisson system is a kinetic-quantum hybrid model describing quasi-lower dimensional electron gases. For this system, we construct a large class of 2D kinetic/1D quantum steady states in a bounded domain as generalized free energy minimizers, and we show their finite subband structure, monotonicity, uniqueness and conditional dynamical stability. Our proof is based on the concentration-compactness principle, but some additional difficulties arise due to lack of compactness originated from the hybrid nature (see Remark 1.9). To overcome the difficulties, we introduce a 3-step refinement of a minimizing sequence by rearrangement and partial minimization problems, and the coercivity lemma for the free energy (Lemma 5.3) is crucially employed.