Existence and stability of stationary solutions to the compressible quantum model

TL;DR

This paper investigates the existence, uniqueness, and nonlinear stability of stationary solutions to a three-dimensional compressible quantum model, providing decay rates under certain initial conditions.

Contribution

It establishes the existence, uniqueness, and nonlinear stability of stationary solutions for the compressible quantum model using weighted $L^2$, $L^ abla$ estimates, and energy methods, which is a novel comprehensive analysis.

Findings

Existence and uniqueness of stationary solutions proven.

Nonlinear stability of solutions demonstrated.

Decay rates of solutions established for specific initial perturbations.

Abstract

In this paper, the compressible quantum model with the given mass source and the external force of general form in three-dimensional whole space is considered. Based on the weighted $L^2$ method and $L^\infty$ estimates, the existence and uniqueness of stationary solutions can be obtained by the contraction mapping principle. By using a general energy method, the nonlinear stability of stationary solutions is studied, and the time decay rates of the solutions are established when the initial perturbation belongs to $\dot{H}^{-s}$ with $0 \leq s < \frac{3}{2}$.