Time periodic solutions for the non-isentropic compressible quantum hydrodynamic equations with viscosity in R^3

Abstract

This article concerns the existence and uniqueness of a time periodic solution for the non-isentropic quantum hydrodynamic equations with viscosity. By applying the Leray-Schauder theory, subtle energy estimates and a limiting method, we obtain the existence of time periodic solutions under some smallness assumptions on the time periodic external force in $\mathbb{R}^3$. The uniqueness can be proved by similar energy estimates. In particular, the quantum effects and the energy equation are taken into account in this paper which play a significant role in the uniform (in the domain R and the positive constant $\epsilon$) estimates, especially in the selection of the norm.

Keywords

• time periodic solutions
• uniform energy estimates
• full quantum hydrodynamic equations with viscosity
• leray-schauder degree theory