Electronic Journal of Differential Equations, 
Vol. 2020 (2020), No. 103, pp. 1-34.

Title: Time periodic solutions for the non-isentropic compressible quantum hydrodynamic
       equations with viscosity in R^3
       
Author: Min Li (Shanxi Univ. of Finance and Economics, Taiyuan, China)

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.
 
Submitted March 29, 2020. Published October 02, 2020.
Math Subject Classifications: 47H11, 35B10, 76Y05, 35Q35, 35G25, 76N10.
Key Words: Time periodic solutions; uniform energy estimates;
           full quantum hydrodynamic equations with viscosity;
           Leray-Schauder degree theory.