Electron. J. Differential Equations, Vol. 2020 (2020), No. 103, pp. 1-34.

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

Min Li

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 2, 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.
DOI: 10.58997/ejde.2020.103

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Min Li
Faculty of Applied Mathematics
Shanxi University of Finance and Economics
Taiyuan 030006, China
email: minlisxcj@163.com

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