Time-periodic solutions for the full quantum Euler equation. (English) Zbl 07442108

Li, Min; Yao, Xianzhong

Math. Methods Appl. Sci. 44, No. 17, 13146-13169 (2021).

Summary: In this paper, we establish the existence and uniqueness of a time-periodic solution to the full compressible quantum Euler equations. First, we prove the existence of time-periodic solutions under some smallness assumptions imposed on the external force in a periodic domain by a regularized approximation scheme and the Leray-Schauder degree theory. Then the result is generalized to \(\mathbb{R}^3\) by adapting a limiting method and a diagonal argument. The uniqueness of the time-periodic solutions is also given. Compared to classical Euler equations, the third-order quantum spatial derivatives are considered which need some elaborated treatments thereof in obtaining the highest-order energy estimates.

MSC:

47H11	Degree theory for nonlinear operators
35B10	Periodic solutions to PDEs
76Y05	Quantum hydrodynamics and relativistic hydrodynamics
35Q35	PDEs in connection with fluid mechanics
35G25	Initial value problems for nonlinear higher-order PDEs
76N10	Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics

Keywords:

full quantum Euler equations; Leray-Schauder degree theory; time-periodic solutions