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

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
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