Quantum Hydrodynamic Equation and Its Mathematical Theory

Quantum hydrodynamics comes from superfluid, superconductivity, semiconductor and so on. Quantum hydrodynamic model describes Helium II superfluid, Bose-Einstein condensation in inert gas, dissipative perturbation of Hamilton-Jacobi system, amplitude and dissipative perturbation of Eikonal quantum wave and so on. Owing to the broad application of quantum hydrodynamic equations, the study of the quantum hydrodynamic equations has aroused the concern of more and more scholars. Based on the above facts, we collected and collated the data of quantum hydrodynamic equations, and studied the concerning mathematical problems.

The main contents of this book are: the derivation and mathematical models of quantum hydrodynamic equations, global existence of weak solutions to the compressible quantum hydrodynamic equations, existence of finite energy weak solutions of inviscid quantum hydrodynamic equations, non-isentropic quantum Navier-Stokes equations with cold pressure, boundary problem of compressible quantum Euler-Poisson equations, asymptotic limit to the bipolar quantum hydrodynamic equations.

量子流體力學源於超流體、超導體、半導體等領域。量子流體力學模型描述了氦II超流體、慣性氣體中的玻色-愛因斯坦凝聚、哈密頓-雅可比系統的耗散擾動、Eikonal量子波的振幅和耗散擾動等。由於量子流體力學方程的廣泛應用，越來越多的學者關注量子流體力學方程的研究。基於上述事實，我們收集和整理了量子流體力學方程的數據，並研究了相關的數學問題。

本書的主要內容包括：量子流體力學方程的推導和數學模型、可壓縮量子流體力學方程的弱解的全局存在性、無黏量子流體力學方程的有限能量弱解的存在性、帶有冷壓力的非等熵量子納維爾-斯托克斯方程、可壓縮量子歐拉-泊松方程的邊界問題、雙極量子流體力學方程的漸近極限。