Table of Contents

Chapter 1 The derivation and mathematical models of quantum hydrodynamic equations
1．1 Isentropic quantum hydrodynamic model
1．2 Non-isentropic quantum hydrodynamic model
1．2．1 Wigner-BGK equation
1．2．2 Non-local momentum equation
1．2．3 Calculation of S1
1．2．4 Calculation of S2
1．2．5 Energy and entropy estimates
1．3 Quantum electron-magnetic model in plasma
1．4 Bipolar quantum hydrodynamic model
1．5 Some plasma equations with quantum effect
1．5．1 Quantum KdV equation
1．5．2 Quantum Zakharov equation

Chapter 2 Global existence of weak solutions to the compressible quantum hydrodynamic equations
2．1 Global existence of weak solutions to one dimensional compressible quantum hydrodynamic equations
2，1．1 Faedo-Galerkin approximation
2．1．2 Existence of the approximate solutions
2．1．3 Existence of weak solutions
2．1．4 Vanishing viscosity limit ε→0
2．2 Global existence of weak solutions to high dimensional compressible quantum hydrodynamic equations
2．2．1 Faedo-Galerkin approximation
2．2．2 A priori estimate
2．2．3 Limit n→∞
2．2．4 Limit δ→0
2．3 Global existence of weak solutions to the compressible quantum hydrodynamic equations with cold pressure
2．3．1 A priori estimate
2．3．2 Global existence of weak solutions
2．3．3 Planck limit

Chapter 3 Existence of finite energy weak solutions of inviseid quantum hydrodynamic equations
3．1 Introduction and main result
3．2 Preliminaries and notations
3．2．1 Notations
3．2．2 Non-linear Schr6dinger equation
3．2．3 Compactness tools
3．2．4 Tools in two-dimension
3．3 Polar decomposition
3．4 Quantum hydrodynamic equations without collision term
3．5 Fractional step method： definition and uniformity
3．6 A priori estimate and convergence
3．7 Further generalization
3．7．1 Case with impurity distribution
3．7．2 Two dimensional case

Chapter 4 Non-isentropic quantum Navier-Stokes equations with cold pressure
4．1 Preliminaries and main result
4．1．1 Preliminaries
4．1．2 Main result
4．2 Approximation
4．3．1 Continuity equation
4．3．2 Internal energy equation
4．3．3 Fixed-point method
4．3．4 A uniform priori estimate and global existence of approximate equations
4．3．5 Entropy estimate
4．3．6 Global existence of first level approximate equations
4．4 Faedo-Galerkin limit
4．4．1 A uniform priori estimate with respect to N
4．4．2 Limit N→∞
4．4．3 Strong convergence of the density and passage to the limit in the continuity equation
4．4．4 Strong convergence of the temperature
4．4．5 Passage to the limit in the internal energy balance equation
4．4．6 Passage to the limit in the total energy balance equation
4．5 B-D entropy inequality
4．6 Artificial viscosity limit ε→0，λ→0
4．6．1 Limit ε→0
4．6．2 Limit λ→0

Chapter 5 Boundary problem of compressible quantum Euler-Poisson equations
5．1 Boundary problem for compressible stationary quantum Euler-Poisson equations
5．1．1 Existence of solutions if h > 0， v > 0
5．1．2 Existence of small solutions for the isothermal equations if h > 0，v = 0
5．1．3 Non-existence of large solutions for the isentropic equations if ε > 0， v=0
5．1．4 Uniqueness of solutions for the isentropic equations if h > 0， v=0
5．1．5 High-dimensional third-order equations
5．2 Initial boundary value problem for compressible quantum Euler-Poisson equations
5．2．1 Existence and uniqueness of the stationary solution
5．2．2 Asymptotic stability of the stationary solution
5．2．3 A priori estimate
5．2．4 Semiclassical limit

Chapter 6 Asymptotic limit to the bipolar quantum hydrodynamic equations
6．1 Semiclassical limit
6．1．1 Main results
6．1．2 Preliminaries
6．1．3 Proof of main results
6．2 The relaxation time limit
6．3 Quasineutral limit
6．4 Time decay
References