In this chapter we introduce some basic notation and definitions. We discuss a few general results on interpolation spaces. The most important one is the Aronszajn-Gagliardo theorem.

General Properties of Interpolation Spaces

In this paper, we mainly review recent results on mathematical theory and
numerical methods for Bose-Einstein condensation (BEC),
based on the Gross-Pitaevskii equation (GPE). Starting from the simplest case with one-component BEC of the weakly interacting bosons, we study the reduction of GPE to lower dimensions, the ground states of BEC including the existence and uniqueness as well as nonexistence results, and the dynamics of GPE including dynamical laws, well-posedness of the Cauchy problem as well as the finite time blow-up. To compute the ground state, the gradient flow with discrete normalization (or imaginary time) method is reviewed and various full discretization methods are presented and compared. To simulate the dynamics, both finite difference methods and time splitting spectral methods are reviewed, and their error estimates are briefly outlined. When the GPE has symmetric properties, we show how to simplify the numerical methods. Then we compare two widely used scalings, i.e. physical scaling (commonly used) and semiclassical scaling, for BEC in strong repulsive interaction regime (Thomas-Fermi regime), and discuss semiclassical limits of the GPE. Extensions of these results for one-component BEC are then
carried out for rotating BEC by GPE with an angular momentum rotation, dipolar BEC by GPE with long range dipole-dipole interaction, and two-component BEC by coupled GPEs. Finally, as a perspective, we show briefly the mathematical models for
spin-1 BEC, Bogoliubov excitation and BEC at finite temperature.

http://osada.csp.escience.cn/dct/attach/Y2xiOmNsYjpwZGY6NjI5NzY=

Mathematical theory and numerical methods for Bose-Einstein condensation

In this paper, we mainly review recent results on mathematical theory and numerical methods for Bose-Einstein condensation (BEC), based on the Gross-Pitaevskii equation (GPE). Starting from the simplest case with one-component BEC of the weakly interacting bosons, we study the reduction of GPE to lower dimensions, the ground states of BEC including the existence and uniqueness as well as nonexistence results, and the dynamics of GPE including dynamical laws, well-posedness of the Cauchy problem as well as the finite time blow-up. To compute the ground state, the gradient flow with discrete normalization (or imaginary time) method is reviewed and various full discretization methods are presented and compared. To simulate the dynamics, both finite difference methods and time splitting spectral methods are reviewed, and their error estimates are briefly outlined. When the GPE has symmetric properties, we show how to simplify the numerical methods. Then we compare two widely used scalings, i.e. physical scaling (commonly used) and semiclassical scaling, for BEC in strong repulsive interaction regime (Thomas-Fermi regime), and discuss semiclassical limits of the GPE. Extensions of these results for one-component BEC are then carried out for rotating BEC by GPE with an angular momentum rotation, dipolar BEC by GPE with long range dipole-dipole interaction, and two-component BEC by coupled GPEs. Finally, as a perspective, we show briefly the mathematical models for spin-1 BEC, Bogoliubov excitation and BEC at finite temperature.

In this paper we study the existence of global solutions to the Cauchy problem of nonlinear SchrSdinger equation by establishing time weight function spaces and using the contraction mapping principle.

Global solutions of nonlinear schrodinger equations

In this paper, we mainly review recent results on mathematical theory and numerical methods for Bose-Einstein condensation (BEC), based on the Gross-Pitaevskii equation (GPE). Starting from the simplest case with one-component BEC of the weakly interacting bosons, we study the reduc- tion of GPE to lower dimensions, the ground states of BEC including the existence and uniqueness as well as nonexistence results, and the dynamics of GPE including dynamical laws, well-posedness of the Cauchy problem as well as the nite time blow-up. To compute the ground state, the gradient ow with discrete normalization (or imaginary time) method is reviewed and vari- ous full discretization methods are presented and compared. To simulate the dynamics, both nite dierence methods and time splitting spectral methods are reviewed, and their error estimates are briey outlined. When the GPE has symmetric properties, we show how to simplify the numerical methods. Then we compare two widely used scalings, i.e. physical scaling (commonly used) and semiclassical scaling, for BEC in strong repulsive interaction regime ( Thomas-Fermi regime), and discuss semiclassical limits of the GPE. Extensions of these results for one-component BEC are then carried out for rotating BEC by GPE with an angular momentum rotation, dipolar BEC by GPE with long range dipole-dipole interaction, and two-component BEC by coupled GPEs. Finally, as a perspective, we show briey the mathematical models for spin-1 BEC, Bogoliubov excitation and BEC at nite temperature.

Mathematical theory and numerical methods for

Fourier Multiplier, Function Spaces Navier-Stokes Equation Strichartz Estimates for Linear Dispersive Equations Local and Global Wellposedness for Nonlinear Dispersive Equations The Low Regularity Theory for the Nonlinear Dispersive Equations Frequency-Uniform Decomposition Method Conservations, Morawetz' Inequalities of NLS Boltzmann Equation without Angular Cutoff.

Harmonic Analysis Method for Nonlinear Evolution Equations, I

Global existence for compressible Navier–Stokes–Poisson equations in three and higher dimensions

http://www.math.ac.cn/kyry/hcc/pub/papers/201409/P020150618516553813608.pdf

Wellposedness for the fourth order nonlinear Schrödinger equations

The Cauchy problem of a multi-dimensional ($d\geqslant 2$) compressible viscous liquid-gas two-phase flow model is concerned in this paper We investigate the global existence and uniqueness of the strong solution for the initial data close to a stable equilibrium and the local in time existence and uniqueness of the solution with general initial data in the framework of Besov spaces A continuation criterion is also obtained for the local solution

/pdf/well-posedness-for-a-multi-dimensional-viscous-liquid-gas-y9i79p29p7.pdf

Well-posedness for a multi-dimensional viscous liquid-gas two-phase flow model

In the present paper, we prove the a priori estimates of Sobolev norms for a free boundary problem of the incompressible inviscid magnetohydrodynamics equations in all physical spatial dimensions n = 2 and 3 by adopting a geometrical point of view used in Christodoulou and Lindblad (Commun Pure Appl Math 53:1536‐1602, 2000), and estimating quantities such as the second fundamental form and the velocity of the free surface. We identify the well-posedness condition thattheouternormalderivativeofthetotalpressureincludingthefluidandmagnetic pressuresisnegativeonthefreeboundary,whichissimilartothephysicalcondition (Taylor sign condition) for the incompressible Euler equations of fluids.

/pdf/a-priori-estimates-for-free-boundary-problem-of-2343cqb631.pdf

Chengchun Hao

Papers

Harmonic Analysis Method for Nonlinear Evolution Equations, I

Global existence for compressible Navier–Stokes–Poisson equations in three and higher dimensions

Wellposedness for the fourth order nonlinear Schrödinger equations

Well-posedness for a multi-dimensional viscous liquid-gas two-phase flow model

A Priori Estimates for Free Boundary Problem of Incompressible Inviscid Magnetohydrodynamic Flows