Chengchun Hao

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

The compressible Navier–Stokes–Poisson system is concerned in the present paper, and the global existence and uniqueness of the strong solution is shown in the framework of hybrid Besov spaces in three and higher dimensions.

Well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in multi-dimensional spaces

We study the well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations i∂t u = −ε u + 2u + P (( ∂ x u ) |α| 2, ( ∂ x ū ) |α| 2 ) , t ∈ R, x ∈ R, where ε ∈ {−1,0,1}, n 2 denotes the spatial dimension and P(·) is a polynomial excluding constant and linear terms.

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

The Cauchy problem of a multi-dimensional (

Global well posedness for the Gross-Pitaevskii equation with an angular momentum rotational term in three dimensions

d\geqslant 2

Global well posedness for the Gross-Pitaevskii equation with an angular momentum rotational term

) 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.

A priori estimates for the free boundary problem of incompressible neo-Hookean elastodynamics

In this paper, we establish the global well posedness of the Cauchy problem for the Gross-Pitaevskii equation with an angular momentum rotational term in which the angular velocity is equal to the isotropic trapping frequency in the space R3.

On the Motion of Free Interface in Ideal Incompressible MHD

In this paper, we establish the global well posedness of the Cauchy problem for the Gross–Pitaevskii equation with a rotational angular momentum term in the space ℝ2. Copyright

Long-time self-similar asymptotic of the macroscopic quantum models

A free boundary problem for the incompressible neo-Hookean elastodynamics is studied in two and three spatial dimensions. The a priori estimates in Sobolev norms of solutions with the physical vacuum condition are established through a geometrical point of view of Christodoulou and Lindblad (2000) [3]. Some estimates on the second fundamental form and velocity of the free surface are also obtained.

MODIFIED SCATTERING FOR BIPOLAR NONLINEAR SCHRODINGER-POISSON EQUATIONS

For the free boundary problem of the plasma–vacuum interface to 3D ideal incompressible magnetohydrodynamics, the a priori estimates of smooth solutions are proved in Sobolev norms by adopting a geometrical point of view and some quantities such as the second fundamental form and the velocity of the free interface are estimated. In the vacuum region, the magnetic fields are described by the div–curl system of pre-Maxwell dynamics, while at the interface the total pressure is continuous and the magnetic fields are tangential to the interface, but we do not need any restrictions on the size of the magnetic fields on the free interface. We introduce the “fictitious particle” endowed with a fictitious velocity field in vacuum to reformulate the problem to a fixed boundary problem under the Lagrangian coordinates. The L2-norms of any order covariant derivatives of the magnetic fields both in vacuum and on the boundaries are bounded in terms of initial data and the second fundamental forms of the free interface and the rigid wall. The estimates of the curl of the electric fields in vacuum are also obtained, which are also indispensable in elliptic estimates.

Well-posedness for the viscous rotating shallow water equations with friction terms

The unipolar and bipolar macroscopic quantum models derived recently, for instance, in the area of charge transport are considered in spatial one-dimensional whole space in the present paper. These models consist of nonlinear fourth-order parabolic equation for unipolar case or coupled nonlinear fourth-order parabolic system for bipolar case. We show for the first time the self-similarity property of the macroscopic quantum models in large time. Namely, we show that there exists a unique global strong solution with strictly positive density to the initial value problem of the macroscopic quantum models which tends to a self-similar wave (which is not the exact solution of the models) in large time at an algebraic time-decay rate.