27 research outputs found
A priori estimates for free boundary problem of incompressible inviscid magnetohydrodynamic flows
In the present paper, we prove the a priori estimates of Sobolev norms for a
free boundary problem of the incompressible inviscid MHD equations in all
physical spatial dimensions and 3 by adopting a geometrical point of view
used in Christodoulou-Lindblad CPAM 2000, and estimating quantities such as the
second fundamental form and the velocity of the free surface. We identify the
well-posedness condition that the outer normal derivative of the total pressure
including the fluid and magnetic pressures is negative on the free boundary,
which is similar to the physical condition (Taylor sign condition) for the
incompressible Euler equations of fluids.Comment: 34 page
Wellposedness of Cauchy problem for the Fourth Order Nonlinear Schr\"odinger Equations in Multi-dimensional Spaces
We study the wellposedness of Cauchy problem for the fourth order nonlinear
Schr\"odinger equations i\partial_t u=-\eps\Delta u+\Delta^2
u+P((\partial_x^\alpha u)_{\abs{\alpha}\ls 2}, (\partial_x^\alpha
\bar{u})_{\abs{\alpha}\ls 2}),\quad t\in \Real, x\in\Real^n, where
\eps\in\{-1,0,1\}, n\gs 2 denotes the spatial dimension and is a
polynomial excluding constant and linear terms.Comment: 28 page
Well-posedness for a multi-dimensional viscous liquid-gas two-phase flow model
The Cauchy problem of a multi-dimensional () 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.Comment: 28 page
Global well-posedness for the Gross-Pitaevskii equation with an angular momentum rotational term in three dimensions
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 \Real^3.Comment: 11 page
Long-time self-similar asymptotic of the macroscopic quantum models
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.Comment: 18 page