Regularity of weak solutions to the Navier-Stokes equations (III)-frequency overlapping

If $u$ is a Leray-Hopf solution to the Navier-Stokes equations with the initial data in $L^2(\mathbb{R}^3)$, then $u$ is regular.Comment: This paper has been withdrawn by the author due to a crucial error in section

Regularity of Leray-Hopf solutions to Navier-Stokes equations

Limit behaviors of blow up solutions for impressible Navier-Stokes equations are obtained.Comment: 12page

Regularity of Leray-Hopf solutions to Navier-Stokes equations (II)-Blow up rate with small L^2(R^3) data

An upper bound of blow up rate for the Navier-Stokes equations with small data in L^2(R^3) is obtained.Comment: 14 pages; This version was submitted to CMP at January 7, 200

On the Dirichlet problem of Landau-Lifshitz-Maxwell equations

We prove the existence and uniqueness of non-trivial stable solutions to Landau-Lifshitz-Maxwell equations with Dirichlet boundary condition for large anisotropies and small domains, where the domains are non-simply connected.Comment: 22 page

Unique determination of a transversely isotropic perturbation in a linearized inverse boundary value problem for elasticity

We consider a linearized inverse boundary value problem for the elasticity system. From the linearized Dirichlet-to-Neumann map at zero frequency, we show that a transversely isotropic perturbation of a homogeneous isotropic elastic tensor can be uniquely determined. From the linearized Dirichlet-to-Neumann map at two distinct positive frequencies, we show that a transversely isotropic perturbation of a homogeneous isotropic density can be identified at the same time

Vortex Lattice in the Planar Bose-Einstein Condensates with Dipolar Interactions

In this letter we investigate the effects of dipole-dipole interactions on the vortex lattices in fast rotating Bose-Einstein condensates. For single planar condensate, we show that the triangular lattice structure will be unfavorable when the s-wave interaction is attractive and exceeds a critical value. It will first change to a square lattice, and then become more and more flat with the increase of s-wave attraction, until the collapse of the condensate. For an array of coupled planar condensates, we discuss how the dipole-dipole interactions between neighboring condensates compete with the quantum tunneling processes, which affects the relative displacement of two neighboring vortex lattices and leads to the loss of phase coherence between different condensates.Comment: 4 pages, 3 figures, published versio

Pairing between Atoms and Molecules in a Boson-Fermion Resonant Mixture

We consider a mixture of fermionic and bosonic atoms nearby interspecies Feshbach resonances, which have been observed recently in $^6$Li-$^{23}$Na mixture by MIT group, and in $^{40}$K-$^{87}$Rb mixture by JILA group. We point out that the fermion-boson bound state, namely the heteronuclear molecules, will coexist with the fermionic atoms in a wide parameter region, and the attraction between fermionic atoms and molecules will lead to the formation of atom-molecule pairs. The pairing structure is studied in detail, and, in particular, we highlight the possible realization of the Fulde-Ferrel-Larkin-Ovchinnikov state in this system.Comment: 4 pages and 5 figure

Regularity criteria in weak spaces for Navier-Stokes equations in R3

In this paper we establish a Serrin type regularity criterion on the gradient of pressure in weak spaces for the Leray-Hopf weak solutions of the Navier-Stokes equations in R3.Comment: 7 page

Paired Superfluidity and Fractionalized Vortices in Spin-orbit Coupled Bosons

In this letter we study finite temperature properties of spin-1/2 interacting bosons with spin-orbit coupling in two dimensions. When the ground state has stripe order, we show that thermal fluctuations will first melt the stripe order and lead to a superfluid of boson pairs if the spin-orbit coupling is isotropic or nearly isotropic. Such a phase supports fractionalized quantum vortices. The Kosterlize-Thouless transition from superfluid to normal state is driven by proliferation of half vortices. When the ground state is a plane wave state, the transition to normal state is driven by conventional Kosterlize-Thouless transition. However, the critical temperature will drop to zero for isotropic spin-orbit coupling.Comment: 4+2 pages, 4 figure

Degeneracy of Many-body Quantum States in an Optical Lattice with a Uniform Magnetic Field

We prove a theorem that shows the degeneracy of many-body states depends on total particle number and flux filling ratio, for particles in a periodic lattice and under a uniform magnetic field. Non-interacting fermions and weakly interacting bosons are given as two examples. For the later case, this phenomena can also be understood in terms of destructive quantum interferences of multiple symmetry related tunneling paths between classical energy minima, which is reminiscent of the spin-parity effect discovered in magnetic molecular cluster. We also show that the quantum ground state of a mesoscopic number of bosons in this system is not a simple mean-field state but a fragmented state even for very weak interactions.Comment: 5 pages, 3 figures, published versio