Global solution and time decay of the Vlasov-Poisson-Landau system in R3

We construct the global unique solution near a global Maxwellian to the Vlasov-Poisson-Landau system in the whole space. The total density of two species of particles decays at the optimal algebraic rates as the Landau equation in the whole space, but the disparity between two species and the electric potential decay at the faster rates as the Vlasov-Poisson-Landau system in a periodic box.Comment: 36 page

Finite time blow-up results for the damped wave equations with arbitrary initial energy in an inhomogeneous medium

In this paper we consider the long time behavior of solutions of the initial value problem for the damped wave equation of the form \begin{eqnarray*} u_{tt}-\rho(x)^{-1}\Delta u+u_t+m^2u=f(u) \end{eqnarray*} with some $\rho(x)$ and $f(u)$ on the whole space $\R^n$ ($n\geq 3$). For the low initial energy case, which is the non-positive initial energy, based on concavity argument we prove the blow up result. As for the high initial energy case, we give out sufficient conditions of the initial datum such that the corresponding solution blows up in finite time.Comment: 15page

Critical Magnetic Number in the MHD Rayleigh-Taylor instability

We reformulate in Lagrangian coordinates the two-phase free boundary problem for the equations of Magnetohydrodynamics in a infinite slab, which is incompressible, viscous and of zero resistivity, as one for the Navier-Stokes equations with a force term induced by the fluid flow map. We study the stabilized effect of the magnetic field for the linearized equations around the steady-state solution by assuming that the upper fluid is heavier than the lower fluid, $i. e.$, the linear Rayleigh-Taylor instability. We identity the critical magnetic number $|B|_c$ by a variational problem. For the cases $(i)$ the magnetic number $\bar{B}$ is vertical in 2D or 3D; $(ii)$ $\bar{B}$ is horizontal in 2D, we prove that the linear system is stable when $|\bar{B}|\ge |B|_c$ and is unstable when $|\bar{B}|<|B|_c$. Moreover, for $|\bar{B}|<|B|_c$ the vertical $\bar{B}$ stabilizes the low frequency interval while the horizontal $\bar{B}$ stabilizes the high frequency interval, and the growth rate of growing modes is bounded.Comment: 25 pages;v2: typo

Sharp nonlinear stability criterion of viscous non-resistive MHD internal waves in 3D

We consider the dynamics of two layers of incompressible electrically conducting fluid interacting with the magnetic field, which are confined within a 3D horizontally infinite slab and separated by a free internal interface. We assume that the upper fluid is heavier than the lower fluid so that the fluids are susceptible to the Rayleigh-Taylor instability. Yet, we show that the viscous and non-resistive problem around the equilibrium is nonlinearly stable provided that the strength of the vertical component of the steady magnetic field, $|\bar B_3|$, is greater than the critical value, $\mathcal{M}_c$, which we identify explicitly. We also prove that the problem is nonlinearly unstable if $|\bar B_3|<\mathcal{M}_c$. Our results indicate that the non-horizontal magnetic field has strong stabilizing effect on the Rayleigh-Taylor instability but the horizontal one does not have in 3D.Comment: 47p

Nonexistence of global solutions of a class of coupled nonlinear Klein-Gordon equations with nonnegative potentials and arbitrary initial energy

In the paper we consider the nonexistence of global solutions of the Cauchy problem for coupled Klein-Gordon equations of the form \begin{eqnarray*} \left\{\begin{array}{l} u_{tt}-\Delta u+m_1^2 u+K_1(x)u=a_1|v|^{q+1}|u|^{p-1}u v_{tt}-\Delta v+m_2^2 u+K_2(x)v=a_2|u|^{p+1}|v|^{q-1}v u(0,x)=u_0; u_t(0,x)=u_1(x) v(0,x)=v_0; v_t(0,x)=v_1(x) \end{array} \right. \end{eqnarray*} on $\R\times\R^n$. Firstly for some special cases of $n=2,3$, we prove the existence of ground state of the corresponding Lagrange-Euler equations of the above equations. Then we establish a blow up result with low initial energy, which leads to instability of standing waves of the system above. Moreover as a byproduct we also discuss the global existence. Next based on concavity method we prove the blow up result for the system with non-positive initial energy in the general case: $n\geq 1$. Finally when the initial energy is given arbitrarily positive, we show that if the initial datum satisfies some conditions, the corresponding solution blows up in a finite time.Comment: 22 page

A sufficient condition for finite time blow up of the nonlinear Klein-Gordon equations with arbitrarily positive initial energy

In this paper we consider the nonexistence of global solutions of a Klein-Gordon equation of the form \begin{eqnarray*} u_{tt}-\Delta u+m^2u=f(u)& (t,x)\in [0,T)\times\R^n. \end{eqnarray*} Here $m\neq 0$ and the nonlinear power $f(u)$ satisfies some assumptions which will be stated later. We give a sufficient condition on the initial datum with arbitrarily high initial energy such that the solution of the above Klein-Gordon equation blows up in a finite time.Comment: 6page

Zero surface tension limit of viscous surface waves

We consider the free boundary problem for a layer of viscous, incompressible fluid in a uniform gravitational field, lying above a rigid bottom and below the atmosphere. For the "semi-small" initial data, we prove the zero surface tension limit of the problem within a local time interval. The unique local strong solution with surface tension is constructed as the limit of a sequence of approximate solutions to a special parabolic regularization. For the small initial data, we prove the global-in-time zero surface tension limit of the problem.Comment: 57pp. arXiv admin note: substantial text overlap with arXiv:1011.5179, arXiv:1109.179

Decay of dissipative equations and negative Sobolev spaces

We develop a general energy method for proving the optimal time decay rates of the solutions to the dissipative equations in the whole space. Our method is applied to classical examples such as the heat equation, the compressible Navier-Stokes equations and the Boltzmann equation. In particular, the optimal decay rates of the higher-order spatial derivatives of solutions are obtained. The negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates. We use a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis

Global well-posedness of an initial-boundary value problem for viscous non-resistive MHD systems

This paper concerns the viscous and non-resistive MHD systems which govern the motion of electrically conducting fluids interacting with magnetic fields. We consider an initial-boundary value problem for both compressible and (nonhomogeneous and homogeneous) incompressible fluids in an infinite flat layer. We prove the global well-posedness of the systems around a uniform magnetic field which is vertical to the layer. Moreover, the solution converges to the steady state at an almost exponential rate as time goes to infinity. Our proof relies on a two-tier energy method for the reformulated systems in Lagrangian coordinates.Comment: 31 pages. Add some more remarks and explanation

Incompressible inviscid resistive MHD surface waves in 2D

We consider the dynamics of a layer of an incompressible electrically conducting fluid interacting with the magnetic field in a two-dimensional horizontally periodic setting. The upper boundary is in contact with the atmosphere, and the lower boundary is a rigid flat bottom. We prove the global well-posedness of the inviscid and resistive problem with surface tension around a non-horizontal uniform magnetic field; moreover, the solution decays to the equilibrium almost exponentially. One of the key observations here is an induced damping structure for the fluid vorticity due to the resistivity and transversal magnetic field.Comment: 36p