arXiv subjects

Yanjin Wang

Publications and source records attributed to Yanjin Wang.

31 records · Page 2Linked to original sources

arXiv2015

The compressible viscous surface-internal wave problem: stability and vanishing surface tension limit

This paper concerns the dynamics of two layers of compressible, barotropic, viscous fluid lying atop one another. The lower fluid is bounded below by a rigid bottom, and the upper fluid is bounded above by a trivial fluid of constant pressure. This is a free boundary problem: the interfaces between the fluids and above the upper fluid are free to move. The fluids are acted on by gravity in the bulk, and at the free interfaces we consider both the case of surface tension and the case of no surface forces. We establish a sharp nonlinear global-in-time stability criterion and give the explicit decay rates to the equilibrium. When the upper fluid is heavier than the lower fluid along the equilibrium interface, we characterize the set of surface tension values in which the equilibrium is nonlinearly stable. Remarkably, this set is non-empty, i.e. sufficiently large surface tension can prevent the onset of the Rayleigh-Taylor instability. When the lower fluid is heavier than the upper fluid, we show that the equilibrium is stable for all non-negative surface tensions and we establish the zero surface tension limit.

math.AP↗

arXiv2012

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.

math.AP↗

arXiv2012

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.

math.AP↗

arXiv2012

On the Rayleigh-Taylor instability for incompressible viscous magnetohydrodynamic equations

We study the Rayleigh-Taylor problem for two incompressible, immiscible, viscous magnetohydrodynamic (MHD) flows, with zero resistivity, surface tension (or without surface tenstion) and special initial magnetic field, evolving with a free interface in the presence of a uniform gravitational field. First, we reformulate in Lagrangian coordinates MHD equations in a infinite slab as one for the Navier-Stokes equations with a force term induced by the fluid flow map. Then we analyze the linearized problem around the steady state which describes a denser immiscible fluid lying above a light one with an free interface separating the two fluids, and both fluids being in (unstable) equilibrium. By a general method of studying a family of modified variational problems, we construct smooth (when restricted to each fluid domain) solutions to the linearized problem that grow exponentially fast in time in Sobolev spaces, thus leading to an global instability result for the linearized problem. Finally, using these pathological solutions, we demonstrate the global instability for the corresponding nonlinear problem in an appropriate sense. In addition, we compute that the so-called critical number indeed is equal $\sqrt{g[\varrho]/2}$.

math.GM↗

arXiv2011

Decay of the Navier-Stokes-Poisson equations

We establish the time decay rates of the solution to the Cauchy problem for the compressible Navier-Stokes-Poisson system via a refined pure energy method. In particular, the optimal decay rates of the higher-order spatial derivatives of the solution are obtained. The $\Dot{H}^{-s}$($0\le s<3/2$) negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates. As a corollary, we also obtain the usual $L^p$--$L^2$($1<p\le 2$) type of the optimal decay rates. Compared to the compressible Navier-Stokes system and the compressible irrotational Euler-Poisson system, our results imply that both the dispersion effect of the electric field and the viscous dissipation contribute to enhance the decay rate of the density. Our proof is based on a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis.

math.AP↗

arXiv2011

Decay of the Vlasov-Poisson-Boltzmann system

We establish the time decay rates of the solutions to the Cauchy problem for the two-species Vlasov-Poisson-Boltzmann system near Maxwellians via a refined pure energy method. The negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates. The total density of two species of particles decays at the optimal algebraic rate as the Boltzmann equation, but the disparity between two species and the electric field decay at an exponential rate. This phenomenon reveals the essential difference when compared to the one-species Vlasov-Poisson-Boltzmann system in which the electric field decays at the optimal algebraic rate or the Vlasov-Boltzmann system in which the disparity between two species decays at the optimal algebraic rate. Our proof is based on a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis, and a reformulation of the problem which well displays the cancelation property of the two-species system.

math.AP↗

arXiv2011

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.

math.AP↗

arXiv2011

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.

math.AP↗

arXiv2011

The viscous surface-internal wave problem: nonlinear Rayleigh-Taylor instability

We consider the free boundary problem for two layers of immiscible, viscous, incompressible fluid in a uniform gravitational field, lying above a rigid bottom in a three-dimensional horizontally periodic setting. The effect of surface tension is either taken into account at both free boundaries or neglected at both. We are concerned with the Rayleigh-Taylor instability, so we assume that the upper fluid is heavier than the lower fluid. When the surface tension at the free internal interface is below a critical value, which we identify, we establish that the problem under consideration is nonlinearly unstable.

math.AP↗

arXiv2011

The viscous surface-internal wave problem: global well-posedness and decay

We consider the free boundary problem for two layers of immiscible, viscous, incompressible fluid in a uniform gravitational field, lying above a general rigid bottom in a three-dimensional horizontally periodic setting. We establish the global well-posedness of the problem both with and without surface tension. We prove that without surface tension the solution decays to the equilibrium state at an almost exponential rate; with surface tension, we show that the solution decays at an exponential rate. Our results include the case in which a heavier fluid lies above a lighter one, provided that the surface tension at the free internal interface is above a critical value, which we identify. This means that sufficiently large surface tension stabilizes the Rayleigh-Taylor instability in the nonlinear setting. As a part of our analysis, we establish elliptic estimates for the two-phase stationary Stokes problem.

math.AP↗

arXiv2007

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}-Δu+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.

math.AP↗

arXiv2007

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}-ρ(x)^{-1}Δu+u_t+m^2u=f(u) \end{eqnarray*} with some $ρ(x)$ 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.

math.AP↗

arXiv2007

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}-Δu+m_1^2 u+K_1(x)u=a_1|v|^{q+1}|u|^{p-1}u v_{tt}-Δv+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.

math.AP↗