On the Dynamical Stability and Instability of Parker Problem

We investigate a perturbation problem for the three-dimensional compressible isentropic viscous magnetohydrodynamic system with zero resistivity in the presence of a modified gravitational force in a vertical strip domain in which the velocity of the fluid is non-slip on the boundary, and focus on the stabilizing effect of the (equilibrium) magnetic field through the non-slip boundary condition. We show that there is a discriminant $\Xi$, depending on the known physical parameters, for the stability/instability of the perturbation problem. More precisely, if $\Xi<0$, then the perturbation problem is unstable, i.e., the Parker instability occurs, while if $\Xi>0$ and the initial perturbation satisfies some relations, then there exists a global (perturbation) solution which decays algebraically to zero in time, i.e., the Parker instability does not happen. The stability results in this paper reveal the stabilizing effect of the magnetic field through the non-slip boundary condition and the importance of boundary conditions upon the Parker instability, and demonstrate that a sufficiently strong magnetic field can prevent the Parker instability from occurring. In addition, based on the instability results, we further rigorously verify the Parker instability under Schwarzschild's or Tserkovnikov's instability conditions in the sense of Hadamard for a horizontally periodic domain.Comment: 51 page

On Linear Instability and Stability of the Rayleigh-Taylor Problem in Magnetohydrodynamics

We investigate the stabilizing effects of the magnetic fields in the linearized magnetic Rayleigh-Taylor (RT) problem of a nonhomogeneous incompressible viscous magnetohydrodynamic fluid of zero resistivity in the presence of a uniform gravitational field in a three-dimensional bounded domain, in which the velocity of the fluid is non-slip on the boundary. By adapting a modified variational method and careful deriving \emph{a priori} estimates, we establish a criterion for the instability/stability of the linearized problem around a magnetic RT equilibrium state. In the criterion, we find a new phenomenon that a sufficiently strong horizontal magnetic field has the same stabilizing effect as that of the vertical magnetic field on growth of the magnetic RT instability. In addition, we further study the corresponding compressible case, i.e., the Parker (or magnetic buoyancy) problem, for which the strength of a horizontal magnetic field decreases with height, and also show the stabilizing effect of a sufficiently large magnetic field.Comment: 33 page

On the Inhibition of Thermal Convection by a Magnetic Field under Zero Resistivity

We investigate the stability and instability of the magnetic Rayleigh--B\'enard problem with zero resistivity. An stability criterion is established, under which the magnetic B\'enard problem is stable. The proof mainly is based on a three-layers energy method and an idea of magnetic inhibition mechanism. The stable result first mathematically verifies Chandrasekhar's assertion in 1955 that the thermal instability can be inhibited by strong magnetic field in magnetohydrodynamic (MHD) fluid with zero resistivity (based on a linearized steady magnetic B\'enard equations). In addition, we also provide an instability criterion, under which the magnetic Rayleigh--B\'enard problem is unstable. The proof mainly is based on the bootstrap instability method by further developing new analysis technique. Our instability result presents that the thermal instability occurs for a small magnetic field.Comment: 4

Global weak solutions to the two-dimensional Navier-Stokes equations of compressible heat-conducting flows with symmetric data and forces

We prove the global existence of weak solutions to the Navier-Stokes equations of compressible heat-conducting fluids in two spatial dimensions with initial data and external forces which are large and spherically symmetric. The solutions will be obtained as the limit of the approximate solutions in an annular domain. We first derive a number of regularity results on the approximate physical quantities in the "fluid region", as well as the new uniform integrability of the velocity and temperature in the entire space-time domain by exploiting the theory of the Orlicz spaces. By virtue of these a priori estimates we then argue in a manner similar to that in [Arch. Rational Mech. Anal. 173 (2004), 297-343] to pass to the limit and show that the limiting functions are indeed a weak solution which satisfies the mass and momentum equations in the entire space-time domain in the sense of distributions, and the energy equation in any compact subset of the "fluid region".Comment: 19 page

Nonlinear Rayleigh-Taylor Instability for Nonhomogeneous Incompressible Viscous Magnetohydrodynamic Flows

We investigate the nonlinear instability of a smooth Rayleigh-Taylor steady-state solution (including the case of heavier density with increasing height) to the three-dimensional incompressible nonhomogeneous magnetohydrodynamic (MHD) equations of zero resistivity in the presence of a uniform gravitational field. We first analyze the linearized equations around the steady-state solution. Then we construct solutions of the linearized problem that grow in time in the Sobolev space $H^k$, thus leading to the linear instability. With the help of the constructed unstable solutions of the linearized problem and a local well-posedness result of smooth solutions to the original nonlinear problem, we establish the instability of the density, the horizontal and vertical velocities in the nonlinear problem. Moreover, when the steady magnetic field is vertical and small, we prove the instability of the magnetic field. This verifies the physical phenomenon: instability of the velocity leads to the instability of the magnetic field through the induction equation.Comment: 46 pages. arXiv admin note: substantial text overlap with arXiv:1205.227

On Multi-dimensional Compressible Flows of Nematic Liquid Crystals with Large Initial Energy in a Bounded Domain

We study the global existence of weak solutions to a multi-dimensional simplified Ericksen-Leslie system for compressible flows of nematic liquid crystals with large initial energy in a bounded domain $\Omega\subset \mathbb{R}^N$, where N=2 or 3. By exploiting a maximum principle, Nirenberg's interpolation inequality and a smallness condition imposed on the $N$-th component of initial direction field \mf{d}_0 to overcome the difficulties induced by the supercritical nonlinearity $|\nabla{\mathbf d}|^2{\mathbf d}$ in the equations of angular momentum, and then adapting a modified three-dimensional approximation scheme and the weak convergence arguments for the compressible Navier-Stokes equations, we establish the global existence of weak solutions to the initial-boundary problem with large initial energy and without any smallness condition on the initial density and velocity.Comment: arXiv admin note: substantial text overlap with arXiv:1210.356

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 $\sqrt{g[\varrho]/2}$.Comment: 34 pages. arXiv admin note: substantial text overlap with arXiv:0911.4703, arXiv:0911.4098 by other author

Derivations and automorphism groups of the original deformative Schr\"odinger-Virasoro algebras

In this paper, we determine the derivation algebra and the automorphism group of the original deformative ${\rm Schr\ddot{o}dinger}$-{\rm Virasoro} algebras which is the semi-direct product Lie algebra of the Witt algebra and its tensor density module ${\rm I^g}(a,b)$.Comment: 33 page

Performance Analysis of Spectrum Handoff for Cognitive Radio Ad Hoc Networks without Common Control Channel under Homogeneous Primary Traffic

Cognitive radio (CR) technology is regarded as a promising solution to the spectrum scarcity problem. Due to the spectrum varying nature of CR networks, unlicensed users are required to perform spectrum handoffs when licensed users reuse the spectrum. In this paper, we study the performance of the spectrum handoff process in a CR ad hoc network under homogeneous primary traffic. We propose a novel three dimensional discrete-time Markov chain to characterize the process of spectrum handoffs and analyze the performance of unlicensed users. Since in real CR networks, a dedicated common control channel is not practical, in our model, we implement a network coordination scheme where no dedicated common control channel is needed. Moreover, in wireless communications, collisions among simultaneous transmissions cannot be immediately detected and the whole collided packets need to be retransmitted, which greatly affects the network performance. With this observation, we also consider the retransmissions of the collided packets in our proposed discrete-time Markov chain. In addition, besides the random channel selection scheme, we study the impact of different channel selection schemes on the performance of the spectrum handoff process. Furthermore, we also consider the spectrum sensing delay in our proposed Markov model and investigate its effect on the network performance. We validate the numerical results obtained from our proposed Markov model against simulation and investigate other parameters of interest in the spectrum handoff scenario. Our proposed analytical model can be applied to various practical network scenarios. It also provides new insights on the process of spectrum handoffs. Currently, no existing analysis has considered the comprehensive aspects of spectrum handoff as what we consider in this paper.Comment: to appear in IEEE INFOCOM 201

A blow-up criterion for compressible viscous heat-conductive flows

We study an initial boundary value problem for the Navier-Stokes equations of compressible viscous heat-conductive fluids in a 2-D periodic domain or the unit square domain. We establish a blow-up criterion for the local strong solutions in terms of the gradient of the velocity only, which coincides with the famous Beale-Kato-Majda criterion for ideal incompressible flows.Comment: Accepted by Acta Mathematica Scienti