10,772 research outputs found
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
An Improved Result on Rayleigh--Taylor Instability of Nonhomogeneous Incompressible Viscous Flows
In [F. Jiang, S. Jiang, On instability and stability of three-dimensional
gravity driven viscous flows in a bounded domain, Adv. Math., 264 (2014)
831--863], Jiang et.al. investigated the instability of Rayleigh--Taylor
steady-state of a three-dimensional nonhomogeneous incompressible viscous flow
driven by gravity in a bounded domain of class . In particular,
they proved the steady-state is nonlinearly unstable under a restrictive
condition of that the derivative function of steady density possesses a
positive lower bound. In this article, by exploiting a standard energy
functional and more-refined analysis of error estimates in the bootstrap
argument, we can show the nonlinear instability result without the restrictive
condition.Comment: 12 page
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 , depending on
the known physical parameters, for the stability/instability of the
perturbation problem. More precisely, if , then the perturbation problem
is unstable, i.e., the Parker instability occurs, while if 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
A Note on Large Time Behavior of Velocity in the Baratropic Compressible Navier-Stokes Equations
Recently, for periodic initial data with initial density allowed to vanish,
Huang and Li [1] establish the global existence of strong and weak solutions
for the two-dimensional compressible Navier{Stokes equations with no
restrictions on the size of initial data provided the bulk viscosity
coefficient is \lambda = \rho^\beta with \beta > 4/3. Moreover, the large-time
behavior of the strong and weak solutions are also obtained, in which the
velocity gradient strongly converges to zero in L^2 norm. In this note, we
further point out that the velocity strongly converges to an equilibrium
velocity in H^1 norm, in which the equilibrium velocity is uniquely determined
by the initial data. Our result can also be regarded a correction for the
result of large-time behavior of velocity in [2].Comment: 5 page
Nonlinear Thermal Instability in Compressible Viscous Flows without Heat Conductivity
We investigate the thermal instability of a smooth equilibrium state, in
which the density function satisfies Schwarzschild's (instability) condition,
to a compressible heat-conducting viscous flow without heat conductivity in the
presence of a uniform gravitational field in a three-dimensional bounded
domain. We show that the equilibrium state is linearly unstable by a modified
variational method. Then, based on the constructed linearly unstable solutions
and a local well-posedness result of classical solutions to the original
nonlinear problem, we further construct the initial data of linearly unstable
solutions to be the one of the original nonlinear problem, and establish an
appropriate energy estimate of Gronwall-type. With the help of the established
energy estimate, we finally show that the equilibrium state is nonlinearly
unstable in the sense of Hadamard by a careful bootstrap instability argument.Comment: 35 pages. arXiv admin note: text overlap with arXiv:1311.436
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 , 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 , where N=2 or 3. By exploiting a maximum principle, Nirenberg's
interpolation inequality and a smallness condition imposed on the -th
component of initial direction field \mf{d}_0 to overcome the difficulties
induced by the supercritical nonlinearity 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
.Comment: 34 pages. arXiv admin note: substantial text overlap with
arXiv:0911.4703, arXiv:0911.4098 by other author
- …