145 research outputs found
Global Solutions of the Equations of 3D Compressible Magnetohydrodynamics with Zero Resistivity
We prove the global-in-time existence of H^2 solutions of the equations of
compressible magnetohydrodynamics with zero magnetic resistivity in three space
dimensions. Initial data are taken to be small in H^2 modulo a constant state
and initial densities are positive and essentially bounded. The present work
generalizes the results obtained by Kawashima.Comment: This paper has been withdrawn by the author due to a crucial error in
the proo
Solutions to a class of forced drift-diffusion equations with applications to the magneto-geostrophic equations
We prove the global existence of classical solutions to a class of forced
drift-diffusion equations with initial data and divergence free drift
velocity , and we obtain
strong convergence of solutions as the viscosity vanishes. We then apply
our results to a family of active scalar equations which includes the three
dimensional magneto-geostrophic MG equation that has been
proposed by Moffatt in the context of magnetostrophic turbulence in the Earth's
fluid core. We prove the existence of a compact global attractor
in for the MG
equations including the critical equation where . Furthermore, we obtain
the upper semicontinuity of the global attractor as vanishes.Comment: 25 page
H\"{o}lder continuity of solutions to the kinematic dynamo equations
We study the propagation of regularity of solutions to a three dimensional
system of linear parabolic PDE known as the kinematic dynamo equations. The
divergence free drift velocity is assumed to be at the critical regularity
level with respect to the natural scaling of the equations.Comment: 10 page
Vanishing diffusion limits and long time behaviour of a class of forced active scalar equations
We investigate the properties of an abstract family of advection diffusion
equations in the context of the fractional Laplacian. Two independent diffusion
parameters enter the system, one via the constitutive law for the drift
velocity and one as the prefactor of the fractional Laplacian. We obtain
existence and convergence results in certain parameter regimes and limits. We
study the long time behaviour of solutions to the general problem and prove the
existence of a unique global attractor. We apply results to two particular
active scalar equations arising in geophysical fluid dynamics, namely the
surface quasigeostrophic equation and the magnetogeostrophic equation
Nonexistence of Self-Similar Singularities in the Ideal Viscoelastic Flow
We prove the nonexistence of finite time self-similar singularities in an
ideal viscoelastic flow in . We exclude the occurrence of Leray-type
self-similar singularities under suitable integrability conditions on velocity
and deformation tensor. We also prove the nonexistence of asymptotically
self-similar singularities in our system. The present work extends the results
obtained by Chae in the case of magnetohydrodynamics (MHD).Comment: 8 page
Some Serrin type blow-up criteria for the three-dimensional viscous compressible flows with large external potential force
We provide a Serrin type blow-up criterion for the 3-D viscous compressible
flows with large external potential force. For the Cauchy problem of the 3-D
compressible Navier-Stokes system with potential force term, it can be proved
that the strong solution exists globally if the velocity satisfies the Serrin's
condition and the sup-norm of the density is bounded. Furthermore, in the case
of isothermal flows with no vacuum, the Serrin's condition on the velocity can
be removed from the claimed criterion.Comment: Some typos and errors are being fixe
Global regularity for the 3D compressible magnetohydrodynamics with general pressure
We address the compressible magnetohydrodynamics (MHD) equations in
and establish a blow-up criterion for the local strong solutions
in terms of the density only. Namely, if the density is away from vacuum
() and the concentration of mass (), then a local strong
solution can be continued globally in time. The results generalise and
strengthen the previous ones in the sense that there is no magnetic field
present in the criterion and the assumption on the pressure is significantly
relaxed. The proof is based on some new a priori estimates for
three-dimensional compressible MHD equations.Comment: arXiv admin note: text overlap with arXiv:2011.0565
Existence and a blow-up criterion of solution to the 3D compressible Navier-Stokes-Poisson equations with finite energy
We study the low-energy solutions to the 3D compressible
Navier-Stokes-Poisson equations. We first obtain the existence of smooth
solutions with small -norm and essentially bounded densities. No smallness
assumption is imposed on the -norm of the initial data. Using a
compactness argument, we further obtain the existence of weak solutions which
may have discontinuities across some hypersurfaces in . We also
provide a blow-up criterion of solutions in terms of the -norm of
density
Large friction limit of the compressible Navier-Stokes equations with Navier Boundary conditions in general three-dimensional domains
In this paper, we study the Navier-Stokes equations of compressible,
barotropic flow posed in a bounded set in with different
boundary conditions. Specifically, we prove that the local-in-time smooth
solution of the Navier-Stokes equations with Navier boundary condition
converges to the smooth solution of the Navier-Stokes equations with no-slip
boundary condition as the Navier friction coefficient tends to infinity
Existence and uniqueness of low-energy weak solutions to the compressible 3D magnetohydrodynamics equations
We prove the existence and uniqueness of weak solutions of the three
dimensional compressible magnetohydrodynamics (MHD) equations. We first obtain
the existence of weak solutions with small -norm which may display
codimension-one discontinuities in density, pressure, magnetic field and
velocity gradient. The weak solutions we consider here exhibit just enough
regularity and structure which allow us to develop uniqueness and continuous
dependence theory for the compressible MHD equations. Our results generalise
and extend those for the intermediate weak solutions of compressible
Navier-Stokes equations
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