31 research outputs found
Higher-Order Global Regularity of an Inviscid Voigt-Regularization of the Three-Dimensional Inviscid Resistive Magnetohydrodynamic Equations
We prove existence, uniqueness, and higher-order global regularity of strong
solutions to a particular Voigt-regularization of the three-dimensional
inviscid resistive Magnetohydrodynamic (MHD) equations. Specifically, the
coupling of a resistive magnetic field to the Euler-Voigt model is introduced
to form an inviscid regularization of the inviscid resistive MHD system. The
results hold in both the whole space \nR^3 and in the context of periodic
boundary conditions. Weak solutions for this regularized model are also
considered, and proven to exist globally in time, but the question of
uniqueness for weak solutions is still open. Since the main purpose of this
line of research is to introduce a reliable and stable inviscid numerical
regularization of the underlying model we, in particular, show that the
solutions of the Voigt regularized system converge, as the regularization
parameter \alpha\maps0, to strong solutions of the original inviscid
resistive MHD, on the corresponding time interval of existence of the latter.
Moreover, we also establish a new criterion for blow-up of solutions to the
original MHD system inspired by this Voigt regularization. This type of
regularization, and the corresponding results, are valid for, and can also be
applied to, a wide class of hydrodynamic models
Frequency decay for Navier-Stokes stationary solutions
We consider stationary Navier-Stokes equations in R 3 with a regular external
force and we prove exponential frequency decay of the solutions. Moreover, if
the external force is small enough, we give a pointwise exponential frequency
decay for such solutions according to the K41 theory. If a damping term is
added to the equation, a pointwise decay is obtained without the smallness
condition over the force
A Blow-Up Criterion for the 3D Euler Equations Via the Euler-Voigt Inviscid Regularization
We propose a new blow-up criterion for the 3D Euler equations of
incompressible fluid flows, based on the 3D Euler-Voigt inviscid
regularization. This criterion is similar in character to a criterion proposed
in a previous work by the authors, but it is stronger, and better adapted for
computational tests. The 3D Euler-Voigt equations enjoy global well-posedness,
and moreover are more tractable to simulate than the 3D Euler equations. A
major advantage of these new criteria is that one only needs to simulate the 3D
Euler-Voigt, and not the 3D Euler equations, to test the blow-up criteria, for
the 3D Euler equations, computationally