20 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
Long-time Behavior of a Two-layer Model of Baroclinic Quasi-geostrophic Turbulence
We study a viscous two-layer quasi-geostrophic beta-plane model that is
forced by imposition of a spatially uniform vertical shear in the eastward
(zonal) component of the layer flows, or equivalently a spatially uniform
north-south temperature gradient. We prove that the model is linearly unstable,
but that non-linear solutions are bounded in time by a bound which is
independent of the initial data and is determined only by the physical
parameters of the model. We further prove, using arguments first presented in
the study of the Kuramoto-Sivashinsky equation, the existence of an absorbing
ball in appropriate function spaces, and in fact the existence of a compact
finite-dimensional attractor, and provide upper bounds for the fractal and
Hausdorff dimensions of the attractor. Finally, we show the existence of an
inertial manifold for the dynamical system generated by the model's solution
operator. Our results provide rigorous justification for observations made by
Panetta based on long-time numerical integrations of the model equations