Davide Catania

Global existence for two regularized MHD models in three space-dimension

The global existence of solutions for the 3D incompressible Euler equations is a major open problem. For the 3D inviscid MHD system, the global existence is an open problem as well. Our main concern in this paper is to understand which kind of regularization, of the form of α-regularization or partial viscous regularization, is capable to provide the global in time solvability for the 3D inviscid MHD system of equations. We consider two different regularized magnetohydrodynamic models for an incompressible fluid. In both cases, we provide a global existence result for the solution of the system. Mathematics Subject Classification (2000). Primary 35Q35; Secondary 76D03.

Length-scale estimates for the 3D simplified Bardina magnetohydrodynamic model

We consider the MHD-α model known as double viscous simplified Bardina MHD (SBMHD) on a 3D periodic box (MHD stands for magnetohydrodynamic). This system is a large eddy simulation model useful to approximate the turbulent behavior of an incompressible homogeneous magnetofluid because of the actual impossibility to handle the MHD model neither analytically nor via direct numerical simulation. In a previous paper (joint work with Secchi), the global existence of strong solutions to the SBMHD has been proved as well as the existence of a global attractor of finite fractal dimension. Upper bounds for such dimension are provided both in terms of the modified Grashof number and the dissipation length associated to the mean rate of energy dissipation. In this paper, we commute the same bound in an estimate in terms of the modified Reynolds number R. This result is useful because the classical Kolmogorov theory of turbulence is expressed using the Reynolds number. The global attractor estimate in terms of R is c...

Existence and Stability for the 3D Linearized Constant-Coefficient Incompressible Current-Vortex Sheets

We consider the free boundary problem for current-vortex sheets in ideal incompressible magnetohydrodynamics. The problem of current-vortex sheets arises naturally, for instance, in geophysics and astrophysics. We prove the existence of a unique solution to the constant-coefficient linearized problem and an a priori estimate with no loss of derivatives. This is a preliminary result to the study of linearized variable-coefficient current-vortex sheets, a first step to prove the existence of solutions to the nonlinear problem.

On the well-posedness of the Boussinesq equations with anisotropic filter for turbulent flows

We consider approximate deconvolution models for the Boussinesq equations, based on suitable anisotropic filters. We discuss existence and well-posedness of the solutions, with particular emphasis on the role of the energy (of the model) balance.

On the existence of an inertial manifold for a deconvolution model of the 2D mean Boussinesq equations

We show the existence of an inertial manifold (i.e. a globally invariant, exponentially attracting, finite-dimensional manifold) for the approximate deconvolution model of the 2D mean Boussinesq equations. This model is obtained by means of the Van Cittern approximate deconvolution operators, which is applied to the 2D filtered Boussinesq equations.