Abstract. 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.
We propose a new Large Eddy Simulation (LES) model for the\ud
Boussinesq equations. We consider the motion in a three-dimensional\ud
domain with solid walls, and in a particular geometric setting we\ud
look for solutions which are periodic in the vertical direction and\ud
satisfy homogeneous Dirichlet conditions on the lateral boundary. We\ud
are thus modeling a vertical pipe and one main difficulty is that of\ud
considering regularizations of the equation which are well behaved\ud
also in presence of a boundary. The LES model we consider is then\ud
obtained by introducing a vertical filter, which is the natural one\ud
for the setting that we are considering. The related interior\ud
closure problem is treated in a standard way with a\ud
simplified-Bardina deconvolution model. The most technical\ud
analytical point is related to the fact that anisotropic filters\ud
provide less regularity than the isotropic ones and, in principle,\ud
the density term appearing in the Boussinesq equations may behave\ud
very differently from the velocity. We are able to define an\ud
appropriate notion of regular weak solution, for which we prove\ud
existence, uniqueness, and we also show that the energy associated\ud
to the model is exactly preserved
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.
We consider two Large Eddy Simulation (LES) models for the approximation of large scales of the equations of Magnetohydrodynamics (MHD in the sequel). We study two α-models, which are obtained adapting to the MHD the approach by Stolz and Adams with van Cittert approximate deconvolution operators. First, we prove existence and uniqueness of a regular weak solution for a system with filtering and deconvolution in both equations. Then we study the behavior of solutions as the deconvolution parameter goes to infinity. The main result of this paper is the convergence to a solution of the filtered MHD equations. In the final section we study also the problem with filtering acting only on the velocity equation.
We consider the 3D simplified Bardina turbulence model with horizontal filtering, fractional dissipation, and the presence of a memory term incorporating hereditary effects. We analyze the regularity properties and the dissipative nature of the considered system and, in our main result, we show the existence of a global exponential attractor in a suitable phase space.
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