### A Nonconforming Finite Element Method to Compute the Spectrum of an Operator Relative to the Stability of a Plasma in Toroidal Geometry.

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We address here mathematical models related to the Laser-Plasma Interaction. After a simplified introduction to the physical background concerning the modelling of the laser propagation and its interaction with a plasma, we recall some classical results about the geometrical optics in plasmas. Then we deal with the well known paraxial approximation of the solution of the Maxwell equation; we state a coupling model between the plasma hydrodynamics and the laser propagation. Lastly, we consider the...

The aim of this paper is to derive a general model for reduced viscous and resistive Magnetohydrodynamics (MHD) and to study its mathematical structure. The model is established for arbitrary density profiles in the poloidal section of the toroidal geometry of Tokamaks. The existence of global weak solutions, on the one hand, and the stability of the fundamental mode around initial data, on the other hand, are investigated.

We consider plasma tearing mode instabilities when the resistivity depends on a flux function (ψ), for the plane slab model. This problem, represented by the MHD equations, is studied as a bifurcation problem. For so doing, it is written in the form (I(.)-T(S,.)) = 0, where T(S,.) is a compact operator in a suitable space and S is the bifurcation parameter. In this work, the resistivity is not assumed to be a given quantity (as usually done in previous papers, see [1,2,5,7,8,9,10], but it depends...

In questo lavoro si studia la instabilità gravitazionale di un fluido comprimibile, elettroconduttore, descritto dalle equazioni della magnetofluidodinamica in presenza delle correnti di Hall e di ion slip. Si determina la condizione per la instabilità relativa ad una classe di perturbazioni assialsimmetriche.

We derive a very sharp nonlinear stability result for the problem of thermal convection in a layer of dielectric fluid subject to an alternating current (AC). It is particularly important to note that the size of the initial energy in which we establish global nonlinear stability is not restricted whatsoever, and the Rayleigh-Roberts number boundary coincides with that found by a formal linear instability analysis.