### A free boundary problem arising in magnetohydrodynamic system

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The aim of this work is to establish, from a mathematical point of view, the limit α → +∞ in the system $i{\partial}_{t}E+\nabla (\nabla .E)-{\alpha}^{2}\nabla \times \nabla \times E=-{\left|E\right|}^{2\sigma}E,$ where $E:{\mathbb{R}}^{3}\to {\u2102}^{3}$. This corresponds to an approximation which is made in the context of Langmuir turbulence in plasma Physics. The L2-subcritical σ (that is σ ≤ 2/3) and the H1-subcritical σ (that is σ ≤ 2) are studied. In the physical case σ = 1, the limit is then studied for the ${H}^{1}\left({\mathbb{R}}^{3}\right)$ norm.

This is a report on project initiated with Anne Nouri [3], presently in progress, with the collaboration of Nicolas Besse [2] ([2] is mainly the material of this report) . It concerns a version of the Vlasov equation where the self interacting potential is replaced by a Dirac mass. Emphasis is put on the relations between the linearized version, the full non linear problem and also on natural connections with several other equations of mathematical physic.

Žemlička, Jan: Structure of steady rings. Zemek, Martin: On some aspects of subdifferentiality of functions on Banach spaces. Hlubinka, Daniel: Construction of Markov kernels with application for moment problem solution. Somberg, Petr: Properties of the BGG resolution on the spheres. Krump, Lukáš: Construction of Bernstein-Gelfand-Gelfand for almost hermitian symmetric structures. Kolář, Jan: Simultaneous extension operators. Porosity.

Nel presente lavoro si fanno alcune osservazioni sulle onde magnetoacustiche. Esse, per quanto è a conoscenza dell'autore, non sono presenti nella vastissima letteratura sull'argomento; nonostante la loro semplicità, è sembrato di un qualche interesse segnalarle esplicitamente per il loro contenuto fisico. Specificatamente riguardano: l'effetto Doppler; la quantità di moto; il carattere di onde di condensazione; il carattere conservativo del campo di forza magnetica; l'equipartizione dell'energia....

The present work is devoted to the simulation of a strongly magnetized plasma as a mixture of an ion fluid and an electron fluid. For simplicity reasons, we assume that each fluid is isothermal and is modelized by Euler equations coupled with a term representing the Lorentz force, and we assume that both Euler systems are coupled through a quasi-neutrality constraint of the form ni = ne. The numerical method which is described in the...

The modelling and the numerical resolution of the electrical charging of a spacecraft in interaction with the Earth magnetosphere is considered. It involves the Vlasov-Poisson system, endowed with non standard boundary conditions. We discuss the pros and cons of several numerical methods for solving this system, using as benchmark a simple 1D model which exhibits the main difficulties of the original models.

In this paper, we prove the convergence of the current defined from the Schrödinger-Poisson system with the presence of a strong magnetic field toward a dissipative solution of the Euler equations.

We present a discontinuous Galerkin scheme for the numerical approximation of the one-dimensional periodic Vlasov-Poisson equation. The scheme is based on a Galerkin-characteristics method in which the distribution function is projected onto a space of discontinuous functions. We present comparisons with a semi-Lagrangian method to emphasize the good behavior of this scheme when applied to Vlasov-Poisson test cases.

We consider a strongly magnetized plasma described by a Vlasov-Poisson system with a large external magnetic field. The finite Larmor radius scaling allows to describe its behaviour at very fine scales. We give a new interpretation of the asymptotic equations obtained by Frénod and Sonnendrücker [SIAM J. Math. Anal. 32 (2001) 1227–1247] when the intensity of the magnetic field goes to infinity. We introduce the so-called polarization drift and show that its contribution is not negligible in the...