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Convex Analysisの二,三の進展について

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Electrodynamics Of Continuous Media

The goal of this final chapter is to show how the boundary value problems of mathematical physics can be solved by the methods of the preceding chapters. This will be done by solving a variety of specific problems that illustrate the principal types of problems that were formulated in Chapter 7. Additional applications are developed in the Exercises. The primary solution method is Fourier’s method of separation of variables and the associated Sturm-Liouville theory of Chapter 8.

Boundary Value Problems of Mathematical Physics

Weakly Differentiable Functions

Linear operators part ii (spectral theory)

We consider the free boundary problem for the plasma vacuum interface model in ideal incompressible magneto-hydrodynamics. Under a suitable stability condition on the initial discontinuity, the well-posedness of the linearized problem, around a non constant basic state sufficiently smooth, is investigated. Since the latter amounts to be a non standard initial-boundary value problem of mixed hyperbolic-elliptic type, for its resolution we introduce a fully ”hyperbolic” regularized problem. For the regularized problem, a suitable a priori estimate, uniform with respect to the small parameter of the regularization, is derived in the anisotropic Sobolev space H 1 .

/pdf/well-posedness-of-the-linearized-plasma-vacuum-interface-58o9hggqh8.pdf

Well-posedness of the linearized plasma-vacuum interface problem in ideal incompressible MHD

We consider the free boundary problem for current-vortex sheets in ideal incompressible magneto-hydrodynamics. It is known that current-vortex sheets may be at most weakly (neutrally) stable due to the existence of surface waves solutions to the linearized equations. The existence of such waves may yield a loss of derivatives in the energy estimate of the solution with respect to the source terms. However, under a suitable stability condition satisfied at each point of the initial discontinuity and a flatness condition on the initial front, we prove an a priori estimate in Sobolev spaces for smooth solutions with no loss of derivatives. The result of this paper gives some hope for proving the local existence of smooth current-vortex sheets without resorting to a Nash-Moser iteration. Such result would be a rigorous confirmation of the stabilizing effect of the magnetic field on Kelvin-Helmholtz instabilities, which is well known in astrophysics.

https://hal.archives-ouvertes.fr/hal-00565759/document

A priori Estimates for 3D Incompressible Current-Vortex Sheets

https://alessandro-morando.unibs.it/Slides/slides-files/praga08.pdf

Stability of incompressible current-vortex sheets

The purpose of this paper is to give a compactness-continuity result for the solution of a nonlinear Dirichlet problem in terms of its domain variation. The topology in the family of domains is given by the Hausdorff metric and continuity is obtained under capacity conditions. A generalisation of Sverak's result in iV-dimensions is deduced as a particular case.

Shape optimisation problems governed by nonlinear state equations

In this paper we study the equations of magneto-hydrodynamics for a 2D incompressible ideal fluid in the exterior domain and in the half-plane. We prove the existence of a global classical solution in Holder spaces, by applying Shauder fixed point theorem.

Paola Trebeschi

Papers

Well-posedness of the linearized plasma-vacuum interface problem in ideal incompressible MHD

A priori Estimates for 3D Incompressible Current-Vortex Sheets

Stability of incompressible current-vortex sheets

Shape optimisation problems governed by nonlinear state equations

Global Classical Solutions for MHD System