Paola Trebeschi

A priori Estimates for 3D Incompressible Current-Vortex Sheets

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.

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

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 .

Shape optimisation problems governed by nonlinear state equations

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 Sveraks result in iV-dimensions is deduced as a particular case.

REGULARITY OF SOLUTIONS TO CHARACTERISTIC INITIAL-BOUNDARY VALUE PROBLEMS FOR SYMMETRIZABLE SYSTEMS

We consider the initial-boundary value problem for linear Friedrichs symmetrizable systems with characteristic boundary of constant rank. We assume the existence of the strong L2 solution satisfying a suitable energy estimate, but we do not assume any structural assumption sufficient for existence, such as the fact that the boundary conditions are maximally dissipative or the Kreiss–Lopatinski condition. We show that this is enough in order to get the regularity of solutions, in the natural framework of weighted anisotropic Sobolev spaces, provided the data are sufficiently smooth.

Whitney property in two dimensional Sobolev spaces

For p > 1, we prove that all the functions of W 2,p loc (R 2 ) satisfy the Whitney property; i.e., if u ∈ W 2,p loc (R 2 ) is such that ∇u = 0 (in the sense of capacity) on a connected set K C R 2 , then u is constant on K.

ON THE LOCAL SOLVABILITY FOR A NONLINEAR WEAKLY HYPERBOLIC EQUATION WITH ANALYTIC COEFFICIENTS

We prove the local solvability in C ∞ of the following degenerate hyperbolic quasilinear equation in one space dimension: where the coefficient a(t, x) is nonnegative and real analytic in both variables, while f satisfies suitable Levi conditions. The main tools used are weighted energy estimates and the Nash-Moser Theorem.

Γ-Limit of Periodic Obstacles

We compute the Γ-limit of a sequence obstacle functionals in the case of periodic obstacles.

Existence of approximate current-vortex sheets near the onset of instability

The paper is concerned with the free boundary problem for 2D current-vortex sheets in ideal incompressible magneto-hydrodynamics near the transition point between the linearized stability and instability. In order to study the dynamics of the discontinuity near the onset of the instability, Hunter and Thoo have introduced an asymptotic quadratically nonlinear integro-differential equation for the amplitude of small perturbations of the planar discontinuity. The local-in-time existence of smooth solutions to the Cauchy problem for such amplitude equation was already proven, under a suitable stability condition. However, the solution found there has a loss of regularity (of order two) from the initial data. In the present paper, we are able to obtain an existence result of solutions with optimal regularity, in the sense that the regularity of the initial data is preserved in the motion for positive times.

Data dependence of approximate current-vortex sheets near the onset of instability

The paper is concerned with the free boundary problem for 2D current-vortex sheets in ideal incompressible magneto-hydrodynamics near the transition point between the linearized stability and instability. In order to study the dynamics of the discontinuity near the onset of the instability, Hunter and Thoo have introduced an asymptotic quadratically nonlinear integro-differential equation for the amplitude of small perturbations of the planar discontinuity. The local-in-time existence of smooth solutions to the Cauchy problem for the amplitude equation was already shown. In the present paper we prove the continuous dependence in strong norm of solutions on the initial data. This completes the proof of the well-posedness of the problem in the classical sense of Hadamard.

A New Relaxation Space for Obstacles

The space of obstacles (i.e. p-quasi upper semicontinuous functions) is endowed with a distance which is topologically equivalent to the Γ-convergence. We find the metric completion of this space and we give some application for minimization problems of cost functionals depending on obstacles via their level sets. An element of the completion is a decreasing and γp-continuous on the left mapping RЭt↦μt, where μt are positive Borel measures vanishing on sets of zero p-capacity.