Alessandro Morando

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 .

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.

Regularity of Weakly Well-Posed Characteristic Boundary Value Problems

We study the boundary value problem for a linear first-order partial differential system with characteristic boundary of constant multiplicity. We assume the problem to be “weakly” well posed, in the sense that a unique -solution exists, for sufficiently smooth data, and obeys an a priori energy estimate with a finite loss of tangential/conormal regularity. This is the case of problems that do not satisfy the uniform Kreiss-Lopatinskiĭ condition in the hyperbolic region of the frequency domain. Provided that the data are sufficiently smooth, we obtain the regularity of solutions, in the natural framework of weighted conormal Sobolev spaces.

Stability of contact discontinuities for the nonisentropic Euler equations

SuntoSi studia la stabilità delle discontinuità di contatto per le equazioni di Eulero non isentropiche in dimensione di spazio 2 e 3. Viene presentato un criterio semplice per la stabilità neutrale e l’instabilità violenta.AbstractWe study the stability of contact discontinuities for the nonisentropic Euler equations in two or three space dimensions. A simple criterion predicting neutral stability or violent instability is given.

L p -microlocal regularity for pseudodifferential operators of quasi-homogeneous type†

The authors consider pseudodifferential operators whose symbols have decay at infinity of quasi-homogeneous type and study their behaviour on the wave front set of distributions in weighted Zygmund–Hölder spaces and weighted Sobolev spaces in Lp -framework. Then microlocal properties for solutions to linear partial differential equations with coefficients in weighted Zygmund–Hölder spaces are obtained. †This article is devoted to the special issue ‘Växjö Conference 2008’.

Continuity in Weighted Sobolev Spaces Of Lp Type for Pseudo-Differential Operators with Completely Nonsmooth Symbols

We consider symbolsa(xξ)with a finite number of bounded derivatives with respect to ξ and of weighted Sobolev type inx.Their continuity in weighted Sobolev spaces of sufficiently large order is studied.

Continuity in Quasi-homogeneous Sobolev Spaces for Pseudo-differential Operators with Besov Symbols

In this paper a result of continuity for pseudo-differential operators with non-regular symbols on spaces of quasi-homogeneous type is given. More precisely, the symbols a(x, ξ) take their values in a quasi-homogeneous Besov space with respect to the x variable; moreover a finite number of derivatives with respect to the second variable satisfies, in Besov norm, decay estimates of quasi-homogeneous type.

Continuity in Weighted Besov Spaces for Pseudodifferential Operators with Non-regular Symbols

The authors state and prove a result of continuity in weighted Besov spaces for a class of pseudodifferential operators whose symbol a(x, ξ) admits a finite number of bounded derivatives with respect to ξ and is of weighted Besov type in the x variable.

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.