253 research outputs found
Well-posedness of the plasma-vacuum interface problem
We consider the free boundary problem for the plasma-vacuum interface in
ideal compressible magnetohydrodynamics (MHD). In the plasma region the flow is
governed by the usual compressible MHD equations, while in the vacuum region we
consider the pre-Maxwell dynamics for the magnetic field. At the
free-interface, driven by the plasma velocity, the total pressure is continuous
and the magnetic field on both sides is tangent to the boundary. The
plasma-vacuum system is not isolated from the outside world, because of a given
surface current on the fixed boundary that forces oscillations.
Under a suitable stability condition satisfied at each point of the initial
interface, stating that the magnetic fields on either side of the interface are
not collinear, we show the existence and uniqueness of the solution to the
nonlinear plasma-vacuum interface problem in suitable anisotropic Sobolev
spaces. The proof is based on the results proved in the companion paper
arXiv:1112.3101, about the well-posedness of the homogeneous linearized problem
and the proof of a basic a priori energy estimate. The proof of the resolution
of the nonlinear problem given in the present paper follows from the analysis
of the elliptic system for the vacuum magnetic field, a suitable tame estimate
in Sobolev spaces for the full linearized equations, and a Nash-Moser
iteration.Comment: 58 page
Characteristic boundary value problems: estimates from H1 to L2
Motivated by the study of certain non linear free-boundary value problems for
hyperbolic systems of partial differential equations arising in
Magneto-Hydrodynamics, in this paper we show that an a priori estimate of the
solution to certain boundary value problems, in the conormal Sobolev space
H1_tan, can be transformed into an L2 a priori estimate of the same problem
Stability of the linearized MHD-Maxwell free interface problem
We consider the free boundary problem for the plasma-vacuum interface in
ideal compressible magnetohydrodynamics (MHD). In the plasma region, the flow
is governed by the usual compressible MHD equations, while in the vacuum region
we consider the Maxwell system for the electric and the magnetic fields, in
order to investigate the well-posedness of the problem, in particular in
relation with the electric field in vacuum. At the free interface, driven by
the plasma velocity, the total pressure is continuous and the magnetic field on
both sides is tangent to the boundary.
Under suitable stability conditions satisfied at each point of the
plasma-vacuum interface, we derive a basic a priori estimate for solutions to
the linearized problem. The proof follows by a suitable secondary
symmetrization of the Maxwell equations in vacuum and the energy method.
An interesting novelty is represented by the fact that the interface is
characteristic with variable multiplicity, so that the problem requires a
different number of boundary conditions, depending on the direction of the
front velocity (plasma expansion into vacuum or viceversa). To overcome this
difficulty, we recast the vacuum equations in terms of a new variable which
makes the interface characteristic of constant multiplicity. In particular, we
don't assume that plasma expands into vacuum.Comment: arXiv admin note: substantial text overlap with arXiv:1112.310
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 for the amplitude equation of surface waves
We consider the amplitude equation for nonlinear surface wave solutions of hyperbolic conservation laws. This is an asymptotic nonlocal, Hamiltonian evolution equation with quadratic nonlinearity. For example, this equation describes the propagation of nonlinear Rayleigh waves, surface waves on current-vortex sheets in incompressible MHD and on the incompressible plasma-vacuum interface.
The local-in-time existence of smooth solutions to the Cauchy problem for the amplitude equation in noncanonical variables was shown in a previous article. 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
Weak stability of the plasma-vacuum interface problem
We consider the free boundary problem for the two-dimensional plasma-vacuum
interface in ideal compressible magnetohydrodynamics (MHD). In the plasma
region, the flow is governed by the usual compressible MHD equations, while in
the vacuum region we consider the Maxwell system for the electric and the
magnetic fields. At the free interface, driven by the plasma velocity, the
total pressure is continuous and the magnetic field on both sides is tangent to
the boundary.
We study the linear stability of rectilinear plasma-vacuum interfaces by
computing the Kreiss-Lopatinskii determinant of an associated linearized
boundary value problem. Apart from possible resonances, we obtain that the
piecewise constant plasma-vacuum interfaces are always weakly linearly stable,
independently of the size of tangential velocity, magnetic and electric fields
on both sides of the characteristic discontinuity.
We also prove that solutions to the linearized problem obey an energy
estimate with a loss of regularity with respect to the source terms, both in
the interior domain and on the boundary, due to the failure of the uniform
Kreiss-Lopatinskii condition, as the Kreiss-Lopatinskii determinant associated
with this linearized boundary value problem has roots on the boundary of the
frequency space. In the proof of the a priori estimates, a crucial part is
played by the construction of symmetrizers for a reduced differential system,
which has poles at which the Kreiss-Lopatinskii condition may fail
simultaneously.Comment: 38 page
A Higher-Order Hardy-Type Inequality in Anisotropic Sobolev Spaces
We prove a higher-order inequality of Hardy type for functions in anisotropic Sobolev spaces that vanish at the boundary of the space domain. This is an important calculus tool for the study of initial-boundary-value problems of symmetric hyperbolic systems with characteristic boundary
Nonlinear Stability of Relativistic Vortex Sheets in Three-Dimensional Minkowski Spacetime
We are concerned with the nonlinear stability of vortex sheets for the
relativistic Euler equations in three-dimensional Minkowski spacetime. This is
a nonlinear hyperbolic problem with a characteristic free boundary. In this
paper, we introduce a new symmetrization by choosing appropriate functions as
primary unknowns. A necessary and sufficient condition for the weakly linear
stability of relativistic vortex sheets is obtained by analyzing the roots of
the Lopatinski\u{\i} determinant associated to the constant coefficient
linearized problem. Under this stability condition, we show that the variable
coefficient linearized problem obeys an energy estimate with a loss of
derivatives. The construction of certain weight functions plays a crucial role
in absorbing error terms caused by microlocalization. Based on the weakly
linear stability result, we establish the existence and nonlinear stability of
relativistic vortex sheets under small initial perturbations by a Nash--Moser
iteration scheme.Comment: 105 pages; to appear in: Arch. Ration. Mech. Anal. 201
Regularity of weakly well posed hyperbolic mixed problems with characteristic boundary
We study the mixed initial-boundary value problem for a linear hyperbolic system with characteristic boundary of constant multiplicity. We assume the problem to be “weakly” well posed, in the sense that a unique L^2-solution exists, for sufficiently smooth data, and obeys an a priori energy estimate with a finite loss of conormal regularity. This is the case of problems that do not satisfy the uniform Kreiss–Lopatinskiı̆ condition in the hyperbolic region of the frequency domain. Under the assumption of the loss of one conormal derivative we obtain the regularity of solutions, in the natural framework of weighted anisotropic Sobolev spaces, provided the data are sufficiently smooth
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