78 research outputs found
Asymptotic pointwise behavior for systems of semilinear wave equations in three space dimensions
In connection with the weak null condition, Alinhac introduced a sufficient
condition for global existence of small amplitude solutions to systems of
semilinear wave equations in three space dimensions. We introduce a slightly
weaker sufficient condition for the small data global existence, and we
investigate the asymptotic pointwise behavior of global solutions for systems
satisfying this condition. As an application, the asymptotic behavior of global
solutions under the Alinhac condition is also derived.Comment: 56 pages, the final versio
Global Solutions for Incompressible Viscoelastic Fluids
We prove the existence of both local and global smooth solutions to the
Cauchy problem in the whole space and the periodic problem in the n-dimensional
torus for the incompressible viscoelastic system of Oldroyd-B type in the case
of near equilibrium initial data. The results hold in both two and three
dimensional spaces. The results and methods presented in this paper are also
valid for a wide range of elastic complex fluids, such as magnetohydrodynamics,
liquid crystals and mixture problems.Comment: We prove the existence of global smooth solutions to the Cauchy
problem for the incompressible viscoelastic system of Oldroyd-B type in the
case of near equilibrium initial dat
Wave equation with concentrated nonlinearities
In this paper we address the problem of wave dynamics in presence of
concentrated nonlinearities. Given a vector field on an open subset of
\CO^n and a discrete set Y\subset\RE^3 with elements, we define a
nonlinear operator on L^2(\RE^3) which coincides with the free
Laplacian when restricted to regular functions vanishing at , and which
reduces to the usual Laplacian with point interactions placed at when
is linear and is represented by an Hermitean matrix. We then consider the
nonlinear wave equation and study the
corresponding Cauchy problem, giving an existence and uniqueness result in the
case is Lipschitz. The solution of such a problem is explicitly expressed
in terms of the solutions of two Cauchy problem: one relative to a free wave
equation and the other relative to an inhomogeneous ordinary differential
equation with delay and principal part . Main properties of
the solution are given and, when is a singleton, the mechanism and details
of blow-up are studied.Comment: Revised version. To appear in Journal of Physics A: Mathematical and
General, special issue on Singular Interactions in Quantum Mechanics:
Solvable Model
Constraint and gauge shocks in one-dimensional numerical relativity
We study how different types of blow-ups can occur in systems of hyperbolic
evolution equations of the type found in general relativity. In particular, we
discuss two independent criteria that can be used to determine when such
blow-ups can be expected. One criteria is related with the so-called geometric
blow-up leading to gradient catastrophes, while the other is based upon the
ODE-mechanism leading to blow-ups within finite time. We show how both
mechanisms work in the case of a simple one-dimensional wave equation with a
dynamic wave speed and sources, and later explore how those blow-ups can appear
in one-dimensional numerical relativity. In the latter case we recover the well
known ``gauge shocks'' associated with Bona-Masso type slicing conditions.
However, a crucial result of this study has been the identification of a second
family of blow-ups associated with the way in which the constraints have been
used to construct a hyperbolic formulation. We call these blow-ups ``constraint
shocks'' and show that they are formulation specific, and that choices can be
made to eliminate them or at least make them less severe.Comment: 19 pages, 8 figures and 1 table, revised version including several
amendments suggested by the refere
Strichartz estimates on Schwarzschild black hole backgrounds
We study dispersive properties for the wave equation in the Schwarzschild
space-time. The first result we obtain is a local energy estimate. This is then
used, following the spirit of earlier work of Metcalfe-Tataru, in order to
establish global-in-time Strichartz estimates. A considerable part of the paper
is devoted to a precise analysis of solutions near the trapping region, namely
the photon sphere.Comment: 44 pages; typos fixed, minor modifications in several place
Conservation laws and symmetries of quasilinear radial wave equations in multi-dimensions
Symmetries and conservation laws are studied for two classes of physically
and analytically interesting radial wave equations with power nonlinearities in
multi-dimensions. The results consist of two main classifications: all
symmetries of point type and all conservation laws of a general energy-momentum
type are explicitly determined, including those such as dilations, inversions,
similarity energies and conformal energies that exist only for special powers
or dimensions. In particular, all variational cases (when a Lagrangian
formulation exists) and non-variational cases (when no Lagrangian exists) for
these wave equations are considered. As main results, the classification yields
generalized energies and radial momenta in certain non-variational cases, which
are shown to arise from a new type of Morawetz dilation identity that produces
conservation laws for each of the two wave equations in a different way than
Noether's theorem.Comment: Typos corrected in published version, 38 pages. Lagrangian
functionals now include missing integration over the time variabl
A smooth introduction to the wavefront set
The wavefront set provides a precise description of the singularities of a
distribution. Because of its ability to control the product of distributions,
the wavefront set was a key element of recent progress in renormalized quantum
field theory in curved spacetime, quantum gravity, the discussion of time
machines or quantum energy inequalitites. However, the wavefront set is a
somewhat subtle concept whose standard definition is not easy to grasp. This
paper is a step by step introduction to the wavefront set, with examples and
motivation. Many different definitions and new interpretations of the wavefront
set are presented. Some of them involve a Radon transform.Comment: 29 pages, 7 figure
The Cauchy Problem for the Einstein Equations
Various aspects of the Cauchy problem for the Einstein equations are
surveyed, with the emphasis on local solutions of the evolution equations.
Particular attention is payed to giving a clear explanation of conceptual
issues which arise in this context. The question of producing reduced systems
of equations which are hyperbolic is examined in detail and some new results on
that subject are presented. Relevant background from the theory of partial
differential equations is also explained at some lengthComment: 98 page
On universality of critical behavior in the focusing nonlinear Schr\uf6dinger equation, elliptic umbilic catastrophe and the Tritronqu\ue9e solution to the Painlev\ue9-I equation
We argue that the critical behavior near the point of "gradient catastrophe" of the solution to the Cauchy problem for the focusing nonlinear Schrodinger equation i epsilon Psi(t) + epsilon(2)/2 Psi(xx) + vertical bar Psi vertical bar(2)Psi = 0, epsilon << 1, with analytic initial data of the form Psi( x, 0; epsilon) = A(x)e(i/epsilon) (S(x)) is approximately described by a particular solution to the Painleve-I equation
Theorems on existence and global dynamics for the Einstein equations
This article is a guide to theorems on existence and global dynamics of
solutions of the Einstein equations. It draws attention to open questions in
the field. The local-in-time Cauchy problem, which is relatively well
understood, is surveyed. Global results for solutions with various types of
symmetry are discussed. A selection of results from Newtonian theory and
special relativity that offer useful comparisons is presented. Treatments of
global results in the case of small data and results on constructing spacetimes
with prescribed singularity structure or late-time asymptotics are given. A
conjectural picture of the asymptotic behaviour of general cosmological
solutions of the Einstein equations is built up. Some miscellaneous topics
connected with the main theme are collected in a separate section.Comment: Submitted to Living Reviews in Relativity, major update of Living
Rev. Rel. 5 (2002)
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