165 research outputs found
Global solutions of quasilinear wave equations
We show that a general class of quasilinear wave equations have global
solutions for small initial data as we conjectured in an earlier paper
Well-posedness for the Linearized Motion of an Incompressible Liquid with Free Surface Boundary
We study the problem of the motion of the free surface of a liquid. We prove
existence and stability for the linearized equations
Well-posedness for the linearized motion of a compressible liquid with free surface boundary
We study the problem of the motion of the free surface of a compressible
fluid. We prove existence for the linearized equations
On the asymptotic behavior of solutions to Einstein's vacuum equations in wave coordinates
We give asymptotics for Einstein vacuum equations in wave coordinates with
small asymptotically flat data. We show that the behavior is wave like at null
infinity and homogeneous towards time like infinity. We use the asymptotics to
show that the outgoing null hypersurfaces approach the Schwarzschild ones for
the same mass and that the radiated energy is equal to the initial mass
A remark on Global existence for small initial data of the minimal surface equation in Minkowskian space time
We show that the nonlinear wave equation corresponding to the minimal surface
equation in Minkowski space time has global solutions for sufficiently small
initial data. This is an interesting model in Lorentziann and is also the
equation for a membrane in field theory
Scattering for the Klein-Gordon equation with quadratic and variable coefficient cubic nonlinearities
We study the 1D Klein-Gordon equation with variable coefficient cubic
nonlinearity. This problem exhibits a striking resonant interaction between the
spatial frequencies of the nonlinear coefficients and the temporal oscillations
of the solutions. In the case where the worst of this resonant behavior is
absent, we prove L-Infinity scattering as well as a certain kind of strong
smoothness for the solution at time-like infinity with the help of several new
normal-forms transformations. Some explicit examples are also given which
suggest qualitatively different behavior in the case where the strongest cubic
resonances are present.Comment: 50 pages; revised version corrected typos and added some reference
Global stability of Minkowski space for the Einstein--Vlasov system in the harmonic gauge
Minkowski space is shown to be globally stable as a solution to the massive
Einstein--Vlasov system. The proof is based on a harmonic gauge in which the
equations reduce to a system of quasilinear wave equations for the metric,
satisfying the weak null condition, coupled to a transport equation for the
Vlasov particle distribution function. Central to the proof is a collection of
vector fields used to control the particle distribution function, a function of
both spacetime and momentum variables. The vector fields are derived using a
general procedure, are adapted to the geometry of the solution and reduce to
the generators of the symmetries of Minkowski space when restricted to acting
on spacetime functions. Moreover, when specialising to the case of vacuum, the
proof provides a simplification of previous stability works
A Remark on Long Range Scattering for the nonlinear Klein-Gordon equation
We consider the problem of scattering for the long range critical nonlinear
Klein-Gordon in one space dimension
A sharp counterexample to local existence of low regularity solutions to Einstein's equations in wave coordinates
We are concerned with how regular initial data have to be to ensure local
existence for Einstein's equations in wave coordinates. Klainerman-Rodnianski
and Smith-Tataru showed that there in general is local existence for data in
Sobolev spaces H^s with regularity s>2. We give an example of data in Sobolev
spaces with regularity s=2 for which there is no local solution in this space
Global existence for the Einstein vacuum equations in wave coordinates
We prove global stability of Minkowski space for the Einstein vacuum
equations in harmonic (wave) coordinate gauge for the set of restricted data
coinciding with Schwartzschild solution in the neighborhood of space-like
infinity. The result contradicts previous beliefs that wave coordinates are
"unstable in the large" and provides an alternative approach to the stability
problem originally solved (for unrestricted data, in a different gauge and with
a precise description of the asymptotic behavior at null infinity) by D.
Christodoulou and S. Klainerman.
Using the wave coordinate gauge we recast the Einstein equations as a system
of quasilinear wave equations and, in absence of the classical null condition,
establish a small data global existence result. In our previous work we
introduced the notion of a eak nul condition and showed that the Einstein
equations in wave coordinates satisfy this condition. The result of this paper
relies on this observation and combines it with the vector field method based
on the symmetries of the standard Minkowski space.
In a forthcoming paper we will address the question of stability of Minkowski
space for the Einstein vacuum equations in wave coordinates for all "small"
asymptotically flat data and the case of the Einstein equations coupled to a
scalar field.Comment: 64 pages, no figures, submitted, CMP Minor changes and corrections
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