171 research outputs found

### Global Solutions for the Gravity Water Waves Equation in Dimension 3

We show existence of global solutions for the gravity water waves equation in
dimension 3, in the case of small data. The proof combines energy estimates,
which yield control of L^2 related norms, with dispersive estimates, which give
decay in L^\infty. To obtain these dispersive estimates, we use an analysis in
Fourier space; the study of space and time resonances is then the crucial
point

### Formation of singularities for equivariant 2+1 dimensional wave maps into the two-sphere

In this paper we report on numerical studies of the Cauchy problem for
equivariant wave maps from 2+1 dimensional Minkowski spacetime into the
two-sphere. Our results provide strong evidence for the conjecture that large
energy initial data develop singularities in finite time and that singularity
formation has the universal form of adiabatic shrinking of the degree-one
harmonic map from $\mathbb{R}^2$ into $S^2$.Comment: 14 pages, 5 figures, final version to be published in Nonlinearit

### Singularity Formation in 2+1 Wave Maps

We present numerical evidence that singularities form in finite time during
the evolution of 2+1 wave maps from spherically equivariant initial data of
sufficient energy.Comment: 5 pages, 3 figure

### Renormalization and blow up for charge one equivariant critical wave maps

We prove the existence of equivariant finite time blow up solutions for the
wave map problem from 2+1 dimensions into the 2-sphere. These solutions are the
sum of a dynamically rescaled ground-state harmonic map plus a radiation term.
The local energy of the latter tends to zero as time approaches blow up time.
This is accomplished by first "renormalizing" the rescaled ground state
harmonic map profile by solving an elliptic equation, followed by a
perturbative analysis

### Global existence problem in $T^3$-Gowdy symmetric IIB superstring cosmology

We show global existence theorems for Gowdy symmetric spacetimes with type
IIB stringy matter. The areal and constant mean curvature time coordinates are
used. Before coming to that, it is shown that a wave map describes the
evolution of this system

### Scattering for the Zakharov system in 3 dimensions

We prove global existence and scattering for small localized solutions of the
Cauchy problem for the Zakharov system in 3 space dimensions. The wave
component is shown to decay pointwise at the optimal rate of t^{-1}, whereas
the Schr\"odinger component decays almost at a rate of t^{-7/6}.Comment: Minor changes and referee's comments include

### Slow equivariant lump dynamics on the two sphere

The low-energy, rotationally equivariant dynamics of n CP^1 lumps on S^2 is
studied within the approximation of geodesic motion in the moduli space of
static solutions. The volume and curvature properties of this moduli space are
computed. By lifting the geodesic flow to the completion of an n-fold cover of
the moduli space, a good understanding of nearly singular lump dynamics within
this approximation is obtained.Comment: 12 pages, 3 figure

### A remark on asymptotic completeness for the critical nonlinear Klein-Gordon equation

We give a short proof of asymptotic completeness and global existence for the
cubic Nonlinear Klein-Gordon equation in one dimension. Our approach to dealing
with the long range behavior of the asymptotic solution is by reducing it, in
hyperbolic coordinates to the study of an ODE. Similar arguments extend to
higher dimensions and other long range type nonlinear problems.Comment: To appear in Lett. Math. Phy

### Scattering below critical energy for the radial 4D Yang-Mills equation and for the 2D corotational wave map system

We describe the asymptotic behavior as time goes to infinity of solutions of
the 2 dimensional corotational wave map system and of solutions to the 4
dimensional, radially symmetric Yang-Mills equation, in the critical energy
space, with data of energy smaller than or equal to a harmonic map of minimal
energy. An alternative holds: either the data is the harmonic map and the
soltuion is constant in time, or the solution scatters in infinite time

### Dispersion and collapse of wave maps

We study numerically the Cauchy problem for equivariant wave maps from 3+1
Minkowski spacetime into the 3-sphere. On the basis of numerical evidence
combined with stability analysis of self-similar solutions we formulate two
conjectures. The first conjecture states that singularities which are produced
in the evolution of sufficiently large initial data are approached in a
universal manner given by the profile of a stable self-similar solution. The
second conjecture states that the codimension-one stable manifold of a
self-similar solution with exactly one instability determines the threshold of
singularity formation for a large class of initial data. Our results can be
considered as a toy-model for some aspects of the critical behavior in
formation of black holes.Comment: 14 pages, Latex, 9 eps figures included, typos correcte

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