171 research outputs found

    Global Solutions for the Gravity Water Waves Equation in Dimension 3

    Get PDF
    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

    Full text link
    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 R2\mathbb{R}^2 into S2S^2.Comment: 14 pages, 5 figures, final version to be published in Nonlinearit

    Singularity Formation in 2+1 Wave Maps

    Full text link
    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

    Full text link
    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 T3T^3-Gowdy symmetric IIB superstring cosmology

    Full text link
    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

    Full text link
    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

    Full text link
    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

    Full text link
    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

    Full text link
    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

    Full text link
    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
    • …