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

Long range scattering for the cubic Dirac equation on R1+1\mathbf{R}^{1+1}

We show that the cubic Dirac equation, also known as the Thirring model, scatters at infinity to a linear solution modulo a phase correction.Comment: References updated and minor changes to introductio