Hans Lindblad

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 the Schwarzschild 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 weak null condition and showed that the Einstein equations in harmonic 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.

Counterexamples to local existence for semi-linear wave equations

In this paper we study how much regularity of initial data is needed to ensure existence of a local solution to a semi-linear wave equation. We give counterexamples to local existence for the typical model equations. The counterexamples we construct are sharp, i.e. one does have a local solution if the data has slightly more regularity.

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 a global solution for sufficiently small initial data.

Global Solutions of Quasilinear Wave Equations

We show global small data existence for a class of quasilinear wave equations related to Einsteins equations in harmonic coordinates. These equations do not satisfy the classical null condition and the asymptotic behavior of solutions is not free but the light cones bend at infinity.

The weak null condition for Einstein's equations

Abstract We show that Einsteins equations of General Relativity expressed in wave coordinates satisfy a ‘weak null condition’. In a forthcoming article we will use this to prove a global existence result for Einsteins equations in wave coordinates with small initial data. To cite this article: H. Lindblad, I. Rodnianski, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

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.

The Strauss conjecture on Kerr black hole backgrounds

We examine solutions to semilinear wave equations on black hole backgrounds and give a proof of an analog of the Strauss conjecture on the Schwarzschild and Kerr, with small angular momentum, black hole backgrounds. The key estimates are a class of weighted Strichartz estimates, which are used near infinity where the metrics can be viewed as small perturbations of the Minkowski metric, and a localized energy estimate on the black hole background, which handles the behavior in the remaining compact set.

Scattering and small data completeness for the critical nonlinear Schrödinger equation

We prove asymptotic completeness of one-dimensional NLS with long range nonlinearities. We also prove existence and expansion of asymptotic solutions with large data at infinity.

A REMARK ON LONG RANGE SCATTERING FOR THE NONLINEAR KLEIN–GORDON EQUATION

We consider the scattering problem for the nonlinear Klein–Gordon Equation with long range nonlinearity in one dimension. We prove that for all prescribed asymptotic solutions there is a solution of the equation with such behavior, for some choice of initial data. In the case the nonlinearity has the good sign (repulsive) the result hold for arbitrary size asymptotic data. The method of proof is based on reducing the long range phase effects to an ODE; this is done via an appropriate ansatz. We also find the complete asymptotic expansion of the solutions.

On the Asymptotic Behavior of Solutions to the Einstein Vacuum Equations in Wave Coordinates

We give asymptotics for the 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.