Local Regularity and Long Time Behavior of Solutions on Non-Linear Evolution Equations

Princeton University, Princeton NJ

Investigators

Abstract

Proposal DMS-04006627 Title: Local regularity and long time behavior of solutions of nonlinear evolution equations PI: Igor Rodnianski, Princeton University ABSTRACT The focus of this proposal is the study of the local regularity and global behavior of solutions of the Einstein equations. The local behavior is encoded in the propagation of local regularity in the scale of the L2 based Sobolev spaces. We will attack the L2 curvature conjecture, which asserts that a solution, metric g, of the Einstein vacuum equations Ric(g)=0 can be locally extended as long as its curvature tensor is bounded in L2. The problem is beyond the range of application of the standard Fourier analysis based methods and thus will require development of new analytic tools. The global behavior will be studied in the context of the problem of stability of Minkowski space in the wave coordinate gauge. This particular gauge is used in physics to construct the post-Minkowskian approximations and we hope to provide an additional insight into validity of these expansions. The stability problem belongs to the category of global existence for solutions with small data problems for the equations not satisfying the standard null condition. The Einstein equations of General Relativity provide the main classical description of evolution of the physical space-time continuum. The study of mathematical and physical phenomena arising in General Relativity is of the fundamental importance. While the physical understanding of the subject has made rapid advancement and generated a number of outstanding conjectures the rigorous mathematical picture is yet to emerge. The latter is to a large extent due to a highly nonlinear nature of the Einstein equations and a lack of the mathematical tools to deal with it. The development of a satisfactory mathematical approach lies on the interface between Analysis, Geometry and Partial Differential Equations.