Global Behaviour of Critical Nonlinear PDE

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Global Behavior of Critical Nonlinear PDEAbstract of Proposed ResearchTerence TaoThis project is to study (and hopefully solve) the global regularity question for the Cauchy problem for several well-known, critical, nonlinear, dispersive and wave equations. They include the two-dimensional wave maps equation into hyperbolic space, the mass-critical defocusing nonlinear Schrodinger equation and the mass-critical generalized Korteweg-de Vries equation. The very recent breakthroughs on this area, including Bourgain's induction-on-energy argument and the successes of concentration-compactness methods, as well as the recently completely resolution of the global regularity problem for the energy-critical nonlinear Schrodinger equation suggest that the resolution of these problems are now within reach.Many wave phenomena in physics (e.g. light, water, sound, gravity, etc.) are described using nonlinear partial differential equations. These equations often encode a struggle between dispersion, which acts to spread out the wave and make it decay over time, and nonlinearity, which can instead cause the wave to concentrate and even to develop singularities (or "blow up") in relatively short periods of time. An important class of equations are the "critical" equations, in which the dispersion and the nonlinearity are in some sense exactly balanced against each other. It is generally believed that if the nonlinearity has a "defocusing" nature then the dispersion should eventually "win", and no singularities will form, whereas the converse should be true in the "focusing" case. Until very recently, this intuition was only confirmed for a handful of critical equations but, in the last few years, some powerful new technical tools have been developed which should now allow us to prove results about a much larger range of critical equations. This project will pursue such issues for some specific and well-known equations arising in physics.