Abstract of Proposed Research Terence Tao
This 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.
Many basic physical systems, such as plasmas or shallow water waves, are governed by a class of partial differential equations known as nonlinear dispersive equations. These equations give a mathematical framework to study the struggle between two types of physical phenomena. On the one hand, there is dispersion, which causes disturbances (such as waves on an ocean surface) to radiate away in space, with the amplitude decaying over time. On the other hand, there is nonlinearity, which can cause a large disturbance to interact with itself to create even larger disturbances, and which in some cases could even lead to an infinite amplitude (or velocity) in a finite amount of time, a scenario known as blowup. At the opposite extreme, there are some equations for which blowup cannot occur no matter how large the choice of initial data, and in such cases we say that the equation exhibits global regularity. One of the basic problems in the mathematical study of nonlinear dispersive equations is to determine which equations exhibit blowup, and which ones exhibit global regularity. The answer depends on the so-called criticality of the equation. For some equations, the dispersive effects are stronger than the nonlinear effects at fine scales; such equations are called subcritical, and they tend to exhibit global regularity. For other equations, it is the nonlinear effects that are stronger at fine scales; such equations are called supercritical, and they tend to exhibit very complicated behavior, such as turbulence and blowup, although obtaining a rigorous description of solutions to supercritical equations is still a major and largely unsolved problem in the subject. However, in recent years there has been a lot of progress on understanding the boundary case of critical equations, in which the nonlinearity and dispersion are of equal strength at all spatial scales. Such equations were notoriously difficult to solve for even ten years ago, except in special cases such as the perturbative regime when the solution was very small. However, thanks to the work of many researchers including the PI, we now have a satisfactory general theory of critical dispersive equations, and for most of the basic such equations studied, we can rigorously determine which of them exhibit global regularity and which ones exhibit blowup. Current research in this area is now building upon this foundation by studying more complicated equations, and understanding the extent to which simplified model equations actually emerge as limits of more complicated, but more realistic models. The PI's work in this area focused particularly on two important examples of critical nonlinear PDE, the critical nonlinear Schrodinger equation and the critical wave maps equations. Over the lifetime of this project, both working individually and in collaboration with other mathematicians, the PI obtained global regularity results for many important instances of these equations, and helped develop what is now the standard concentration compactness" method for establishing such results, by focusing attention on the minimal energy (or minimal mass) blowup solution, if such exists. The PI continues to work on related problems involving the Navier-Stokes equations, which are of fundamental importance in fluid dynamics.