Phase Space Analysis of Evolution Equations

The main purpose of the proposed research is to analyse global space time properties of dispersive partial differential equations. There are several aspects of such analysis. First, the global analysis of linear equations is crucial in both local and global problems for nonlinear evolution equations. Second, in global problems one finds many important relations between problems in partial differential equations (PDEs) and the underlying geometry. Equations under consideration include hyperbolic equations, hyperbolic systems with or without multiplicities, single and coupled Schrodinger type equations, relativistic equations, Klein-Gordon, KdV and many others. Such equations are all called dispersive equations because there are many similarities in the behaviour of their solutions exhibiting instances of the so-called dispersion (of energy, moments, singularities, or of other information).Local qualitative properties of linear equations have been studied for decades with many important and fascinating discoveries. However, for their nonlinear versions one needs global quantitative information on the behaviour of linearised equations, and here almost no results are available in general. The proposed project suggests a new unified approach to these problems based on the new area of ``global microlocal analysis'' which deals with global properties of so-called Fourier integral operators (FIOs) and which allows to go far beyond the known spectral and other methods.These operators (FIOs) have been used in the local theories for over 35 years and proved to be very efficient since they encode many analytic and geometric properties of equations. For example, solutions to Cauchy problems for hyperbolic equations, transformations operators between different types of dispersive equations, etc., can all be reduced to the form of Fourier integral operators or their relevant extensions. The first aim of this project is to analyse required global (space and time) properties of Fourier integral type operators. These properties have been successfully studied so far in a number of special cases only under very restrictive conditions on the operator (partly because they were not realised in the form of FIOs). However, recent research indicates that it should be possible to treat the general case of nondegenerate Fourier integral operators by combining recent developments in the local regularity theory with new approaches for establishing global estimates. Global estimates for these operators are of crucial importance for nonlinear problems but were largely unapproachable in the past.It is expected that the new approach described in this proposal will allow me to deal with equations with variable coefficients which is nowadays one of the main challenges of the whole area. Present methods coming from spectral theory or from harmonic analysis generally fail when dealing with variable coefficients. At the same time the approach that I propose here is very well suited for it. In fact, already for some classes of equations it allowed to recover and improve most of the results that can be obtained with other approaches, and go far beyond!Another part of the project is to use all this as well as other recently discovered ideas and techniques to investigate dispersive, Strichartz, and smoothing estimates for dispersive equations with variable coefficients and lower order terms, and relations between them. The obtained results will be applied to local and global well-posedness questions of nonlinear hyperbolic, Schrodinger and other dispersive equations.It is important and challenging research with deep implications in theories of linear and nonlinear dispersive equations and their relation to geometry and other areas. The research will be undertaken at the Mathematics Department of Imperial College, while collaboration with other mathematicians on some aspects of this project is expected.