DispersiveWiki:About
Welcome to DispersiveWiki!
This collection of web pages is concerned with the local and global well-posedness of various non-linear dispersive and wave equations. An equation is locally well-posed (LWP) if, for any data in a given regularity class, there exists a time of existence T and a unique solution to the Cauchy problem for that data which depends continuously on the data (with respect to the original regularity class). We usually expect the solution to have some additional regularity properties (and the uniqueness result is usually phrased assuming those additional regularity properties). An equation is globally well-posed (GWP) if one can take T arbitrarily large.
The ambition of these pages is to try to summarize the state of the art concerning the local and global well-posedness of common dispersive and wave equations, particularly with regard to the question of low regularity data. We'll try also to collect references.html a bibliography for these results, with hyper-links whenever available. As secondary goals, we hope to compile a little bit of background about each of these equations, pose some interesting open problems, address some related problems (persistence of regularity, scattering, polynomial growth of norms, nature of blowup, stability of special solutions, etc.), and collect some survey articles on the general theory of LWP and GWP for these equations. However, to stop the project from getting completely out of control, we will initially concentrate on the LWP and GWP results for low regularity data. As such, the results gathered here are only a small fraction of the vast amount of work done on these equations.
The ultimate aim is for these pages will be complete, 100% accurate, and up-to-date. At present, they are far from being so in all three respects. Undoubtedly many important contributions have been omitted, misquoted, or misattributed, and one should always check the claims found here against the original source material whenever possible. If you discover an error of any sort, please #email e-mail us!
Any suggestions, notifications of new papers, and/or corrections are very welcome, and can be sent #email by e-mail. Anyone who wishes to submit some discussion or background for an equation or problem, or to pose some interesting conjectures or open problems, is very welcome to do so, and their contribution will be attributed appropriately.
Contact information
These pages are maintained jointly by [Jim Colliander], [Mark Keel], [Gigliola Staffilani], [Hideo Takaoka], and [Terry Tao]. Technical issues concerning web-page problems, etc. should be addressed to [Terry Tao].
