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 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 have initially concentrated 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 don't hesitate to correct it!
More generally, any suggestions, notifications of new papers, and/or corrections are very welcome. 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 is encouraged to visit the Community Portal and Current events pages, where current issues and projects are being discussed.
- This wiki was derived from an earlier HTML version maintained jointly by the I-team, initially created in Oct 2000, and based in turn on an earlier prototype used internally by the I-team. That version has not been updated since roughly 2004 and should now be considered obsolete.
- In July 2006 James Colliander, Marco de la Cruz-Heredia and Emile Leblanc set up the Linux box which hosts the Dispersive wiki. The HTML version was passed through an HTML to wiki converter. The conversion required substantial editing of links and mathematical expressions. Mathematical formulae have been edited to exploit the Texvc (TeX validator and coder) capabilities of the MediaWiki software.
- On 31 October 2008, James Colliander and Marco de la Cruz-Heredia moved the dispersive wiki to a new linux box. The underlying database and mediawiki software is now upgraded to current versions. We retain the old machine and plan to eventually set it up as a backup machine.
- The dispersive wiki will sometimes close between 04:00a.m. EST and 04:30a.m. EST for backup operations. Colliand 19:19, 31 October 2008 (UTC)