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Dispersive PDE Wiki

Local and global well-posedness for non-linear dispersive and wave equations

Maintained by Jim Colliander, Mark Keel, Gigliola Staffilani, Hideo Takaoka, and Terry Tao

Disclaimer: Although we have tried our best to make all attributions accurate, it is inevitable that there are some omissions and misattributions in this page. These pages should be considered as a work in progress. Please [#email notify us] of any errors!

Purpose of this page

Bibliography

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The big three: Wave, Schrodinger, KdV

Wave equations	Schrodinger equations	KdV equations
Wave estimates wave:Linear estimates wave:Bilinear estimates wave:Multilinear estimates	Schrodinger estimates schrodinger:Linear estimates schrodinger:Bilinear estimates schrodinger:Trilinear estimates schrodinger:Multilinear estimates	kdv:Linear estimates kdv:Bilinear estimates kdv:Trilinear estimates kdv:Multilinear estimates
[wave:semilinear Semilinear NLW/NLKG]] wave:Scattering for NLW references:Scattering for NLKG wave:Sine-Gordon		kdv:gKdV on R^+
[[wave:nlw-2 Quadratic NLW/NLKG]	[[schrodinger:Quadratic_NLS_on_R NLS-2 on R] schrodinger:Quartic_NLS_on_R^2 NLS-2 on R^2 schrodinger:Quadratic_NLS_on_R^3 NLS-2 on R^3 schrodinger:Quadratic_NLS_on_T NLS-2 on T schrodinger:Quadratic_NLS_on_T^2 NLS-2 on T^2	kdv:kdv KdV (gKdV-1) kdv:kdv on R KdV on R kdv:kdv on T KdV on T kdv:kdv on R+ KdV on R^+
Cubic NLW/NLKG wave:nlw-3 on R NLW-3 on R wave:nlw-3 on R^2 NLW-3 on R^2 wave:nlw-3 on R^3 NLW-3 on R^3	Cubic NLS schrodinger:Cubic_NLS_on_R NLS-3 on R schrodinger:Cubic_NLS_on_R^2 NLS-3 on R^2 schrodinger:Cubic_NLS_on_R^3 NLS-3 on R^3 schrodinger:Cubic_NLS_on_R^4 NLS-3 on R^4 schrodinger:Cubic_NLS_on_T NLS-3 on T schrodinger:Cubic_NLS_on_T^2 NLS-3 on T^2 schrodinger:Cubic_NLS_on_T^4 NLS-3 on T^4 schrodinger:Cubic_NLS_on_RxT NLS-3 on R x T schrodinger:Cubic_NLS_on_S^6 NLS-3 on S^6	kdv:mkdv Modified KdV (gKdV-2) kdv:mKdV on R mKdV on R kdv:mKdV on T mKdV on T
	Quartic NLS schrodinger:Quartic_NLS_on_R NLS-4 on R schrodinger:Quartic_NLS_on_T NLS-4 on T schrodinger:Quartic_NLS_on_R^2 NLS-4 on R^2	gKdV-3 kdv:gKdV_3 on R gKdV-3 on R kdv:gKdV_3 on T gKdV-3 on T
Quintic NLW/NLKG wave:nlw-5 on R NLW-5 on R wave:nlw-5 on R^2 NLW-5 on R^2 wave:nlw-5 on R^3 NLW-5 on R^3	Quintic NLS schrodinger:Quintic_NLS_on_R NLS-5 on R schrodinger:Quintic_NLS_on_R^2 NLS-5 on R^2 schrodinger:Quintic_NLS_on_R^3 NLS-5 on R^3 schrodinger:Quintic_NLS_on_T NLS-5 on T	gKdV-4 kdv:gKdV_4 on R gKdV-4 on R kdv:gKdV_4 on T gKdV-4 on T
Septic NLW/NLKG wave.htm#nlw-7 on R NLW-7 on R wave.htm#nlw-7 on R^2 NLW-7 on R^2 wave.htm#nlw-7 on R^3 NLW-7 on R^3	Septic NLS schrodinger:Septic_NLS_on_R NLS-7 on R schrodinger:Septic_NLS_on_R^2 NLS-7 on R^2 schrodinger:Septic_NLS_on_R^3 NLS-7 on R^3
wave:gwp d-nlw GWP for D-NLW wave:gwp d-nlkg GWP for D-NLKG
wave:mkg MKG wave:mkg on R MKG on R wave:mkg on R^2 MKG on R^2 wave:mkg on R^3 MKG on R^3 wave:mkg on R^4 MKG on R^4 wave:mkg on R^5+ MKG on R^5+ wave:YM Yang Mills references:YM on R^2 YM on R^2 wave:YM on R^3 YM on R^3 wave:YM on R^4 YM on R^4 wave:ym on R^5+ YM on R^5+ wave:YMH on R^3 YMH on R^3 wave:Maxwell-Dirac Maxwell-Dirac wave:Dirac-Klein-Gordon Dirac-Klein-Gordon wave:two-speed DNLW Two-speed DNLW-2	Quadratic DNLS schrodinger:Maxwell-Schrodinger Maxwell-Schrodinger
	Cubic DNLS schrodinger:dnls-3 on R DNLS-3 on R
wave:wm Wave maps wave:wm on R WM on R wave:wm on R^2 WM on R^2 wave:two-speed DDNLW Two-speed DDNLW	DDNLS schrodinger:smaps Schrodinger Maps
wave:gwp qnlw GWP for QNLW wave:Einstein Einstein equations	Quasilinear NLS

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misc.html Other equations:

• Schrodinger-Ginsberg-Landau
• misc:Vlasov-Maxwell Vlasov-Maxwell
• misc:KP The Kadomtsev-Petvisasvhili equations
  • misc:KP-I Kadomtsev-Petviashvili I
  • misc:KP-II Kadomtsev-Petviashvili II
• misc:Zakharov Zakharov
  • misc:Zakharov-1 Zakharov on R and T
  • misc:Zakharov-2 Zakharov on R^2
  • misc:Zakharov-3 Zakharov on R^3
  • Klein-Gordon-Zakharov
  • misc:WS Other Wave-Schrodinger systems
    • Ishimori
    • Davey-Stewartson
    • Yukawa

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Purpose of this page:

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.

Thanks to Oliver Schnuere, we have now found [some tools to represent (some) mathematical symbols in HTML]. We will slowly begin prettifying these pages accordingly. Further suggestions as to how to improve the presentation are still appreciated, though.

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Contact:

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].