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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 Purpose of this page]

[references.html Bibliography]

The big three: Wave, Schrodinger, KdV

[wave.html Wave equations]

[schrodinger.html Schrodinger equations]

[kdv.html#overview KdV equations]

[wave.html#wave Wave estimates]

  • [wave.html#linear Linear estimates]
  • [wave.html#bilinear Bilinear estimates]
  • [wave.html#multilinear Multilinear estimates]

[schrodinger.html#Schrodinger Schrodinger estimates]

  • [schrodinger.html#linear Linear estimates]
  • [schrodinger.html#bilinear Bilinear estimates]
  • [schrodinger.html#trilinear Trilinear estimates]
  • [schrodinger.html#multilinear Multilinear estimates]

[kdv.html#Airy Airy estimates]

  • [kdv.html#kdv linear Linear estimates]
  • [kdv.html#kdv bilinear Bilinear estimates]
  • [kdv.html#kdv trilinear Trilinear estimates]
  • [kdv.html#kdv multilinear Multilinear estimates]

[wave.html#semilinear Semilinear NLW/NLKG]

  • [wave.html#scat nlw Scattering for NLW]
  • [references.html#scat nlkg Scattering for NLKG]
  • [wave.html#Sine-Gordon Sine-Gordon]

[schrodinger.html#nls Semilinear NLS]

[kdv.html#overview gKdV]

  • [kdv.html#gKdV on R+ gKdV on R^+]

[wave.html#nlw-2 Quadratic NLW/NLKG]

[schrodinger.html#Quadratic_NLS Quadratic NLS]

  • [schrodinger.html#Quadratic_NLS_on_R NLS-2 on R]
  • [schrodinger.html#Quartic_NLS_on_R^2 NLS-2 on R^2]
  • [schrodinger.html#Quadratic_NLS_on_R^3 NLS-2 on R^3]
  • [schrodinger.html#Quadratic_NLS_on_T NLS-2 on T]
  • [schrodinger.html#Quadratic_NLS_on_T^2 NLS-2 on T^2]

[kdv.html#kdv KdV] (gKdV-1)

  • [kdv.html#kdv on R KdV on R]
  • [kdv.html#kdv on T KdV on T]
  • [kdv.html#kdv on R+ KdV on R^+]


  • [wave.html#nlw-3 on R NLW-3 on R]
  • [wave.html#nlw-3 on R^2 NLW-3 on R^2]
  • [wave.html#nlw-3 on R^3 NLW-3 on R^3]

Cubic NLS

  • [schrodinger.html#Cubic_NLS_on_R NLS-3 on R]
  • [schrodinger.html#Cubic_NLS_on_R^2 NLS-3 on R^2]
  • [schrodinger.html#Cubic_NLS_on_R^3 NLS-3 on R^3]
  • [schrodinger.html#Cubic_NLS_on_R^4 NLS-3 on R^4]
  • [schrodinger.html#Cubic_NLS_on_T NLS-3 on T]
  • [schrodinger.html#Cubic_NLS_on_T^2 NLS-3 on T^2]
  • [schrodinger.html#Cubic_NLS_on_T^4 NLS-3 on T^4]
  • [schrodinger.html#Cubic_NLS_on_RxT NLS-3 on R x T]
  • [schrodinger.html#Cubic_NLS_on_S^6 NLS-3 on S^6]

[kdv.html#mkdv Modified KdV] (gKdV-2)

  • [kdv.html#mKdV on R mKdV on R]
  • [kdv.html#mKdV on T mKdV on T]

[wave.html#nlw-4 Quartic NLW/NLKG]

Quartic NLS

  • [schrodinger.html#Quartic_NLS_on_R NLS-4 on R]
  • [schrodinger.html#Quartic_NLS_on_T NLS-4 on T]
  • [schrodinger.html#Quartic_NLS_on_R^2 NLS-4 on R^2]


  • [kdv.html#gKdV_3 on R gKdV-3 on R]
  • [kdv.html#gKdV_3 on T gKdV-3 on T]

Quintic NLW/NLKG

  • [wave.html#nlw-5 on R NLW-5 on R]
  • [wave.html#nlw-5 on R^2 NLW-5 on R^2]
  • [wave.html#nlw-5 on R^3 NLW-5 on R^3]

Quintic NLS

  • [schrodinger.html#Quintic_NLS_on_R NLS-5 on R]
  • [schrodinger.html#Quintic_NLS_on_R^2 NLS-5 on R^2]
  • [schrodinger.html#Quintic_NLS_on_R^3 NLS-5 on R^3]
  • [schrodinger.html#Quintic_NLS_on_T NLS-5 on T]


  • [kdv.html#gKdV_4 on R gKdV-4 on R]
  • [kdv.html#gKdV_4 on T gKdV-4 on T]


  • [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.html#Septic_NLS_on_R NLS-7 on R]
  • [schrodinger.html#Septic_NLS_on_R^2 NLS-7 on R^2]
  • [schrodinger.html#Septic_NLS_on_R^3 NLS-7 on R^3]

[wave.html#dnlw D-NLW]

  • [wave.html#gwp d-nlw GWP for D-NLW]
  • [wave.html#gwp d-nlkg GWP for D-NLKG]

[schrodinger.html#d-nls D-NLS]

[kdv.html#hierarchy The KdV hierachy]

[wave.html#dnlw-2 Quadratic DNLW]

  • [wave.html#mkg MKG]
    • [wave.html#mkg on R MKG on R]
    • [wave.html#mkg on R^2 MKG on R^2]
    • [wave.html#mkg on R^3 MKG on R^3]
    • [wave.html#mkg on R^4 MKG on R^4]
    • [wave.html#mkg on R^5+ MKG on R^5+]
  • [wave.html#YM Yang Mills]
    • [references.html#YM on R^2 YM on R^2]
    • [wave.html#YM on R^3 YM on R^3]
    • [wave.html#YM on R^4 YM on R^4]
    • [wave.html#ym on R^5+ YM on R^5+]
    • [wave.html#YMH on R^3 YMH on R^3]
  • [wave.html#Maxwell-Dirac Maxwell-Dirac]
  • [wave.html#Dirac-Klein-Gordon Dirac-Klein-Gordon]
  • [wave.html#two-speed DNLW Two-speed DNLW-2]

Quadratic DNLS

  • [schrodinger.html#Maxwell-Schrodinger Maxwell-Schrodinger]

Cubic DNLS

  • [schrodinger.html#dnls-3 on R DNLS-3 on R]

[wave.html#ddnlw DDNLW]

  • [wave.html#wm Wave maps]
    • [wave.html#wm on R WM on R]
    • [wave.html#wm on R^2 WM on R^2]
  • [wave.html#two-speed DDNLW Two-speed DDNLW]


  • [schrodinger.html#smaps Schrodinger Maps]

[wave.html#Quasilinear Quasilinear NLW]

  • [wave.html#gwp qnlw GWP for QNLW]
  • [wave.html#Einstein Einstein equations]

Quasilinear NLS

[schrodinger.html#Hartree Hartree equation]

[kdv.html#Benjamin-Ono Benjamin-Ono equation]

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

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.

What is well posedness?

As stated above, by well posedness in H^s we generally mean that there exists a unique solution u for some time T for each set of initial data in H^s, which stays in H^s and depends continuously on the initial data as a map from H^s to H^s.However, there are a couple subtleties involved here.

·Existence.For classical (smooth) solutions it is clear what it means for a solution to exist; for rough solutions one usually asks (as a bare minimum) for a solution to exist in the sense of distributions.(One may sometimes have to write the equation in conservation form before one can make sense of a distribution).It is possible for negative regularity solutions to exist if there is a sufficient amount of local smoothing available.

·Uniqueness.There are many different notions of uniqueness.One common one is uniqueness in the class of limits of smooth solutions.Another is uniqueness assuming certain spacetime regularity assumptions on the solution.A stronger form of uniqueness is in the class of all H^s functions.Stronger still is uniqueness in the class of all distributions for which the equation makes sense.

·Time of existence.In subcritical situations the time of existence typically depends only on the H^s norm of the initial data, or at a bare minimum one should get a fixed non-zero time of existence for data of sufficiently small norm.When combined with a conservation law this can often be extended to global existence.In critical situations one typically obtains global existence for data of small norm, and local existence for data of large norm but with a time of existence depending on the profile of the data (in particular, the frequencies where the norm is largest) and not just on the norm itself.

·Continuity.There are many different ways the solution map can be continuous from H^s to H^s.One of the strongest is real analyticity (which is what is commonly obtained by iteration methods).Weaker than this are various types of C^k continuity (C^1, C^2, C^3, etc.).If the solution map is C^k, then this implies that the k^th derivative at the origin is in H^s, which roughly corresponds to some iterate (often the k^th iterate) lying in H^s.Weaker than this is Lipschitz continuity, and weaker than that is uniform continuity.Finally, there is just plain old continuity.Interestingly, several examples have emerged recently in which one form of continuity holds but not another; in particular we now have several examples (critical wave maps, low-regularity periodic KdV and mKdV, Benjamin-Ono, quasilinear wave equations, ...) where the solution map is continuous but not uniformly continuous.


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.

Getting started