Difference between revisions of "Main Page"
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== Dispersive PDE Wiki ==  == Dispersive PDE Wiki ==  
<div class="Section1">  
==Local and global wellposedness for nonlinear dispersive and wave equations==  
<center>Maintained by [http://www.math.berkeley.edu/~colliand/ Jim Colliander], [http://www.math.caltech.edu/people/keel.html Mark Keel], [mailto:gigliola@math.stanford.edu Gigliola Staffilani], [mailto:takaoka@math.sci.hokudai.ac.jp Hideo Takaoka], and [http://www.math.ucla.edu/~tao Terry Tao]</center>  
''Disclaimer''<nowiki>: 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 </nowiki>[#email notify us] of any errors! <br />  
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[#purpose Purpose of this page]  
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[references.html Bibliography]  
}  
<div class="MsoNormal" style="textalign: center"><center>  
  
</center></div>  
<center>The big three: Wave, Schrodinger, KdV</center>  
{ style="width: 100.0%; msocellspacing: 1.5pt; msopaddingalt: 0in 0in 0in 0in" width="100%" border="1"  
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[wave.html Wave equations]  
 style="padding: .75pt .75pt .75pt .75pt"   
[schrodinger.html Schrodinger equations]  
 style="padding: .75pt .75pt .75pt .75pt"   
[kdv.html#overview KdV equations]  
  
 style="padding: .75pt .75pt .75pt .75pt"   
[wave.html#wave Wave estimates]  
* [wave.html#linear Linear estimates]  
* [wave.html#bilinear Bilinear estimates]  
* [wave.html#multilinear Multilinear estimates]  
 style="padding: .75pt .75pt .75pt .75pt"   
[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]  
 style="padding: .75pt .75pt .75pt .75pt"   
[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]  
  
 style="padding: .75pt .75pt .75pt .75pt"   
[wave.html#semilinear Semilinear NLW/NLKG]  
* [wave.html#scat nlw Scattering for NLW]  
* [references.html#scat nlkg Scattering for NLKG]  
* [wave.html#SineGordon SineGordon]  
 style="padding: .75pt .75pt .75pt .75pt"   
[schrodinger.html#nls Semilinear NLS]  
 style="padding: .75pt .75pt .75pt .75pt"   
[kdv.html#overview gKdV]  
* [kdv.html#gKdV on R+ gKdV on R^+]  
  
 style="padding: .75pt .75pt .75pt .75pt"   
[wave.html#nlw2 Quadratic NLW/NLKG]  
 style="padding: .75pt .75pt .75pt .75pt"   
[schrodinger.html#Quadratic_NLS Quadratic NLS]  
* [schrodinger.html#Quadratic_NLS_on_R NLS2 on R]  
* [schrodinger.html#Quartic_NLS_on_R^2 NLS2 on R^2]  
* [schrodinger.html#Quadratic_NLS_on_R^3 NLS2 on R^3]  
* [schrodinger.html#Quadratic_NLS_on_T NLS2 on T]  
* [schrodinger.html#Quadratic_NLS_on_T^2 NLS2 on T^2]  
 style="padding: .75pt .75pt .75pt .75pt"   
[kdv.html#kdv KdV] (gKdV1)  
* [kdv.html#kdv on R KdV on R]  
* [kdv.html#kdv on T KdV on T]  
* [kdv.html#kdv on R+ KdV on R^+]  
  
 style="padding: .75pt .75pt .75pt .75pt"   
Cubic NLW/NLKG  
* [wave.html#nlw3 on R NLW3 on R]  
* [wave.html#nlw3 on R^2 NLW3 on R^2]  
* [wave.html#nlw3 on R^3 NLW3 on R^3]  
 style="padding: .75pt .75pt .75pt .75pt"   
Cubic NLS  
* [schrodinger.html#Cubic_NLS_on_R NLS3 on R]  
* [schrodinger.html#Cubic_NLS_on_R^2 NLS3 on R^2]  
* [schrodinger.html#Cubic_NLS_on_R^3 NLS3 on R^3]  
* [schrodinger.html#Cubic_NLS_on_R^4 NLS3 on R^4]  
* [schrodinger.html#Cubic_NLS_on_T NLS3 on T]  
* [schrodinger.html#Cubic_NLS_on_T^2 NLS3 on T^2]  
* [schrodinger.html#Cubic_NLS_on_T^4 NLS3 on T^4]  
* [schrodinger.html#Cubic_NLS_on_RxT NLS3 on R x T]  
* [schrodinger.html#Cubic_NLS_on_S^6 NLS3 on S^6]  
 style="padding: .75pt .75pt .75pt .75pt"   
[kdv.html#mkdv Modified KdV] (gKdV2)  
* [kdv.html#mKdV on R mKdV on R]  
* [kdv.html#mKdV on T mKdV on T]  
  
 style="padding: .75pt .75pt .75pt .75pt"   
[wave.html#nlw4 Quartic NLW/NLKG]  
 style="padding: .75pt .75pt .75pt .75pt"   
Quartic NLS  
* [schrodinger.html#Quartic_NLS_on_R NLS4 on R]  
* [schrodinger.html#Quartic_NLS_on_T NLS4 on T]  
* [schrodinger.html#Quartic_NLS_on_R^2 NLS4 on R^2]  
 style="padding: .75pt .75pt .75pt .75pt"   
gKdV3  
* [kdv.html#gKdV_3 on R gKdV3 on R]  
* [kdv.html#gKdV_3 on T gKdV3 on T]  
  
 style="padding: .75pt .75pt .75pt .75pt"   
Quintic NLW/NLKG  
* [wave.html#nlw5 on R NLW5 on R]  
* [wave.html#nlw5 on R^2 NLW5 on R^2]  
* [wave.html#nlw5 on R^3 NLW5 on R^3]  
 style="padding: .75pt .75pt .75pt .75pt"   
Quintic NLS  
* [schrodinger.html#Quintic_NLS_on_R NLS5 on R]  
* [schrodinger.html#Quintic_NLS_on_R^2 NLS5 on R^2]  
* [schrodinger.html#Quintic_NLS_on_R^3 NLS5 on R^3]  
* [schrodinger.html#Quintic_NLS_on_T NLS5 on T]  
 style="padding: .75pt .75pt .75pt .75pt"   
gKdV4  
* [kdv.html#gKdV_4 on R gKdV4 on R]  
* [kdv.html#gKdV_4 on T gKdV4 on T]  
  
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Septic NLW/NLKG  
* [wave.htm#nlw7 on R NLW7 on R]  
* [wave.htm#nlw7 on R^2 NLW7 on R^2]  
* [wave.htm#nlw7 on R^3 NLW7 on R^3]  
 style="padding: .75pt .75pt .75pt .75pt"   
Septic NLS  
* [schrodinger.html#Septic_NLS_on_R NLS7 on R]  
* [schrodinger.html#Septic_NLS_on_R^2 NLS7 on R^2]  
* [schrodinger.html#Septic_NLS_on_R^3 NLS7 on R^3]  
 style="padding: .75pt .75pt .75pt .75pt"   
  
 style="padding: .75pt .75pt .75pt .75pt"   
[wave.html#dnlw DNLW]  
* [wave.html#gwp dnlw GWP for DNLW]  
* [wave.html#gwp dnlkg GWP for DNLKG]  
 style="padding: .75pt .75pt .75pt .75pt"   
[schrodinger.html#dnls DNLS]  
 style="padding: .75pt .75pt .75pt .75pt"   
[kdv.html#hierarchy The KdV hierachy]  
  
 style="padding: .75pt .75pt .75pt .75pt"   
[wave.html#dnlw2 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#MaxwellDirac MaxwellDirac]  
* [wave.html#DiracKleinGordon DiracKleinGordon]  
* [wave.html#twospeed DNLW Twospeed DNLW2]  
 style="padding: .75pt .75pt .75pt .75pt"   
Quadratic DNLS  
* [schrodinger.html#MaxwellSchrodinger MaxwellSchrodinger]  
 style="padding: .75pt .75pt .75pt .75pt"   
  
 style="padding: .75pt .75pt .75pt .75pt"   
 style="padding: .75pt .75pt .75pt .75pt"   
Cubic DNLS  
* [schrodinger.html#dnls3 on R DNLS3 on R]  
 style="padding: .75pt .75pt .75pt .75pt"   
  
 style="padding: .75pt .75pt .75pt .75pt"   
[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#twospeed DDNLW Twospeed DDNLW]  
 style="padding: .75pt .75pt .75pt .75pt"   
DDNLS  
* [schrodinger.html#smaps Schrodinger Maps]  
 style="padding: .75pt .75pt .75pt .75pt"   
  
 style="padding: .75pt .75pt .75pt .75pt"   
[wave.html#Quasilinear Quasilinear NLW]  
* [wave.html#gwp qnlw GWP for QNLW]  
* [wave.html#Einstein Einstein equations]  
 style="padding: .75pt .75pt .75pt .75pt"   
Quasilinear NLS  
 style="padding: .75pt .75pt .75pt .75pt"   
  
 style="padding: .75pt .75pt .75pt .75pt"   
 style="padding: .75pt .75pt .75pt .75pt"   
[schrodinger.html#Hartree Hartree equation]  
 style="padding: .75pt .75pt .75pt .75pt"   
[kdv.html#BenjaminOno BenjaminOno equation]  
}  
<div class="MsoNormal" style="textalign: center"><center>  
  
</center></div>  
<center>[misc.html Other equations]<nowiki>:</nowiki></center>  
* SchrodingerGinsbergLandau  
* [misc.html#VlasovMaxwell VlasovMaxwell]  
* [misc.html#KP The KadomtsevPetvisasvhili equations]  
** [misc.html#KPI KadomtsevPetviashvili I]  
** [misc.html#KPII KadomtsevPetviashvili II]  
* [misc.html#Zakharov Zakharov]  
** [misc.html#Zakharov1 Zakharov on R and T]  
** [misc.html#Zakharov2 Zakharov on R^2]  
** [misc.html#Zakharov3 Zakharov on R^3]  
** KleinGordonZakharov  
** [misc.html#WS Other WaveSchrodinger systems]  
*** Ishimori  
*** DaveyStewartson  
*** Yukawa  
<div class="MsoNormal" style="textalign: center"><center>  
  
</center></div>  
<center>Purpose of this page:</center>  
This collection of web pages is concerned with the local and global wellposedness of various nonlinear dispersive and wave equations. An equation is locally wellposed (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 wellposed (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 wellposedness 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 hyperlinks 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 uptodate. 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 email us]<nowiki>! </nowiki>  
Any suggestions, notifications of new papers, and/or corrections are very welcome, and can be sent [#email by email]. 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 [http://www.univie.ac.at/future.media/moe/formeln.html 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.  
<div class="MsoNormal" style="textalign: center"><center>  
  
</center></div>  
<center>What is well posedness?</center>  
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.  
<font face="Symbol">·</font>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.  
<font face="Symbol">·</font>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.  
<font face="Symbol">·</font>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 nonzero 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.  
<font face="Symbol">·</font>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, lowregularity periodic KdV and mKdV, BenjaminOno, quasilinear wave equations, ...) where the solution map is continuous but not uniformly continuous.  
<div class="MsoNormal" style="textalign: center"><center>  
  
</center></div>  
<center>Contact:</center>  
These pages are maintained jointly by [mailto:colliand@math.berkeley.edu Jim Colliander], [mailto:keel@cco.caltech.edu Mark Keel], [mailto:gigliola@math.stanford.edu Gigliola Staffilani], [mailto:takaoka@math.sci.hokudai.ac.jp Hideo Takaoka], and [mailto:tao@math.ucla.edu Terry Tao]. Technical issues concerning webpage problems, etc. should be addressed to [mailto:tao@math.ucla.edu Terry Tao]. <br /><br />  
</div>  
== Getting started ==  == Getting started == 
Revision as of 02:10, 20 July 2006
MediaWiki has been successfully installed.
Consult the User's Guide for information on using the wiki software.
Dispersive PDE Wiki
Local and global wellposedness for nonlinear dispersive and wave equations
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] 
[wave.html Wave equations] 
[schrodinger.html Schrodinger equations] 
[kdv.html#overview KdV equations] 
[wave.html#wave Wave estimates]

[schrodinger.html#Schrodinger Schrodinger estimates]

[kdv.html#Airy Airy estimates]

[wave.html#semilinear Semilinear NLW/NLKG]

[schrodinger.html#nls Semilinear NLS] 
[kdv.html#overview gKdV]

[wave.html#nlw2 Quadratic NLW/NLKG] 
[schrodinger.html#Quadratic_NLS Quadratic NLS]

[kdv.html#kdv KdV] (gKdV1)

Cubic NLW/NLKG

Cubic NLS

[kdv.html#mkdv Modified KdV] (gKdV2)

[wave.html#nlw4 Quartic NLW/NLKG] 
Quartic NLS

gKdV3

Quintic NLW/NLKG

Quintic NLS

gKdV4

Septic NLW/NLKG

Septic NLS


[wave.html#dnlw DNLW]

[schrodinger.html#dnls DNLS] 
[kdv.html#hierarchy The KdV hierachy] 
[wave.html#dnlw2 Quadratic DNLW]

Quadratic DNLS


Cubic DNLS


[wave.html#ddnlw DDNLW]

DDNLS


[wave.html#Quasilinear Quasilinear NLW]

Quasilinear NLS 

[schrodinger.html#Hartree Hartree equation] 
[kdv.html#BenjaminOno BenjaminOno equation] 
 SchrodingerGinsbergLandau
 [misc.html#VlasovMaxwell VlasovMaxwell]
 [misc.html#KP The KadomtsevPetvisasvhili equations]
 [misc.html#KPI KadomtsevPetviashvili I]
 [misc.html#KPII KadomtsevPetviashvili II]
 [misc.html#Zakharov Zakharov]
 [misc.html#Zakharov1 Zakharov on R and T]
 [misc.html#Zakharov2 Zakharov on R^2]
 [misc.html#Zakharov3 Zakharov on R^3]
 KleinGordonZakharov
 [misc.html#WS Other WaveSchrodinger systems]
 Ishimori
 DaveyStewartson
 Yukawa
This collection of web pages is concerned with the local and global wellposedness of various nonlinear dispersive and wave equations. An equation is locally wellposed (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 wellposed (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 wellposedness 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 hyperlinks 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 uptodate. 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 email us]!
Any suggestions, notifications of new papers, and/or corrections are very welcome, and can be sent [#email by email]. 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.
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 nonzero 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, lowregularity periodic KdV and mKdV, BenjaminOno, 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 webpage problems, etc. should be addressed to Terry Tao.