Preprints in Partial Differential Equations


Local/global well-posedness for dispersive/wave equations

Papers, and projects close to completion

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Endpoint Strichartz Estimates

Mark Keel

Amer. J. Math., 120 (1998), 955-980

dvi + Figures 12345

ps.Z

Low regularity semi-linear wave equations

 

Comm. PDE 24 (1999), 599—630

math.AP/9709222 

Slides: dvi + Figures12

Small data blowup for semilinear Klein-Gordon equations

Mark Keel

Amer. J. Math. 121 (1999), 629-669

dvi + Figure 1 

ps.Z

Local and global well-posedness of wave maps in R^{1+1} for rough data

Mark Keel

IMRN 21 (1998), 1117-1156

math.AP/9807171 

Slides

Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation

 

Comm. PDE 25 (2000), 1471-1485

math.AP/9811168

Ill-posedness for one-dimensional wave maps at the critical regularity

 

Amer. J. Math., 122 (2000), 451-463

math.AP/9811169

Local well-posedness for the Yang-Mills equation below the energy norm

 

JDE  189 (2003), 366-382

math.AP/0005064

Almost conservation laws and global rough solutions to a nonlinear Schrodinger equation

Jim Colliander 

Mark Keel

Gigliola Staffilani 

Hideo Takaoka

Math. Res. Letters 9 (2002), 659-682.

math.AP/0203218 

Slides 

ANU notes

Multilinear weighted convolution of L^2 functions, and applications to non-linear dispersive equations

 

Amer. J. Math. 123 (2001), 839-908

math.AP/0005001

Global well-posedness result for KdV in Sobolev spaces of negative index

Jim Colliander 

Mark Keel 

Gigliola Staffilani 

Hideo Takaoka

EJDE 2001 (2001) No 26, 1-7

math.AP/0101261 

ANU notes 

Chicago notes

Sharp global well-posedness results for periodic and non-periodic KdV and modified KdV on R and T

Jim Colliander 

Mark Keel 

Gigliola Staffilani 

Hideo Takaoka

J. Amer. Math. Soc. 16 (2003), 705-749.

math.AP/0110045 

ANU notes 

Chicago notes

Multi-linear estimates for periodic KdV equations, and applications

Jim Colliander 

Mark Keel 

Gigliola Staffilani 

Hideo Takaoka

J. Funct. Anal. 211 (2004), 173-218

math.AP/0110049

Global well-posedness for the Schrodinger equations with derivative

Jim Colliander 

Mark Keel 

Gigliola Staffilani 

Hideo Takaoka

Siam J. Math. 33 (2001), 649-669

math.AP/0101263

Global regularity of wave maps I.  Small critical Sobolev norm in high dimension

 

IMRN 7 (2001), 299-328

math.AP/0010068 

Slides

Global regularity of wave maps II.  Small energy in two dimensions

 

Comm. Math. Phys. 224 (2001), 443-544

math.AP/0011173 

Slides

A refined global well-posedness for the Schrodinger equations with derivative

Jim Colliander 

Mark Keel 

Gigliola Staffilani 

Hideo Takaoka

Siam J. Math. 34 (2002), 64-86.

math.AP/0110026

Resonant decompositions and the I-method for cubic nonlinear Schrodinger on R^2

Jim Colliander 

Mark Keel 

Gigliola Staffilani 

Hideo Takaoka

Disc. Cont. Dynam. Systems A 21 (2008), 665-686

math.AP/0704.2730

discussion

Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm

Jim Colliander 

Mark Keel 

Gigliola Staffilani 

Hideo Takaoka

Discrete Cont. Dynam. Systems 9 (2003), 31-54

math.AP/0206218

Polynomial growth and orbital instability bounds for $L^2$-subcritical NLS below the energy norm

Jim Colliander 

Mark Keel 

Gigliola Staffilani 

Hideo Takaoka

Comm. Pure Appl. Anal. 2 (2003), 33-50

math.AP/0212113

Global existence and scattering for rough solutions of a nonlinear Schrodinger equation in R^3

Jim Colliander 

Mark Keel 

Gigliola Staffilani 

Hideo Takaoka

CPAM 57 (2004), 987-1014

math.AP/0301260

A physical approach to wave equation bilinear estimates

Sergiu Klainerman 

Igor Rodnianski

J. Anal. Math. 87 (2002), 299—336

math.AP/0106091

A singularity removal theorem for Yang-Mills fields in higher dimensions

Gang Tian

J. Amer. Math. Soc. 17 (2004), 557-593.

math.DG/0209352

Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocussing equations

Michael Christ 

Jim Colliander

Amer. J. Math. 125 (2003), 1235-1293

math.AP/0203044 

Chicago notes

Global regularity for the Maxwell-Klein-Gordon equation in high dimensions

Igor Rodnianski

Comm. Math. Phys. 251 (2004), 377-426

math.AP/0309353

Symplectic nonsqueezing of the KdV flow

Jim Colliander 

Mark Keel 

Gigliola Staffilani 

Hideo Takaoka

Acta Math. 195 (2005), 197-252

math.AP/0412381

Chicago notes

Upper and lower bounds for Dirichlet eigenfunctions

Andrew Hassell

Math. Res. Letters 9 (2002), 289-305

math.AP/0202140

Short version

Ill-posedness for nonlinear Schrodinger and wave equations

Michael Christ 

Jim Colliander

to appear, Annales IHP

math.AP/0311048

Local and global well-posedness for nonlinear dispersive equations

 

Proc. Centre Math. Appl. Austral. Nat. Univ. 40 (2002), 19-48

dvi

Existence globale et diffusion pour l'équation de Schrödinger nonlinéaire répulsive cubique sur R^3 en dessous l'espace d'énergie

Jim Colliander 

Mark Keel 

Gigliola Staffilani 

Hideo Takaoka

Journées "Équations aux Dérivées Partielles" (Forges-les-Eaux, 2002), Exp. No. X, 14 pp., Univ. Nantes, Nantes,2002

ps

Global well-posedness and scattering in the energy space for the critical nonlinear Schrodinger equation in R^3

("Project Gopher")

Jim Colliander 

Mark Keel 

Gigliola Staffilani 

Hideo Takaoka

Annals of Math. 167 (2007), 767-865

[A survey article is in Contemp. Math. 439, "Recent Developments in Nonlinear Partial Differential Equations: The second symposium on Analysis and PDEs June 7-10 2004, Purdue University, West Lafayette Indiana", D. Danielli, Ed., pp. 69-80.  American Mathematial Society, Providence RI 2007]

math.AP/0402129

Long-time decay estimates for Schrodinger equations on manifolds

Igor Rodnianski

Mathematical aspects of nonlinear dispersive equations, 223-253, Ann. of Math. Stud. 163, Princeton University Press, Princeton NJ 2007

math.AP/0412416

A Strichartz inequality for the Schrodinger equation on non-trapping asymptotically conic manifolds

Andrew Hassell 

Jared Wunsch

Comm. PDE 30 (2004), 157-205

math.AP/0312225

Global well-posedness of the Benjamin-Ono equation in H^1(R)

 

J. Hyperbolic Diff. Eq. 1 (2004) 27-49

math.AP/0307289

Instability of the periodic nonlinear Schrodinger equation

Michael Christ 

Jim Colliander

Submitted,

math.AP/0311227

On the asymptotic behavior of large radial data for a focusing non-linear Schr\"odinger equation

 

Dynamics of PDE 1 (2004), 1-48

math.AP/0309428

Global well-posedness and scattering for the higher-dimensional energy-critical non-linear Schrodinger equation for radial data

 

New York Journal of Mathematics 11 (2005), 57-80

math.AP/0402130

Sharp Strichartz estimates on non-trapping asymptotically conic manifolds

Andrew Hassell 

Jared Wunsch

Amer. J. Math. 128 (2006), 963—1024.

math.AP/0408273

Geometric renormalization of large energy wave maps

 

Journees “Equations aux derives partielles”, Forges les Eaux, 7-11 June 2004, XI 1-32

math.AP/0411354

Stability of energy-critical nonlinear Schr\"odinger equations in high dimensions

Monica Visan

Electron. J. Diff. Eq. Vol. 2005 (2005), No. 118, 1-28.

math.AP/0507005

Sharp well-posedness and ill-posedness results for a quadratic non-linear Schr\"odinger equation

Ioan Bejenaru

J. Funct. Anal. Vol. 233 (2006), 228-259

math.AP/0508210

Velocity averaging, kinetic formulations, and regularizing effects in quasilinear PDE.

Eitan Tadmor

CPAM 61 (2007), 1-34

math.AP/0511054

The nonlinear Schr\”odinger equation with combined power-type nonlinearities

Monica Visan

Xiaoyi Zhang

Comm. PDE 32 (2007), 1281-1343.

math.AP/0511070

Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions

 

Dynamics of PDE 3 (2006), 93-110

math.AP/0601164

Scattering for the quartic generalised Korteweg-de Vries equation

 

J. Diff. Eq. 232 (2007), 623—651

math.AP/0605357

Global regularity for a logarithmically supercritical defocusing nonlinear wave equation for spherically symmetric data

 

J. Hyperbolic Diff. Eq. 4 (2007),  259-266

math.AP/0606145

Two remarks on the generalised Korteweg-de Vries equation

 

Discrete Cont. Dynam. Systems 18 (2007), 1-14

math.AP/0606236

A pseudoconformal compactification of the nonlinear Schrodinger equation and applications

 

New York J. Math. 15 (2009), 265--282. 

math.AP/0606254

Global behaviour of nonlinear dispersive and wave equations

 

Current Developments in Mathematics 2006, International Press.  255-340.

math.AP/0608293

Minimal-mass blowup solutions of the mass-critical NLS

Monica Visan

Xiaoyi Zhang

Forum Mathematicum 20 (2008), 881-919

math.AP/0609690

Global well-posedness and scattering for the mass-critical nonlinear Schr\”odinger equation for radial data in high dimensions

Monica Visan

Xiaoyi Zhang

Duke Math J. 140 (2007), 165-202

math.AP/0609692

A counterexample to an endpoint bilinear Strichartz inequality

 

Electron. J. Diff. Eq. 2006 (2006) 151, 1—6.

math.AP/0609849

A (concentration-)compact attractor for high-dimensional non-linear Schrödinger equations

 

Dynamics of PDE 4 (2007), 1-53

math.AP/0611402

A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order

Michael Christ 

Jim Colliander

J. Funct. Anal  254 (2007), 368-395

math.AP/0612457 

The cubic nonlinear Schrödinger equation in two dimensions with radial data

Rowan Killip

Monica Visan

J. Eur. Math. Soc. (JEMS) 11 (2009), no. 6, 1203--1258. 

arXiv:0707.3188

discussion

A quantitative formulation of the global regularity problem for the periodic Navier-Stokes equation

Dynamics of PDE 4 (2007), 293--302.

arXiv:0710.1604

discussion

Why are solitons stable?

Bull. Amer. Math. Soc. 46 (2009), 1-33.

arXiv:0802.2408

discussion

A global compact attractor for high-dimensional defocusing non-linear Schrödinger equations with potential

Dynamics of PDE 5 (2008), 101—116.

arXiv:0805.1544

discussion

Global regularity of wave maps III.  Large energy from $R^{1+2}$ to hyperbolic spaces.

("Project Heatwave", part 1 of 5.)

To be submitted, pending revision.

arXiv:0805.4666

discussion

Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy class

("Project Heatwave", part 2 of 5.)

To be submitted, pending revision.

arXiv:0806.3592

discussion

Global existence and uniqueness results for weak solutions of the focusing mass-critical non-linear Schrödinger equation

 Anal. PDE 2 (2009), no. 1, 61--81.

arXiv:0807.2676

discussion

Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrodinger equation

Jim Colliander 

Mark Keel 

Gigliola Staffilani 

Hideo Takaoka

Inventiones Math.181 (2010), 39-113

arXiv:0808.1742

discussion

Global regularity of wave maps V. Large data local well-posedness in the energy class

("Project Heatwave", part 3 of 5.)

 

To be submitted, pending revision.

arXiv:0808.0368

discussion

The high exponent limit p \to \infty for the one-dimensional nonlinear wave equation

 

Anal. PDE 2 (2009), no. 2, 235--259.

arXiv:0901.3548

discussion

An inverse theorem for the bilinear L^2 Strichartz estimate for the wave equation

 

To be submitted, pending revision.

arXiv:0904.2880

discussion

Global regularity of wave maps VI.  Abstract theory of minimal-energy blowup solutions

(“Project Heatwave”, part 4 of 5.)

 

To be submitted, pending revision.

arXiv:0906.2883

discussion

Global regularity of wave maps VII.  Control of delocalised or dispersed solutions

(“Project Heatwave”, part 5 of 5.)

 

To be submitted, pending revision.

arXiv:0908.0776

discussion

Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation

 

Analysis & PDE 2 (2009), 361-366

arXiv:0906.3070

discussion

Operator splitting for the KdV equation

Helge Holden

Kenneth Karlsen

Nils Risebro

Math. Comp. 80 (2011) 821-846. 

arXiv:0906.4902

discussion

Global well-posedness for the Maxwell-Klein-Gordon equation below the energy norm

Mark Keel

Tristan Roy

To appearDiscrete Cont. Dynam. Systems

arXiv:0910.1850

discussion

Slides

Asymptotic decay for a one-dimensional nonlinear wave equation

Hans Lindblad

Analysis & PDE 5-2 (2012), 411--422.

arXiv:1011.0949

discussion

Effective limiting absorption principles, and applications

Igor Rodnianski

SubmittedComm. Math. Phys. 

arXiv:1105.0873

discussion

Localisation and compactness properties of the Navier-Stokes global regularity problem

To appear ,  Analysis & PDE

arXiv:1108:1165

discussion

Concentration compactness for critical wave maps, by Joachim Krieger and Wilhelm Schlag. EMS Monographs in Mathematics, European Mathematical Society, 2012, 499pp, ISBN 978-3-03719-106-4

Submitted, Bull. Amer. Math. Soc.

PDF

Some further PDE-related preprints can be found in my Kakeya/restriction preprints page.  

Short stories

 

Counterexamples to endpoints of  n=3 wave equation Strichartz

Existence questions for non-linear wave equations

A null form estimate from an improved Strichartz estimate

An algebra for critical regularity solutions to the free wave equation

The division problem for critical regularity wave maps

The wave bestiary

Inverse scattering for the Dirac equation

The non-linear Fourier transform

Low regularity behavior of KdV/mKdV

Viriel, Morawetz, and interaction Morawetz inequalities

An informal summary of Bourgain's radial critical NLS result

An informal summary of Grillakis's radial critical NLS result

Progress in nonlinear wave equations (slides)

Modulation stability – a very simple example

Informal derivation of Schrodinger’s equation

The Kenig-Merle scattering result for the energy-critical focusing NLS

Nash-Moser iteration

Gauges for the Schrodinger map

Why global regularity for Navier-Stokes is hard

John’s blowup theorem for the nonlinear wave equation

Hassell's proof of scarring for the Bunimovich stadium

Concentration compactness and the profile decomposition

What is a gauge?

An explicitly solvable nonlinear wave equation

A physical space proof of the bilinear Strichartz and local smoothing estimates for the Schrodinger equation

Open questions

Princeton Companion to Mathematics

Miscellaneous

Back to my preprints page.