Partial differential equations

Preprints in Partial Differential Equations

Local/global well-posedness for dispersive/wave equations

Papers, and projects close to completion

Title	With	Status	Download
Endpoint Strichartz Estimates	Mark Keel	Amer. J. Math., 120 (1998), 955-980	dvi + Figures 1 2 3 4 5 ps.Z
Low regularity semi-linear wave equations		Comm. PDE 24 (1999), 599—630	math.AP/9709222 Slides: dvi + Figures 1 2
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
Global well-posedness for large data for the Maxwell-Klein-Gordon equation below the energy norm	Mark Keel	in preparation	Slides
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	to appear, Acta Math.	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), ???-??? [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	to appear, IAS conference proceedings	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		Submitted, New York Journal of Mathematics	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	To appear, Forum Mathematicum	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	To appear, Duke Math J.	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	Submitted, Annals of Math.	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?		Submitted, Bull. Amer. Math. Soc.	arXiv:0802.2408 discussion
A global compact attractor for high-dimensional defocusing non-linear Schrödinger equations with potential		Submitted, Dynamics of PDE	arXiv:0805.1544 discussion
Global regularity of wave maps III. Large energy from $R^{1+2}$ to hyperbolic spaces. ("Project Heatwave", part 1 of 4.)		Submitted,	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 4.)		Submitted,	arXiv:0806.3592 discussion

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

Short stories

These are generally very short, toy versions of real results due to other people, and are not publication-quality. Caveat emptor. All files other than figures are in dvi format. Unlike the preprints, these articles are fluid and subject to new developments. Please let me know if you have any comments, references, etc. on any of them.

Disclaimer: Many of the notes here are based on papers written by other people. My intention here is not to try to "beat" these authors' work in any way, but rather to isolate the main ingredients of the argument, which are often very beautiful, and try to present them in as simple and brief a context as possible (often sacrificing generality, rigour, and/or details in order to do this). Certainly I do not view these notes as worthy of publication in a refereed journal, and are definitely inferior to the original article in every single aspect, with the possible exception of brevity.

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

Open questions

Princeton Companion to Mathematics

Miscellaneous

Conformal compactification applet (1999)
Tom Wolff Memorial Lectures on non-linear dispersive equations (2003).
Nonlinear dispersive equations (2004). This is intended for incoming UCLA grad students.
My book “Nonlinear dispersive equations: local and global analysis” is now in press; see this page for more details.

Back to my preprints page.