## 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 12345ps.Z Low regularity semi-linear wave equations Comm. PDE 24 (1999), 599—630 Slides: dvi + Figures12 Small data blowup for semilinear Klein-Gordon equations Mark Keel Amer. J. Math. 121 (1999), 629-669 ps.Z Local and global well-posedness of wave maps in R^{1+1} for rough data Mark Keel IMRN 21 (1998), 1117-1156 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 Mark KeelHideo Takaoka Math. Res. Letters 9 (2002), 659-682. 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 Hideo Takaoka EJDE 2001 (2001) No 26, 1-7 Chicago notes Sharp global well-posedness results for periodic and non-periodic KdV and modified KdV on R and T Hideo Takaoka J. Amer. Math. Soc. 16 (2003), 705-749. Chicago notes Multi-linear estimates for periodic KdV equations, and applications Hideo Takaoka J. Funct. Anal. 211 (2004), 173-218 math.AP/0110049 Global well-posedness for the Schrodinger equations with derivative 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 Slides Global regularity of wave maps II.  Small energy in two dimensions Comm. Math. Phys. 224 (2001), 443-544 Slides A refined global well-posedness for the Schrodinger equations with derivative 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 Hideo Takaoka Disc. Cont. Dynam. Systems A 21 (2008), 665-686 math.AP/0704.2730discussion Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm 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 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 Hideo Takaoka CPAM 57 (2004), 987-1014 math.AP/0301260 A physical approach to wave equation bilinear estimates 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 Jim Colliander Amer. J. Math. 125 (2003), 1235-1293 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 Hideo Takaoka Acta Math. 195 (2005), 197-252 math.AP/0412381Chicago notes Upper and lower bounds for Dirichlet eigenfunctions Andrew Hassell Math. Res. Letters 9 (2002), 289-305 math.AP/0202140Short version Ill-posedness for nonlinear Schrodinger and wave equations 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 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") 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 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 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 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 VisanXiaoyi 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 VisanXiaoyi 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 VisanXiaoyi 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 Jim Colliander J. Funct. Anal  254 (2007), 368-395 math.AP/0612457 The cubic nonlinear Schrödinger equation in two dimensions with radial data Rowan KillipMonica Visan J. Eur. Math. Soc. (JEMS) 11 (2009), no. 6, 1203--1258. arXiv:0707.3188discussion A quantitative formulation of the global regularity problem for the periodic Navier-Stokes equation Dynamics of PDE 4 (2007), 293--302. arXiv:0710.1604discussion Why are solitons stable? Bull. Amer. Math. Soc. 46 (2009), 1-33. arXiv:0802.2408discussion A global compact attractor for high-dimensional defocusing non-linear Schrödinger equations with potential Dynamics of PDE 5 (2008), 101—116. arXiv:0805.1544discussion 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:0805.4666discussion 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:0806.3592discussion 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:0807.2676discussion Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrodinger equation Hideo Takaoka Inventiones Math.181 (2010), 39-113 arXiv:0808.1742discussion 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:0808.0368discussion The high exponent limit p \to \infty for the one-dimensional nonlinear wave equation Anal. PDE 2 (2009), no. 2, 235--259. arXiv:0901.3548discussion An inverse theorem for the bilinear L^2 Strichartz estimate for the wave equation To be submitted, pending revision. arXiv:0904.2880discussion 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:0906.2883discussion Global regularity of wave maps VII.  Control of delocalised or dispersed solutions(“Project Heatwave”, part 5 of 5.) To be submitted, pending revision. arXiv:0908.0776discussion Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation Analysis & PDE 2 (2009), 361-366 arXiv:0906.3070discussion Operator splitting for the KdV equation Helge HoldenKenneth KarlsenNils Risebro Math. Comp. 80 (2011) 821-846. arXiv:0906.4902discussion Global well-posedness for the Maxwell-Klein-Gordon equation below the energy norm Mark KeelTristan Roy To appear, Discrete Cont. Dynam. Systems arXiv:0910.1850discussionSlides Asymptotic decay for a one-dimensional nonlinear wave equation Hans Lindblad arXiv:1011.0949discussion Effective limiting absorption principles, and applications Igor Rodnianski Submitted, Comm. Math. Phys. arXiv:1105.0873discussion Localisation and compactness properties of the Navier-Stokes global regularity problem To appear ,  Analysis & PDE arXiv:1108:1165discussion 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 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

### Miscellaneous

Back to my preprints page.