Sharp global well-posedness for KdV and modified KdV on $\mathbb R$ and $\mathbb T$
HTML articles powered by AMS MathViewer
- by J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao
- J. Amer. Math. Soc. 16 (2003), 705-749
- DOI: https://doi.org/10.1090/S0894-0347-03-00421-1
- Published electronically: January 29, 2003
- HTML | PDF | Request permission
Abstract:
The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown to be globally well-posed in all $L^2$-based Sobolev spaces $H^s$ where local well-posedness is presently known, apart from the $H^{\frac {1}{4}} (\mathbb {R} )$ endpoint for mKdV and the $H^{-\frac {3}{4}}$ endpoint for KdV. The result for KdV relies on a new method for constructing almost conserved quantities using multilinear harmonic analysis and the available local-in-time theory. Miura’s transformation is used to show that global well-posedness of modified KdV is implied by global well-posedness of the standard KdV equation.References
- Michael Beals, Self-spreading and strength of singularities for solutions to semilinear wave equations, Ann. of Math. (2) 118 (1983), no. 1, 187–214. MR 707166, DOI 10.2307/2006959
- Björn Birnir, Carlos E. Kenig, Gustavo Ponce, Nils Svanstedt, and Luis Vega, On the ill-posedness of the IVP for the generalized Korteweg-de Vries and nonlinear Schrödinger equations, J. London Math. Soc. (2) 53 (1996), no. 3, 551–559. MR 1396718, DOI 10.1112/jlms/53.3.551
- Björn Birnir, Gustavo Ponce, and Nils Svanstedt, The local ill-posedness of the modified KdV equation, Ann. Inst. H. Poincaré C Anal. Non Linéaire 13 (1996), no. 4, 529–535 (English, with English and French summaries). MR 1404320, DOI 10.1016/S0294-1449(16)30112-3
- J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A 278 (1975), no. 1287, 555–601. MR 385355, DOI 10.1098/rsta.1975.0035
- L. Kantorovitch, The method of successive approximations for functional equations, Acta Math. 71 (1939), 63–97. MR 95, DOI 10.1007/BF02547750
- Jean Bourgain, Approximation of solutions of the cubic nonlinear Schrödinger equations by finite-dimensional equations and nonsqueezing properties, Internat. Math. Res. Notices 2 (1994), 79–88. MR 1264931, DOI 10.1155/S1073792894000103
- J. Bourgain, Aspects of long time behaviour of solutions of nonlinear Hamiltonian evolution equations, Geom. Funct. Anal. 5 (1995), no. 2, 105–140. MR 1334864, DOI 10.1007/BF01895664
- Jean Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, Internat. Math. Res. Notices 6 (1996), 277–304. MR 1386079, DOI 10.1155/S1073792896000207
- J. Bourgain, Periodic Korteweg de Vries equation with measures as initial data, Selecta Math. (N.S.) 3 (1997), no. 2, 115–159. MR 1466164, DOI 10.1007/s000290050008
- J. Bourgain, Refinements of Strichartz’ inequality and applications to $2$D-NLS with critical nonlinearity, Internat. Math. Res. Notices 5 (1998), 253–283. MR 1616917, DOI 10.1155/S1073792898000191
- Lennart Carleson and Per Sjölin, Oscillatory integrals and a multiplier problem for the disc, Studia Math. 44 (1972), 287–299. (errata insert). MR 361607, DOI 10.4064/sm-44-3-287-299
- Amy Cohen, Existence and regularity for solutions of the Korteweg-de Vries equation, Arch. Rational Mech. Anal. 71 (1979), no. 2, 143–175. MR 525222, DOI 10.1007/BF00248725 CCT1 M. Christ, J. Colliander, and T. Tao Asymptotics, frequency modulation and low regularity ill-posedness for canonical defocusing equations. To appear Amer. J. Math., 2002. CKSTTDNLS2 J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao. A refined global wellposedness result for Schrödinger equations with derivative. To appear SIAM J. Math. Anal., 2002.
- J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Global well-posedness for KdV in Sobolev spaces of negative index, Electron. J. Differential Equations (2001), No. 26, 7. MR 1824796
- J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal. 33 (2001), no. 3, 649–669. MR 1871414, DOI 10.1137/S0036141001384387 CKSTTGKdV J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao. Multilinear estimates for periodic KdV equations and applications. To appear J. Funct. Anal., 2002.
- J. Colliander, G. Staffilani, and H. Takaoka, Global wellposedness for KdV below $L^2$, Math. Res. Lett. 6 (1999), no. 5-6, 755–778. MR 1739230, DOI 10.4310/MRL.1999.v6.n6.a13
- Charles Fefferman, A note on spherical summation multipliers, Israel J. Math. 15 (1973), 44–52. MR 320624, DOI 10.1007/BF02771772
- German Fonseca, Felipe Linares, and Gustavo Ponce, Global well-posedness for the modified Korteweg-de Vries equation, Comm. Partial Differential Equations 24 (1999), no. 3-4, 683–705. MR 1683054, DOI 10.1080/03605309908821438 FLP2002 G. Fonseca, F. Linares, and G. Ponce. Global existence for the critical generalized KdV equation. Preprint, 2002.
- J. Ginibre, An introduction to nonlinear Schrödinger equations, Nonlinear waves (Sapporo, 1995) GAKUTO Internat. Ser. Math. Sci. Appl., vol. 10, Gakk\B{o}tosho, Tokyo, 1997, pp. 85–133. MR 1602772
- J. Ginibre, Y. Tsutsumi, and G. Velo, Existence and uniqueness of solutions for the generalized Korteweg de Vries equation, Math. Z. 203 (1990), no. 1, 9–36. MR 1030705, DOI 10.1007/BF02570720
- Jean Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d’espace (d’après Bourgain), Astérisque 237 (1996), Exp. No. 796, 4, 163–187 (French, with French summary). Séminaire Bourbaki, Vol. 1994/95. MR 1423623 Grunrock A. Grünrock. A bilinear Airy-estimate with application to gKdV-3. Preprint, 2001.
- Helmut Hofer and Eduard Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 1994. MR 1306732, DOI 10.1007/978-3-0348-8540-9
- J.-L. Joly, G. Métivier, and J. Rauch, A nonlinear instability for $3\times 3$ systems of conservation laws, Comm. Math. Phys. 162 (1994), no. 1, 47–59. MR 1272766, DOI 10.1007/BF02105186
- Tosio Kato, The Cauchy problem for the Korteweg-de Vries equation, Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. I (Paris, 1978/1979) Res. Notes in Math., vol. 53, Pitman, Boston, Mass.-London, 1981, pp. 293–307. MR 631399
- Markus Keel and Terence Tao, Local and global well-posedness of wave maps on $\mathbf R^{1+1}$ for rough data, Internat. Math. Res. Notices 21 (1998), 1117–1156. MR 1663216, DOI 10.1155/S107379289800066X KeelTaoMKG M. Keel and T. Tao. Global well-posedness for large data for the Maxwell-Klein-Gordon equation below the energy norm. Preprint, 2000.
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), no. 4, 527–620. MR 1211741, DOI 10.1002/cpa.3160460405
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9 (1996), no. 2, 573–603. MR 1329387, DOI 10.1090/S0894-0347-96-00200-7
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J. 106 (2001), no. 3, 617–633. MR 1813239, DOI 10.1215/S0012-7094-01-10638-8
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Global well-posedness for semi-linear wave equations, Comm. Partial Differential Equations 25 (2000), no. 9-10, 1741–1752. MR 1778778, DOI 10.1080/03605300008821565
- S. B. Kuksin, On squeezing and flow of energy for nonlinear wave equations, Geom. Funct. Anal. 5 (1995), no. 4, 668–701. MR 1345018, DOI 10.1007/BF01902057
- Sergej B. Kuksin, Infinite-dimensional symplectic capacities and a squeezing theorem for Hamiltonian PDEs, Comm. Math. Phys. 167 (1995), no. 3, 531–552. MR 1316759, DOI 10.1007/BF02101534
- S. Klainerman and M. Machedon, Smoothing estimates for null forms and applications, Internat. Math. Res. Notices 9 (1994), 383ff., approx. 7 pp.}, issn=1073-7928, review= MR 1301438, doi=10.1155/S1073792894000425, DOI 10.1155/S1073792894000425 MM Y. Martel and F. Merle. Blow up in finite time and dynamics of blow up solutions for the $L^2$ critical generalized KdV equation. J. Amer. Math. Soc. 15(3):617–664, 2002. FMcomm F. Merle. Personal communication. 2002.
- John J. Benedetto and Hans Heinig, Fourier transform inequalities with measure weights, Adv. Math. 96 (1992), no. 2, 194–225. MR 1196988, DOI 10.1016/0001-8708(92)90055-P
- Yves Meyer and Ronald Coifman, Wavelets, Cambridge Studies in Advanced Mathematics, vol. 48, Cambridge University Press, Cambridge, 1997. Calderón-Zygmund and multilinear operators; Translated from the 1990 and 1991 French originals by David Salinger. MR 1456993
- Robert M. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Mathematical Phys. 9 (1968), 1202–1204. MR 252825, DOI 10.1063/1.1664700
- Robert M. Miura, The Korteweg-de Vries equation: a survey of results, SIAM Rev. 18 (1976), no. 3, 412–459. MR 404890, DOI 10.1137/1018076
- Robert M. Miura, Errata: “The Korteweg-deVries equation: a survey of results” (SIAM Rev. 18 (1976), no. 3, 412–459), SIAM Rev. 19 (1977), no. 4, vi. MR 467039, DOI 10.1137/1019101 NTT K. Nakanishi, H. Takaoka, and Y. Tsutsumi. Counterexamples to bilinear estimates related to the KdV equation and the nonlinear Schrödinger equation. Methods of Appl. Anal. 8(4):569–578, 2001.
- Jeffrey Rauch and Michael Reed, Nonlinear microlocal analysis of semilinear hyperbolic systems in one space dimension, Duke Math. J. 49 (1982), no. 2, 397–475. MR 659948 RosalesThesis R. R. Rosales. I. Exact solution of some nonlinear evolution equations, II. The similarity solution for the Korteweg-de Vries equation and the related Painlevé transcendent. PhD thesis, California Institute of Technology, 1977.
- Hideo Takaoka, Global well-posedness for the Kadomtsev-Petviashvili II equation, Discrete Contin. Dynam. Systems 6 (2000), no. 2, 483–499. MR 1739371, DOI 10.3934/dcds.2000.6.483
- Terence Tao, Multilinear weighted convolution of $L^2$-functions, and applications to nonlinear dispersive equations, Amer. J. Math. 123 (2001), no. 5, 839–908. MR 1854113, DOI 10.1353/ajm.2001.0035
- N. Tzvetkov, Global low-regularity solutions for Kadomtsev-Petviashvili equation, Differential Integral Equations 13 (2000), no. 10-12, 1289–1320. MR 1787069
Bibliographic Information
- J. Colliander
- Affiliation: Department of Mathematics, University of Toronto, Toronto, ON Canada, M5S 3G3
- M. Keel
- Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455
- G. Staffilani
- Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305-2125
- Address at time of publication: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02138
- MR Author ID: 614986
- H. Takaoka
- Affiliation: Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
- Address at time of publication: Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan
- T. Tao
- Affiliation: Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90095-1555
- MR Author ID: 361755
- ORCID: 0000-0002-0140-7641
- Received by editor(s): November 7, 2001
- Received by editor(s) in revised form: November 19, 2002
- Published electronically: January 29, 2003
- Additional Notes: The first author is supported in part by N.S.F. Grant DMS 0100595.
The second author is supported in part by N.S.F. Grant DMS 9801558
The third author is supported in part by N.S.F. Grant DMS 9800879 and by a grant from the Sloan Foundation
The fourth author is supported in part by J.S.P.S. Grant No. 13740087
The last author is a Clay Prize Fellow and is supported in part by grants from the Packard and Sloan Foundations - © Copyright 2003 American Mathematical Society
- Journal: J. Amer. Math. Soc. 16 (2003), 705-749
- MSC (2000): Primary 35Q53, 42B35, 37K10
- DOI: https://doi.org/10.1090/S0894-0347-03-00421-1
- MathSciNet review: 1969209