## Sharp global well-posedness for KdV and modified KdV on $$\mathbb R$$ and $$\mathbb T$$.(English)Zbl 1025.35025

The authors consider the $$\mathbb R$$-valued Korteweg-de Vries (KdV) equation $u_t+u_{xxx}+uu_x =0,\quad u:{\mathbb R}\times[0,T]\mapsto {\mathbb R}.$ They prove the global well-posedness (GWP) of KdV for initial data in $$H^s({\mathbb R})$$, $$s>-3/4$$. The local well-posedness of KdV in $$H^s$$ for $$s>-3/4$$ was obtained by C. E. Kenig, G. Ponce and L. Vega [J. Am. Math. Soc. 9, 573-603 (1996; Zbl 0848.35114)].
The range of $$s$$ is sharp; M. Christ, J. Colliander and T. Tao [Am. J. Math. 125, No. 6, 1235–1293 (2003; Zbl 1048.35101)] proved the ill-posedness of KdV for $$s<-3/4$$ in the sense that the solution operator is not uniformly continuous with respect to the $$H^s$$ norm.
The authors also prove various GWP results for periodic KdV and modified KdV equations.
The paper is well written and highly readable. The introduction includes a detailed history of related results as well as a useful heuristic discussion of their method.

### MSC:

 35Q53 KdV equations (Korteweg-de Vries equations) 42B35 Function spaces arising in harmonic analysis 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)

### Citations:

Zbl 0848.35114; Zbl 1048.35101
Full Text:

### References:

 [1] Michael Beals, Self-spreading and strength of singularities for solutions to semilinear wave equations, Ann. of Math. (2) 118 (1983), no. 1, 187 – 214. · Zbl 0522.35064 [2] 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. · Zbl 0855.35112 [3] Björn Birnir, Gustavo Ponce, and Nils Svanstedt, The local ill-posedness of the modified KdV equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), no. 4, 529 – 535 (English, with English and French summaries). · Zbl 0858.35130 [4] 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. · Zbl 0306.35027 [5] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), no. 2, 107 – 156. , https://doi.org/10.1007/BF01896020 J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal. 3 (1993), no. 3, 209 – 262. , https://doi.org/10.1007/BF01895688 J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), no. 2, 107 – 156. , https://doi.org/10.1007/BF01896020 J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal. 3 (1993), no. 3, 209 – 262. · Zbl 0787.35098 [6] 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. · Zbl 0818.35112 [7] J. Bourgain, Aspects of long time behaviour of solutions of nonlinear Hamiltonian evolution equations, Geom. Funct. Anal. 5 (1995), no. 2, 105 – 140. · Zbl 0879.35024 [8] 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. · Zbl 0934.35166 [9] J. Bourgain, Periodic Korteweg de Vries equation with measures as initial data, Selecta Math. (N.S.) 3 (1997), no. 2, 115 – 159. · Zbl 0891.35138 [10] J. Bourgain, Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity, Internat. Math. Res. Notices 5 (1998), 253 – 283. · Zbl 0917.35126 [11] Lennart Carleson and Per Sjölin, Oscillatory integrals and a multiplier problem for the disc, Studia Math. 44 (1972), 287 – 299. (errata insert). Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, III. · Zbl 0215.18303 [12] Amy Cohen, Existence and regularity for solutions of the Korteweg-de Vries equation, Arch. Rational Mech. Anal. 71 (1979), no. 2, 143 – 175. · Zbl 0415.35069 [13] 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. [14] 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. · Zbl 1034.35120 [15] 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. · Zbl 0967.35119 [16] 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. · Zbl 1002.35113 [17] 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. · Zbl 1062.35109 [18] J. Colliander, G. Staffilani, and H. Takaoka, Global wellposedness for KdV below \?², Math. Res. Lett. 6 (1999), no. 5-6, 755 – 778. · Zbl 0959.35144 [19] Charles Fefferman, A note on spherical summation multipliers, Israel J. Math. 15 (1973), 44 – 52. · Zbl 0262.42007 [20] 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. · Zbl 0930.35154 [21] G. Fonseca, F. Linares, and G. Ponce. Global existence for the critical generalized KdV equation. Preprint, 2002. · Zbl 1034.35121 [22] J. Ginibre, An introduction to nonlinear Schrödinger equations, Nonlinear waves (Sapporo, 1995) GAKUTO Internat. Ser. Math. Sci. Appl., vol. 10, Gakkōtosho, Tokyo, 1997, pp. 85 – 133. · Zbl 0891.35146 [23] 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. · Zbl 0662.35114 [24] 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. [25] A. Grünrock. A bilinear Airy-estimate with application to gKdV-3. Preprint, 2001. · Zbl 1212.35412 [26] Helmut Hofer and Eduard Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 1994. · Zbl 0805.58003 [27] 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. · Zbl 0820.35093 [28] 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. [29] Markus Keel and Terence Tao, Local and global well-posedness of wave maps on \?\textonesuperior $$^{+}$$\textonesuperior for rough data, Internat. Math. Res. Notices 21 (1998), 1117 – 1156. · Zbl 0999.58013 [30] M. Keel and T. Tao. Global well-posedness for large data for the Maxwell-Klein-Gordon equation below the energy norm. Preprint, 2000. [31] 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. · Zbl 0808.35128 [32] 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. · Zbl 0848.35114 [33] 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. · Zbl 1034.35145 [34] 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. · Zbl 0961.35092 [35] S. B. Kuksin, On squeezing and flow of energy for nonlinear wave equations, Geom. Funct. Anal. 5 (1995), no. 4, 668 – 701. · Zbl 0834.35086 [36] Sergej B. Kuksin, Infinite-dimensional symplectic capacities and a squeezing theorem for Hamiltonian PDEs, Comm. Math. Phys. 167 (1995), no. 3, 531 – 552. · Zbl 0827.35121 [37] 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,. · Zbl 0832.35096 [38] 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. · Zbl 0996.35064 [39] F. Merle. Personal communication. 2002. [40] John J. Benedetto and Hans Heinig, Fourier transform inequalities with measure weights, Adv. Math. 96 (1992), no. 2, 194 – 225. · Zbl 0772.42005 [41] 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. · Zbl 0945.42015 [42] Robert M. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Mathematical Phys. 9 (1968), 1202 – 1204. , https://doi.org/10.1063/1.1664700 Robert M. Miura, Clifford S. Gardner, and Martin D. Kruskal, Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion, J. Mathematical Phys. 9 (1968), 1204 – 1209. · Zbl 0283.35019 [43] Robert M. Miura, The Korteweg-de Vries equation: a survey of results, SIAM Rev. 18 (1976), no. 3, 412 – 459. · Zbl 0333.35021 [44] 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. · Zbl 0373.35011 [45] 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. · Zbl 1011.35119 [46] 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. · Zbl 0503.35055 [47] 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. [48] Hideo Takaoka, Global well-posedness for the Kadomtsev-Petviashvili II equation, Discrete Contin. Dynam. Systems 6 (2000), no. 2, 483 – 499. · Zbl 1021.35099 [49] Terence Tao, Multilinear weighted convolution of \?²-functions, and applications to nonlinear dispersive equations, Amer. J. Math. 123 (2001), no. 5, 839 – 908. · Zbl 0998.42005 [50] N. Tzvetkov, Global low-regularity solutions for Kadomtsev-Petviashvili equation, Differential Integral Equations 13 (2000), no. 10-12, 1289 – 1320. · Zbl 0977.35125
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