Real-valued non-analytic solutions for the generalized Korteweg-de Vries equation

Himonas, A.; Petronilho, Gerson

doi:10.1090/S0002-9939-2011-10983-3

Real-valued non-analytic solutions for the generalized Korteweg-de Vries equation
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by A. Alexandrou Himonas and Gerson Petronilho PDF

Proc. Amer. Math. Soc. 139 (2011), 2759-2766 Request permission

Abstract:

In both the periodic and non-periodic cases, non-analytic in time solutions to the Cauchy problem of the gKdV equation are constructed with real-valued analytic initial data when $k$ is not a multiple of four. In the case that $k=4\ell$, that is, the non-linearity is of the form $u^{4\ell }\partial _xu$, where $\ell$ is a positive integer, then non-analytic in time solutions are available only for complex-valued initial data.

References

Jerry L. Bona, Zoran Grujić, and Henrik Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation, Ann. Inst. H. Poincaré C Anal. Non Linéaire 22 (2005), no. 6, 783–797 (English, with English and French summaries). MR 2172859, DOI 10.1016/j.anihpc.2004.12.004
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
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
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. MR 1215780, DOI 10.1007/BF01895688
Jean Bourgain, On the Cauchy problem for periodic KdV-type equations, Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993), 1995, pp. 17–86 (English, with English and French summaries). MR 1364878
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
Michael Christ, James Colliander, and Terrence Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math. 125 (2003), no. 6, 1235–1293. MR 2018661, DOI 10.1353/ajm.2003.0040
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\Bbb R$ and $\Bbb T$, J. Amer. Math. Soc. 16 (2003), no. 3, 705–749. MR 1969209, DOI 10.1090/S0894-0347-03-00421-1
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Multilinear estimates for periodic KdV equations, and applications, J. Funct. Anal. 211 (2004), no. 1, 173–218. MR 2054622, DOI 10.1016/S0022-1236(03)00218-0
Anne De Bouard, Nakao Hayashi, and Keiichi Kato, Gevrey regularizing effect for the (generalized) Korteweg-de Vries equation and nonlinear Schrödinger equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 12 (1995), no. 6, 673–725 (English, with English and French summaries). MR 1360541, DOI 10.1016/S0294-1449(16)30148-2
J. Ginibre and Y. Tsutsumi, Uniqueness of solutions for the generalized Korteweg-de Vries equation, SIAM J. Math. Anal. 20 (1989), no. 6, 1388–1425. MR 1019307, DOI 10.1137/0520091
Jennifer Gorsky and A. Alexandrou Himonas, Construction of non-analytic solutions for the generalized KdV equation, J. Math. Anal. Appl. 303 (2005), no. 2, 522–529. MR 2122235, DOI 10.1016/j.jmaa.2004.08.055
Jennifer Gorsky and A. Alexandrou Himonas, On analyticity in space variable of solutions to the KdV equation, Geometric analysis of PDE and several complex variables, Contemp. Math., vol. 368, Amer. Math. Soc., Providence, RI, 2005, pp. 233–247. MR 2126473, DOI 10.1090/conm/368/06782
Zoran Grujić and Henrik Kalisch, Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions, Differential Integral Equations 15 (2002), no. 11, 1325–1334. MR 1920689
H. Hannah, A. Himonas and G. Petronilho, Gevrey regularity for the periodic gKdV equation. J. Differential Equations 250 (2011), 2581–2600.
Nakao Hayashi, Analyticity of solutions of the Korteweg-de Vries equation, SIAM J. Math. Anal. 22 (1991), no. 6, 1738–1743. MR 1129407, DOI 10.1137/0522107
T. Kappeler and P. Topalov, Global wellposedness of KdV in $H^{-1}(\Bbb T,\Bbb R)$, Duke Math. J. 135 (2006), no. 2, 327–360. MR 2267286, DOI 10.1215/S0012-7094-06-13524-X
T. Kappeler and P. Topalov, Global well-posedness of mKdV in $L^2(\Bbb T,\Bbb R)$, Comm. Partial Differential Equations 30 (2005), no. 1-3, 435–449. MR 2131061, DOI 10.1081/PDE-200050089
Tosio Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in applied mathematics, Adv. Math. Suppl. Stud., vol. 8, Academic Press, New York, 1983, pp. 93–128. MR 759907
Tosio Kato and Kyūya Masuda, Nonlinear evolution equations and analyticity. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), no. 6, 455–467 (English, with French summary). MR 870865, DOI 10.1016/S0294-1449(16)30377-8
Keiichi Kato and Takayoshi Ogawa, Analyticity and smoothing effect for the Korteweg de Vries equation with a single point singularity, Math. Ann. 316 (2000), no. 3, 577–608. MR 1752786, DOI 10.1007/s002080050345
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
D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. 5 39 (1895), 422–443.
J. C. Saut and R. Temam, Remarks on the Korteweg-de Vries equation, Israel J. Math. 24 (1976), no. 1, 78–87. MR 454425, DOI 10.1007/BF02761431
Anders Sjöberg, On the Korteweg-de Vries equation: existence and uniqueness, J. Math. Anal. Appl. 29 (1970), 569–579. MR 410135, DOI 10.1016/0022-247X(70)90068-5
Terence Tao, Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics, vol. 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. Local and global analysis. MR 2233925, DOI 10.1090/cbms/106
Shigeo Tarama, Analyticity of solutions of the Korteweg-de Vries equation, J. Math. Kyoto Univ. 44 (2004), no. 1, 1–32. MR 2062705, DOI 10.1215/kjm/1250283580
E. Trubowitz, The inverse problem for periodic potentials, Comm. Pure Appl. Math. 30 (1977), no. 3, 321–337. MR 430403, DOI 10.1002/cpa.3160300305