Dispersion dynamics for the defocusing generalized Korteweg-de Vries equation

Steinerberger, Stefan

doi:10.1090/S0002-9939-2014-12285-4

Dispersion dynamics for the defocusing generalized Korteweg-de Vries equation
HTML articles powered by AMS MathViewer

by Stefan Steinerberger PDF

Proc. Amer. Math. Soc. 143 (2015), 789-800 Request permission

Abstract:

We study dispersion for the defocusing gKdV equation. It is expected that it is not possible for the bulk of the $L^2-$mass to concentrate in a small interval for a long time. We study a variance-type functional exploiting Tao’s monotonicity formula in the spirit of earlier work by Tao, as well as Kwon and Shao, and quantify its growth in terms of sublevel estimates.

References

James Colliander, Manoussos Grillakis, and Nikolaos Tzirakis, Improved interaction Morawetz inequalities for the cubic nonlinear Schrödinger equation on $\Bbb R^2$, Int. Math. Res. Not. IMRN 23 (2007), Art. ID rnm090, 30. MR 2377216
Anne de Bouard and Yvan Martel, Non existence of $L^2$-compact solutions of the Kadomtsev-Petviashvili II equation, Math. Ann. 328 (2004), no. 3, 525–544. MR 2036335, DOI 10.1007/s00208-003-0498-6
Benjamin G. Dodson, Global well-posedness for the defocusing, quintic nonlinear Schrödinger equation in one dimension for low regularity data, Int. Math. Res. Not. IMRN 4 (2012), 870–893. MR 2889161, DOI 10.1093/imrn/rnr037
Benjamin G. Dodson, Global well-posedness and scattering for the defocusing, mass - critical generalized KdV equation, arXiv:1304.8025
Hans G. Feichtinger, Compactness in translation invariant Banach spaces of distributions and compact multipliers, J. Math. Anal. Appl. 102 (1984), no. 2, 289–327. MR 755964, DOI 10.1016/0022-247X(84)90173-2
Jean Ginibre and Giorgio Velo, Quadratic Morawetz inequalities and asymptotic completeness in the energy space for nonlinear Schrödinger and Hartree equations, Quart. Appl. Math. 68 (2010), no. 1, 113–134. MR 2598884, DOI 10.1090/S0033-569X-09-01141-9
Rowan Killip, Soonsik Kwon, Shuanglin Shao, and Monica Visan, On the mass-critical generalized KdV equation, Discrete Contin. Dyn. Syst. 32 (2012), no. 1, 191–221. MR 2837059, DOI 10.3934/dcds.2012.32.191
S. Kwon and S. Shao, Nonexistence of soliton-like solutions for defocusing generalized KdV equations, arXiv:1205.0849
Celine Laurent and Yvan Martel, Smoothness and exponential decay of $L^2$-compact solutions of the generalized KdV equations, Comm. Partial Differential Equations 29 (2004), no. 1-2, 157–171. MR 2038148, DOI 10.1081/PDE-120028848
Hans Lindblad and Terence Tao, Asymptotic decay for a one-dimensional nonlinear wave equation, Anal. PDE 5 (2012), no. 2, 411–422. MR 2970713, DOI 10.2140/apde.2012.5.411
Yvan Martel and Frank Merle, A Liouville theorem for the critical generalized Korteweg-de Vries equation, J. Math. Pures Appl. (9) 79 (2000), no. 4, 339–425. MR 1753061, DOI 10.1016/S0021-7824(00)00159-8
Cathleen S. Morawetz, The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math. 14 (1961), 561–568. MR 132908, DOI 10.1002/cpa.3160140327
Robert L. Pego, Compactness in $L^2$ and the Fourier transform, Proc. Amer. Math. Soc. 95 (1985), no. 2, 252–254. MR 801333, DOI 10.1090/S0002-9939-1985-0801333-9
Fabrice Planchon and Luis Vega, Bilinear virial identities and applications, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 2, 261–290 (English, with English and French summaries). MR 2518079, DOI 10.24033/asens.2096
Terence Tao, Two remarks on the generalised Korteweg-de Vries equation, Discrete Contin. Dyn. Syst. 18 (2007), no. 1, 1–14. MR 2276483, DOI 10.3934/dcds.2007.18.1
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