Remarks on global a priori estimates for the nonlinear Schrödinger equation

Colliander, J.; Grillakis, M.; Tzirakis, N.

doi:10.1090/S0002-9939-2010-10487-2

Remarks on global a priori estimates for the nonlinear Schrödinger equation
HTML articles powered by AMS MathViewer

by J. Colliander, M. Grillakis and N. Tzirakis PDF

Proc. Amer. Math. Soc. 138 (2010), 4359-4371 Request permission

Abstract:

We present a unified approach for obtaining global a priori estimates for solutions of nonlinear defocusing Schrödinger equations with defocusing nonlinearities. The estimates are produced by contracting the local momentum conservation law with appropriate vector fields. The corresponding law is written for defocusing equations of tensored solutions. In particular, we obtain a new estimate in two dimensions. We bound the restricted $L_t^4L_{\gamma }^4$ Strichartz norm of the solution on any curve $\gamma$ in $\mathbb R^2$. For the specific case of a straight line we upgrade this estimate to a weighted Strichartz estimate valid in the full plane.

References

J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc. 12 (1999), no. 1, 145–171. MR 1626257, DOI 10.1090/S0894-0347-99-00283-0
Thierry Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. MR 2002047, DOI 10.1090/cln/010
J. Colliander, M. Grillakis, and N. Tzirakis, Tensor products and correlation estimates with applications to nonlinear Schrödinger equations, Comm. Pure Appl. Math. 62 (2009), no. 7, 920–968. MR 2527809, DOI 10.1002/cpa.20278
James Colliander, Justin Holmer, Monica Visan, and Xiaoyi Zhang, Global existence and scattering for rough solutions to generalized nonlinear Schrödinger equations on $\Bbb R$, Commun. Pure Appl. Anal. 7 (2008), no. 3, 467–489. MR 2379437, DOI 10.3934/cpaa.2008.7.467
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\Bbb R^3$, Comm. Pure Appl. Math. 57 (2004), no. 8, 987–1014. MR 2053757, DOI 10.1002/cpa.20029
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\Bbb R^3$, Ann. of Math. (2) 167 (2008), no. 3, 767–865. MR 2415387, DOI 10.4007/annals.2008.167.767
J. Ginibre and G. Velo, Scattering theory in the energy space for a class of Hartree equations, Nonlinear wave equations (Providence, RI, 1998) Contemp. Math., vol. 263, Amer. Math. Soc., Providence, RI, 2000, pp. 29–60. MR 1777634, DOI 10.1090/conm/263/04190
J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl. (9) 64 (1985), no. 4, 363–401. MR 839728
Manoussos G. Grillakis, On nonlinear Schrödinger equations, Comm. Partial Differential Equations 25 (2000), no. 9-10, 1827–1844. MR 1778782, DOI 10.1080/03605300008821569
Jeng Eng Lin and Walter A. Strauss, Decay and scattering of solutions of a nonlinear Schrödinger equation, J. Functional Analysis 30 (1978), no. 2, 245–263. MR 515228, DOI 10.1016/0022-1236(78)90073-3
Cathleen S. Morawetz, Time decay for the nonlinear Klein-Gordon equations, Proc. Roy. Soc. London Ser. A 306 (1968), 291–296. MR 234136, DOI 10.1098/rspa.1968.0151
Cathleen S. Morawetz and Walter A. Strauss, Decay and scattering of solutions of a nonlinear relativistic wave equation, Comm. Pure Appl. Math. 25 (1972), 1–31. MR 303097, DOI 10.1002/cpa.3160250103
Kenji Nakanishi, Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions $1$ and $2$, J. Funct. Anal. 169 (1999), no. 1, 201–225. MR 1726753, DOI 10.1006/jfan.1999.3503
Kenji Nakanishi, Energy scattering for Hartree equations, Math. Res. Lett. 6 (1999), no. 1, 107–118. MR 1682697, DOI 10.4310/MRL.1999.v6.n1.a8
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, 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
Terence Tao, Monica Visan, and Xiaoyi Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1281–1343. MR 2354495, DOI 10.1080/03605300701588805