Spatial behavior for NLS and applications to scattering

Cazenave, Thierry; Naumkin, Ivan

Séminaire Laurent Schwartz — EDP et applications (2017-2018), Exposé no. 1, 11 p.

We review recent results on the nonlinear Schrödinger equation

$$i{u}_t+\Delta u+\lambda {\left|u\right|}^{\alpha }u=0$$

where $\lambda \in ℂ$ and $\alpha >0$. In any space dimension $N\ge 1$ and for any $\alpha >0$, we construct a class of (arbitrarily large) initial values for which there exists a local solution. Moreover, if $\alpha >2/N$, we construct a class of (arbitrarily large) initial values for which there exists a global solution that scatters as $t\to \infty$. If $\alpha =2/N$ and $\Im \lambda \le 0$, we construct a class of (arbitrarily large) initial values for which there exists a global solution, of which we give a precise asymptotic expansion as $t\to \infty$ (of modified scattering type). These results rely on the construction of solutions that do not vanish, so as to avoid any issue related to the lack of regularity of the nonlinearity at $u=0$. To study the asymptotic behavior, we apply the pseudo-conformal transformation. This yields the desired asymptotic behavior if $\alpha >2/N$. In the case $\alpha =2/N$, a further step is required, and we estimate the solutions by allowing a certain growth of the Sobolev norms, which depends on the order of regularity through a cascade of exponents.

[1] Barab J.E. Nonexistence of asymptotically free solutions for a nonlinear Schrödinger equation, J. Math. Phys. 25 (1984), no. 11, 3270–3273. | DOI | Zbl

[2] Carles R. Geometric optics and long range scattering for one-dimensional nonlinear Schrödinger equations. Comm. Math. Phys. 220 (2001), no. 1, 41–67. | DOI | Zbl

[3] Cazenave T., Correia S., Dickstein F. and Weissler F.B. A Fujita-type blowup result and low energy scattering for a nonlinear Schrödinger equation. São Paulo J. Math. Sci. 9 (2015), no. 2, 146–161. | DOI | Zbl

[4] Cazenave T., Dickstein F. and Weissler F.B. Non-regularity in Hölder and Sobolev spaces of solutions to the semilinear heat and Schrödinger equations. Nagoya Math. J. 226 (2017), 44–70. | DOI | Zbl

[5] Cazenave T., Fang D. and Han Z. Local well-posedness for the $H^2$-critical nonlinear Schrödinger equation. Trans. Amer. Math. Soc. 368 (2016), no. 11, 7911–7934. | DOI | Zbl

[6] Cazenave T. and Naumkin I. Local existence, global existence, and scattering for the nonlinear Schrödinger equation. Commun. Contemp. Math. 19 (2017), no. 2, 1650038, 20 pp. | DOI | Zbl

[7] Cazenave T. and Naumkin I. Modified scattering for the critical nonlinear Schrödinger equation. J. Funct. Anal. 274 (2018), 402–432. | DOI | Zbl

[8] Cazenave T. and Weissler F. B. The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal. 14 (1990), no. 10, 807–836. | DOI | Zbl

[9] Cazenave T. and Weissler F. B. Some remarks on the nonlinear Schrödinger equation in the critical case, in Nonlinear Semigroups, Partial Differential Equations, and Attractors, T.L. Gill and W.W. Zachary (eds.) Lecture Notes in Math. 1394, Springer, New York, 1989, 18–29. | DOI | Zbl

[10] Cazenave T. and Weissler F. B. Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys. 147 (1992), 75–100. (link: http://projecteuclid.org/euclid.cmp/1104250527) | DOI | Zbl

[11] Fang D. and Han Z. On the well-posedness for NLS in $H^s$. J. Funct. Anal. 264 (2013), no. 6, 1438–1455. | DOI | Zbl

[12] Ginibre J., Ozawa T. and Velo G. On the existence of the wave operators for a class of nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 60 (1994), no. 2, 211–239. (link: http://archive.numdam.org/article/AIHPA_1994__60_2_211_0.pdf) | Numdam | Zbl

[13] Ginibre J. and Velo G. On a class of nonlinear Schrödinger equations. II. Scattering theory, general case, J. Funct. Anal. 32, no. 1 (1979), 33–71. | DOI | Zbl

[14] Ginibre J. and Velo G. On the global Cauchy problem for some nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 309–323. (link: http://www.numdam.org/item?id=AIHPC_1984__1_4_309_0) | DOI | Numdam | Zbl

[15] Hayashi N. and Naumkin P.I. Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations. Amer. J. Math. 120 (1998), no. 2, 369–389. | DOI | MR | Zbl

[16] Hayashi N. and Naumkin P.I. Large time behavior for the cubic nonlinear Schrödinger equation. Canad. J. Math. 54 (2002), no. 5, 1065–1085. | DOI | Zbl

[17] Hayashi N. and Naumkin P.I. Logarithmic time decay for the cubic nonlinear Schrödinger equations. Int. Math. Res. Not. IMRN 2015, no. 14, 5604–5643. | DOI | Zbl

[18] Hayashi N., Naumkin P.I. , Shimomura A. and Tonegawa S. Modified wave operators for nonlinear Schrödinger equations in one and two dimensions. Electron. J. Differential Equations 2004, No. 62, 16 pp. (link: http://ejde.math.txstate.edu/Volumes/2004/62/hayashi.pdf) | Zbl

[19] Kato T. Nonlinear Schrödinger equations, in Schrödinger Operators (Sønderborg, 1988), Lecture Notes in Phys. 345, Springer, Berlin, 1989, 218–263. | DOI

[20] Kato T. On nonlinear Schrödinger equations. II. $H^s$-solutions and unconditional well-posedness. J. Anal. Math. 67 (1995), 281–306. | DOI | Zbl

[21] Kita N. and Wada T. Sharp asymptotic behavior of solutions to nonlinear Schrödinger equations in one space dimension. Funkcial. Ekvac. 45 (2002), no. 1, 53–69. (link: http://fe.math.kobe-u.ac.jp/FE/FullPapers/vol45/fe45-1-3.pdf) | Zbl

[22] Linares F., Miyazaki H., and Ponce G. On a class of solutions to the generalized KdV type equation. Preprint, 2018, arXiv:1802.07345 [math.AP]. | arXiv | DOI | MR | Zbl

[23] Linares F., Ponce G., and Santos G.N. On a class of solutions to the generalized derivative Schrödinger equations. Preprint, 2017, arXiv:1712.00663 [math.AP]. | arXiv | DOI | Zbl

[24] Nakamura M. and Ozawa T. Nonlinear Schrödinger equations in the Sobolev space of critical order, J. Funct. Anal. 155 (1998), no. 2, 364–380. | DOI | Zbl

[25] Nakanishi K. and Ozawa T. Remarks on scattering for nonlinear Schrödinger equations, NoDEA Nonlinear Differential Equations Appl. 9 (2002), no. 1, 45–68. | DOI | Zbl

[26] Ozawa T. Long range scattering for the nonlinear Schrödinger equation in one space dimension, Comm. Math. Phys. 139 (1991), no. 3, 479–493. (link: http://projecteuclid.org/euclid.cmp/1104203467) | DOI | Zbl

[27] Pecher H. Solutions of semilinear Schrödinger equations in $H^s$, Ann. Inst. H. Poincaré Phys. Théor. 67 (1997), no. 3, 259–296. (link: http://www.numdam.org/item?id=AIHPA_1997__67_3_259_0) | Numdam | Zbl

[28] Shimomura A. Asymptotic behavior of solutions for Schrödinger equations with dissipative nonlinearities, Comm. Partial Differential Equations 31 (2006), no. 7-9, 1407–1423. | DOI | Zbl

[29] Shimomura A. and Tonegawa S. Long-range scattering for nonlinear Schrödinger equations in one and two space dimensions. Differential Integral Equations 17 (2004), no. 1-2, 127–150. (link: http://projecteuclid.org/euclid.die/1356060476) | Zbl

[30] Strauss W.A. On weak solutions of semi-linear hyperbolic equations, An. Acad. Brasil. Ciên. 42 (1970), 645–651. | Zbl

[31] Strauss W.A. Nonlinear scattering theory, in Scattering Theory in Mathematical Physics, NATO Advanced Study Institutes Series Volume 9, 53-78. Springer Netherlands, 1974. | DOI

[32] Strauss W.A. Nonlinear invariant wave equations, in Invariant wave equations (Proc. “Ettore Majorana" Internat. School of Math. Phys., Erice, 1977), Lecture Notes in Phys. 73, Springer, 1978, 197–249.

[33] Strauss W.A. Nonlinear scattering theory at low energy, J. Funct. Anal. 41 (1981), no. 1, 110–133. | DOI | MR | Zbl

[34] Tsutsumi Y. $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac. 30 (1987), no. 1, 115–125. (link: http://www.math.sci.kobe-u.ac.jp/~fe/) | Zbl

[35] Visciglia N. On the decay of solutions to a class of defocusing NLS, Math. Res. Lett. 16 (2009), no. 5, 919–926. | DOI | MR