Unconditional local well-posedness for periodic NLS

[1] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear

evolution equations, I, Schr¨

odinger equations, Geom. Funct. Anal. 3 (1993), 107–156.

[2] J. Bourgain, On Strichartz’s inequalities and the nonlinear Schr¨

odinger equation on irrational tori, Mathematical aspects of nonlinear dispersive equations, 1–20, Ann. of Math. Stud., 163, Princeton Univ. Press,

Princeton, NJ, 2007.

[3] J. Bourgain, Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces, Israel

J. Math. 193 (2013), no. 1, 441–458.

[4] J. Bourgain and C. Demeter, The proof of the l2 decoupling conjecture, Ann. of Math. (2) 182 (2015), no. 1,

351–389.

[5] F. Catoire and W.-M. Wang, Bounds on Sobolev norms for the defocusing nonlinear Schr¨

odinger equation

on general flat tori, Commun. Pure Appl. Anal. 9 (2010), no. 2, 483–491.

[6] X. Chen and J. Holmer, The derivation of the T3 energy-critical NLS from quantum many-body dynamics,

Invent. Math. 217 (2019), no. 2, 433–547.

[7] M. Christ, J. Colliander, and T. Tao, Instability of the periodic nonlinear Schr¨

odinger equation, preprint

(2003). Available at arXiv:math/0311227

[8] 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.

[9] C. Demeter, Incidence theory and restriction estimates, preprint (2014). Available at arXiv:1401.1873

A Self-archived copy in

Kyoto University Research Information Repository

https://repository.kulib.kyoto-u.ac.jp

18

N. KISHIMOTO

[10] S. Demirbas, Local well-posedness for 2-D Schr¨

odinger equation on irrational tori and bounds on Sobolev

norms, Commun. Pure Appl. Anal. 16 (2017), no. 5, 1517–1530.

[11] G. Furioli and E. Terraneo, Besov spaces and unconditional well-posedness for the nonlinear Schr¨

odinger

equation in H˙ s (Rn ), Commun. Contemp. Math.

[12] Z. Guo, S. Kwon, and T. Oh, Poincar´e-Dulac normal form reduction for unconditional well-posedness of the

periodic cubic NLS, Comm. Math. Phys. 322 (2013), no. 1, 19–48.

[13] Z. Guo and T. Oh, Non-existence of solutions for the periodic cubic NLS below L2 , Int. Math. Res. Not.

IMRN 2018, no. 6, 1656–1729.

[14] Z. Guo, T. Oh, and Y. Wang, Strichartz estimates for Schr¨

odinger equations on irrational tori, Proc. Lond.

Math. Soc. (3) 109 (2014), no. 4, 975–1013.

[15] Z. Han and D. Fang, On the unconditional uniqueness for NLS in H˙ s , SIAM J. Math. Anal. 45 (2013), no.

3, 1505–1526.

[16] S. Herr and V. Sohinger, Unconditional uniqueness results for the nonlinear Schr¨

odinger equation, Commun.

Contemp. Math. 21 (2019), no. 7, 1850058, 33 pp.

[17] S. Herr, D. Tataru, and N. Tzvetkov, Global well-posedness of the energy-critical nonlinear Schr¨

odinger

equation with small initial data in H 1 (T3 ), Duke Math. J. 159 (2011), no. 2, 329–349.

[18] S. Herr, D. Tataru, and N. Tzvetkov, Strichartz estimates for partially periodic solutions to Schr¨

odinger

equations in 4d and applications, J. Reine Angew. Math. 690 (2014), 65–78.

[19] V. Jarn´ık, Uber die Gitterpunkte auf konvexen Kurven (German), Math. Z. 24 (1926), 500–518.

[20] T. Kato, On nonlinear Schr¨

odinger equations. II. H s -solutions and unconditional well-posedness, J. Anal.

Math. 67 (1995), 281–306.

[21] R. Killip and M. Vi¸san, Scale invariant Strichartz estimates on tori and applications, Math. Res. Lett. 23

(2016), no. 2, 445–472.

[22] N. Kishimoto, Remark on the periodic mass critical nonlinear Schr¨

odinger equation, Proc. Amer. Math. Soc.

142 (2014), no. 8, 2649–2660.

[23] N. Kishimoto, Unconditional uniqueness of solutions for nonlinear dispersive equations, preprint (2019).

Available at arXiv:1911.04349

[24] G.E. Lee, Local wellposedness for the critical nonlinear Schr¨

odinger equation on T3 , Discrete Contin. Dyn.

Syst. 39 (2019), no. 5, 2763–2783.

[25] L. Molinet, On ill-posedness for the one-dimensional periodic cubic Schr¨

odinger equation, Math. Res. Lett.

16 (2009), no. 1, 111–120.

[26] K.M. Rogers, Unconditional well-posedness for subcritical NLS in H s , C. R. Math. Acad. Sci. Paris 345

(2007), no. 7, 395–398.

[27] N. Strunk, Strichartz estimates for Schr¨

odinger equations on irrational tori in two and three dimensions, J.

Evol. Equ. 14 (2014), no. 4-5, 829–839.

[28] Y. Wang, Periodic nonlinear Schr¨

odinger equation in critical H s (Tn ) spaces, SIAM J. Math. Anal. 45 (2013),

no. 3, 1691–1703.

[29] Y.Y.S. Win and Y. Tsutsumi, Unconditional uniqueness of solution for the Cauchy problem of the nonlinear

Schr¨

odinger equation, Hokkaido Math. J. 37 (2008), no. 4, 839–859.

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