May  2019, 39(5): 2763-2783. doi: 10.3934/dcds.2019116

Local wellposedness for the critical nonlinear Schrödinger equation on $ \mathbb{T}^3 $

Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095, USA

Received  June 2018 Revised  October 2018 Published  January 2019

Fund Project: This work was supported by NSF grants DMS-1265868, DMS-1600942 (principal investigator: Rowan Killip) and DMS-1500707 (principal investigator: Monica Vişan).

For $ p\geq 2 $, we prove local wellposedness for the nonlinear Schrödinger equation $ (i\partial _t + \Delta)u = \pm|u|^pu $ on $ \mathbb{T}^3 $ with initial data in $ H^{s_c}(\mathbb{T}^3) $, where $ \mathbb{T}^3 $ is a rectangular irrational $ 3 $-torus and $ s_c = \frac{3}{2} - \frac{2}{p} $ is the scaling-critical regularity. This extends work of earlier authors on the local Cauchy theory for NLS on $ \mathbb{T}^3 $ with power nonlinearities where $ p $ is an even integer.

Citation: Gyu Eun Lee. Local wellposedness for the critical nonlinear Schrödinger equation on $ \mathbb{T}^3 $. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2763-2783. doi: 10.3934/dcds.2019116
References:
[1]

J. M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup., 14 (1981), 209-246.  doi: 10.24033/asens.1404.

[2]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156.  doi: 10.1007/BF01896020.

[3]

J. Bourgain and C. Demeter, The proof of the $l^2$ decoupling conjecture, Ann. of Math. (2), 182 (2015), 351-389.  doi: 10.4007/annals.2015.182.1.9.

[4]

F. M. Christ and M. I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109.  doi: 10.1016/0022-1236(91)90103-C.

[5]

Z. GuoT. Oh and Y. Wang, Strichartz estimates for Schrödinger equations on irrational tori, Proc. Lond. Math. Soc., 109 (2014), 975-1013.  doi: 10.1112/plms/pdu025.

[6]

M. HadacS. Herr and H. Koch, Well-posedness and scattering for the KP-Ⅱ equation in a critical space, Ann. I. H. Poincare, 26 (2009), 917-941.  doi: 10.1016/j.anihpc.2008.04.002.

[7]

S. HerrD. Tataru and N. Tzvetkov, Strichartz estimates for partially periodic solutions to Schrödinger equations in 4d and applications, J. Reine Angew. Math, 2014 (2012), 65-78.  doi: 10.1515/crelle-2012-0013.

[8]

S. HerrD. Tataru and N. Tzvetkov, Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in $H^1(\mathbb T^3)$, Duke Math. J., 159 (2011), 329-349.  doi: 10.1215/00127094-1415889.

[9]

A. D. Ionescu and B. Pausauder, The energy-critical defocusing NLS on $\mathbb T^3$, Duke Math. J., 161 (2012), 1581-1612.  doi: 10.1215/00127094-1593335.

[10]

R. Killip and M. Vişan, Scale invariant Strichartz estimates on tori and applications, Math. Res. Lett., 23 (2016), 445-472.  doi: 10.4310/MRL.2016.v23.n2.a8.

[11]

R. Killip and M. Vişan, Nonlinear Schrödinger equations at critical regularity, in "Evolution Equations", Clay Math. Proc., 17, (eds. D. Ellwood, I. Rodnianski, G. Staffilani and J. Wunsch), Amer. Math. Soc., Providence, RI, (2013), 325–437.

[12]

H. Koch, D. Tataru and M. Vişan, Dispersive Equations and Nonlinear Waves, Oberwolfach Seminars 45, Springer, Basel, 2014.

[13]

N. Strunk, Strichartz estimates for Schrödinger equations on irrational tori in two and three dimensions, J. Evol. Equ., 14 (2014), 829-839.  doi: 10.1007/s00028-014-0240-8.

[14]

M. Taylor, Tools for PDE: Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Amer. Math. Soc., Providence, RI, 2000.

show all references

References:
[1]

J. M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup., 14 (1981), 209-246.  doi: 10.24033/asens.1404.

[2]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156.  doi: 10.1007/BF01896020.

[3]

J. Bourgain and C. Demeter, The proof of the $l^2$ decoupling conjecture, Ann. of Math. (2), 182 (2015), 351-389.  doi: 10.4007/annals.2015.182.1.9.

[4]

F. M. Christ and M. I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109.  doi: 10.1016/0022-1236(91)90103-C.

[5]

Z. GuoT. Oh and Y. Wang, Strichartz estimates for Schrödinger equations on irrational tori, Proc. Lond. Math. Soc., 109 (2014), 975-1013.  doi: 10.1112/plms/pdu025.

[6]

M. HadacS. Herr and H. Koch, Well-posedness and scattering for the KP-Ⅱ equation in a critical space, Ann. I. H. Poincare, 26 (2009), 917-941.  doi: 10.1016/j.anihpc.2008.04.002.

[7]

S. HerrD. Tataru and N. Tzvetkov, Strichartz estimates for partially periodic solutions to Schrödinger equations in 4d and applications, J. Reine Angew. Math, 2014 (2012), 65-78.  doi: 10.1515/crelle-2012-0013.

[8]

S. HerrD. Tataru and N. Tzvetkov, Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in $H^1(\mathbb T^3)$, Duke Math. J., 159 (2011), 329-349.  doi: 10.1215/00127094-1415889.

[9]

A. D. Ionescu and B. Pausauder, The energy-critical defocusing NLS on $\mathbb T^3$, Duke Math. J., 161 (2012), 1581-1612.  doi: 10.1215/00127094-1593335.

[10]

R. Killip and M. Vişan, Scale invariant Strichartz estimates on tori and applications, Math. Res. Lett., 23 (2016), 445-472.  doi: 10.4310/MRL.2016.v23.n2.a8.

[11]

R. Killip and M. Vişan, Nonlinear Schrödinger equations at critical regularity, in "Evolution Equations", Clay Math. Proc., 17, (eds. D. Ellwood, I. Rodnianski, G. Staffilani and J. Wunsch), Amer. Math. Soc., Providence, RI, (2013), 325–437.

[12]

H. Koch, D. Tataru and M. Vişan, Dispersive Equations and Nonlinear Waves, Oberwolfach Seminars 45, Springer, Basel, 2014.

[13]

N. Strunk, Strichartz estimates for Schrödinger equations on irrational tori in two and three dimensions, J. Evol. Equ., 14 (2014), 829-839.  doi: 10.1007/s00028-014-0240-8.

[14]

M. Taylor, Tools for PDE: Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, Amer. Math. Soc., Providence, RI, 2000.

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