# American Institute of Mathematical Sciences

September  2017, 16(5): 1517-1530. doi: 10.3934/cpaa.2017072

## Local well-posedness for 2-D Schrödinger equation on irrational tori and bounds on Sobolev norms

 Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC V6T1Z2 Canada

Received  July 2013 Revised  February 2017 Published  May 2017

Fund Project: The author was partially supported by NSF grants DMS-0900865 and DMS-0901222.

In this paper we consider the cubic Schrödinger equation in two space dimensions on irrational tori. Our main result is an improvement of the Strichartz estimates on irrational tori. Using this estimate we obtain a local well-posedness result in $H^{s}$ for $s>\frac{131}{416}$. We also obtain improved growth bounds for higher order Sobolev norms.

Citation: Seckin Demirbas. Local well-posedness for 2-D Schrödinger equation on irrational tori and bounds on Sobolev norms. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1517-1530. doi: 10.3934/cpaa.2017072
##### References:
 [1] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geometric and Functional Analysis, 3 (1993), 209-262.  doi: 10.1007/BF01895688. [2] J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, International Mathematics Research Notices, 6 (1996), 277-304.  doi: 10.1155/S1073792896000207. [3] J. Bourgain, On Strichartz's inequalities and the nonlinear Schrödinger equation on irrational tori, Mathematical Aspects of Nonlinear Dispersive Equations. Ann. of Math. Stud., 163 (2007), 1-20. [4] N. Burq, P. Gérard and N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, American Journal of Mathematics, 126 (2004), 569-605. [5] N. Burq, P. Gérard and N. Tzvetkov, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Inventiones mathematicae, 159 (2005), 187-223.  doi: 10.1007/s00222-004-0388-x. [6] F. Catoire and W-M. Wang, Bounds on Sobolev norms for the nonlinear Schrödinger equation on general tori, Communications in Pure and Applied Analysis, 9 (2010), 483-491.  doi: 10.3934/cpaa.2010.9.483. [7] D. De Silva, N. Pavlovic, G. Staffilani and N. Tzirakis, Global well-posedness for a periodic nonlinear Schrödinger equation in 1-D and 2-D, Discrete and Continuous Dynamical Systems, 19 (2007), 37-65.  doi: 10.3934/dcds.2007.19.37. [8] J. Ginibre, Le probléme de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace, Inventiones Mathematicae, 37 (1995), 163-187. [9] Z. Guo, T. Oh and Y. Wang, Strichartz estimates for Schödinger equations on irrational tori, arXiv: 1306.4973, (2013). doi: 10.1112/plms/pdu025. [10] Z. Hani, B. Pausader, N. Tzvetkov and N. Visciglia, Modified scattering for the cubic Schrödinger equation on product spaces and applications, arXiv: 1311.2275, 2013. doi: 10.1017/fmp. 2015. 5. [11] M. N. Huxley, Exponential sums and lattice points Ⅲ, Proceedings of the London Mathematical Society, 87 (2003), 591-609.  doi: 10.1112/S0024611503014485. [12] V. Jarník, Über die Gitterpunkte auf konvexen Kurven, Mathematische Zeitschrift, 24 (1926), 500-518.  doi: 10.1007/BF01216795. [13] V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to Nonlinear Schrodinger Equations on $S^1$, Differential and Integral Equations, 24 (2011), 653-718. [14] V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to 2D Hartree equations, Discrete and Continuous Dynamical Systems, 32 (2012), 3733-3771.  doi: 10.3934/dcds.2012.32.3733. [15] G. Staffilani, Quadratic forms for a 2-D semilinear Schrödinger equation, Duke Mathematical Journal, 86 (1997), 79-107.  doi: 10.1215/S0012-7094-97-08603-8. [16] N. Strunk, Strichartz estimates for Schrödinger equations on irrational tori in two and three dimensions J. Evol. Equ. 14 (2014) 829. doi: 10.1007/s00028-014-0240-8. [17] T. Tao, Nonlinear dispersive equations: local and global analysis Amer Mathematical Society 106 (2006). doi: 10.1090/cbms/106. [18] S. Zhong, The growth in time of higher Sobolev norms of solutions to Schrödinger equations on compact Riemannian manifolds, Journal of Differential Equations, 245 (2008), 359-376.  doi: 10.1016/j.jde.2008.03.008.

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##### References:
 [1] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geometric and Functional Analysis, 3 (1993), 209-262.  doi: 10.1007/BF01895688. [2] J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, International Mathematics Research Notices, 6 (1996), 277-304.  doi: 10.1155/S1073792896000207. [3] J. Bourgain, On Strichartz's inequalities and the nonlinear Schrödinger equation on irrational tori, Mathematical Aspects of Nonlinear Dispersive Equations. Ann. of Math. Stud., 163 (2007), 1-20. [4] N. Burq, P. Gérard and N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, American Journal of Mathematics, 126 (2004), 569-605. [5] N. Burq, P. Gérard and N. Tzvetkov, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Inventiones mathematicae, 159 (2005), 187-223.  doi: 10.1007/s00222-004-0388-x. [6] F. Catoire and W-M. Wang, Bounds on Sobolev norms for the nonlinear Schrödinger equation on general tori, Communications in Pure and Applied Analysis, 9 (2010), 483-491.  doi: 10.3934/cpaa.2010.9.483. [7] D. De Silva, N. Pavlovic, G. Staffilani and N. Tzirakis, Global well-posedness for a periodic nonlinear Schrödinger equation in 1-D and 2-D, Discrete and Continuous Dynamical Systems, 19 (2007), 37-65.  doi: 10.3934/dcds.2007.19.37. [8] J. Ginibre, Le probléme de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace, Inventiones Mathematicae, 37 (1995), 163-187. [9] Z. Guo, T. Oh and Y. Wang, Strichartz estimates for Schödinger equations on irrational tori, arXiv: 1306.4973, (2013). doi: 10.1112/plms/pdu025. [10] Z. Hani, B. Pausader, N. Tzvetkov and N. Visciglia, Modified scattering for the cubic Schrödinger equation on product spaces and applications, arXiv: 1311.2275, 2013. doi: 10.1017/fmp. 2015. 5. [11] M. N. Huxley, Exponential sums and lattice points Ⅲ, Proceedings of the London Mathematical Society, 87 (2003), 591-609.  doi: 10.1112/S0024611503014485. [12] V. Jarník, Über die Gitterpunkte auf konvexen Kurven, Mathematische Zeitschrift, 24 (1926), 500-518.  doi: 10.1007/BF01216795. [13] V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to Nonlinear Schrodinger Equations on $S^1$, Differential and Integral Equations, 24 (2011), 653-718. [14] V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to 2D Hartree equations, Discrete and Continuous Dynamical Systems, 32 (2012), 3733-3771.  doi: 10.3934/dcds.2012.32.3733. [15] G. Staffilani, Quadratic forms for a 2-D semilinear Schrödinger equation, Duke Mathematical Journal, 86 (1997), 79-107.  doi: 10.1215/S0012-7094-97-08603-8. [16] N. Strunk, Strichartz estimates for Schrödinger equations on irrational tori in two and three dimensions J. Evol. Equ. 14 (2014) 829. doi: 10.1007/s00028-014-0240-8. [17] T. Tao, Nonlinear dispersive equations: local and global analysis Amer Mathematical Society 106 (2006). doi: 10.1090/cbms/106. [18] S. Zhong, The growth in time of higher Sobolev norms of solutions to Schrödinger equations on compact Riemannian manifolds, Journal of Differential Equations, 245 (2008), 359-376.  doi: 10.1016/j.jde.2008.03.008.
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