# 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 & 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [8] J. Ginibre, Le probléme de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace, Inventiones Mathematicae, 37 (1995), 163-187. Google Scholar [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. Google Scholar [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. Google Scholar [11] M. N. Huxley, Exponential sums and lattice points Ⅲ, Proceedings of the London Mathematical Society, 87 (2003), 591-609. doi: 10.1112/S0024611503014485. Google Scholar [12] V. Jarník, Über die Gitterpunkte auf konvexen Kurven, Mathematische Zeitschrift, 24 (1926), 500-518. doi: 10.1007/BF01216795. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [17] T. Tao, Nonlinear dispersive equations: local and global analysis Amer Mathematical Society 106 (2006). doi: 10.1090/cbms/106. Google Scholar [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. Google Scholar

show all references

##### 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [8] J. Ginibre, Le probléme de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace, Inventiones Mathematicae, 37 (1995), 163-187. Google Scholar [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. Google Scholar [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. Google Scholar [11] M. N. Huxley, Exponential sums and lattice points Ⅲ, Proceedings of the London Mathematical Society, 87 (2003), 591-609. doi: 10.1112/S0024611503014485. Google Scholar [12] V. Jarník, Über die Gitterpunkte auf konvexen Kurven, Mathematische Zeitschrift, 24 (1926), 500-518. doi: 10.1007/BF01216795. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [17] T. Tao, Nonlinear dispersive equations: local and global analysis Amer Mathematical Society 106 (2006). doi: 10.1090/cbms/106. Google Scholar [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. Google Scholar
 [1] F. Catoire, W. M. Wang. Bounds on Sobolev norms for the defocusing nonlinear Schrödinger equation on general flat tori. Communications on Pure & Applied Analysis, 2010, 9 (2) : 483-491. doi: 10.3934/cpaa.2010.9.483 [2] Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for a periodic nonlinear Schrödinger equation in 1D and 2D. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 37-65. doi: 10.3934/dcds.2007.19.37 [3] Massimo Cicognani, Michael Reissig. Well-posedness for degenerate Schrödinger equations. Evolution Equations & Control Theory, 2014, 3 (1) : 15-33. doi: 10.3934/eect.2014.3.15 [4] Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027 [5] Takafumi Akahori. Low regularity global well-posedness for the nonlinear Schrödinger equation on closed manifolds. Communications on Pure & Applied Analysis, 2010, 9 (2) : 261-280. doi: 10.3934/cpaa.2010.9.261 [6] Nobu Kishimoto. Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1123-1143. doi: 10.3934/cpaa.2008.7.1123 [7] Zihua Guo, Yifei Wu. Global well-posedness for the derivative nonlinear Schrödinger equation in $H^{\frac 12} (\mathbb{R} )$. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 257-264. doi: 10.3934/dcds.2017010 [8] Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for the $L^2$ critical nonlinear Schrödinger equation in higher dimensions. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1023-1041. doi: 10.3934/cpaa.2007.6.1023 [9] Jun-ichi Segata. Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1093-1105. doi: 10.3934/dcds.2010.27.1093 [10] Jaime Angulo, Carlos Matheus, Didier Pilod. Global well-posedness and non-linear stability of periodic traveling waves for a Schrödinger-Benjamin-Ono system. Communications on Pure & Applied Analysis, 2009, 8 (3) : 815-844. doi: 10.3934/cpaa.2009.8.815 [11] Benjamin Dodson. Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when $n = 3$ via a linear-nonlinear decomposition. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1905-1926. doi: 10.3934/dcds.2013.33.1905 [12] G. Fonseca, G. Rodríguez-Blanco, W. Sandoval. Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1327-1341. doi: 10.3934/cpaa.2015.14.1327 [13] Yonggeun Cho, Gyeongha Hwang, Soonsik Kwon, Sanghyuk Lee. Well-posedness and ill-posedness for the cubic fractional Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2863-2880. doi: 10.3934/dcds.2015.35.2863 [14] Myeongju Chae, Soonsik Kwon. The stability of nonlinear Schrödinger equations with a potential in high Sobolev norms revisited. Communications on Pure & Applied Analysis, 2016, 15 (2) : 341-365. doi: 10.3934/cpaa.2016.15.341 [15] Luiz Gustavo Farah. Local solutions in Sobolev spaces and unconditional well-posedness for the generalized Boussinesq equation. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1521-1539. doi: 10.3934/cpaa.2009.8.1521 [16] Ming Wang. Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5763-5788. doi: 10.3934/dcds.2016053 [17] Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831 [18] Igor Chueshov, Alexey Shcherbina. Semi-weak well-posedness and attractors for 2D Schrödinger-Boussinesq equations. Evolution Equations & Control Theory, 2012, 1 (1) : 57-80. doi: 10.3934/eect.2012.1.57 [19] Changxing Miao, Bo Zhang. Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 181-200. doi: 10.3934/dcds.2007.17.181 [20] Chengchun Hao. Well-posedness for one-dimensional derivative nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2007, 6 (4) : 997-1021. doi: 10.3934/cpaa.2007.6.997

2018 Impact Factor: 0.925