# American Institute of Mathematical Sciences

May  2018, 17(3): 1071-1101. doi: 10.3934/cpaa.2018052

## Scattering for the two dimensional NLS with (full) exponential nonlinearity

 UCLA Department of Mathematics, 520 Portola Plaza, Math Sciences Building 6363, Los Angeles, CA 90095-1555, USA

* Corresponding author

Received  October 2016 Revised  August 2017 Published  January 2018

Fund Project: The first author is supported by NSF grant DMS 1265868 and DMS-1500707

We obtain global well-posedness, scattering, and global
 $L_t^4H_{x}^{1,4}$
spacetime bounds for energy-space solutions to the energy-subcritical nonlinear Schrödinger equation
 $iu_t+Δ u = u(e^{4π |u|^2}-1)$
in two spatial dimensions. Our approach is perturbative; we view our problem as a perturbation of the mass-critical NLS to employ the techniques of Tao-Visan-Zhang from [25]. This permits us to combine the known spacetime estimates for mass-critical NLS proved by Dodson [12] and the work of [15] and [14] to prove corresponding spacetime estimates which imply scattering.
Citation: A. Adam Azzam. Scattering for the two dimensional NLS with (full) exponential nonlinearity. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1071-1101. doi: 10.3934/cpaa.2018052
##### References:
 [1] S. Adachi and K. Tanaka, Trudinger type inequalities in ${{\bf{R}}^{N}}$ and their best exponents, Proc. Amer. Math. Soc., 128 (2000), 2051-2057. Google Scholar [2] H. Bahouri, S. Ibrahim and G. Perelman, Scattering for the critical 2-D NLS with exponential growth, Differential Integral Equations, 27 (2014), 233-268. Google Scholar [3] A. Biryuk, An optimal limiting 2D Sobolev inequality, Proc. Amer. Math. Soc., 138 (2010), 1461-1470. Google Scholar [4] J. Bourgain, Scattering in the energy space and below for 3D NLS, J. Anal. Math., 75 (1998), 267-297. Google Scholar [5] J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171. Google Scholar [6] T. Cazenave, Semilinear Schrödinger Equations, vol. 10 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. Google Scholar [7] T. Cazenave and F. B. Weissler, Some remarks on the nonlinear Schrödinger equation in the subcritical case, in New Methods and Results in Nonlinear Field Equations (Bielefeld, 1987), vol. 347 of Lecture Notes in Phys., Springer, Berlin, 1989, 59-69. Google Scholar [8] T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836. Google Scholar [9] J. Colliander, M. Grillakis and N. Tzirakis, Tensor products and correlation estimates with applications to nonlinear Schrödinger equations, Comm. Pure Appl. Math., 62 (2009), 920-968. Google Scholar [10] J. Colliander, S. Ibrahim, M. Majdoub and N. Masmoudi, Energy critical NLS in two space dimensions, J. Hyperbolic Differ. Equ., 6 (2009), 549-575. Google Scholar [11] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\Bbb R^3$, Ann. of Math. (2), 167 (2008), 767-865. Google Scholar [12] B. Dodson, Global well-posedness and scattering for the defocusing, $l^{2}$-critical, nonlinear schrödinger equation when $d = 2$, Duke Math. J., 165 (2016), 3435-3516. Google Scholar [13] S. Ibrahim, M. Majdoub and N. Masmoudi, Double logarithmic inequality with a sharp constant, Proc. Amer. Math. Soc., 135 (2007), 87-97 (electronic). Google Scholar [14] S. Ibrahim, M. Majdoub, N. Masmoudi and K. Nakanishi, Scattering for the two-dimensional energy-critical wave equation, Duke Math. J., 150 (2009), 287-329. Google Scholar [15] S. Ibrahim, M. Majdoub, N. Masmoudi and K. Nakanishi, Scattering for the two-dimensional NLS with exponential nonlinearity, Nonlinearity, 25 (2012), 1843-1849. Google Scholar [16] S. Ibrahim, N. Masmoudi and K. Nakanishi, Trudinger -moser inequality on the whole plane with the exact growth condition, Journal of the European Mathematical Society, (), 819-835. Google Scholar [17] M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. Google Scholar [18] R. Killip, T. Tao and M. Visan, The cubic nonlinear Schrödinger equation in two dimensions with radial data, J. Eur. Math. Soc. (JEMS), 11 (2009), 1203-1258. Google Scholar [19] R. Killip and M. Visan, Evolution Equations, vol. 17 of Clay Mathematics Proceedings, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2013, Lecture notes from the Clay Mathematics Institute Summer School held at the Eidgenössische Technische Hochschule (ETH), Zürich, June 23-July 18,2008.Google Scholar [20] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092. Google Scholar [21] K. Nakanishi, Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions 1 and 2, J. Funct. Anal., 169 (1999), 201-225. Google Scholar [22] F. Planchon and L. Vega, Bilinear virial identities and applications, Ann. Sci. Éc. Norm. Supér. (4), 42 (2009), 261-290. Google Scholar [23] B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\Bbb R^2$, J. Funct. Anal., 219 (2005), 340-367. Google Scholar [24] E. Ryckman and M. Visan, Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in $\Bbb R^{1+4}$, Amer. J. Math., 129 (2007), 1-60. Google Scholar [25] T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343. Google Scholar [26] N. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483. Google Scholar [27] M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J., 138 (2007), 281-374. Google Scholar

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##### References:
 [1] S. Adachi and K. Tanaka, Trudinger type inequalities in ${{\bf{R}}^{N}}$ and their best exponents, Proc. Amer. Math. Soc., 128 (2000), 2051-2057. Google Scholar [2] H. Bahouri, S. Ibrahim and G. Perelman, Scattering for the critical 2-D NLS with exponential growth, Differential Integral Equations, 27 (2014), 233-268. Google Scholar [3] A. Biryuk, An optimal limiting 2D Sobolev inequality, Proc. Amer. Math. Soc., 138 (2010), 1461-1470. Google Scholar [4] J. Bourgain, Scattering in the energy space and below for 3D NLS, J. Anal. Math., 75 (1998), 267-297. Google Scholar [5] J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171. Google Scholar [6] T. Cazenave, Semilinear Schrödinger Equations, vol. 10 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. Google Scholar [7] T. Cazenave and F. B. Weissler, Some remarks on the nonlinear Schrödinger equation in the subcritical case, in New Methods and Results in Nonlinear Field Equations (Bielefeld, 1987), vol. 347 of Lecture Notes in Phys., Springer, Berlin, 1989, 59-69. Google Scholar [8] T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836. Google Scholar [9] J. Colliander, M. Grillakis and N. Tzirakis, Tensor products and correlation estimates with applications to nonlinear Schrödinger equations, Comm. Pure Appl. Math., 62 (2009), 920-968. Google Scholar [10] J. Colliander, S. Ibrahim, M. Majdoub and N. Masmoudi, Energy critical NLS in two space dimensions, J. Hyperbolic Differ. Equ., 6 (2009), 549-575. Google Scholar [11] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\Bbb R^3$, Ann. of Math. (2), 167 (2008), 767-865. Google Scholar [12] B. Dodson, Global well-posedness and scattering for the defocusing, $l^{2}$-critical, nonlinear schrödinger equation when $d = 2$, Duke Math. J., 165 (2016), 3435-3516. Google Scholar [13] S. Ibrahim, M. Majdoub and N. Masmoudi, Double logarithmic inequality with a sharp constant, Proc. Amer. Math. Soc., 135 (2007), 87-97 (electronic). Google Scholar [14] S. Ibrahim, M. Majdoub, N. Masmoudi and K. Nakanishi, Scattering for the two-dimensional energy-critical wave equation, Duke Math. J., 150 (2009), 287-329. Google Scholar [15] S. Ibrahim, M. Majdoub, N. Masmoudi and K. Nakanishi, Scattering for the two-dimensional NLS with exponential nonlinearity, Nonlinearity, 25 (2012), 1843-1849. Google Scholar [16] S. Ibrahim, N. Masmoudi and K. Nakanishi, Trudinger -moser inequality on the whole plane with the exact growth condition, Journal of the European Mathematical Society, (), 819-835. Google Scholar [17] M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. Google Scholar [18] R. Killip, T. Tao and M. Visan, The cubic nonlinear Schrödinger equation in two dimensions with radial data, J. Eur. Math. Soc. (JEMS), 11 (2009), 1203-1258. Google Scholar [19] R. Killip and M. Visan, Evolution Equations, vol. 17 of Clay Mathematics Proceedings, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2013, Lecture notes from the Clay Mathematics Institute Summer School held at the Eidgenössische Technische Hochschule (ETH), Zürich, June 23-July 18,2008.Google Scholar [20] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092. Google Scholar [21] K. Nakanishi, Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions 1 and 2, J. Funct. Anal., 169 (1999), 201-225. Google Scholar [22] F. Planchon and L. Vega, Bilinear virial identities and applications, Ann. Sci. Éc. Norm. Supér. (4), 42 (2009), 261-290. Google Scholar [23] B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in $\Bbb R^2$, J. Funct. Anal., 219 (2005), 340-367. Google Scholar [24] E. Ryckman and M. Visan, Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in $\Bbb R^{1+4}$, Amer. J. Math., 129 (2007), 1-60. Google Scholar [25] T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343. Google Scholar [26] N. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483. Google Scholar [27] M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J., 138 (2007), 281-374. Google Scholar
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