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January  2012, 32(1): 191-221. doi: 10.3934/dcds.2012.32.191

## On the mass-critical generalized KdV equation

 1 Mathematics Department, University of California, Los Angeles, Box 951555, Los Angeles, CA 90095, United States, United States 2 Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, 335 Gwahangno, Yuseong-gu, Daejeon, Korea 305-701, South Korea 3 Institute for Mathematics and its Applications, University of Minnesota, 207 Church St. SE Minneapolis, MN 55455, United States

Received  August 2010 Revised  December 2010 Published  September 2011

We consider the mass-critical generalized Korteweg--de Vries equation $$(\partial_t + \partial_{xxx})u=\pm \partial_x(u^5)$$ for real-valued functions $u(t,x)$. We prove that if the global well-posedness and scattering conjecture for this equation failed, then, conditional on a positive answer to the global well-posedness and scattering conjecture for the mass-critical nonlinear Schrödinger equation $(-i\partial_t + \partial_{xx})u=\pm (|u|^4u)$, there exists a minimal-mass blowup solution to the mass-critical generalized KdV equation which is almost periodic modulo the symmetries of the equation. Moreover, we can guarantee that this minimal-mass blowup solution is either a self-similar solution, a soliton-like solution, or a double high-to-low frequency cascade solution.
Citation: Rowan Killip, Soonsik Kwon, Shuanglin Shao, Monica Visan. On the mass-critical generalized KdV equation. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 191-221. doi: 10.3934/dcds.2012.32.191
##### References:
 [1] P. Begout and A. Vargas, Mass concentration phenomena for the $L^2$-critical nonlinear Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 5257-5282. doi: 10.1090/S0002-9947-07-04250-X. [2] J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171. doi: 10.1090/S0894-0347-99-00283-0. [3] R. Carles and S. Keraani, On the role of quadratic oscillations in nonlinear Schrödinger equation II, the $L^2$-critical case, Trans. Amer. Math. Soc., 359 (2007), 33-62. doi: 10.1090/S0002-9947-06-03955-9. [4] T. Cazenave and F. B. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case, in Nonlinear Semigroups, Partial Differential Equations and Attractors, Lecture Notes in Math., 1394 (1989), 18-29. doi: 10.1007/BFb0086749. [5] T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836. doi: 10.1016/0362-546X(90)90023-A. [6] M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math., 125 (2003), 1235-1293. doi: 10.1353/ajm.2003.0040. [7] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation, Math. Res. Lett., 9 (2002), 659-682. [8] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbbR^3$, Ann. of Math., 167 (2008), 767-865. doi: 10.4007/annals.2008.167.767. [9] D. de Silva, N. Pavlovič, G. Staffilani and N. Tzirakis, Global well-posedness and polynomial bounds for the defocusing $L^{2}$ -critical nonlinear Schrödinger equation in $\mathbbR$, Comm. Partial Differential Equations, 33 (2008), 1395-1429. [10] L. G. Farah, Global rough solutions to the critical generalized KdV equation, J. Differential Equations, 249 (2010), 1968-1985. [11] G. Fonseca, F. Linares and G. Ponce, Global existence for the critical generalized KdV equation, Proc. Amer. Math. Soc., 131 (2003), 1847-1855. doi: 10.1090/S0002-9939-02-06871-5. [12] G. Grillakis, On nonlinear Schrödinger equations, Comm. Partial Differential Equations, 25 (2000), 1827-1844. [13] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in applied mathematics, Adv. Math. Suppl. Stud., 8 (1983), 93-128. [14] C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675. doi: 10.1007/s00222-006-0011-4. [15] C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40 (1991), 33-69. doi: 10.1512/iumj.1991.40.40003. [16] C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405. [17] S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations, 175 (2001), 353-392. [18] S. Keraani, On the blow-up phenomenon of the critical nonlinear Schrödinger equation, J. Funct. Anal., 235 (2006), 171-192. doi: 10.1016/j.jfa.2005.10.005. [19] 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. doi: 10.4171/JEMS/180. [20] R. Killip and M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math., 132 (2010), 361-424. doi: 10.1353/ajm.0.0107. [21] R. Killip and M. Visan, "Nonlinear Schrödinger Equations at Critical Regularity," Lecture notes prepared for Clay Mathematics Institute Summer School, Zürich, Switzerland, 2008. [22] R. Killip, M. Visan and X. Zhang, The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, Anal. PDE, 1 (2008), 229-266. doi: 10.2140/apde.2008.1.229. [23] F. Merle, Existence of blow-up solutions in the energy space for the critical generalized KdV equation, J. Amer. Math. Soc., 14 (2001), 555-578. doi: 10.1090/S0894-0347-01-00369-1. [24] F. Merle and L. Vega, Compactness at blow-up time for $L^2$ solutions of the critical nonlinear Schrödinger equation in 2D, Internat. Math. Res. Notices, (1998), 399-425. doi: 10.1155/S1073792898000270. [25] E. Ryckman and M. Visan, Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in $\mathbbR^{1+4}$, Amer. J. Math., 129 (2007), 1-60. [26] S. Shao, Maximizers for the Strichartz inequalities and the Sobolev-Strichartz inequalities for the Schrödinger equation, Electron. J. Differential Equations, (2009). [27] S. Shao, The linear profile decomposition for the Airy equation and the existence of maximizers for the Airy Strichartz inequality, Anal. PDE, 2 (2009), 83-117. doi: 10.2140/apde.2009.2.83. [28] E. M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals," Princeton Mathematical Series, 43, 1993. [29] T. Tao, Global well-posedness and scattering for the higher-dimensional energy-critical non-linear Schrödinger equation for radial data, New York J. of Math., 11 (2005), 57-80. [30] T. Tao, "Nonlinear Dispersive Equations. Local and Global Analysis," CBMS Regional Conference Series in Mathematics, 106, 2006. [31] T. Tao, Two remarks on the generalised Korteweg-de Vries equation, Discrete and Continuous Dynamical Systems, 18 (2007), 1-14. doi: 10.3934/dcds.2007.18.1. [32] T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343. [33] T. Tao, M. Visan and X. Zhang, Minimal-mass blow-up solutions of the mass-critical NLS, Forum Math., 20 (2008), 881-919. doi: 10.1515/FORUM.2008.042. [34] T. Tao, M. Visan and X. Zhang, Global well-posedness and scattering for the mass-critical nonlinear Schrödinger equation for radial data in high dimensions, Duke Math. J., 140 (2007), 165-202. doi: 10.1215/S0012-7094-07-14015-8. [35] N. Tzirakis, The Cauchy problem for the semilinear quintic Schrödinger equation in one dimension, Differential Integral Equations, 18 (2005), 947-960. [36] M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J., 138 (2007), 281-374. doi: 10.1215/S0012-7094-07-13825-0. [37] M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (): 1982. [38] X. Zhang, On the Cauchy problem of 3-D energy-critical Schrödinger equations with subcritical perturbations, J. Differential Equations, 230 (2006), 422-445.

show all references

##### References:
 [1] P. Begout and A. Vargas, Mass concentration phenomena for the $L^2$-critical nonlinear Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 5257-5282. doi: 10.1090/S0002-9947-07-04250-X. [2] J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171. doi: 10.1090/S0894-0347-99-00283-0. [3] R. Carles and S. Keraani, On the role of quadratic oscillations in nonlinear Schrödinger equation II, the $L^2$-critical case, Trans. Amer. Math. Soc., 359 (2007), 33-62. doi: 10.1090/S0002-9947-06-03955-9. [4] T. Cazenave and F. B. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case, in Nonlinear Semigroups, Partial Differential Equations and Attractors, Lecture Notes in Math., 1394 (1989), 18-29. doi: 10.1007/BFb0086749. [5] T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836. doi: 10.1016/0362-546X(90)90023-A. [6] M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math., 125 (2003), 1235-1293. doi: 10.1353/ajm.2003.0040. [7] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation, Math. Res. Lett., 9 (2002), 659-682. [8] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbbR^3$, Ann. of Math., 167 (2008), 767-865. doi: 10.4007/annals.2008.167.767. [9] D. de Silva, N. Pavlovič, G. Staffilani and N. Tzirakis, Global well-posedness and polynomial bounds for the defocusing $L^{2}$ -critical nonlinear Schrödinger equation in $\mathbbR$, Comm. Partial Differential Equations, 33 (2008), 1395-1429. [10] L. G. Farah, Global rough solutions to the critical generalized KdV equation, J. Differential Equations, 249 (2010), 1968-1985. [11] G. Fonseca, F. Linares and G. Ponce, Global existence for the critical generalized KdV equation, Proc. Amer. Math. Soc., 131 (2003), 1847-1855. doi: 10.1090/S0002-9939-02-06871-5. [12] G. Grillakis, On nonlinear Schrödinger equations, Comm. Partial Differential Equations, 25 (2000), 1827-1844. [13] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in applied mathematics, Adv. Math. Suppl. Stud., 8 (1983), 93-128. [14] C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675. doi: 10.1007/s00222-006-0011-4. [15] C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40 (1991), 33-69. doi: 10.1512/iumj.1991.40.40003. [16] C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405. [17] S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations, 175 (2001), 353-392. [18] S. Keraani, On the blow-up phenomenon of the critical nonlinear Schrödinger equation, J. Funct. Anal., 235 (2006), 171-192. doi: 10.1016/j.jfa.2005.10.005. [19] 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. doi: 10.4171/JEMS/180. [20] R. Killip and M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math., 132 (2010), 361-424. doi: 10.1353/ajm.0.0107. [21] R. Killip and M. Visan, "Nonlinear Schrödinger Equations at Critical Regularity," Lecture notes prepared for Clay Mathematics Institute Summer School, Zürich, Switzerland, 2008. [22] R. Killip, M. Visan and X. Zhang, The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, Anal. PDE, 1 (2008), 229-266. doi: 10.2140/apde.2008.1.229. [23] F. Merle, Existence of blow-up solutions in the energy space for the critical generalized KdV equation, J. Amer. Math. Soc., 14 (2001), 555-578. doi: 10.1090/S0894-0347-01-00369-1. [24] F. Merle and L. Vega, Compactness at blow-up time for $L^2$ solutions of the critical nonlinear Schrödinger equation in 2D, Internat. Math. Res. Notices, (1998), 399-425. doi: 10.1155/S1073792898000270. [25] E. Ryckman and M. Visan, Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in $\mathbbR^{1+4}$, Amer. J. Math., 129 (2007), 1-60. [26] S. Shao, Maximizers for the Strichartz inequalities and the Sobolev-Strichartz inequalities for the Schrödinger equation, Electron. J. Differential Equations, (2009). [27] S. Shao, The linear profile decomposition for the Airy equation and the existence of maximizers for the Airy Strichartz inequality, Anal. PDE, 2 (2009), 83-117. doi: 10.2140/apde.2009.2.83. [28] E. M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals," Princeton Mathematical Series, 43, 1993. [29] T. Tao, Global well-posedness and scattering for the higher-dimensional energy-critical non-linear Schrödinger equation for radial data, New York J. of Math., 11 (2005), 57-80. [30] T. Tao, "Nonlinear Dispersive Equations. Local and Global Analysis," CBMS Regional Conference Series in Mathematics, 106, 2006. [31] T. Tao, Two remarks on the generalised Korteweg-de Vries equation, Discrete and Continuous Dynamical Systems, 18 (2007), 1-14. doi: 10.3934/dcds.2007.18.1. [32] T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343. [33] T. Tao, M. Visan and X. Zhang, Minimal-mass blow-up solutions of the mass-critical NLS, Forum Math., 20 (2008), 881-919. doi: 10.1515/FORUM.2008.042. [34] T. Tao, M. Visan and X. Zhang, Global well-posedness and scattering for the mass-critical nonlinear Schrödinger equation for radial data in high dimensions, Duke Math. J., 140 (2007), 165-202. doi: 10.1215/S0012-7094-07-14015-8. [35] N. Tzirakis, The Cauchy problem for the semilinear quintic Schrödinger equation in one dimension, Differential Integral Equations, 18 (2005), 947-960. [36] M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J., 138 (2007), 281-374. doi: 10.1215/S0012-7094-07-13825-0. [37] M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (): 1982. [38] X. Zhang, On the Cauchy problem of 3-D energy-critical Schrödinger equations with subcritical perturbations, J. Differential Equations, 230 (2006), 422-445.
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