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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 & Continuous Dynamical Systems - A, 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.  doi: 10.1090/S0002-9947-07-04250-X.  Google Scholar [2] J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case,, J. Amer. Math. Soc., 12 (1999), 145.  doi: 10.1090/S0894-0347-99-00283-0.  Google Scholar [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.  doi: 10.1090/S0002-9947-06-03955-9.  Google Scholar [4] T. Cazenave and F. B. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case,, in Nonlinear Semigroups, 1394 (1989), 18.  doi: 10.1007/BFb0086749.  Google Scholar [5] T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$,, Nonlinear Anal., 14 (1990), 807.  doi: 10.1016/0362-546X(90)90023-A.  Google Scholar [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.  doi: 10.1353/ajm.2003.0040.  Google Scholar [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.   Google Scholar [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.  doi: 10.4007/annals.2008.167.767.  Google Scholar [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.   Google Scholar [10] L. G. Farah, Global rough solutions to the critical generalized KdV equation,, J. Differential Equations, 249 (2010), 1968.   Google Scholar [11] G. Fonseca, F. Linares and G. Ponce, Global existence for the critical generalized KdV equation,, Proc. Amer. Math. Soc., 131 (2003), 1847.  doi: 10.1090/S0002-9939-02-06871-5.  Google Scholar [12] G. Grillakis, On nonlinear Schrödinger equations,, Comm. Partial Differential Equations, 25 (2000), 1827.   Google Scholar [13] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation,, Studies in applied mathematics, 8 (1983), 93.   Google Scholar [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.  doi: 10.1007/s00222-006-0011-4.  Google Scholar [15] C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations,, Indiana Univ. Math. J., 40 (1991), 33.  doi: 10.1512/iumj.1991.40.40003.  Google Scholar [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.  doi: 10.1002/cpa.3160460405.  Google Scholar [17] S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations,, J. Differential Equations, 175 (2001), 353.   Google Scholar [18] S. Keraani, On the blow-up phenomenon of the critical nonlinear Schrödinger equation,, J. Funct. Anal., 235 (2006), 171.  doi: 10.1016/j.jfa.2005.10.005.  Google Scholar [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.  doi: 10.4171/JEMS/180.  Google Scholar [20] R. Killip and M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher,, Amer. J. Math., 132 (2010), 361.  doi: 10.1353/ajm.0.0107.  Google Scholar [21] R. Killip and M. Visan, "Nonlinear Schrödinger Equations at Critical Regularity,", Lecture notes prepared for Clay Mathematics Institute Summer School, (2008).   Google Scholar [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.  doi: 10.2140/apde.2008.1.229.  Google Scholar [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.  doi: 10.1090/S0894-0347-01-00369-1.  Google Scholar [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.  doi: 10.1155/S1073792898000270.  Google Scholar [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.   Google Scholar [26] S. Shao, Maximizers for the Strichartz inequalities and the Sobolev-Strichartz inequalities for the Schrödinger equation,, Electron. J. Differential Equations, (2009).   Google Scholar [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.  doi: 10.2140/apde.2009.2.83.  Google Scholar [28] E. M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals,", Princeton Mathematical Series, 43 (1993).   Google Scholar [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.   Google Scholar [30] T. Tao, "Nonlinear Dispersive Equations. Local and Global Analysis,", CBMS Regional Conference Series in Mathematics, 106 (2006).   Google Scholar [31] T. Tao, Two remarks on the generalised Korteweg-de Vries equation,, Discrete and Continuous Dynamical Systems, 18 (2007), 1.  doi: 10.3934/dcds.2007.18.1.  Google Scholar [32] T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities,, Comm. Partial Differential Equations, 32 (2007), 1281.   Google Scholar [33] T. Tao, M. Visan and X. Zhang, Minimal-mass blow-up solutions of the mass-critical NLS,, Forum Math., 20 (2008), 881.  doi: 10.1515/FORUM.2008.042.  Google Scholar [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.  doi: 10.1215/S0012-7094-07-14015-8.  Google Scholar [35] N. Tzirakis, The Cauchy problem for the semilinear quintic Schrödinger equation in one dimension,, Differential Integral Equations, 18 (2005), 947.   Google Scholar [36] M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions,, Duke Math. J., 138 (2007), 281.  doi: 10.1215/S0012-7094-07-13825-0.  Google Scholar [37] M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (): 1982.   Google Scholar [38] X. Zhang, On the Cauchy problem of 3-D energy-critical Schrödinger equations with subcritical perturbations,, J. Differential Equations, 230 (2006), 422.   Google Scholar

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.  doi: 10.1090/S0002-9947-07-04250-X.  Google Scholar [2] J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case,, J. Amer. Math. Soc., 12 (1999), 145.  doi: 10.1090/S0894-0347-99-00283-0.  Google Scholar [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.  doi: 10.1090/S0002-9947-06-03955-9.  Google Scholar [4] T. Cazenave and F. B. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case,, in Nonlinear Semigroups, 1394 (1989), 18.  doi: 10.1007/BFb0086749.  Google Scholar [5] T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$,, Nonlinear Anal., 14 (1990), 807.  doi: 10.1016/0362-546X(90)90023-A.  Google Scholar [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.  doi: 10.1353/ajm.2003.0040.  Google Scholar [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.   Google Scholar [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.  doi: 10.4007/annals.2008.167.767.  Google Scholar [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.   Google Scholar [10] L. G. Farah, Global rough solutions to the critical generalized KdV equation,, J. Differential Equations, 249 (2010), 1968.   Google Scholar [11] G. Fonseca, F. Linares and G. Ponce, Global existence for the critical generalized KdV equation,, Proc. Amer. Math. Soc., 131 (2003), 1847.  doi: 10.1090/S0002-9939-02-06871-5.  Google Scholar [12] G. Grillakis, On nonlinear Schrödinger equations,, Comm. Partial Differential Equations, 25 (2000), 1827.   Google Scholar [13] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation,, Studies in applied mathematics, 8 (1983), 93.   Google Scholar [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.  doi: 10.1007/s00222-006-0011-4.  Google Scholar [15] C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations,, Indiana Univ. Math. J., 40 (1991), 33.  doi: 10.1512/iumj.1991.40.40003.  Google Scholar [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.  doi: 10.1002/cpa.3160460405.  Google Scholar [17] S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations,, J. Differential Equations, 175 (2001), 353.   Google Scholar [18] S. Keraani, On the blow-up phenomenon of the critical nonlinear Schrödinger equation,, J. Funct. Anal., 235 (2006), 171.  doi: 10.1016/j.jfa.2005.10.005.  Google Scholar [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.  doi: 10.4171/JEMS/180.  Google Scholar [20] R. Killip and M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher,, Amer. J. Math., 132 (2010), 361.  doi: 10.1353/ajm.0.0107.  Google Scholar [21] R. Killip and M. Visan, "Nonlinear Schrödinger Equations at Critical Regularity,", Lecture notes prepared for Clay Mathematics Institute Summer School, (2008).   Google Scholar [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.  doi: 10.2140/apde.2008.1.229.  Google Scholar [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.  doi: 10.1090/S0894-0347-01-00369-1.  Google Scholar [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.  doi: 10.1155/S1073792898000270.  Google Scholar [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.   Google Scholar [26] S. Shao, Maximizers for the Strichartz inequalities and the Sobolev-Strichartz inequalities for the Schrödinger equation,, Electron. J. Differential Equations, (2009).   Google Scholar [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.  doi: 10.2140/apde.2009.2.83.  Google Scholar [28] E. M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals,", Princeton Mathematical Series, 43 (1993).   Google Scholar [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.   Google Scholar [30] T. Tao, "Nonlinear Dispersive Equations. Local and Global Analysis,", CBMS Regional Conference Series in Mathematics, 106 (2006).   Google Scholar [31] T. Tao, Two remarks on the generalised Korteweg-de Vries equation,, Discrete and Continuous Dynamical Systems, 18 (2007), 1.  doi: 10.3934/dcds.2007.18.1.  Google Scholar [32] T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities,, Comm. Partial Differential Equations, 32 (2007), 1281.   Google Scholar [33] T. Tao, M. Visan and X. Zhang, Minimal-mass blow-up solutions of the mass-critical NLS,, Forum Math., 20 (2008), 881.  doi: 10.1515/FORUM.2008.042.  Google Scholar [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.  doi: 10.1215/S0012-7094-07-14015-8.  Google Scholar [35] N. Tzirakis, The Cauchy problem for the semilinear quintic Schrödinger equation in one dimension,, Differential Integral Equations, 18 (2005), 947.   Google Scholar [36] M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions,, Duke Math. J., 138 (2007), 281.  doi: 10.1215/S0012-7094-07-13825-0.  Google Scholar [37] M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (): 1982.   Google Scholar [38] X. Zhang, On the Cauchy problem of 3-D energy-critical Schrödinger equations with subcritical perturbations,, J. Differential Equations, 230 (2006), 422.   Google Scholar
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