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Mass concentration for the $L^2$-critical nonlinear Schrödinger equations of higher orders
1. | Department of Applied Mathematics, Hankyong National University, Ansong 456-749, South Korea |
2. | Department of Mathematics, Chosun University, Gwangju 501-759, South Korea |
3. | School of Mathematical Sciences, Seoul National University, Seoul 151-742, South Korea |
References:
[1] |
P. Bégout and A. Vargas, Mass concentration Phenomena for the $L^2$-critical for the nonlinear Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 5257-5282.
doi: 10.1090/S0002-9947-07-04250-X. |
[2] |
J. Bergh and J. Löfström, "Interpolation Spaces," Springer, New York, 1976. |
[3] |
J. Bourgain, Refinements of Strichartz' inequality and applications to $2$D-NLS with critical nonlinearity, Int. Math. Res. Not., 5 (1998), 253-283.
doi: 10.1155/S1073792898000191. |
[4] |
M. Chae, S. Hong, J. Kim, S. Lee and C. W. Yang, On mass concentration for the $L^2$-critical nonlinear Schrödinger equations, Comm. Partial Differential Equations, 34 (2009), 486-505.
doi: 10.1080/03605300902812426. |
[5] |
M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425.
doi: 10.1006/jfan.2000.3687. |
[6] |
G. Fibich, B. Ilan and G. Papanicolaou, Self-focusing with fourth order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462.
doi: 10.1137/S0036139901387241. |
[7] |
V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339.
doi: 10.1103/PhysRevE.53.R1336. |
[8] |
V. I. Karpman and A. G. Shagalov, Stability of soliton described by nonlinear Schrödinger type equations with higher-order dispersion, Phys. D., 144 (2000), 194-210.
doi: 10.1016/S0167-2789(00)00078-6. |
[9] |
M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039. |
[10] |
S. Lee and A. Vargas, Sharp null form estimates for the wave equation, Amer. J. Math., 130 (2008), 1279-1326.
doi: 10.1353/ajm.0.0024. |
[11] |
F. Merle and Y. Tsutsumi, $L^2$ concentration of blow up solutions for the nonlinear Schrödinger equation with critical power nonlinearity, J. Diff. Eq., 84 (1990), 205-214. |
[12] |
A. Moyua, A. Vargas and L. Vega, Schrödinger maximal function and restriction properties of the Fourier transform, Int. Math. Res. Not., 16 (1996), 793-815.
doi: 10.1155/S1073792896000499. |
[13] |
H. Nawa, Mass concentration phenomenon for the nonlinear Schrödinger equation with the critical power nonlinearity, Funkcial. Ekvac., 35 (1992), 1-18. |
[14] |
B. Pausader, "Problèmes Bien Posés et Diffusion pour Des équations Non Linéaires Dispersives D'ordre Quatre," Ph.d dissertation, Université de Cergy Pontoise, 2008. |
[15] |
K. Rogers and A. Vargas, A refinement of the Strichartz inequality on the saddle and applications, J. Funct. Anal., 241 (2006), 212-231.
doi: 10.1016/j.jfa.2006.04.026. |
[16] |
E. M. Stein, "Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals," Princeton Univ. Press, 1993. |
[17] |
T. Tao, A sharp bilinear restrictions estimate for paraboloids, Geom. Funct. Anal., 13 (2003), 1359-1384.
doi: 10.1007/s00039-003-0449-0. |
[18] |
T. Tao, "Nonlinear Dispersive Equations," CBMS Regional Conf. Ser. in Math. Amer. Math. Soc., 106, 2006. |
[19] |
T. Tao, A. Vargas and L. Vega, A bilinear approach to the restriction and Kakeya conjectures, J. Amer. Math. Soc., 11 (1998), 967-1000.
doi: 10.1090/S0894-0347-98-00278-1. |
show all references
References:
[1] |
P. Bégout and A. Vargas, Mass concentration Phenomena for the $L^2$-critical for the nonlinear Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 5257-5282.
doi: 10.1090/S0002-9947-07-04250-X. |
[2] |
J. Bergh and J. Löfström, "Interpolation Spaces," Springer, New York, 1976. |
[3] |
J. Bourgain, Refinements of Strichartz' inequality and applications to $2$D-NLS with critical nonlinearity, Int. Math. Res. Not., 5 (1998), 253-283.
doi: 10.1155/S1073792898000191. |
[4] |
M. Chae, S. Hong, J. Kim, S. Lee and C. W. Yang, On mass concentration for the $L^2$-critical nonlinear Schrödinger equations, Comm. Partial Differential Equations, 34 (2009), 486-505.
doi: 10.1080/03605300902812426. |
[5] |
M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425.
doi: 10.1006/jfan.2000.3687. |
[6] |
G. Fibich, B. Ilan and G. Papanicolaou, Self-focusing with fourth order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462.
doi: 10.1137/S0036139901387241. |
[7] |
V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339.
doi: 10.1103/PhysRevE.53.R1336. |
[8] |
V. I. Karpman and A. G. Shagalov, Stability of soliton described by nonlinear Schrödinger type equations with higher-order dispersion, Phys. D., 144 (2000), 194-210.
doi: 10.1016/S0167-2789(00)00078-6. |
[9] |
M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039. |
[10] |
S. Lee and A. Vargas, Sharp null form estimates for the wave equation, Amer. J. Math., 130 (2008), 1279-1326.
doi: 10.1353/ajm.0.0024. |
[11] |
F. Merle and Y. Tsutsumi, $L^2$ concentration of blow up solutions for the nonlinear Schrödinger equation with critical power nonlinearity, J. Diff. Eq., 84 (1990), 205-214. |
[12] |
A. Moyua, A. Vargas and L. Vega, Schrödinger maximal function and restriction properties of the Fourier transform, Int. Math. Res. Not., 16 (1996), 793-815.
doi: 10.1155/S1073792896000499. |
[13] |
H. Nawa, Mass concentration phenomenon for the nonlinear Schrödinger equation with the critical power nonlinearity, Funkcial. Ekvac., 35 (1992), 1-18. |
[14] |
B. Pausader, "Problèmes Bien Posés et Diffusion pour Des équations Non Linéaires Dispersives D'ordre Quatre," Ph.d dissertation, Université de Cergy Pontoise, 2008. |
[15] |
K. Rogers and A. Vargas, A refinement of the Strichartz inequality on the saddle and applications, J. Funct. Anal., 241 (2006), 212-231.
doi: 10.1016/j.jfa.2006.04.026. |
[16] |
E. M. Stein, "Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals," Princeton Univ. Press, 1993. |
[17] |
T. Tao, A sharp bilinear restrictions estimate for paraboloids, Geom. Funct. Anal., 13 (2003), 1359-1384.
doi: 10.1007/s00039-003-0449-0. |
[18] |
T. Tao, "Nonlinear Dispersive Equations," CBMS Regional Conf. Ser. in Math. Amer. Math. Soc., 106, 2006. |
[19] |
T. Tao, A. Vargas and L. Vega, A bilinear approach to the restriction and Kakeya conjectures, J. Amer. Math. Soc., 11 (1998), 967-1000.
doi: 10.1090/S0894-0347-98-00278-1. |
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