A Smoothing Property for the ${L}^{2}$-Critical NLS Equations and an Application to Blowup Theory

Keraani, Sahbi; Vargas, Ana

doi:10.1016/j.anihpc.2008.03.001

A Smoothing Property for the

L^{2}

-Critical NLS Equations and an Application to Blowup Theory

Keraani, Sahbi ; Vargas, Ana

Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 3, pp. 745-762

MR Zbl | 1 citation dans Numdam

DOI : 10.1016/j.anihpc.2008.03.001

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     author = {Keraani, Sahbi and Vargas, Ana},
     title = {A {Smoothing} {Property} for the ${L}^{2}${-Critical} {NLS} {Equations} and an {Application} to {Blowup} {Theory}},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {745--762},
     year = {2009},
     publisher = {Elsevier},
     volume = {26},
     number = {3},
     doi = {10.1016/j.anihpc.2008.03.001},
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Keraani, Sahbi; Vargas, Ana. A Smoothing Property for the ${L}^{2}$-Critical NLS Equations and an Application to Blowup Theory. Annales de l'I.H.P. Analyse non linéaire, Tome 26 (2009) no. 3, pp. 745-762. doi: 10.1016/j.anihpc.2008.03.001

Bibliographie
Cité par

[1] Bégout P., Vargas A., Mass Concentration Phenomena for the $L^{2}$ -Critical Nonlinear Schrödinger Equation, Trans. Amer. Math. Soc. 359 (2007) 5257-5282. | Zbl | MR

[2] Bergh J., Löfström J., Interpolation Spaces. an Introduction, Grundlehren der Mathematischen Wissenschaften, vol. 223, Springer-Verlag, Berlin, New York, 1976. | Zbl | MR

[3] Bony J.-M., Calcul Symbolique Et Propagation Des Singularités Pour Les Équations Aux Dérivées Partielles Non Linéaires, Ann. Sci. École Norm. Sup. (4) 14 (2) (1981) 209-246. | Zbl | MR | Numdam

[4] Bourgain J., On the Restriction and the Multiplier Problem in $R^{3}$ , in: Geometrics Aspects of Functional Analysis, Springer Lecture Notes in Math., vol. 1469, 1991, pp. 179-191. | Zbl | MR

[5] Bourgain J., Refinements of Strichartz Inequality and Applications to 2D-NLS With Critical Nonlinearity, IMRN 5 (1998) 253-283. | Zbl | MR

[6] Carles R., Keraani S., Quadratic Oscillations in NLS II. the $L^{2}$ -Critical Case, Trans. Amer. Math. Soc. 359 (1) (2007) 33-62. | Zbl | MR

[7] Cazenave T., Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York, 2003, American Mathematical Society, Providence, RI. | Zbl | MR

[8] Colliander J., Keel M., Staffilani G., Takaoka H., Tao T., Almost Conservation Laws and Global Rough Solutions to a Nonlinear Schrödinger Equation, Math. Res. Lett. 9 (2002) 659-682. | Zbl | MR

[9] Colliander J., Raynor S., Sulem C., Wright J. D., Ground State Mass Concentration in the $L^{2}$ -Critical Nonlinear Schrödinger Equation Below $H^{1}$ , Math. Res. Lett. 12 (2-3) (2005) 357-375. | Zbl | MR

[10] Constantin P., Saut J. C., Local Smoothing Properties of Dispersive Equations, J. Amer. Math. Soc. 1 (1) (April 1988). | Zbl | MR

[11] Cordoba A., The Kakeya Maximal Function and the Spherical Summation Multipliers, Amer. J. Math. 99 (1) (1977) 1-22. | Zbl | MR

[12] D. De Silva, N. Pavlovic, G. Staffilani, N. Tzirakis, Global well-posedness and polynomial bounds for the $L^{2}$ -critical nonlinear Schrödinger equation in $R$ , Preprint.

[13] D. De Silva, N. Pavlovic, G. Staffilani, N. Tzirakis, Global well-posedness for the $L^{2}$ -critical nonlinear Schrödinger equation in higher dimensions, Preprint, 2006. | Zbl

[14] Y. Fang, M. Grillakis, On the global existence of rough solutions of the cubic defocusing Schrödinger equation in $R^{2 + 1}$ , Preprint, 2006. | Zbl | MR

[15] Fefferman C., Inequalities for Strongly Singular Convolution Operators, Acta Math. 124 (1970) 9-36. | Zbl | MR

[16] Fefferman C., A Note on Spherical Summation Multipliers, Israel J. Math. 15 (1973) 44-52. | Zbl | MR

[17] J. Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace (d'après Bourgain) (in French. French summary) (The Cauchy problem for periodic semilinear PDE in space variables (after Bourgain)), Seminaire Bourbaki, vol. 1994/95. | Zbl | Numdam

[18] Hmidi T., Keraani S., Blowup Theory for the Critical Nonlinear Schrödinger Equations Revisited, Int. Math. Res. Not. 5 (2005) 2815-2828. | Zbl | MR

[19] Hmidi T., Keraani S., Remarks on the Blowup for the $L^{2}$ -Critical Nonlinear Schrödinger Equations, SIAM J. Math. Anal. 38 (4) (2006) 1035-1047. | Zbl | MR

[20] Keraani S., On the Blow Up Phenomenon of the Critical Nonlinear Schrödinger Equation, J. Funct. Anal. 235 (1) (2006) 171-192. | Zbl | MR

[21] Merle F., Determination of Blowup Solutions With Minimal Mass for Nonlinear Schrödinger Equations With Critical Power, Duke Math. J. 69 (2) (1993) 203-254. | Zbl | MR

[22] Merle F., Construction of Solutions With Exactly K Blowup Points for Nonlinear Schrödinger Equations With Critical Nonlinearity, Commun. Math. Phys. 129 (2) (1990) 223-240. | Zbl | MR

[23] Merle F., Blow-Up Phenomena for Critical Nonlinear Schrödinger and Zakharov Equations, in: Proceedings of the International Congress of Mathematicians, Vol. III, Berlin, 1998, Doc. Math., Extra vol. III, 1998, pp. 57-66. | Zbl | MR

[24] Merle F., Raphael P., On a Sharp Lower Bound on the Blow-Up Rate for the $L^{2}$ Critical Nonlinear Schrödinger Equation, J. Amer. Math. Soc. 19 (1) (2006) 37-90. | Zbl | MR

[25] Merle F., Raphael P., On Universality of Blow-Up Profile for $L^{2}$ Critical Nonlinear Schrödinger Equation, Invent. Math. 156 (3) (2004) 565-672. | Zbl | MR

[26] Merle F., Tsutsumi Y., $L^{2}$ Concentration of Blowup Solutions for the Nonlinear Schröinger Equation With Critical Power Nonlinearity, J. Differential Equations 84 (2) (1990) 205-214. | Zbl | MR

[27] Merle F., Vega L., Compactness at Blowup Time for $L^{2}$ Solutions of the Critical Nonlinear Schrödinger Equations in 2D, IMRN 8 (1998) 399-425. | Zbl | MR

[28] Moyua A., Vargas A., Vega L., Schrödinger Maximal Function and Restriction Properties of the Fourier Transform, Int. Math. Res. Not. 16 (1996) 793-815. | Zbl | MR

[29] Moyua A., Vargas A., Vega L., Restriction Theorems and Maximal Operators Related to Oscillatory Integrals in $R^{3}$ , Duke Math. J. 96 (1999) 547-574. | Zbl | MR

[30] Sjölin P., Regularity of Solutions to the Schrödinger Equations, Duke Math. J. 55 (1987) 699-715. | Zbl | MR

[31] Sulem C., Sulem P.-L., The Nonlinear Schrödinger Equation. Self-Focusing and Wave Collapse, Applied Mathematical Sciences, vol. 139, Springer-Verlag, New York, 1999. | Zbl | MR

[32] Tao T., A Sharp Bilinear Restrictions Estimate for Paraboloids, Geom. Funct. Anal. 13 (6) (2003) 1359-1384. | Zbl | MR

[33] Tao T., Visan M., Zhang X., Global Well-Posedness and Scattering for the Mass-Critical Nonlinear Schrödinger Equation for Radial Data in High Dimensions, Duke Math. J. 140 (1) (2007) 165-202. | MR

[34] T. Tao, M. Visan, X. Zhang, Minimal-mass blowup solutions of the mass-critical NLS, Forum Mathematicum, in press. | Zbl

[35] Tzirakis N., Mass Concentration Phenomenon for the Quintic Nonlinear Schrödinger Equation in 1D, SIAM J. Math. Anal. 37 (6) (2006) 1923-1946, (electronic). | Zbl | MR

[36] Vega L., Schrödinger Equation: Pointwise Convergence to the Initial Data, Proc. Amer. Math. Soc. 102 (4) (1988) 874-878. | Zbl | MR

[37] Visan M., Zhang X., On the Blowup for the $L^{2}$ -Critical Focusing Nonlinear Schrödinger Equation in Higher Dimensions Below the Energy Class, SIAM J. Math. Anal. 39 (1) (2007) 34-56. | Zbl | MR

[38] Weinstein M. I., Nonlinear Schrödinger Equations and Sharp Interpolation Estimates, Commun. Math. Phys. 87 (1983) 567. | Zbl | MR

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