Blow up of the critical norm for some radial $L^2$ super critical non linear Schrödinger equations

Raphaël, Pierre

Blow up of the critical norm for some radial

L^{2}

super critical non linear Schrödinger equations

Raphaël, Pierre ¹

¹ Université de Paris-Sud, Département de Mathématiques F - 91405 Orsay cedex

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2005-2006), Exposé no. 18, 15 p.

Affiliations des auteurs :

Raphaël, Pierre ¹

¹ Université de Paris-Sud, Département de Mathématiques F - 91405 Orsay cedex

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     author = {Rapha\"el, Pierre},
     title = {Blow up of the critical norm for some radial $L^2$ super critical non linear {Schr\"odinger} equations},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
     note = {talk:18},
     pages = {1--15},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2005-2006},
     mrnumber = {2276083},
     language = {en},
     url = {http://archive.numdam.org/item/SEDP_2005-2006____A18_0/}
}

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Raphaël, Pierre. Blow up of the critical norm for some radial $L^2$ super critical non linear Schrödinger equations. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2005-2006), Exposé no. 18, 15 p. http://archive.numdam.org/item/SEDP_2005-2006____A18_0/

Bibliographie
Cité par

[1] Cazenave, Th.; Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, 10, NYU, CIMS, AMS 2003. | MR | Zbl

[2] Cazenave, Th.; Weissler, F., Some remarks on the nonlinear Schrödinger equation in the critical case. Nonlinear semigroups, partial differential equations and attractors (Washington, DC, 1987), 18–29, Lecture Notes in Math., 1394, Springer, Berlin, 1989. | MR | Zbl

[3] Escauriaza, L.; Seregin, G. A.; Svverak, V., $L_{3, \infty}$ -solutions of Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk 58 (2003), no. 2(350), 3–44; translation in Russian Math. Surveys 58 (2003), no. 2, 211–250. | MR | Zbl

[4] Ginibre, J.; Velo, G., On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal. 32 (1979), no. 1, 1–32. | MR | Zbl

[5] Glangetas, L.; Merle, F., A geometrical approach of existence of blow-up solutions in $H^{1}$ for nonlinear Schrödinger equation, prepublication Univ. P.M. Curie, R95031.

[6] Kopell, N.; Landman, M., Spatial structure of the focusing singularity of the nonlinear Schrödinger equation: a geometrical analysis, SIAM J. Appl. Math. 55 (1995), no. 5, 1297–1323. | MR | Zbl

[7] Kato, T.; Ponce, G., Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 7, 891–907. | MR | Zbl

[8] Martel, Y.; Merle, F., Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation, Ann. of Math. (2) 155 (2002), no. 1, 235–280. | MR | Zbl

[9] Merle, F.; Raphaël, P., On Universality of Blow up Profile for $L^{2}$ critical nonlinear Schrödinger equation, Invent. Math. 156, 565-672 (2004). | MR | Zbl

[10] Merle, F.; Raphaël, P., On a sharp lower bound on the blow-up rate for the $L^{2}$ critical nonlinear Schrödinger equation, J. Amer. Math. Soc. 19 (2006), no. 1, 37–90. | MR | Zbl

[11] Blow up of the critical norm for some radial $L^{2}$ super critical non linear Schrödinger equations, preprint 2006.

[12] Ogawa, T.; Tsutsumi, Y., Blow-up of $H^{1}$ solution for the nonlinear Schrödinger equation, J. Differential Equations 92 (1991), no. 2, 317–330. | MR | Zbl

[13] Raphaël, P., Existence and stability of a solution blowing up on a sphere for a $L^{2}$ supercritical nonlinear Schrödinger equation, to appear in Duke Math. Jour. | Zbl

[14] Zakharov, V.E.; Shabat, A.B., Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linear media, Sov. Phys. JETP 34 (1972), 62—69. | MR