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

Pierre Raphaël[1]

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

Séminaire Équations aux dérivées partielles (2005-2006)

  • Volume: 2005-2006, page 1-15

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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 2005-2006 (2005-2006): 1-15. <http://eudml.org/doc/11130>.

@article{Raphaël2005-2006,
affiliation = {Université de Paris-Sud, Département de Mathématiques F - 91405 Orsay cedex},
author = {Raphaël, Pierre},
journal = {Séminaire Équations aux dérivées partielles},
keywords = {critical norm; Liouville theorem},
language = {eng},
pages = {1-15},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Blow up of the critical norm for some radial $L^2$ super critical non linear Schrödinger equations},
url = {http://eudml.org/doc/11130},
volume = {2005-2006},
year = {2005-2006},
}

TY - JOUR
AU - Raphaël, Pierre
TI - Blow up of the critical norm for some radial $L^2$ super critical non linear Schrödinger equations
JO - Séminaire Équations aux dérivées partielles
PY - 2005-2006
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2005-2006
SP - 1
EP - 15
LA - eng
KW - critical norm; Liouville theorem
UR - http://eudml.org/doc/11130
ER -

References

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  1. Cazenave, Th.; Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, 10, NYU, CIMS, AMS 2003. Zbl1055.35003MR2002047
  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. Zbl0694.35170MR1021011
  3. Escauriaza, L.; Seregin, G. A.; Svverak, V., L 3 , -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. Zbl1064.35134MR1992563
  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. Zbl0396.35028MR533218
  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. Zbl0808.35137
  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. Zbl0836.34041MR1349311
  7. Kato, T.; Ponce, G., Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 7, 891–907. Zbl0671.35066MR951744
  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. Zbl1005.35081MR1888800
  9. Merle, F.; Raphaël, P., On Universality of Blow up Profile for L 2 critical nonlinear Schrödinger equation, Invent. Math. 156, 565-672 (2004). Zbl1067.35110MR2061329
  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. Zbl1075.35077MR2169042
  11. Blow up of the critical norm for some radial L 2 super critical non linear Schrödinger equations, preprint 2006. Zbl1111.35084
  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. Zbl0739.35093MR1120908
  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. Zbl1117.35077
  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. MR406174

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