On the blowup theory for the critical nonlinear Schrödinger equations
Keraani, Sahbi1
1 IRMAR, Université de Rennes 1 Campus de Beaulieu 35042 Rennes cedex France
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2004-2005), Exposé no. 21, 8 p.
Keraani, Sahbi 1

1 IRMAR, Université de Rennes 1 Campus de Beaulieu 35042 Rennes cedex France 
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     title = {On the blowup theory for the critical nonlinear {Schr\"odinger} equations},
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Keraani, Sahbi. On the blowup theory for the critical nonlinear Schrödinger equations. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2004-2005), Exposé no. 21, 8 p. http://www.numdam.org/item/SEDP_2004-2005____A21_0/

[1] H. Bahouri and P. Gérard: High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1999), no. 1, 131–175. | MR | Zbl

[2] V. Banica: Remarks on the blow-up for the Schrödinger equation with critical mass on a plane domain, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004), no. 1, 139–170. | Numdam | MR | Zbl

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

[4] P.  Gérard: Description du défaut de compacité de l’injection de Sobolev, ESAIM.COCV, Vol 3, (1998) 213-233. | Numdam | Zbl

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

[6] R.T. Glassey : On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18 (1977), no. 9, 1794–1797. | MR | Zbl

[7] S. Keraani: On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations 175 (2001), no. 2, 353–392. | MR | Zbl

[8] M.K.  Kwong: uniqueness of positive solutions to Δu-u+u p =0, in ℝ n , Arch. Rat. Mech. Anal 105 (1989), n. 3, 243-266. | MR | Zbl

[9] P-L. Lions: The concentration-compactness principle in the calculus of variations. The compact case. Part 1, Ann. Inst. Henri Poincaré, Analyse non linéaire 1 (1984), 109-145. | Numdam | MR

[10] F.  Merle: Determination of blowup solutions with minimal mass for nonlinear Schrödinger equations with critical power, Duke Math. J. 69, (1993),no. 2, 203-254. | MR | Zbl

[11] —: Construction of solutions with exactly k blowup points for nonlinear Schrödinger equations with critical nonlinearity, Comm.Math. Phys. 129 (1990), no.2, 223-240. | MR | Zbl

[12] —: Blow-up phenomena for critical nonlinear Schrödinger and Zakharov equations. Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998). Doc. Math. 1998, Extra Vol. III, 57–66 | MR | Zbl

[13] F.  Merle, Y.  Tsutsumi: L 2 concentration of blowup solutions for the nonlinear Schrödinger equation with critical power nonlinearity, J. Differential Equations 84 (1990), no. 2, 205–214. | MR | Zbl

[14] C.  Sulem, P-L.  Sulem: The nonlinear Schrödinger equation. Self-focusing and wave collapse, Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999. | MR | Zbl

[15] Y.  Tsutsumi: Rate of L 2 concentration of blowup solutions for the nonlinear Schr̦dinger equation with critical power, Nonlinear Anal. 15 (1990), no. 8, 719–724. | MR | Zbl

[16] M. I.  Weinstein: Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Physi. 87 (1983), 567-567. | MR | Zbl

[17] —: On the structure of singularities in solutions to the nonlinear dispersive evolution equations, Comm. Partial Differential Equations, 11 (1984), 545-565. | Zbl

[18] —: The nonlinear Schrödinger equation—singularity formation, stability and dispersion. The connection between infinite-dimensional and finite-dimensional dynamical systems, Contemp. Math., 99, Amer. Math. Soc., Providence, RI, 1989, 213–232. | MR | Zbl