Recent progress on the blow-up problem of Zakharov equations
Merle, Frank
Journées équations aux dérivées partielles, (1995), p. 1-7 / Harvested from Numdam
Publié le : 1995-01-01
@article{JEDP_1995____A20_0,
     author = {Merle, Frank},
     title = {Recent progress on the blow-up problem of Zakharov equations},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     year = {1995},
     pages = {1-7},
     mrnumber = {96j:35235},
     language = {en},
     url = {http://dml.mathdoc.fr/item/JEDP_1995____A20_0}
}
Merle, Frank. Recent progress on the blow-up problem of Zakharov equations. Journées équations aux dérivées partielles,  (1995), pp. 1-7. http://gdmltest.u-ga.fr/item/JEDP_1995____A20_0/

[AA1] H. Added, S. Added, Existence globale de solutions fortes pour les équations de la turbulence de Langmuir en dimension 2, C.R. Acad. Sci. Paris 200, (1984) 551-554. | MR 86g:35163 | Zbl 0575.35080

[AA2] H. Added, S. Added, Equations of Langmuir turbulence and nonlinear Schrödinger equations : Smoothness and approximation, J. Funct. Anal. 79, (1988) 183-210. | MR 89h:35273 | Zbl 0655.76044

[BeL] H. Berestycki, P.L. Lions, Nonlinear scalar field equations, I Existence of ground state; II Existence of infinitely many solutions, Arch. Rational Mech. Anal. 82, (1983) 313-375. | MR 84h:35054a | Zbl 0533.35029

[Bo1] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Part I Schrödinger equations, G.A.F.A. 3, (1993) 107-178. | MR 95d:35160a | Zbl 0787.35097

[Bo2] J. Bourgain, On the Cauchy and invariant measure problem for the periodic Zakharov system, Duke Math. J. 76, (1994) 175-202. | MR 95h:35212 | Zbl 0821.35120

[CaW] T. Cazenave, F. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case, in nonlinear semigroups, partial equations, and attractors, T.L. Gill and Zachary (eds.) Lect. Notes in Math. 347 Springer (1989) 18-29. | Zbl 0694.35170

[GV] J. Ginibre, G. Velo, On a class of nonlinear Schrödinger equations I, II The Cauchy problem, general case, J. Funct. Anal. 32, (1979) 1-71. | MR 82c:35057 | Zbl 0396.35028

[G1M1] L. Glangetas, F. Merle, Existence of self-similar blow-up solution for the Zakharov equation in dimension two, Commu. Math. Phys. 160, (1994) 173-215. | MR 95e:35195 | Zbl 0808.35137

[G1M2] L. Glangetas, F. Merle, Concentration properties of blow-up solutions and instability results for the Zakharov equation in dimension two, Commu. Math. Phys. 160, (1994) 349-389. | MR 95e:35196 | Zbl 0808.35138

[Gla] R.T. Glassey, On the blowing-up of solutions to the Cauchy problem for the nonlinear Schrödinger equation, J. Math. Phys. 18, (1977) 1794-1797. | MR 57 #842 | Zbl 0372.35009

[Ka] T. Kato, On nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, Physique Théorique 49, (1987) 113-129. | Numdam | MR 88f:35133 | Zbl 0632.35038

[KePVg] C. Kenig, G. Ponce, L. Vega, On the Zakharov and Zakharov-Schulman systems, J. Funct. Anal. 127, (1995) 204-234. | MR 96a:35197 | Zbl 0823.35158

[Kw] M.A. Kwong, Uniqueness of positive solutions of u = Δu + up in RN, Arch. Rational Mech. Anal. 105, (1989) 243-266. | MR 90d:35015 | Zbl 0676.35032

[LPSSW] M. Landman, G.C. Papanicolaou, C. Sulem, P.L. Sulem, X.P. Wang, Stability of isotropic self-similar dynamics for scalar collapse, Phys.Rev. A 46, (1992) 7869-7876.

[LPSS] M. Landman, G.C. Papanicolaou, C. Sulem, P.L. Sulem, Rate of the blow-up for solutions of the nonlinear Schrödinger equation in critical dimension, Phys. Rev. A 38, (1988) 3837-3843. | MR 89k:35218

[M1] F. Merle, Determination of blow-up solutions with minimal mass for Schrödinger equation with critical power, Duke J. 69, (1993) 427-454. | MR 94b:35262 | Zbl 0808.35141

[M2] F. Merle, Blow-up results of the viriel type for Zakharov equations, Commun. Math. Phys. (to appear). | Zbl 0858.35117

[M3] F. Merle, Lower bounds for the blow-up rate of solutions of Zakharov equations in dimension two, preprint.

[M4] F. Merle, Asymptotics for L2 minimal blow-up solutions of critical nonlinear Schrödinger equation, Anal. I.H.P. Analyse non linéaire (to appear). | Numdam | Zbl 0862.35013

[MT] F. Merle, Y. Tsutsumi, L2 concentration of blow-up solutions for the nonlinear Schrödinger equation with the critical power nonlinearity, J. Diff. Equ. 84, (1990), 205-214. | MR 91e:35194 | Zbl 0722.35047

[OT1] T. Ozawa, Y. Tsutsumi, The nonlinear Schrödinger limit and the initial layer of the Zakharov equations, preprint. | Zbl 0754.35113

[OT2] T. Ozawa, Y. Tsutsumi, Existence of smoothing effect of solutions for the Zakharov equations, preprint. | Zbl 0842.35116

[PSSW] G.C. Papanicolaou, C. Sulem, P.L. Sulem, X.P. Wang, Singular solutions of the Zakharov equations for the Langmuir turbulence, Phys. Fluids B3, (1991) 969-980. | MR 91k:76159

[SoSyZ] V.V. Sobolev, V.S. Synach, V.E. Zakharov, Character of the singularity and stochastic phenomena in self-focussing, Zh. Eksp. Theor. Fiz., Pis'ma Red 14, (1974) 173-176.

[St] W.A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys. 55, (1977) 149-162. | MR 56 #12616 | Zbl 0356.35028

[SS] C. Sulem, P.L. Sulem, Quelques résultats de régularité pour les équations de la turbulence de Langmuir, C.R.Acad. Sci. Paris 289, (1979) 173-176. | MR 80i:35165 | Zbl 0431.35077

[W1] M.I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys. 87, (1983) 567-576. | MR 84d:35140 | Zbl 0527.35023

[W2] M.I. Weinstein, On the structure and formation of singularities in solutions to the nonlinear dispersive evolution equations, Commun. Partial Diff. Equ. 11, (1986) 545-565. | MR 87i:35026 | Zbl 0596.35022