Recent progress on the blow-up problem of Zakharov equations
Journées équations aux dérivées partielles (1995), article no. 20, 7 p.
@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},
     eid = {20},
     publisher = {Ecole polytechnique},
     year = {1995},
     mrnumber = {96j:35235},
     language = {en},
     url = {http://archive.numdam.org/item/JEDP_1995____A20_0/}
}
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%T Recent progress on the blow-up problem of Zakharov equations
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%F JEDP_1995____A20_0
Merle, Frank. Recent progress on the blow-up problem of Zakharov equations. Journées équations aux dérivées partielles (1995), article  no. 20, 7 p. http://archive.numdam.org/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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

[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

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

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

[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

[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 | Zbl

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

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

[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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl