Recent progress on the blow-up problem of Zakharov equations

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1. [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. Zbl0575.35080MR86g:35163
2. [AA2] H. Added, S. Added, Equations of Langmuir turbulence and nonlinear Schrödinger equations : Smoothness and approximation, J. Funct. Anal. 79, (1988) 183-210. Zbl0655.76044MR89h:35273
3. [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. Zbl0533.35029MR84h:35054a
4. [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. Zbl0787.35097MR95d:35160a
5. [Bo2] J. Bourgain, On the Cauchy and invariant measure problem for the periodic Zakharov system, Duke Math. J. 76, (1994) 175-202. Zbl0821.35120MR95h:35212
6. [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. Zbl0694.35170
7. [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. Zbl0396.35028MR82c:35057
8. [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. Zbl0808.35137MR95e:35195
9. [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. Zbl0808.35138MR95e:35196
10. [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. Zbl0372.35009MR57 #842
11. [Ka] T. Kato, On nonlinear Schrödinger equations, Ann. Inst. Henri Poincaré, Physique Théorique 49, (1987) 113-129. Zbl0632.35038MR88f:35133
12. [KePVg] C. Kenig, G. Ponce, L. Vega, On the Zakharov and Zakharov-Schulman systems, J. Funct. Anal. 127, (1995) 204-234. Zbl0823.35158MR96a:35197
13. [Kw] M.A. Kwong, Uniqueness of positive solutions of u = Δu + up in RN, Arch. Rational Mech. Anal. 105, (1989) 243-266. Zbl0676.35032MR90d:35015
14. [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. 
15. [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. MR89k:35218
16. [M1] F. Merle, Determination of blow-up solutions with minimal mass for Schrödinger equation with critical power, Duke J. 69, (1993) 427-454. Zbl0808.35141MR94b:35262
17. [M2] F. Merle, Blow-up results of the viriel type for Zakharov equations, Commun. Math. Phys. (to appear). Zbl0858.35117
18. [M3] F. Merle, Lower bounds for the blow-up rate of solutions of Zakharov equations in dimension two, preprint. 
19. [M4] F. Merle, Asymptotics for L2 minimal blow-up solutions of critical nonlinear Schrödinger equation, Anal. I.H.P. Analyse non linéaire (to appear). Zbl0862.35013
20. [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. Zbl0722.35047MR91e:35194
21. [OT1] T. Ozawa, Y. Tsutsumi, The nonlinear Schrödinger limit and the initial layer of the Zakharov equations, preprint. Zbl0754.35113
22. [OT2] T. Ozawa, Y. Tsutsumi, Existence of smoothing effect of solutions for the Zakharov equations, preprint. Zbl0842.35116
23. [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. MR91k:76159
24. [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. 
25. [St] W.A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys. 55, (1977) 149-162. Zbl0356.35028MR56 #12616
26. [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. Zbl0431.35077MR80i:35165
27. [W1] M.I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys. 87, (1983) 567-576. Zbl0527.35023MR84d:35140
28. [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. Zbl0596.35022MR87i:35026