Lower bounds for the blowup rate of solutions of the Zakharov equation in dimension two

Communications on Pure and Applied Mathematics

Volumn 49, Issue 8, 1996, Pages 765-794

  Merle, Frank ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030510597     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1097-0312(199608)49:8<765::AID-CPA1>3.0.CO;2-6     Document Type: Article

References (27)

1. Added, H., and Added, S., Existence globale de solutions fortes pour les équations de la turbulence de Langmuir en dimension 2, C. R. Acad. Sci. Paris Sér. I Math. 200, 1984, pp. 551-554.
2. Added, H., and Added, S., Equations of Langmuir turbulence and nonlinear Schrödinger equation: smoothness and approximation, J. Funct. Anal. 79, 1988, pp. 183-210.
3. Berestycki, H., and Lions, P. L., Nonlinear scalar field equations, I. Existence of ground state; II. Existence of infinitely many solutions, Arch. Rational Mech. Anal. 82, 1983, pp. 313-375.
4. Bourgain, J., On the Cauchy and invariant measure problem for the periodic Zakharov system, Duke Math. J. 76, 1994, pp. 175-202.
5. S, Nonlinear Anal. 14, 1990, pp. 807-836.
6. Ginibre, J., and Velo, G., On a class of nonlinear Schrödinger equations: I. Scattering theory, general case; II. The Cauchy problem, general case, J. Funct. Anal. 32, 1979, pp. 1-71.
7. Glangetas, L., and Merle, F., Existence of self-similar blow-up solutions for Zakharov equation in dimension two: Part I, Comm. Math. Phys. 160, 1994, pp. 173-215.
8. Glangetas, L., and Merle, F., Concentration properties of blow-up solutions and instability results for Zakharov equation in dimension two: Part II, Comm. Math. Phys. 160, 1994, pp. 349-389.
9. Glassey, R. T., On the blowing-up of solutions to the Cauchy problem for the nonlinear Schrödinger equation, J. Math. Phys. 18, 1977, pp. 1794-1797.
10. Kato, T., On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 49, 1987, pp. 113-129.
11. N, Arch. Rational Mech. Anal. 105, 1989, pp. 243-266.
12. Landman, M., Papanicolaou, G. C., Sulem, C., and Sulem, P. L., Rate of blow-up for solutions of the nonlinear Schrödinger equation in critical dimension, Phys. Rev. A 38, 1988, pp. 3837-3843.
13. Landman, M., Papanicolaou, G. C., Sulem, C., Sulem, P. L., and Wang, X. P., Stability of isotropic self-similar dynamics for scalar collapse, Phys. Rev. A 46, 1992, pp. 7869-7876.
14. Lieb, E. H., On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math. 74, 1983, pp. 441-448.
15. Merle, F., Determination of blow-up solutions with minimal mass for Schrödinger equation with critical power, Duke Math. J. 69(2), 1993, pp. 427-454.
16. Merle, F., Blow-up results of virial type for Zakharov equations, Comm. Math. Phys., to appear.
17. Nawa, H., "Mass concentration" phenomenon for nonlinear Schrödinger equation with the critical power nonlinearity, Kodai Math. J. 13, 1990, pp. 333-348.
18. Nawa, H., Asymptotic profiles of blow-up solutions of the nonlinear Schrödinger equation with critical power nonlinearity, J. Math. Soc. Japan 46, 1994, pp. 557-586.
19. Ozawa, T., and Tsutsumi, Y., Existence of smoothing effect of solutions for the Zakharov equations, Differential Integral Equations 5, 1992, pp. 329-361.
20. Ozawa, T., and Tsutsumi, Y., The nonlinear Schrödinger limit and the initial layer of the Zakharov equations, Publ. Res. Inst. Math. Sci. 28, 1992, pp. 329-361.
21. Papanicolaou, G. C., Sulem, C., Sulem, P. L., and Wang, X. P., Singular solutions of the Zakharov equations for Langmuir turbulence, Phys. Fluids B 3, 1991, pp. 969-980.
22. Sobolev, V. V., Synach, V. S., and Zakharov, V. E., Character of the singularity and stochastic phenomena in self-focusing, Zh. Èksper. Teoret. Fiz. 14, 1971, pp. 390-393.
23. Strauss, W. A., Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55, 1977, pp. 149-162.
24. Sulem, C., and Sulem, P. L., Quelques résultats de régularité pour les équations de la turbulence de Langmuir, C. R. Acad. Sci. Paris Sér. A-B 289, 1979, pp. 173-176.
25. Weinstein, M. I., Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87, 1983, pp. 567-576.
26. Weinstein, M. I., On the structure and formation of singularities in solutions to the nonlinear dispersive evolution equations, Comm. Partial Differential Equations 11, 1986, pp. 545-565.
27. Zakharov, V. E., Collapse of Langmuir waves, Soviet Phys. JETP 35, 1972, pp. 908-914.