Ill-posedness for the Zakharov system with generalized nonlinearity
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- by H. A. Biagioni and F. Linares PDF
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Abstract:
We study the ill-posedness question for the one-dimensional Zakharov system and a generalization of it in one and higher dimensions. Our point of reference is the criticality criteria introduced by Ginibre, Tsutsumi and Velo (1997) to establish local well-posedness.References
- Hélène Added and Stéphane Added, Existence globale de solutions fortes pour les équations de la turbulence de Langmuir en dimension $2$, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 12, 551–554 (French, with English summary). MR 770444
- Hélène Added and Stéphane Added, Equations of Langmuir turbulence and nonlinear Schrödinger equation: smoothness and approximation, J. Funct. Anal. 79 (1988), no. 1, 183–210. MR 950090, DOI 10.1016/0022-1236(88)90036-5
- Rudolph E. Langer, The boundary problem of an ordinary linear differential system in the complex domain, Trans. Amer. Math. Soc. 46 (1939), 151–190 and Correction, 467 (1939). MR 84, DOI 10.1090/S0002-9947-1939-0000084-7
- H. A. Biagioni and F. Linares, Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations, Trans. Amer. Math. Soc. 353 (2001), no. 9, 3649–3659. MR 1837253, DOI 10.1090/S0002-9947-01-02754-4
- Björn Birnir, Carlos E. Kenig, Gustavo Ponce, Nils Svanstedt, and Luis Vega, On the ill-posedness of the IVP for the generalized Korteweg-de Vries and nonlinear Schrödinger equations, J. London Math. Soc. (2) 53 (1996), no. 3, 551–559. MR 1396718, DOI 10.1112/jlms/53.3.551
- Björn Birnir, Gustavo Ponce, and Nils Svanstedt, The local ill-posedness of the modified KdV equation, Ann. Inst. H. Poincaré C Anal. Non Linéaire 13 (1996), no. 4, 529–535 (English, with English and French summaries). MR 1404320, DOI 10.1016/S0294-1449(16)30112-3
- J. Bourgain and J. Colliander, On wellposedness of the Zakharov system, Internat. Math. Res. Notices 11 (1996), 515–546. MR 1405972, DOI 10.1155/S1073792896000359
- J. Colliander, Wellposedness for Zakharov systems with generalized nonlinearity, J. Differential Equations 148 (1998), no. 2, 351–363. MR 1643187, DOI 10.1006/jdeq.1998.3445
- J. Ginibre, Y. Tsutsumi, and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal. 151 (1997), no. 2, 384–436. MR 1491547, DOI 10.1006/jfan.1997.3148
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, On the Zakharov and Zakharov-Schulman systems, J. Funct. Anal. 127 (1995), no. 1, 204–234. MR 1308623, DOI 10.1006/jfan.1995.1009
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J. 106 (2001), no. 3, 617–633. MR 1813239, DOI 10.1215/S0012-7094-01-10638-8
- Tohru Ozawa and Yoshio Tsutsumi, Existence and smoothing effect of solutions for the Zakharov equations, Publ. Res. Inst. Math. Sci. 28 (1992), no. 3, 329–361. MR 1184829, DOI 10.2977/prims/1195168430
- Walter A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), no. 2, 149–162. MR 454365
- Catherine Sulem and Pierre-Louis Sulem, Quelques résultats de régularité pour les équations de la turbulence de Langmuir, C. R. Acad. Sci. Paris Sér. A-B 289 (1979), no. 3, A173–A176 (French, with English summary). MR 552204
- Ya Ping Wu, Orbital stability of solitary waves of Zakharov system, J. Math. Phys. 35 (1994), no. 5, 2413–2422. MR 1271934, DOI 10.1063/1.530512
- V.E. Zakharov, The collapse of Langmuir waves, Sov. Phys. JETP 35 (1972), 908-914.
Additional Information
- H. A. Biagioni
- Affiliation: Departamento de Matemática, IMECC-UNICAMP, 13081-970, Campinas, SP, Brasil
- Email: hebe@ime.unicamp.br
- F. Linares
- Affiliation: Instituto de Matemática Pura e Aplicada, 22460-320, Rio de Janeiro, Brasil
- MR Author ID: 343833
- Email: linares@impa.br
- Received by editor(s): June 15, 2001
- Received by editor(s) in revised form: April 28, 2002
- Published electronically: February 6, 2003
- Communicated by: David S. Tartakoff
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 3113-3121
- MSC (2000): Primary 35Q55, 35Q51
- DOI: https://doi.org/10.1090/S0002-9939-03-06898-9
- MathSciNet review: 1993221