Ill-posedness For The Zakharov System With Generalized Nonlestearity

We study the ill-posednees 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.1311031133121Added, H., Added, S., Existence globale de solutions fortes pour les équations de la turbulence de Langmuir em dimension 2 (1984) C. R. Acad. Sci. Paris, 299, pp. 551-554. , MR 86g:35163Added, H., Added, S., Equations of Langmuir turbulence and nonlinear Schrödinger equation: Smoothness and approximation (1988) J. Funct. Anal., 79, pp. 183-210. , MR 89h:35273Berestycki, H., Lyons, P.L., Nonlinear scalar field equations (1983) Arch. Rational Mech. Anal., 82, pp. 313-376. , MR 84h:35054aMR 84h:35054bBiagioni, H.A., Linares, F., Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equation (2001) Trans. Amer. Math. Soc., 353, pp. 3649-3659. , MR2002e:35215Birnir, B., Kenig, C.E., Ponce, G., Svanstedt, N., Vega, L., On the ill-posedness of the IVP for the generalized Korteweg-de Vries and nonlinear Schrödinger equations (1996) J. London Math. Soc. (2), 53, pp. 551-559. , MR 97d:35233Birnir, B., Ponce, G., Svanstedt, N., The local ill-posedness of the modified KdV equation (1996) Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (4), pp. 529-535. , MR 97e:35152Bourgain, J., Colliander, J., On well-posedness of the Zakharov system (1996) Int. Math. Res. Not., 11, pp. 515-546. , MR 97h:35206Colliander, J., Wellposedness for Zakharov systems with generalized nonlinearity (1998) J. Differential Equations, 148 (2), pp. 351-363. , MR 99h:35196Ginibre, J., Tsutsumi, Y., Velo, G., On the Cauchy problem for the Zakharov system (1997) J. Funct. Anal, 151, pp. 384-436. , MR 2000c:35220Kenig, C.E., Ponce, G., Vega, L., On the Zakharov and Zakharov-Schulman systems (1995) J. Funct. Anal., 127, pp. 204-234. , MR 96a:35197Kenig, C.E., Ponce, G., Vega, L., On ill-posedness of some canonical dispersive equations (2001) Duke Math. J., 106, pp. 617-633. , MR 2002c:35265Ozawa, T., Tsutsumi, Y., Existence and smoothing effect of solutions for the Zakharov equations (1992) RIMS Kyoto Univ., 28, pp. 329-361. , MR 93k:35246Strauss, W., Existence of solitary waves in higher dimensions (1977) Comm. Math. Phys., 55, pp. 149-162. , MR 56:12616Sulem, C., Sulem, P.L., Quelques résultats de régularité pour les équations de la turbulence de Langmuir (1979) C. R. Acad. Sci. Paris, 289, pp. 173-176. , MR 80i:35165Wu, Y., Orbital stability of solitary waves of Zakharov system (1994) J. Math. Phys., 35, pp. 2413-2422. , MR 95e:35170Zakharov, V.E., The collapse of Langmuir waves (1972) Sov. Phys. JETP, 35, pp. 908-91