Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations. (English) Zbl 0970.35154
Summary: Ill-posedness is established for the initial value problem (IVP) associated to the derivative nonlinear Schrödinger equation for data in \(H^s(\mathbb{R})\), \(s<1/2\). This result implies that the best result concerning local well-posedness for the IVP is in \(H^s(\mathbb{R}), s\geq 1/2\). It is also shown that the IVP associated to the generalized Benjamin-Ono equation for data below the scaling is in fact ill-posed.
MSC:
| 35R25 | Ill-posed problems for PDEs |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q51 | Soliton equations |
References:
| [1] | T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech. 29 (1967), 559-592. · Zbl 0147.46502 |
| [2] | Björn Birnir, Gustavo Ponce, and Nils Svanstedt, The local ill-posedness of the modified KdV equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), no. 4, 529 – 535 (English, with English and French summaries). · Zbl 0858.35130 |
| [3] | 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. · Zbl 0855.35112 · doi:10.1112/jlms/53.3.551 |
| [4] | Bo Ling Guo and Ya Ping Wu, Orbital stability of solitary waves for the nonlinear derivative Schrödinger equation, J. Differential Equations 123 (1995), no. 1, 35 – 55. · Zbl 0844.35116 · doi:10.1006/jdeq.1995.1156 |
| [5] | J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), no. 2, 107 – 156. , https://doi.org/10.1007/BF01896020 J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal. 3 (1993), no. 3, 209 – 262. , https://doi.org/10.1007/BF01895688 J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), no. 2, 107 – 156. , https://doi.org/10.1007/BF01896020 J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal. 3 (1993), no. 3, 209 – 262. · Zbl 0787.35098 · doi:10.1007/BF01895688 |
| [6] | Nakao Hayashi and Tohru Ozawa, On the derivative nonlinear Schrödinger equation, Phys. D 55 (1992), no. 1-2, 14 – 36. · Zbl 0741.35081 · doi:10.1016/0167-2789(92)90185-P |
| [7] | Tosio Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in applied mathematics, Adv. Math. Suppl. Stud., vol. 8, Academic Press, New York, 1983, pp. 93 – 128. · Zbl 0549.34001 |
| [8] | Carlos E. Kenig, Gustavo Ponce, and Luis Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J. 71 (1993), no. 1, 1 – 21. · Zbl 0787.35090 · doi:10.1215/S0012-7094-93-07101-3 |
| [9] | Carlos E. Kenig, Gustavo Ponce, and Luis Vega, On the generalized Benjamin-Ono equation, Trans. Amer. Math. Soc. 342 (1994), no. 1, 155 – 172. · Zbl 0804.35105 |
| [10] | H. Beirão da Veiga, Singular limits in fluidynamics, Rend. Sem. Mat. Univ. Padova 94 (1995), 55 – 69. · Zbl 0853.76073 |
| [11] | -, On ill-posedness of some canonical dispersive equations, preprint 1999. |
| [12] | Rafael José Iório Jr., On the Cauchy problem for the Benjamin-Ono equation, Comm. Partial Differential Equations 11 (1986), no. 10, 1031 – 1081. · Zbl 0608.35030 · doi:10.1080/03605308608820456 |
| [13] | Koji Mio, Tatsuki Ogino, Kazuo Minami, and Susumu Takeda, Modified nonlinear Schrödinger equation for Alfvén waves propagating along the magnetic field in cold plasmas, J. Phys. Soc. Japan 41 (1976), no. 1, 265 – 271. · Zbl 1334.76181 · doi:10.1143/JPSJ.41.265 |
| [14] | Hiroaki Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan 39 (1975), no. 4, 1082 – 1091. · Zbl 1334.76027 · doi:10.1143/JPSJ.39.1082 |
| [15] | T. Ozawa, On the nonlinear Schrödinger equations of derivative type, Indiana Univ. Math. J. 45 (1996), no. 1, 137 – 163. · Zbl 0859.35117 · doi:10.1512/iumj.1996.45.1962 |
| [16] | T. Ozawa and Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear Schrödinger equations, Differential Integral Equations 11 (1998), no. 2, 201 – 222. · Zbl 1008.35070 |
| [17] | Gustavo Ponce, On the well posedness of some nonlinear evolution equations, Nonlinear waves (Sapporo, 1995) GAKUTO Internat. Ser. Math. Sci. Appl., vol. 10, Gakkōtosho, Tokyo, 1997, pp. 393 – 424. · Zbl 0899.35042 |
| [18] | Gustavo Ponce, On the global well-posedness of the Benjamin-Ono equation, Differential Integral Equations 4 (1991), no. 3, 527 – 542. · Zbl 0732.35038 |
| [19] | Hideo Takaoka, Well-posedness for the one-dimensional nonlinear Schrödinger equation with the derivative nonlinearity, Adv. Differential Equations 4 (1999), no. 4, 561 – 580. · Zbl 0951.35125 |
| [20] | -, Well-posedness for the one-dimensional nonlinear Schrödinger equation with the derivative nonlinearity, preprint (2000). |
| [21] | A. S. Bratus\(^{\prime}\) and A. D. Myshkis, Extremum problems for Laplacian eigenvalues with free boundary, Nonlinear Anal. 19 (1992), no. 9, 815 – 831. · Zbl 0776.35038 · doi:10.1016/0362-546X(92)90053-H |
| [22] | M. Wadadi, H. Sanuki, K. Konno, Y.-H. Ichikawa, Circular polarized nonlinear Alfvén waves - a new type of nonlinear evolution equation in plasma physics. Conference on the Theory and Application of Solitons (Tucson, Ariz., 1976), Rocky Mountain J. Math. 8 (1978), 323-331. |
| [23] | Michael I. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation, Comm. Partial Differential Equations 12 (1987), no. 10, 1133 – 1173. · Zbl 0657.73040 · doi:10.1080/03605308708820522 |
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