Low regularity well-posedness for KP-I equations: the dispersion-generalized case. (English) Zbl 1522.35449
Local-in-time well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili I equation in \(\mathbb{R}^2\) is studied in the framework of anisotropic Sobolev spaces. The data-to-solution mapping is shown to be nonanalytic (neither \(C^2\)). Methods used include Strichartz inequalities as well as a nonlinear version of Loomis-Whitney inequality for convolutions.
Reviewer: Piotr Biler (Wrocław)
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35R11 | Fractional partial differential equations |
| 42B37 | Harmonic analysis and PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
Keywords:
Kadomtsev-Petviashvili I equation; generalized dispersion; well-posedness; anisotropic Sobolev spacesReferences:
| [1] | Bejenaru, I.; Herr, S.; Holmer, J.; Tataru, D., On the 2D Zakharov system with L^2-Schrödinger data, Nonlinearity, 22, 1063-89 (2009) · Zbl 1173.35651 · doi:10.1088/0951-7715/22/5/007 |
| [2] | Bejenaru, I.; Herr, S.; Tataru, D., A convolution estimate for two-dimensional hypersurfaces, Rev. Mat. Iberoam., 26, 707-28 (2010) · Zbl 1203.42033 · doi:10.4171/RMI/615 |
| [3] | Bennett, J.; Carbery, A.; Wright, J., A non-linear generalisation of the Loomis-Whitney inequality and applications, Math. Res. Lett., 12, 443-57 (2005) · Zbl 1106.26020 · doi:10.4310/MRL.2005.v12.n4.a1 |
| [4] | Bourgain, J., Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal., 3, 107-56 (1993) · Zbl 0787.35097 · doi:10.1007/BF01896020 |
| [5] | Guo, B.; Huo, Z.; Fang, S., Low regularity for the fifth order Kadomtsev-Petviashvili-I type equation, J. Differ. Equ., 263, 5696-726 (2017) · Zbl 1400.35050 · doi:10.1016/j.jde.2017.06.032 |
| [6] | Guo, Z.; Peng, L.; Wang, B., On the local regularity of the KP-I equation in anisotropic Sobolev space, J. Math. Pures Appl., 94, 414-32 (2010) · Zbl 1200.35257 · doi:10.1016/j.matpur.2010.03.012 |
| [7] | Hadac, M., Well-posedness for the Kadomtsev-Petviashvili II equation and generalisations, Trans. Am. Math. Soc., 360, 6555-72 (2008) · Zbl 1157.35094 · doi:10.1090/S0002-9947-08-04515-7 |
| [8] | Herr, S.; Sanwal, A.; Schippa, R., Low regularity well-posedness of KP-I equations: the three-dimensional case (2022) |
| [9] | Ionescu, A. D.; Kenig, C. E.; Tataru, D., Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math., 173, 265-304 (2008) · Zbl 1188.35163 · doi:10.1007/s00222-008-0115-0 |
| [10] | Kim, K.; Schippa, R., Low regularity well-posedness for generalized Benjamin-Ono equations on the circle, J. Hyperbolic Differ. Equ., 18, 931-84 (2021) · Zbl 1490.35071 · doi:10.1142/S0219891621500272 |
| [11] | Kinoshita, S.; Schippa, R., Loomis-Whitney-type inequalities and low regularity well-posedness of the periodic Zakharov-Kuznetsov equation, J. Funct. Anal., 280, 54 (2021) · Zbl 1458.35374 · doi:10.1016/j.jfa.2020.108904 |
| [12] | Koch, H.; Tzvetkov, N., On finite energy solutions of the KP-I equation, Math. Z., 258, 55-68 (2008) · Zbl 1387.35530 · doi:10.1007/s00209-007-0156-x |
| [13] | Li, J.; Xiao, J., Well-posedness of the fifth order Kadomtsev-Petviashvili I equation in anisotropic Sobolev spaces with nonnegative indices, J. Math. Pures Appl., 90, 338-52 (2008) · Zbl 1154.35016 · doi:10.1016/j.matpur.2008.06.005 |
| [14] | Linares, F.; Pilod, D.; Saut, J-C, The Cauchy problem for the fractional Kadomtsev-Petviashvili equations, SIAM J. Math. Anal., 50, 3172-209 (2018) · Zbl 1420.35320 · doi:10.1137/17M1145379 |
| [15] | Molinet, L.; Saut, J-C; Tzvetkov, N., Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation, Duke Math. J., 115, 353-84 (2002) · Zbl 1033.35103 · doi:10.1215/S0012-7094-02-11525-7 |
| [16] | Molinet, L.; Saut, J. C.; Tzvetkov, N., Global well-posedness for the KP-I equation on the background of a non-localized solution, Commun. Math. Phys., 272, 775-810 (2007) · Zbl 1160.35065 · doi:10.1007/s00220-007-0243-1 |
| [17] | Saut, J-C, Remarks on the generalized Kadomtsev-Petviashvili equations, Indiana Univ. Math. J., 42, 1011-26 (1993) · Zbl 0814.35119 · doi:10.1512/iumj.1993.42.42047 |
| [18] | Saut, J. C.; Tzvetkov, N., The Cauchy problem for higher-order KP equations, J. Differ. Equ., 153, 196-222 (1999) · Zbl 0927.35098 · doi:10.1006/jdeq.1998.3534 |
| [19] | Saut, J. C.; Tzvetkov, N., The Cauchy problem for the fifth order KP equations, J. Math. Pures Appl., 79, 307-38 (2000) · Zbl 0961.35137 · doi:10.1016/S0021-7824(00)00156-2 |
| [20] | Schippa, R2019Short-time Fourier transform restriction phenomena and applications to nonlinear dispersive equationsPhD ThesisUniversität Bielefeld |
| [21] | Tao, T., Nonlinear Dispersive Equations. Local and Global Analysis (CBMS Regional Conference Series in Mathematics vol 106) (2006), American Mathematical Society · Zbl 1106.35001 |
| [22] | Tom, M. M., On a generalized Kadomtsev-Petviashvili equation, Mathematical Problems in The Theory Of Water Waves (Luminy, 1995) (Contemporary Mathematics vol 200), pp 193-210 (1996), American Mathematical Society · Zbl 0861.35103 |
| [23] | Yan, W.; Li, Y.; Huang, J.; Duan, J., The Cauchy problem for a two-dimensional generalized Kadomtsev-Petviashvili-I equation in anisotropic Sobolev spaces, Anal. Appl., 18, 469-522 (2020) · Zbl 1434.35171 · doi:10.1142/S0219530519500180 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
