A Note on C^2 Ill-posedness Results for the Zakharov System in Arbitrary Dimension
DOI:
https://doi.org/10.5540/tcam.2023.024.03.00505Keywords:
Zakharov System, $C^2$ ill-posednessAbstract
This work is concerned with the Cauchy problem for a Zakharov system with initial data in Sobolev spaces H^k(\R^d)×H^l(\R^d)×H^l−1(\R^d).We recall the well-posedness and ill-posedness results known to date and establish new ill-posedness results.We prove C^2 ill-posedness for some new indices (k, l) ∈ \R^2. Moreover, our results are valid in arbitrary dimension. We believe that our detailed proofs are built on a methodical approach and can be adapted to obtain similar results for other systems and equations.
References
V. E. Zakharov, 'Collapse of LangmuirWaves', Soviet Journal of Experimental and Theoretical Physics, vol. 35, p. 908, Jan. 1972.
N. Tzvetkov, "Remark on the local ill-posedness for kdv equation", Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, vol. 329, no. 12, pp. 1043-1047, 1999.
J. Bourgain, "Periodic korteweg de vries equation with measures as initial data", Sel. Math., New Ser., vol. 3, pp. 115-159, 1997.
L. Domingues, "Sharp well-posedness results for the Schrödinger-Benjamin-Ono system", Advances in Differential Equations, vol. 21, no. 1/2, pp. 31-54, 2016.
J. Ginibre, Y. Tsutsumi, and G. Velo, "On the cauchy problem for the zakharov system", Journal of Functional Analysis, vol. 151, pp. 384-436, 1997.
J. Bourgain and J. Colliander, "On wellposedness of the Zakharov system", International Mathematics Research Notices, vol. 1996, pp. 515-546, 06 1996.
J. Bourgain, "Fourier transform restriction phenomena for certain lattice subsets and application to the nonlinear evolution equations. i. schrödinger equations. ii. kdv-equation.", Geom. Funct. Anal., vol. 3, pp. 107-156, 209-262, 1993.
H. Biagioni and F. Linares, "Ill-posedness for the zakharov system with generalized nonlinearity", Proc. Amer. Math. Soc., vol. 131, pp. 3113-3121, 2003.
J. Holmer, "Local ill-posedness of the 1d zakharov system", Electronic Journal of Differential Equations, vol. 2007, 02 2007.
I. Bejenaru, S. Herr, J. Holmer, and D. Tataru, "On the 2d zakharov system with $L^2$-schrödinger data", Nonlinearity, vol. 22, pp. 1063-1089, 2009.
S. H. I. Bejenaru, Z. Guo and K. Nakanishi, "Well-posedness and scattering for the zakharov system in four dimensions", Anal. PDE, vol. 8, pp. 2029-2055, 2015.
I. Kato and K. Tsugawa, "Scattering and well-posedness for the Zakharov system at a critical space in four and more spatial dimensions" ,Differential and Integral Equations, vol. 30, no. 9/10, pp. 763 - 794, 2017.
H. Pecher, "Global well-posedness below energy space for the 1-dimensional zakharov system", International Mathematics Research Notices, pp. 1027- 1056, vol. 01, 2001.
H. Pecher, "Global solutions with innite energy for the one-dimensional zakharov system", Electronic Journal of Differential Equations, vol. 2005, pp. 1-18, vol. 04, 2005.
D. Fang, H. Pecher, and S. Zhong, "Low regularity global well-posedness for the two-dimensional zakharov system", vol. 29, no. 3, pp. 265-282, 2009.
N. Kishimoto, "Resonant decomposition and the i-method for the twodimensional zakharov system", Discrete and Continuous Dynamical Systems, vol. 33, pp. 4095 - 4122, 03 2012.
I. Bejenaru and T. Tao, "Sharp well-posedness and ill-posedness results for a quadratic non-linear schrödinger equation", Journal of Functional Analysis, vol. 233, no. 1, pp. 228-259, 2006.
N. Kishimoto, "local well-posedness for the zakharov system on multidimensional torus", Journal d'Analyse Mathématique, vol. 119, pp. 213-253, 09 2011.
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish in this journal agree to the following terms:
Authors retain copyright and grant the journal the right of first publication, with the work simultaneously licensed under the Creative Commons Attribution License that allows the sharing of the work with acknowledgment of authorship and initial publication in this journal.
Authors are authorized to assume additional contracts separately, for non-exclusive distribution of the version of the work published in this journal (eg, publish in an institutional repository or as a book chapter), with acknowledgment of authorship and initial publication in this journal.
Authors are allowed and encouraged to publish and distribute their work online (eg, in institutional repositories or on their personal page) at any point before or during the editorial process, as this can generate productive changes as well as increase impact and the citation of the published work (See The effect of open access).
This is an open access journal which means that all content is freely available without charge to the user or his/her institution. Users are allowed to read, download, copy, distribute, print, search, or link to the full texts of the articles, or use them for any other lawful purpose, without asking prior permission from the publisher or the
author. This is in accordance with the BOAI definition of open access
Intellectual Property
All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License under attribution BY.