A Note on C^2 Ill-posedness Results for the Zakharov System in Arbitrary Dimension

Autores

DOI:

https://doi.org/10.5540/tcam.2023.024.03.00505

Palavras-chave:

Zakharov System, $C^2$ ill-posedness

Resumo

This work is concerned with the Cauchy problem for a Zakharov system with initial data in Sobolev spaces H^k(\R^dH^l(\R^dH^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 (kl) ∈ \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.

Biografia do Autor

L. Domingues, Universidade Federal do Espírito Santo

Departamento de Matemática Aplicada, CEUNES/UFES

R. Santos, Universidade Federal do Rio de Janeiro

IPoli, Centro Multidisciplinar UFRJ-Macaé

Referências

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.

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Publicado

2023-07-20

Como Citar

Domingues, L., & Santos, R. (2023). A Note on C^2 Ill-posedness Results for the Zakharov System in Arbitrary Dimension. Trends in Computational and Applied Mathematics, 24(3), 505–519. https://doi.org/10.5540/tcam.2023.024.03.00505

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