Home Pointwise convergence and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation
Article
Licensed
Unlicensed Requires Authentication

Pointwise convergence and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation

  • Wei Yan EMAIL logo , Weimin Wang and Xiangqian Yan
Published/Copyright: April 24, 2024

Abstract

This paper is devoted to studying the pointwise convergence problem and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation. Firstly, we present an alternative proof of Theorem 1.5 of Linares and Ramos [Maximal function estimates and local well-posedness for the generalized Zakharov–Kuznetsov equation, SIAM J. Math. Anal. 53 (2021), 1, 914–936] and Theorem 1.8 of Linares and Ramos [The Cauchy problem for the L 2 -critical generalized Zakharov–Kuznetsov equation in dimension 3, Comm. Partial Differential Equations 46 (2021), 9, 1601–1627]. Secondly, we give an alternative proof of Theorem 1.1 of Ribaud and Vento [A note on the Cauchy problem for the 2D generalized Zakharov–Kuznetsov equations, C. R. Math. Acad. Sci. Paris 350 (2012), 9–10, 499–503] and present the nonlinear smoothing and uniform convergence of two-dimensional generalized Zakharov–Kuznetsov equation. Thirdly, we give an alternative proof of Theorem 1.4 of Linares and Ramos [Maximal function estimates and local well-posedness for the generalized Zakharov–Kuznetsov equation, SIAM J. Math. Anal. 53 (2021), 1, 914–936]. Finally, we study the nonlinear smoothing and uniform convergence of 𝑛-dimensional generalized Zakharov–Kuznetsov equation with n 3 .

MSC 2020: 42B25; 35Q53; 42B37

Acknowledgements

We are deeply indebted to the reviewers for their valuable suggestions, which greatly improved our original paper.

  1. Communicated by: Christopher D. Sogge

References

[1] M. Ben-Artzi, H. Koch and J.-C. Saut, Dispersion estimates for third order equations in two dimensions, Comm. Partial Differential Equations 28 (2003), no. 11–12, 1943–1974. 10.1081/PDE-120025491Search in Google Scholar

[2] A. Benedek and R. Panzone, The space L p , with mixed norm, Duke Math. J. 28 (1961), 301–324. 10.1215/S0012-7094-61-02828-9Search in Google Scholar

[3] D. Bhattacharya, L. G. Farah and S. Roudenko, Global well-posedness for low regularity data in the 2d modified Zakharov–Kuznetsov equation, J. Differential Equations 268 (2020), no. 12, 7962–7997. 10.1016/j.jde.2019.11.092Search in Google Scholar

[4] J. L. Bona and J.-C. Saut, Dispersive blowup of solutions of generalized Korteweg–de Vries equations, J. Differential Equations 103 (1993), no. 1, 3–57. 10.1006/jdeq.1993.1040Search in Google Scholar

[5] M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal. 179 (2001), no. 2, 409–425. 10.1006/jfan.2000.3687Search in Google Scholar

[6] E. Compaan, A smoothing estimate for the nonlinear Schrödinger equation, UIUC research experience for graduate students report, 2013. Search in Google Scholar

[7] E. Compaan, R. Lucà and G. Staffilani, Pointwise convergence of the Schrödinger flow, Int. Math. Res. Not. IMRN 2021 (2021), no. 1, 599–650. 10.1093/imrn/rnaa036Search in Google Scholar

[8] S. Correia and J. D. Silva, Nonlinear smoothing for dispersive PDE: A unified approach, J. Differential Equations 269 (2020), no. 5, 4253–4285. 10.1016/j.jde.2020.03.038Search in Google Scholar

[9] L. Cossetti, L. Fanelli and F. Linares, Uniqueness results for Zakharov–Kuznetsov equation, Comm. Partial Differential Equations 44 (2019), no. 6, 504–544. 10.1080/03605302.2019.1581803Search in Google Scholar

[10] X. Du, A sharp Schrödinger maximal estimate in R 2 , Dissertation, 2017. 10.4007/annals.2017.186.2.5Search in Google Scholar

[11] B. Erdoğan and N. Tzirakis, Smoothing and global attractors for the Zakharov system on the torus, Anal. PDE 6 (2013), no. 3, 723–750. 10.2140/apde.2013.6.723Search in Google Scholar

[12] M. B. Erdoğan, T. B. Gürel and N. Tzirakis, Smoothing for the fractional Schrödinger equation on the torus and the real line, Indiana Univ. Math. J. 68 (2019), no. 2, 369–392. 10.1512/iumj.2019.68.7618Search in Google Scholar

[13] M. B. Erdoğan and N. Tzirakis, Global smoothing for the periodic KdV evolution, Int. Math. Res. Not. IMRN 2013 (2013), no. 20, 4589–4614. 10.1093/imrn/rns189Search in Google Scholar

[14] L. G. Farah, F. Linares and A. Pastor, A note on the 2D generalized Zakharov–Kuznetsov equation: Llocal, global, and scattering results, J. Differential Equations 253 (2012), no. 8, 2558–2571. 10.1016/j.jde.2012.05.019Search in Google Scholar

[15] R. L. Frank and J. Sabin, Extremizers for the Airy–Strichartz inequality, Math. Ann. 372 (2018), no. 3–4, 1121–1166. 10.1007/s00208-018-1695-7Search in Google Scholar

[16] L. Grafakos, Classical Fourier Analysis, 2nd ed., Grad. Texts in Math. 249, Springer, New York, 2008. 10.1007/978-0-387-09432-8Search in Google Scholar

[17] A. Grünrock, New applications of the Fourier restriction norm method to wellposedness problems for nonlinear evolution equations, Dissertation, Universität Wuppertal, Wuppertal, 2002. Search in Google Scholar

[18] A. Grünrock, A remark on the modified Zakharov–Kuznetsov equation in three space dimensions, Math. Res. Lett. 21 (2014), no. 1, 127–131. 10.4310/MRL.2014.v21.n1.a10Search in Google Scholar

[19] A. Grünrock, On the generalized Zakharov–Kuznetsov equation at critical regularity, preprint (2015), https://arxiv.org/abs/1509.09146. Search in Google Scholar

[20] S. Herr and S. Kinoshita, The Zakharov–Kuznetsov equation in high dimensions: Small initial data of critical regularity, J. Evol. Equ. 21 (2021), no. 2, 2105–2121. 10.1007/s00028-021-00671-9Search in Google Scholar

[21] S. Herr and S. Kinoshita, Subcritical well-posedness results for the Zakharov–Kuznetsov equation in dimension three and higher, Ann. Inst. Fourier (Grenoble) 73 (2023), no. 3, 1203–1267. 10.5802/aif.3547Search in Google Scholar

[22] Z. Huo and Y. Jia, Dyadic bilinear estimates and applications to the well-posedness for the 2D Zakharov–Kuznetsov equation in the endpoint space H 1 / 4 , Forum Math. 32 (2020), no. 6, 1575–1598. 10.1515/forum-2020-0003Search in Google Scholar

[23] T. Kato, Well-posedness for the generalized Zakharov–Kuznetsov equation on modulation spaces, J. Fourier Anal. Appl. 23 (2017), no. 3, 612–655. 10.1007/s00041-016-9480-zSearch in Google Scholar

[24] T. Kato, The Cauchy problem for the generalized Zakharov–Kuznetsov equation on modulation spaces, J. Differential Equations 264 (2018), no. 5, 3402–3444. 10.1016/j.jde.2017.11.020Search in Google Scholar

[25] C. E. Kenig, G. Ponce and L. 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. 10.1215/S0012-7094-93-07101-3Search in Google Scholar

[26] S. Kinoshita, Global well-posedness for the Cauchy problem of the Zakharov–Kuznetsov equation in 2D, Ann. Inst. H. Poincaré C Anal. Non Linéaire 38 (2021), no. 2, 451–505. 10.1016/j.anihpc.2020.08.003Search in Google Scholar

[27] S. Kinoshita, Well-posedness for the Cauchy problem of the modified Zakharov–Kuznetsov equation, Funkcial. Ekvac. 65 (2022), no. 2, 139–158. 10.1619/fesi.65.139Search in Google Scholar

[28] F. Linares and A. Pastor, Well-posedness for the two-dimensional modified Zakharov–Kuznetsov equation, SIAM J. Math. Anal. 41 (2009), no. 4, 1323–1339. 10.1137/080739173Search in Google Scholar

[29] F. Linares and A. Pastor, Local and global well-posedness for the 2D generalized Zakharov–Kuznetsov equation, J. Funct. Anal. 260 (2011), no. 4, 1060–1085. 10.1016/j.jfa.2010.11.005Search in Google Scholar

[30] F. Linares and J. P. G. Ramos, Maximal function estimates and local well-posedness for the generalized Zakharov–Kuznetsov equation, SIAM J. Math. Anal. 53 (2021), no. 1, 914–936. 10.1137/20M1344524Search in Google Scholar

[31] F. Linares and J. P. G. Ramos, The Cauchy problem for the L 2 -critical generalized Zakharov–Kuznetsov equation in dimension 3, Comm. Partial Differential Equations 46 (2021), no. 9, 1601–1627. 10.1080/03605302.2021.1888119Search in Google Scholar

[32] F. Linares and M. Scialom, On the smoothing properties of solutions to the modified Korteweg–de Vries equation, J. Differential Equations 106 (1993), no. 1, 141–154. 10.1006/jdeq.1993.1103Search in Google Scholar

[33] A. J. Mendez, C. Muñoz, F. Poblete and J. C. Pozo, On local energy decay for large solutions of the Zakharov–Kuznetsov equation, Comm. Partial Differential Equations 46 (2021), no. 8, 1440–1487. 10.1080/03605302.2021.1881793Search in Google Scholar

[34] L. Molinet and D. Pilod, Bilinear Strichartz estimates for the Zakharov–Kuznetsov equation and applications, Ann. Inst. H. Poincaré C Anal. Non Linéaire 32 (2015), no. 2, 347–371. 10.1016/j.anihpc.2013.12.003Search in Google Scholar

[35] D. Pornnopparath, Small data well-posedness for derivative nonlinear Schrödinger equations, J. Differential Equations 265 (2018), no. 8, 3792–3840. 10.1016/j.jde.2018.05.016Search in Google Scholar

[36] F. Ribaud and S. Vento, A note on the Cauchy problem for the 2D generalized Zakharov–Kuznetsov equations, C. R. Math. Acad. Sci. Paris 350 (2012), no. 9–10, 499–503. 10.1016/j.crma.2012.05.007Search in Google Scholar

[37] F. Ribaud and S. Vento, Well-posedness results for the three-dimensional Zakharov–Kuznetsov equation, SIAM J. Math. Anal. 44 (2012), no. 4, 2289–2304. 10.1137/110850566Search in Google Scholar

[38] B. Wang, L. Han and C. Huang, Global well-posedness and scattering for the derivative nonlinear Schrödinger equation with small rough data, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 6, 2253–2281. 10.1016/j.anihpc.2009.03.004Search in Google Scholar

[39] W. Yan, Y. J. Xie, W. M. Wang and Y. T. Zhang, The Cauchy problem for the two-dimensional modified Zakharov–Kuznetsov equation with rough data, J. Differential Equations, to appear. Search in Google Scholar

[40] V. E. Zakharov and E. A. Kuznetsov, On three dimensional solitons, Sov. Phys. JETP 39 (1974), 285–286. Search in Google Scholar

Received: 2023-12-17
Published Online: 2024-04-24
Published in Print: 2025-02-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 15.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2023-0463/html
Scroll to top button