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 
                  
                     
Acknowledgements
We are deeply indebted to the reviewers for their valuable suggestions, which greatly improved our original paper.
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            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-120025491Suche in Google Scholar
[2] 
A. Benedek and R. Panzone,
The space 
                  
                     
[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.2019.11.092Suche 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.1993.1040Suche 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.2000.3687Suche in Google Scholar
[6] E. Compaan, A smoothing estimate for the nonlinear Schrödinger equation, UIUC research experience for graduate students report, 2013. Suche 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/rnaa036Suche 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.2020.03.038Suche 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. 10.1080/03605302.2019.1581803Suche in Google Scholar
[10] 
X. Du,
A sharp Schrödinger maximal estimate in 
                  
                     
                        
[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.2013.6.723Suche 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.2019.68.7618Suche 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/rns189Suche 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.2012.05.019Suche 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/s00208-018-1695-7Suche in Google Scholar
[16] L. Grafakos, Classical Fourier Analysis, 2nd ed., Grad. Texts in Math. 249, Springer, New York, 2008. 10.1007/978-0-387-09432-8Suche 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. Suche 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.a10Suche in Google Scholar
[19] A. Grünrock, On the generalized Zakharov–Kuznetsov equation at critical regularity, preprint (2015), https://arxiv.org/abs/1509.09146. Suche 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/s00028-021-00671-9Suche 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.3547Suche 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 
                  
                     
[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-zSuche 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.2017.11.020Suche 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/S0012-7094-93-07101-3Suche 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.2020.08.003Suche 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.65.139Suche 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. 10.1137/080739173Suche 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.2010.11.005Suche 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. 10.1137/20M1344524Suche in Google Scholar
[31] 
F. Linares and J. P. G. Ramos,
The Cauchy problem for the 
                  
                     
[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.1993.1103Suche 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. 10.1080/03605302.2021.1881793Suche 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.2013.12.003Suche 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.2018.05.016Suche 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.2012.05.007Suche 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. 10.1137/110850566Suche 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.2009.03.004Suche 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. Suche in Google Scholar
[40] V. E. Zakharov and E. A. Kuznetsov, On three dimensional solitons, Sov. Phys. JETP 39 (1974), 285–286. Suche in Google Scholar
© 2024 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Triangles with one fixed side–length, a Furstenberg-type problem, and incidences in finite vector spaces
- Existence and multiplicity of solutions for fractional Schrödinger-p-Kirchhoff equations in ℝ N
- Estimates of Picard modular cusp forms
- Some q-supercongruences from a q-analogue of Watson's 3 F 2 summation
- Existence of strong solutions for one-dimensional reflected mixed stochastic delay differential equations
- Free groups generated by two unipotent maps
- Degenerate Schrödinger--Kirchhoff {(p,N)}-Laplacian problem with singular Trudinger--Moser nonlinearity in ℝ N
- Colored multizeta values in positive characteristic
- Simultaneous nonvanishing of central L-values with large level
- Laplace convolutions of weighted averages of arithmetical functions
- Cohomological properties of maximal pro-p Galois groups that are preserved under profinite completion
- On the geometric trace of a generalized Selberg trace formula
- Elementary properties of free lattices
- Weighted estimates for product singular integral operators in Journé’s class on RD-spaces
- Small generators of abelian number fields
- Pointwise convergence and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation
- Weighted bilinear multiplier theorems in Dunkl setting via singular integrals
Artikel in diesem Heft
- Frontmatter
- Triangles with one fixed side–length, a Furstenberg-type problem, and incidences in finite vector spaces
- Existence and multiplicity of solutions for fractional Schrödinger-p-Kirchhoff equations in ℝ N
- Estimates of Picard modular cusp forms
- Some q-supercongruences from a q-analogue of Watson's 3 F 2 summation
- Existence of strong solutions for one-dimensional reflected mixed stochastic delay differential equations
- Free groups generated by two unipotent maps
- Degenerate Schrödinger--Kirchhoff {(p,N)}-Laplacian problem with singular Trudinger--Moser nonlinearity in ℝ N
- Colored multizeta values in positive characteristic
- Simultaneous nonvanishing of central L-values with large level
- Laplace convolutions of weighted averages of arithmetical functions
- Cohomological properties of maximal pro-p Galois groups that are preserved under profinite completion
- On the geometric trace of a generalized Selberg trace formula
- Elementary properties of free lattices
- Weighted estimates for product singular integral operators in Journé’s class on RD-spaces
- Small generators of abelian number fields
- Pointwise convergence and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation
- Weighted bilinear multiplier theorems in Dunkl setting via singular integrals