
Previous Article
Wellposedness for regularized nonlinear dispersive wave equations
 DCDS Home
 This Issue

Next Article
A twophase problem for capillarygravity waves and the BenjaminOno equation
Sharp wellposedness results for the BBM equation
1.  Department of Mathematics, Statistics and Computer Science, The University of Illinois at Chicago , 851 S. Morgan Street MC 249, Chicago, Illinois 606077045 
2.  Laboratoire Paul Painlevé, Bâtiment M2, Cité scientifique, 59655 Villeneuve D'ascq Cedex, France 
$ u_{t}+u_{x}+u u_{x}u_{x x t} = 0 $
was derived as a model for the unidirectional propagation of longcrested, surface water waves. It arises in other contexts as well, and is generally understood as an alternative to the Kortewegde Vries equation. Considered here is the initialvalue problem wherein $u$ is specified everywhere at a given time $t = 0$, say, and inquiry is then made into its further development for $t>0$. It is proven that this initialvalue problem is globally well posed in the $L^2$based Sobolev class $H^s$ if $s \geq 0$. Moreover, the map that associates the relevant solution to given initial data is shown to be smooth. On the other hand, if $s < 0$, it is demonstrated that the correspondence between initial data and putative solutions cannot be even of class $C^2$. Hence, it is concluded that the BBM equation cannot be solved by iteration of a bounded mapping leading to a fixed point in $H^s$based spaces for $s < 0$. One is thus led to surmise that the initialvalue problem for the BBM equation is not even locally well posed in $H^s$ for negative values of $s$.
[1] 
Jerry L. Bona, Hongqiu Chen, ChunHsiung Hsia. Wellposedness for the BBMequation in a quarter plane. Discrete & Continuous Dynamical Systems  S, 2014, 7 (6) : 11491163. doi: 10.3934/dcdss.2014.7.1149 
[2] 
Ming Wang. Sharp global wellposedness of the BBM equation in $L^p$ type Sobolev spaces. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 57635788. doi: 10.3934/dcds.2016053 
[3] 
Ricardo A. Pastrán, Oscar G. Riaño. Sharp wellposedness for the ChenLee equation. Communications on Pure & Applied Analysis, 2016, 15 (6) : 21792202. doi: 10.3934/cpaa.2016033 
[4] 
Didier Pilod. Sharp wellposedness results for the KuramotoVelarde equation. Communications on Pure & Applied Analysis, 2008, 7 (4) : 867881. doi: 10.3934/cpaa.2008.7.867 
[5] 
Tadahiro Oh, Yuzhao Wang. On global wellposedness of the modified KdV equation in modulation spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 29712992. doi: 10.3934/dcds.2020393 
[6] 
Takamori Kato. Global wellposedness for the Kawahara equation with low regularity. Communications on Pure & Applied Analysis, 2013, 12 (3) : 13211339. doi: 10.3934/cpaa.2013.12.1321 
[7] 
Zhaoyang Yin. Wellposedness, blowup, and global existence for an integrable shallow water equation. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 393411. doi: 10.3934/dcds.2004.11.393 
[8] 
SeungYeal Ha, Jinyeong Park, Xiongtao Zhang. A global wellposedness and asymptotic dynamics of the kinetic Winfree equation. Discrete & Continuous Dynamical Systems  B, 2020, 25 (4) : 13171344. doi: 10.3934/dcdsb.2019229 
[9] 
Hideo Takaoka. Global wellposedness for the KadomtsevPetviashvili II equation. Discrete & Continuous Dynamical Systems, 2000, 6 (2) : 483499. doi: 10.3934/dcds.2000.6.483 
[10] 
Thomas Y. Hou, Congming Li. Global wellposedness of the viscous Boussinesq equations. Discrete & Continuous Dynamical Systems, 2005, 12 (1) : 112. doi: 10.3934/dcds.2005.12.1 
[11] 
Boris Kolev. Local wellposedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167189. doi: 10.3934/jgm.2017007 
[12] 
Lin Shen, Shu Wang, Yongxin Wang. The wellposedness and regularity of a rotating blades equation. Electronic Research Archive, 2020, 28 (2) : 691719. doi: 10.3934/era.2020036 
[13] 
A. Alexandrou Himonas, Curtis Holliman. On wellposedness of the DegasperisProcesi equation. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 469488. doi: 10.3934/dcds.2011.31.469 
[14] 
Nils Strunk. Wellposedness for the supercritical gKdV equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 527542. doi: 10.3934/cpaa.2014.13.527 
[15] 
Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp wellposedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (2) : 487504. doi: 10.3934/cpaa.2018027 
[16] 
Wei Yan, Yimin Zhang, Yongsheng Li, Jinqiao Duan. Sharp wellposedness of the Cauchy problem for the rotationmodified KadomtsevPetviashvili equation in anisotropic Sobolev spaces. Discrete & Continuous Dynamical Systems, 2021 doi: 10.3934/dcds.2021097 
[17] 
Nobu Kishimoto, Minjie Shan, Yoshio Tsutsumi. Global wellposedness and existence of the global attractor for the KadomtsevPetviashvili Ⅱ equation in the anisotropic Sobolev space. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 12831307. doi: 10.3934/dcds.2020078 
[18] 
Luc Molinet, Francis Ribaud. On global wellposedness for a class of nonlocal dispersive wave equations. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 657668. doi: 10.3934/dcds.2006.15.657 
[19] 
Jinkai Li, Edriss Titi. Global wellposedness of strong solutions to a tropical climate model. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 44954516. doi: 10.3934/dcds.2016.36.4495 
[20] 
DanAndrei Geba, Kenji Nakanishi, Sarada G. Rajeev. Global wellposedness and scattering for Skyrme wave maps. Communications on Pure & Applied Analysis, 2012, 11 (5) : 19231933. doi: 10.3934/cpaa.2012.11.1923 
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]