-
Previous Article
Examinations on a three-dimensional differentiable vector field that equals its own curl
- CPAA Home
- This Issue
-
Next Article
Gevrey asymptotic theory for singular first order linear partial differential equations of nilpotent type — Part I —
Some results about a bidimensional version of the generalized BO
| 1. | Departamento de Matematica, IMECC-UNICAMP, 13081-970, Campinas, SP, Brazil |
$u_t-H^{(x)}u_{x y}+u^p u_y=0, \quad t\in \mathbb R,\quad (x,y)\in \mathbb R^2,$
we use the method of parabolic regularization to prove local well-posedness in the spaces $H^s(\mathbb R^2), \quad s>2$ and in the weighted spaces $\mathcal F_r^s=H^s(\mathbb R^2) \cap L^2((1+x^2+y^2)^rdxdy), \quad s>2,\quad r\in [0,1]$ and $\mathcal F_{1,k}^k=H^k(\mathbb R^2) \cap L^2((1+x^2+y^{2k})dxdy), \quad k\in\mathbb N, \quad k\geq 3. \quad $ As in the case of BO there is lack of persistence for both the linear and nonlinear equations (for $p$ odd) in $\mathcal F_2^s$. That leads to unique continuation principles in a natural way. By standard methods based on $L^p-L^q$ estimates of the associated group we obtain global well-posedness for small initial data and nonlinear scattering for $p\geq 3,\quad s>3$. Nonexistence of square integrable solitary waves of the form $u(x,y,t)=v(x,y-ct),\quad c>0, \quad p\in \{1,2\}$ is obtained using the results about existence of solitary waves of the BO and variational methods.
| [1] |
Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215 |
| [2] |
Amin Esfahani, Steve Levandosky. Solitary waves of the rotation-generalized Benjamin-Ono equation. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 663-700. doi: 10.3934/dcds.2013.33.663 |
| [3] |
Nakao Hayashi, Pavel Naumkin. On the reduction of the modified Benjamin-Ono equation to the cubic derivative nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 237-255. doi: 10.3934/dcds.2002.8.237 |
| [4] |
Jerry Bona, Hongqiu Chen. Solitary waves in nonlinear dispersive systems. Discrete and Continuous Dynamical Systems - B, 2002, 2 (3) : 313-378. doi: 10.3934/dcdsb.2002.2.313 |
| [5] |
Dongfeng Yan. KAM Tori for generalized Benjamin-Ono equation. Communications on Pure and Applied Analysis, 2015, 14 (3) : 941-957. doi: 10.3934/cpaa.2015.14.941 |
| [6] |
Jerry Bona, H. Kalisch. Singularity formation in the generalized Benjamin-Ono equation. Discrete and Continuous Dynamical Systems, 2004, 11 (1) : 27-45. doi: 10.3934/dcds.2004.11.27 |
| [7] |
Kenta Ohi, Tatsuo Iguchi. A two-phase problem for capillary-gravity waves and the Benjamin-Ono equation. Discrete and Continuous Dynamical Systems, 2009, 23 (4) : 1205-1240. doi: 10.3934/dcds.2009.23.1205 |
| [8] |
Sondre Tesdal Galtung. A convergent Crank-Nicolson Galerkin scheme for the Benjamin-Ono equation. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1243-1268. doi: 10.3934/dcds.2018051 |
| [9] |
Jason Murphy, Kenji Nakanishi. Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1507-1517. doi: 10.3934/dcds.2020328 |
| [10] |
Agnid Banerjee. A note on the unique continuation property for fully nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2015, 14 (2) : 623-626. doi: 10.3934/cpaa.2015.14.623 |
| [11] |
H. Kalisch. Stability of solitary waves for a nonlinearly dispersive equation. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 709-717. doi: 10.3934/dcds.2004.10.709 |
| [12] |
Ola I. H. Maehlen. Solitary waves for weakly dispersive equations with inhomogeneous nonlinearities. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4113-4130. doi: 10.3934/dcds.2020174 |
| [13] |
Lufang Mi, Kangkang Zhang. Invariant Tori for Benjamin-Ono Equation with Unbounded quasi-periodically forced Perturbation. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 689-707. doi: 10.3934/dcds.2014.34.689 |
| [14] |
G. Fonseca, G. Rodríguez-Blanco, W. Sandoval. Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1327-1341. doi: 10.3934/cpaa.2015.14.1327 |
| [15] |
Thomas Kappeler, Riccardo Montalto. Normal form coordinates for the Benjamin-Ono equation having expansions in terms of pseudo-differential operators. Discrete and Continuous Dynamical Systems, 2022, 42 (9) : 4127-4201. doi: 10.3934/dcds.2022048 |
| [16] |
Robert Schippa. On the Cauchy problem for higher dimensional Benjamin-Ono and Zakharov-Kuznetsov equations. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5189-5215. doi: 10.3934/dcds.2020225 |
| [17] |
Eddye Bustamante, José Jiménez Urrea, Jorge Mejía. The Cauchy problem for a family of two-dimensional fractional Benjamin-Ono equations. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1177-1203. doi: 10.3934/cpaa.2019057 |
| [18] |
Luc Molinet, Francis Ribaud. Well-posedness in $ H^1 $ for generalized Benjamin-Ono equations on the circle. Discrete and Continuous Dynamical Systems, 2009, 23 (4) : 1295-1311. doi: 10.3934/dcds.2009.23.1295 |
| [19] |
Khaled El Dika. Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 583-622. doi: 10.3934/dcds.2005.13.583 |
| [20] |
Juan Belmonte-Beitia, Vladyslav Prytula. Existence of solitary waves in nonlinear equations of Schrödinger type. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 1007-1017. doi: 10.3934/dcdss.2011.4.1007 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]