Abstract
We prove existence of solutions for the Benjamin—Ono equation with data in H s(ℝ), s > 0. Thanks to conservation laws, this yields global solutions for H 21 (ℝ) data, which is the natural “finite energy” class. Moreover, unconditional uniqueness is obtained in L ∞t (H 21 (ℝ)), which includes weak solutions, while for s > 203 , uniqueness holds in a suitable space.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
T. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech. 29(1967), 559–592.
N. Burq, P. Gérard, and N. Tzvetkov. Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math. 126(2004), 569–605.
J. Ginibre and G. Velo. Smoothing properties and existence of solutions for the generalized Benjamin—Ono equation, J. Differential Equations 93(1991), 150–212.
N. Hayashi and T. Ozawa. Remarks on nonlinear Schrödinger equations in one space dimension, Differential Integral Equations 7(2): 453–461, 1994.
A. D. Ionescu and C. E. Kenig, Global well-posedness of the Benjamin—Ono equation in low regularity spaces, Preprint, arXiv: math. AP/0508632, 2005.
C. E. Kenig and K. D. Koenig, On the local well-posedness of the Benjamin—Ono and modified Benjamin—Ono equations, Math. Res. Lett. 10(2003), 879–895.
H. Koch and N. Tzvetkov, On the local well-posedness of the Benjamin—Ono equation in H s(ℝ), Int. Math. Res. Not. 26(2003), 1449–1464.
H. Koch and N. Tzvetkov, Nonlinear wave interactions for the Benjamin—Ono equation, Preprint, arXiv:math.AP/0411434, 2004.
L. Molinet, J. C. Saut, and N. Tzvetkov, Ill-posedness issues for the Benjamin—Ono and related equations, SIAM J. Math. Anal. 33(2001), 982–988 (electronic).
H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan 39(1975), 1082–1091.
G. Ponce, On the global well-posedness of the Benjamin—Ono equation, Differential Integral Equations 4(1991), 527–542.
T. Tao, Global well-posedness of the Benjamin—Ono equation in H 1(ℝ), J. Hyperbolic Differ. Equ. 1(2004), 27–49.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Birkhäuser Boston
About this chapter
Cite this chapter
Burq, N., Planchon, F. (2006). The Benjamin—Ono equation in energy space. In: Bove, A., Colombini, F., Del Santo, D. (eds) Phase Space Analysis of Partial Differential Equations. Progress in Nonlinear Differential Equations and Their Applications, vol 69. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4521-2_5
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4521-2_5
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4511-3
Online ISBN: 978-0-8176-4521-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)