Osaka Journal of Mathematics

Well-posedness of the generalized Benjamin-Ono-Burgers equations in Sobolev spaces of negative order

Masanori Otani

Full-text: Open access

Abstract

We study the well-posedness issue of the generalized Benjamin-Ono-Burgers (gBO-B) equations. We solve the initial-value problem (IVP) of the gBO-B equations with data below $L^2 (\mathbf{R})$. Our proof is based on the method of L. Molinet and F. Ribaud, which is analogous to that of J. Bourgain, and C.E. Kenig, G. Ponce, and L. Vega. It is known that such a method cannot be applied to the Benjamin-Ono equation.

Article information

Source
Osaka J. Math., Volume 43, Number 4 (2006), 935-965.

Dates
First available in Project Euclid: 11 December 2006

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1165850043

Mathematical Reviews number (MathSciNet)
MR2303557

Zentralblatt MATH identifier
1145.35305

Subjects
Primary: 35A07 35M10: Equations of mixed type 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10] 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction [See also 35Q30]

Citation

Otani, Masanori. Well-posedness of the generalized Benjamin-Ono-Burgers equations in Sobolev spaces of negative order. Osaka J. Math. 43 (2006), no. 4, 935--965. https://projecteuclid.org/euclid.ojm/1165850043


Export citation

References

  • D. Bekiranov: The initial-value problem for the generalized Burgers' equation, Diff. Int. Eqns. 9 (1996), 1253--1265.
  • D. Bekiranov, T. Ogawa and G. Ponce: Interaction equations for short and long dispersive waves, J. Funct. Anal. 158 (1998), 357--388.
  • H.A. Biagioni and F. Linares: Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations, Trans. Amer. Math. Soc. 353 (2001), 3649--3659.
  • J. Bourgain: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part II The KDV-equation, Geom. Funct. Anal. 3 (1993), 209--262.
  • N. Burq and F. Planchon: On the well-posedness for the Benjamin-Ono equation, arXiv: math.AP/0509096.
  • M. Christ, J. Colliander and T. Tao: Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocsing equations, Amer. J. Math. 125 (2003), 1235--1293.
  • J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao: Sharp global well-posedness for KdV and modified KdV on $\mathbb{R}$ and $\mathbb{T}$, J. Amer. Math. Soc. 16 (2003), 705--749.
  • J. Colliander, C. Kenig and G. Staffilani: Local well-posedness for dispersion-generalized Benjamin-Ono equations, Diff. Int. Eqns. 16 (2003), 1441--1472.
  • D.B. Dix: Nonuniqueness and uniqueness in the initial-value problem for Burgers' equation, SIAM J. Math. Anal. 27 (1996), 708--724.
  • J. Ginibre, Y. Tsutsumi and G. Velo: On the Cauchy problem for the Zakharov System, J. Funct. Anal. 151 (1997), 384--436.
  • A. Grünrock: An improved local wellposedness result for the modified KdV-equation, Internat. Math. Res. Not. 2004 (2004), 3287--3308.
  • S. Herr: Well-posedness for equations of Benjamin-Ono type, Illinois J. Math., to appear.
  • S. Herr: An improved bilinear estimates for Benjamin-Ono type equations, arXiv: math.AP/0509218.
  • A.D. Ionescu and C.E. Kenig: Global well-posedness of the Benjamin-Ono equation in low-regularity spaces, arXiv: math.AP/0508632.
  • K. Kato: On the existence of solutions to the Benjamin-Ono equation, in preparation.
  • 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.
  • C.E. Kenig, G. Ponce and L. Vega: A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9 (1996), 573--603.
  • N. Kita and J. Segata: Time local well-posedness for the Benjamin-Ono equation with large initial data, Publ. Res. Inst. Math. Sci. 42 (2006), 143--171.
  • H. Koch and N. Tzvetkov: Nonlinear wave interactions for the Benjamin-Ono equation, Internat. Math. Res. Not. 2005 (2005), 1833--1847.
  • L. Molinet: Global well-posedness in $L^2$ for the periodic Benjamin-Ono equation, arXiv: math.AP/0601217.
  • L. Molinet and F. Ribaud: The Cauchy problem for dissipative Korteweg-de Vries equations in Sobolev spaces of negative order, Indiana Univ. Math. J. 50 (2001), 1745--1776.
  • L. Molinet and F. Ribaud: The global Cauchy problem in Bourgain's-type spaces for a dispersive dissipative semilinear equation, SIAM J. Math. Anal. 33 (2002), 1269--1296.
  • L. Molinet and F. Ribaud: On the low regularity of the Korteweg-de Vries-Burgers equation, Internat. Math. Res. Not. 2002 (2002), 1979--2005.
  • 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.
  • M. Otani: Bilinear estimates with applications to the generalized Benjamin-Ono-Burgers equations, Diff. Int. Eqns. 18 (2005), 1397--1426.
  • T. Tao: Global well-posedness of the Benjamin-Ono equation in $H^1(\mathbf{R})$, J. Hyperbolic Diff. Eqns. 1 (2004), 27--49.