Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Cora Sadosky: Her Mathematics, Mentorship, and Professional Contributions Read chapter Read first chapter

2016 | OriginalPaper | Chapter

On the IVP for the k-Generalized Benjamin–Ono Equation

Author : Gustavo Ponce

Published in: Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory (Volume 1)

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

We shall study special properties of solutions to the IVP associated to the k-generalized Benjamin–Ono equation. We shall compare them with those for the k-generalized Korteweg-de Vries equation and for the k-generalized dispersive Benjamin–Ono equation. Also we shall discuss some open questions appearing in this subject.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Metric Characterizations of Some Classes of Banach Spaces
Literature
1.
2.
M.J. Ablowitz, A.S. Fokas, The inverse scattering transform for the Benjamin-Ono equation, a pivot for multidimensional problems. Stud. Appl. Math. 68, 1–10 (1983)MathSciNetCrossRefMATH
3.
C.J. Amick, J.F. Toland, Uniqueness and related analytic properties for the Benjamin-Ono equation - a nonlinear Neumann problem in the plane. Acta Math. 167, 107–126 (1991)MathSciNetCrossRefMATH
4.
B. Bajsanski, R.R. Coifman, Singular Integrals. Proceedings of Symposia in Pure Mathematics (American Mathematical Society, Providence, RI, 1966), pp. 1–17
5.
T.B. Benjamin, Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29, 559–592 (1967)CrossRefMATH
6.
H.A. Biagioni, F. Linares, Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations. Trans. Am. Math. Soc. 353, 3649–3659 (2001)CrossRefMATH
7.
J. Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Geom. Funct. Anal. 3, 107–156/209–262 (1993)
8.
9.
10.
R.R. Coifman, Y. Meyer, Au delá des opérateurs pseudodifférentiels, Asterisque 57, Société Mathemátique de France (1973)
11.
12.
L. Dawson, H. McGahagan, G. Ponce, On the decay properties of solutions to a class of Schrödinger equations. Proc. Am. Math. Soc. 136, 2081–2090 (2008)CrossRefMATH
13.
J. Duoandikoetxea, Fourier Analysis. Graduate Studies in Mathematics, vol. 29 (American Mathematical Society, Providence, RI, 2000)
14.
15.
16.
G. Fonseca, F. Linares, G. Ponce, The IVP for the Benjamin-Ono equation in weighted Sobolev spaces II. J. Funct. Anal. 262, 2031–2049 (2012)MathSciNetCrossRefMATH
17.
J. Garcia-Cuerva, J.L. Rubio de Francia, Weighted Norm Inequalities and Related topics (North Holland, Amsterdam, 1985)MATH
18.
19.
S. Herr, A. Ionescu, C.E. Kenig, H. Koch, A para-differential renormalization technique for nonlinear dispersive equations. Commun. Partial Differ. Equ. 35, 1827–1875 (2010)MathSciNetCrossRefMATH
20.
R. Hunt, B. Muckenhoupt, R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform. Trans. Am. Math. Soc. 176, 227–251 (1973)MathSciNetCrossRefMATH
21.
A.D. Ionescu, C.E. Kenig, Global well-posedness of the Benjamin-Ono equation on low-regularity spaces. J. Am. Math. Soc. 20 (3), 753–798 (2007)MathSciNetCrossRefMATH
22.
23.
R.J. Iorio, Unique continuation principle for the Benjamin-Ono equation. Differ. Integr. Equ. 16, 1281–1291 (2003)MathSciNetMATH
24.
P. Isaza, F. Linares, G. Ponce, On the propagation of regularity and decay of solutions to the k-generalized Korteweg-de Vries equation. Commun Partial Differ. Equ. 40, 1336–1364 (2015)MathSciNetCrossRefMATH
25.
P. Isaza, F. Linares, G. Ponce, On the propagation of regularity in solutions to the Benjamin-Ono equation. J. Funct. Anal. 270, 976–1000 (2016)MathSciNetCrossRefMATH
26.
T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation. Adv. Math. Suppl. Stud. Stud. Appl. Math. 8, 93–128 (1983)MathSciNet
27.
28.
C.E. Kenig, K.D. Koenig, On the local well-posedness of the Benjamin-Ono and modified Benjamin-Ono equations. Math. Res. Lett. 10, 879–895 (2003)MathSciNetCrossRefMATH
29.
C.E. Kenig, H. Takaoka, Global well-posedness of the modified Benjamin-Ono equation with initial data in H 1∕2. Int. Math. Res. Not. 9, 1–44 (2006). Art. ID 95702
30.
C.E. Kenig, G. Ponce, L. Vega, Oscillatory integrals and regularity of dispersive equations. Indiana Univ. Math. J. 40, 33–69 (1991)MathSciNetCrossRefMATH
31.
C.E. Kenig, G. Ponce, L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation. J. Am. Math. Soc. 4, 323–347 (1991)MathSciNetCrossRefMATH
32.
C.E. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via contraction principle. Commun. Pure Appl. Math. 46, 527–620 (1993)MathSciNetCrossRefMATH
33.
34.
H. Koch, N. Tzvetkov, On the local well-posedness of the Benjamin-Ono equation on \(H^{s}(\mathbb{R})\). Int. Math. Res. Not. 26, 1449–1464 (2003)MathSciNetCrossRefMATH
35.
36.
F. Linares, G. Ponce, Introduction to Nonlinear Dispersive Equations (Springer, New York, 2009)MATH
37.
Y. Martel, F. Merle, Blow up in finite time and dynamics of blow up solutions for the L 2-critical generalized KdV equation. J. Am. Math. Soc. 15, 617–664 (2002)MathSciNetCrossRefMATH
38.
39.
L. Molinet, F. Ribaud, Well-posedness results for the generalized Benjamin-Ono equation with small initial data. J. Math. Pures et Appl. 83, 277–311 (2004)MathSciNetCrossRefMATH
40.
L. Molinet, F. Ribaud, Well-posedness results for the Benjamin-Ono equation with arbitrary large initial data. Int. Math. Res. Not. 70, 3757–3795 (2004)MathSciNetCrossRefMATH
41.
L. Molinet, J.C. Saut, N. Tzvetkov, Ill- posedness issues for the Benjamin-Ono and related equations. SIAM J. Math. Anal. 33, 982–988 (2001)MathSciNetCrossRefMATH
42.
43.
J. Nahas, G. Ponce, On the persistent properties of solutions to semi-linear Schrodinger equation. Commun. Partial Differ. Equ. 34, 1–20 (2009)MathSciNetCrossRefMATH
44.
45.
S. Petermichl, The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical A p characteristic. Am. J. Math. 129, 1355–1375 (2007)MathSciNetCrossRefMATH
46.
G. Ponce, On the global well-posedness of the Benjamin-Ono equation. Differ. Integr. Equ. 4, 527–542 (1991)MathSciNetMATH
47.
C. Sadosky, Interpolation of Operators and Singular Integrals (Marcel Dekker Inc, New York, 1979)MATH
48.
49.
E.M. Stein, Harmonic Analysis (Princeton University Press, Princeton, 1993)
50.
R.S. Strichartz, Restriction of Fourier transforms to quadratic surface and decay of solutions of wave equations. Duke Math. J. 44, 705–714 (1977)MathSciNetCrossRefMATH
51.
52.
S. Vento, Sharp well-posedness results for the generalized Benjamin-Ono equations with higher nonlinearity. Differ. Integr. Equ. 22, 425–446 (2009)MathSciNetMATH
53.
S. Vento, Well-posedness of the generalized Benjamin-Ono equations with arbitrary large initial data in the critical space. Int. Math. Res. Not. 2, 297–319 (2010)MathSciNetMATH
Metadata
Title
On the IVP for the k-Generalized Benjamin–Ono Equation
Author
Gustavo Ponce
Copyright Year
2016
Book
Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory (Volume 1)

Print ISBN: 978-3-319-30959-0

Electronic ISBN: 978-3-319-30961-3

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-30961-3_16

Premium Partner

    Image Credits