Hostname: page-component-84b7d79bbc-4hvwz Total loading time: 0 Render date: 2024-07-25T14:39:47.788Z Has data issue: false hasContentIssue false
  • English
  • Français

Dilation method and smoothing effects of solutions to the Benjamin–Ono equation

Published online by Cambridge University Press:  14 November 2011

Nako Hayashi ,
Keiichi Kato  and
Tohru Ozawa
Nako Hayashi
Affiliation:
Department of Mathematics, Faculty of Engineering, Gunma University, Kiryu 376, Japan
Keiichi Kato
Affiliation:
Department of Mathematics, Faculty of Science, Osaka University, Toyonaka 560, Japan
Tohru Ozawa
Affiliation:
Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 060., Japan
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we study smoothing effects of solutions to the Benjamin–Ono equation

where H is the Hilbcrt transform defined by

We prove that if φ ∈ H2 and (x∂x)4φ then the solution u of(BO) belongs to

, where

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 2 , 1996 , pp. 273 - 285
Copyright
Copyright © Royal Society of Edinburgh 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Abdelouhab, L., Bona, J. L.. Felland, M. and Saut, J. C.. Nonlocal models for nonlinear dispersive waves. Phys. D 40 (1989), 360–92.CrossRefGoogle Scholar
2Benjamin, T. B.. Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29 (1967), 559–92.CrossRefGoogle Scholar
3de Bouard, A., Hayashi, N. and Kato, K.. Gevrey regularizing effect for the (generalized) Korteweg–deVries equation and nonlinear Schrödinger equations. Ann. Inst. H. Poincaré, Anal, nonlin. (to appear).Google Scholar
4Case, K. M.. Benjamin–Ono related equations and their solutions. Proc. Nat. Acad. Sci. U.S.A. 76 (1976), 1–3.CrossRefGoogle Scholar
5Ginibre, J. and Velo, G.. On a class of nonlinear Schrödinger equation III. Ann. Inst. H. Poincaré Phys. Theorique 28 (1978). 287–316.Google Scholar
6Ginibre, J. and Velo, G.. On a class of nonlinear Schrodinger equation I. II. J. Fund. Anal. 32 ( 1979). 1–32; 33–71.CrossRefGoogle Scholar
7Ginibre, J. and Velo, G.. Smoothing properties and existence of solutions for the generalized Benjamin-Ono equation. J. Differential Equations 93 (1991), 150–212.CrossRefGoogle Scholar
8Hayashi, N. and Kato, K.. Regularity in time of solutions to nonlinear Schrodinger equations J. Funct Anal 128 (1995), 253–77.CrossRefGoogle Scholar
9Hayashi, N.. Kato, K. and Ozawa, T.. Dilation method and smoothing effect of the Schrödinger evolution group. Reviews in Math. Phys. (to appear).Google Scholar
10Hayashi, N., Nakamitsu, K. and Tsutsumi, M.. On solutions of the initial value problem for the nonlinear Schrödinger equation in one space dimension. Math. Z. 192 (1986), 637–50.CrossRefGoogle Scholar
11Hayashi, N., Nakamitsu, K. and Tsutsumi, M.. On solutions of the initial value problem for the nonlinear Schrödinger equation. J. Fund. Anal. 71 (1987). 218–45.CrossRefGoogle Scholar
12Hayashi, N., Nakamitsu, K. and Tsutsumi, M.. Nonlinear Schrödinger equations in weighted Sobolev spaces. Funkeial. Ekvac. 31 (1988), 363–81.Google Scholar
13Iorio, R. J.. On the Cauchy problem for the Benjamin Ono equation. Comm. Partial Differential Equations 11 (1986), 1031–81.Google Scholar
14Iorio, R. J.. The Benjamin-Ono equations in weighted Sobolev spaces. J. Math Anal. Appl. 157 (1991), 577–90.CrossRefGoogle Scholar
15Kato, T.. Quasilinear equations of evolution with applications to partial differential equations. Lecture Notes in Mathematics 44, 25–70 (Berlin: Springer. 1975).Google Scholar
16Kato, T.. On the Cauchy problem for the (generalized) Korteweg de Vries equation. Adv. in Math. Suppt. Stud., Stud. Appl Math. 8 (1983), 93–128.Google Scholar
17Kenig, C. E., Ponce, G. and Vega, L.. Well-posedness and scattering results forthe generalised Korteweg de Vries equation via contraction principle. Comm. Pure Appl. Math. 48 (1993), 527–620.CrossRefGoogle Scholar
18Kenig, C. E., Ponce, G. and Vega, L.. On the generalized Benjamin-Ono equation. Trans. Amer. Math. Soc. 342 (1994), 155–72.CrossRefGoogle Scholar
19Ono, H.. Algebraic solitary waves in stratified fluids. J. Phys. Soc. Japan 39(1975), 1082–91.CrossRefGoogle Scholar
20Ponce, G.. Regularity of solutions to nonlinear dispersive equations. J. Differential Equations 78 (1989), 122–35.CrossRefGoogle Scholar
21Ponce, G.. Smoothing properties of solutions to the Benjamin-Ono equation. Lecture Notes in Pure and Appl Math., 667–679 (New York: Dekker, 1990).Google Scholar
22Ponce, G.. On the global well-posedness of the Benjamin-Ono equation. Differential Integral Equations 4 (1991), 527–42.CrossRefGoogle Scholar
23Tom, M. M.. Smoothing properties of some weak solutions of the Benjamin-Ono equation. Differential Integral Equations 3 (1990), 683–94.CrossRefGoogle Scholar
24Tsutsumi, M.. Weighted Sobolev spaces, and rapidly decreasing solutions of some nonlinear dispersive wave equations. J. Differentia1 Equations 42 (1981). 260–81.CrossRefGoogle Scholar