Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T11:59:58.907Z Has data issue: false hasContentIssue false
  • English
  • Français

Global well-posedness and conservation laws for the water wave interaction equation

Published online by Cambridge University Press:  14 November 2011

Takayoshi Ogawa
Takayoshi Ogawa
Affiliation:
Department of Mathematics, University of California Santa Barbara, Santa Barbara, CA 93106, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

Interaction equations of long and short water wave are considered. It is shown that the Cauchy problem for

is locally well posed in the largest space where the three conservations

can be justified. Here E(u,v) is the energy functional associated to the system. By these conservation laws, we establish the global well-posedness of the system in the largest class of initial data.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 2 , 1997 , pp. 369 - 384
Copyright
Copyright © Royal Society of Edinburgh 1997

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

1Bekiranov, D., Ogawa, T. and Ponce, G.. On the well-posedness of Benney's interaction equation of short and long waves. Adv. in Differential Equations 1 (1996), 919–37.CrossRefGoogle Scholar
2Benney, D. J.. A general theory for interactions between short and long waves. Stud. Appl. Math. 56 (1977), 81–94.CrossRefGoogle Scholar
3Benney, D. J.. Instabilities associated with forced nonlinear waves. Stud. Appl. Math. 60 (1979), 27–41.CrossRefGoogle Scholar
4Bergh, J. and Löfström, J.. Interpolation Spaces (Berlin: Springer, 1970).Google Scholar
5Cazenave, T. and Weissler, F. B.. The Cauchy problem for the critical nonlinear Schrödinger equation in Hs. Nonlinear Anal. 100 (1990), 807–36.CrossRefGoogle Scholar
6Constantin, P. and Saut, J. C.. Local smoothing properties of dispersive equations. J. Amer. Math. Soc. 1 (1988), 413–46.CrossRefGoogle Scholar
7Ginibre, J. and Velo, G.. On a class of Schrödinger equations. III. Ann. Inst. H. Poincaré, Phys. Théor. 28(1978), 287–316.Google Scholar
8Ginibre, J. and Velo, G.. The global Cauchy problem for the nonlinear Schrödinger equation, revisited. Ann. H. Poincaré, Anal. Non Linéaire 2 (1985), 309–27.CrossRefGoogle Scholar
9Kato, T.. On nonlinear Schrödinger equations. Ann. H. Poincaré, Phys. Théor. 46 (1987), 113–29.Google Scholar
10Kenig, C. E., Ponce, G. and Vega, L.. Oscillatory integrals and regularity of dispersive equations. Indiana Univ. Math. J. 40 (1991), 33–69.CrossRefGoogle Scholar
11Kenig, C. E., Ponce, G. and Vega, L.. Small solutions to nonlinear Schrödinger equations. Ann. Inst. H. Poincaré, Anal. Non Linéaire 10 (1993), 255–88.CrossRefGoogle Scholar
12Kenig, C. E., Ponce, G. and Vega, L.. Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction mapping principle. Comm. Pure Appl. Math. 46 (1993), 527–620.CrossRefGoogle Scholar
13Kenig, C. E., Ponce, G. and Vega, L.. On the generalized Benjamin–Ono equation. Trans. Amer. Math. Soc. 342 (1994), 155–72.CrossRefGoogle Scholar
14Laurencot, Ph.. On a nonlinear Schrödinger equation arising in the theory of water waves. Nonlinear Anal. 24 (1995), 509–27.CrossRefGoogle Scholar
15Ma, Yan-Chow. The complete solution of the long-wave-short-wave resonance equations. Stud. Appl. Math. 59(1978), 201–21.CrossRefGoogle Scholar
16Sjölin, P.. Regularity of solutions to the Schrödinger equations. Duke Math. J. 55 (1987), 699–715.CrossRefGoogle Scholar
17Stein, E. M. and Weiss, G.. Introduction to Fourier Analysis in Euclidean Spaces (Princeton: NJ: Princeton University Press, 1971).Google Scholar
18Strichartz, R. S.. Restriction of Fourier transform to quadratic surfaces and decay solutions of wave equations. Duke Math. J. 44 (1977), 705–14.CrossRefGoogle Scholar
19Tsutsumi, M. and Hatano, S.. Well-posedness of the Cauchy problem for the long-wave-short-wave resonance equations. Nonlinear Anal. 22 (1994), 155–71.CrossRefGoogle Scholar
20Tsutsumi, M. and Hatano, S.. Well-posedness of the Cauchy problem for Benney's first equations of long wave-short wave interactions. Funkcial. Ekvac. 37 (1994), 289–316.Google Scholar
21Tsutsumi, Y.. Global solutions for nonlinear Schrödinger equations. Nonlinear Anal. 11 (1987), 1143–54.CrossRefGoogle Scholar
22Vega, L.. The Schrödinger equation: pointwise convergence to the initial data. Proc. Amer. Math. Soc. 102 (1988), 874–8.Google Scholar
23Yajima, K.. Existence of solutions for Schrödinger evolution equations. Comm. Math. Phys. 110 (1987), 415–26.CrossRefGoogle Scholar
24Yajima, N. and Oikawa, M.. Formation and interaction of sonic Langmuri soliton. Progr. Theoret. Phys. 56 (1974), 1719–39.CrossRefGoogle Scholar