Home Modified energy method and applications for the well-posedness for the higher-order Benjamin–Ono equation and the higher-order intermediate long wave equation
Article
Licensed
Unlicensed Requires Authentication

Modified energy method and applications for the well-posedness for the higher-order Benjamin–Ono equation and the higher-order intermediate long wave equation

  • Boling Guo and Zhaohui Huo EMAIL logo
Published/Copyright: October 6, 2019

Abstract

In this paper, the well-posedness of the higher-order Benjamin–Ono equation

ut+(uxx)+uxxx=uux-x(uxu+(uxu))

is considered. The modified energy method is introduced to consider the equation. It is shown that the Cauchy problem of the higher-order Benjamin–Ono equation is locally well-posed in H3/4 without using the gauge transformation. Moreover, the well-posedness of the higher-order intermediate long wave equation

ut+𝒢δ(uxx)+uxxx=uux-x(u𝒢δxu+𝒢δ(uxu)),𝒢δ=x-1i(coth(δξ))x,

is considered. It is shown that the Cauchy problem of the higher-order intermediate long wave equation is locally well-posed in H3/4.

MSC 2010: 35E15; 35Q53

Communicated by Christopher D. Sogge


Award Identifier / Grant number: 11471323

Award Identifier / Grant number: 11571254

Award Identifier / Grant number: 11771444

Funding statement: The second author is supported by NSFC (grant No. 11471323, No. 11571254 and No. 11771444).

Acknowledgements

We deeply thank the referee and editor for their suggestions and corrections.

References

[1] L. Abdelouhab, J. L. Bona, M. Felland and J.-C. Saut, Nonlocal models for nonlinear, dispersive waves, Phys. D 40 (1989), no. 3, 360–392. 10.1016/0167-2789(89)90050-XSearch in Google Scholar

[2] T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech. 29 (1967), 559–592. 10.1017/S002211206700103XSearch in Google Scholar

[3] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), no. 3, 107–156. 10.1007/BF01896020Search in Google Scholar

[4] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal. 3 (1993), no. 3, 209–262. 10.1007/BF01895688Search in Google Scholar

[5] W. Craig, P. Guyenne and H. Kalisch, Hamiltonian long-wave expansions for free surfaces and interfaces, Comm. Pure Appl. Math. 58 (2005), no. 12, 1587–1641. 10.1002/cpa.20098Search in Google Scholar

[6] Z. Guo, Local well-posedness for dispersion generalized Benjamin–Ono equations in Sobolev spaces, J. Differential Equations 252 (2012), no. 3, 2053–2084. 10.1016/j.jde.2011.10.012Search in Google Scholar

[7] B. L. Guo and S. B. Tan, Long time behavior for the equation of finite-depth fluids, Comm. Math. Phys. 163 (1994), no. 1, 1–15. 10.1007/BF02101732Search in Google Scholar

[8] A. D. Ionescu and C. E. Kenig, Global well-posedness of the Benjamin–Ono equation in low-regularity spaces, J. Amer. Math. Soc. 20 (2007), no. 3, 753–798. 10.1090/S0894-0347-06-00551-0Search in Google Scholar

[9] A. D. Ionescu, C. E. Kenig and D. Tataru, Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math. 173 (2008), no. 2, 265–304. 10.1007/s00222-008-0115-0Search in Google Scholar

[10] R. I. Joseph, Solitary waves in a finite depth fluid, J. Phys. A 10 (1977), no. 12, 225–227. 10.1088/0305-4470/10/12/002Search in Google Scholar

[11] C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg–de Vries equation in Sobolev spaces of negative indices, Duke Math. J. 71 (1993), no. 1, 1–21. 10.1215/S0012-7094-93-07101-3Search in Google Scholar

[12] C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9 (1996), no. 2, 573–603. 10.1090/S0894-0347-96-00200-7Search in Google Scholar

[13] H. Koch and N. Tzvetkov, Nonlinear wave interactions for the Benjamin-Ono equation, Int. Math. Res. Not. 2005 (2005), no. 30, 1833–1847. 10.1155/IMRN.2005.1833Search in Google Scholar

[14] Y. Kodama, J. Satsuma and M. J. Ablowitz, Nonlinear intermediate long-wave equation: Analysis and method of solution, Phys. Rev. Lett. 46 (1981), no. 11, 687–690. 10.1103/PhysRevLett.46.687Search in Google Scholar

[15] F. Linares, D. Pilod and G. Ponce, Well-posedness for a higher-order Benjamin–Ono equation, J. Differential Equations 250 (2011), no. 1, 450–475. 10.1016/j.jde.2010.08.022Search in Google Scholar

[16] L. Molinet and D. Pilod, Global well-posedness and limit behavior for a higher-order Benjamin–Ono equation, Comm. Partial Differential Equations 37 (2012), no. 11, 2050–2080. 10.1080/03605302.2012.683846Search in Google Scholar

[17] L. Molinet, J. C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin–Ono and related equations, SIAM J. Math. Anal. 33 (2001), no. 4, 982–988. 10.1137/S0036141001385307Search in Google Scholar

[18] C. Muscalu, J. Pipher, T. Tao and C. Thiele, Bi-parameter paraproducts, Acta Math. 193 (2004), no. 2, 269–296. 10.1007/BF02392566Search in Google Scholar

[19] H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan 39 (1975), no. 4, 1082–1091. 10.1143/JPSJ.39.1082Search in Google Scholar

[20] D. Pilod, On the Cauchy problem for higher-order nonlinear dispersive equations, J. Differential Equations 245 (2008), no. 8, 2055–2077. 10.1016/j.jde.2008.07.017Search in Google Scholar

[21] J. Satsuma, M. J. Ablowitz and Y. Kodama, On an internal wave equation describing a stratified fluid with finite depth, Phys. Lett. A 73 (1979), no. 4, 283–286. 10.1016/0375-9601(79)90534-6Search in Google Scholar

[22] J.-C. Saut, Sur quelques généralisations de l’équation de Korteweg–de Vries, J. Math. Pures Appl. (9) 58 (1979), no. 1, 21–61. Search in Google Scholar

[23] T. Tao, Multilinear weighted convolution of L2-functions, and applications to nonlinear dispersive equations, Amer. J. Math. 123 (2001), no. 5, 839–908. 10.1353/ajm.2001.0035Search in Google Scholar

[24] T. Tao, Global well-posedness of the Benjamin–Ono equation in H1(𝐑), J. Hyperbolic Differ. Equ. 1 (2004), no. 1, 27–49. 10.1142/S0219891604000032Search in Google Scholar

[25] Y. L. Zhou and B. L. Guo, Initial value problems for a nonlinear singular integral-differential equation of deep water, Partial Differential Equations (Tianjin 1986), Lecture Notes in Math. 1306, Springer, Berlin (1988), 278–290. 10.1007/BFb0082940Search in Google Scholar

Received: 2019-05-17
Revised: 2019-08-22
Published Online: 2019-10-06
Published in Print: 2020-01-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 17.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2019-0133/html
Scroll to top button