Abstract
In this paper, the well-posedness of the higher-order Benjamin–Ono equation
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
is considered.
It is shown that the Cauchy problem of the higher-order intermediate long wave equation is locally well-posed in
Funding source: National Natural Science Foundation of China
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
[24]
T. Tao,
Global well-posedness of the Benjamin–Ono equation in
[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
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- On the distribution of zeros of derivatives of the Riemann ξ-function
- Representations of constant socle rank for the Kronecker algebra
- Abstract bivariant Cuntz semigroups II
- Curves on Segre threefolds
- 𝒩(p, q, s)-type spaces in the unit ball of ℂn(II): Carleson measure and its application
- On a class of critical Robin problems
- On the description of multidimensional normal Hausdorff operators on Lebesgue spaces
- Spectral asymptotics for Krein–Feller operators with respect to 𝑉-variable Cantor measures
- Schneider–Siegel theorem for a family of values of a harmonic weak Maass form at Hecke orbits
- Modified energy method and applications for the well-posedness for the higher-order Benjamin–Ono equation and the higher-order intermediate long wave equation
- Some remarks on products of sets in the Heisenberg group and in the affine group
- Codimension growth of central polynomials of Lie algebras
- Unramified Whittaker functions for certain Brylinski–Deligne covering groups
- Tilting classes over commutative rings
Articles in the same Issue
- Frontmatter
- On the distribution of zeros of derivatives of the Riemann ξ-function
- Representations of constant socle rank for the Kronecker algebra
- Abstract bivariant Cuntz semigroups II
- Curves on Segre threefolds
- 𝒩(p, q, s)-type spaces in the unit ball of ℂn(II): Carleson measure and its application
- On a class of critical Robin problems
- On the description of multidimensional normal Hausdorff operators on Lebesgue spaces
- Spectral asymptotics for Krein–Feller operators with respect to 𝑉-variable Cantor measures
- Schneider–Siegel theorem for a family of values of a harmonic weak Maass form at Hecke orbits
- Modified energy method and applications for the well-posedness for the higher-order Benjamin–Ono equation and the higher-order intermediate long wave equation
- Some remarks on products of sets in the Heisenberg group and in the affine group
- Codimension growth of central polynomials of Lie algebras
- Unramified Whittaker functions for certain Brylinski–Deligne covering groups
- Tilting classes over commutative rings