×

zbMATH — the first resource for mathematics

Local and global well-posedness for the Ostrovsky equation. (English) Zbl 1089.35061
Summary: We consider the initial value problem for \[ \partial_tu- \beta\partial_x^3u- \gamma\partial_x^{-1}u+ uu_x=0, \quad x,t\in\mathbb R, \] where \(u\) is a real valued function, \(\beta\) and \(\gamma\) are real numbers such that \(\beta\cdot\gamma\neq 0\) and \(\partial_x^{-1}f= ((i\xi)^{-1} \widehat{f}(\xi))^\vee\).
This equation differs from the Korteweg-de Vries equation in a nonlocal term. Nevertheless, we obtain local well-posedness in \(X_s= \{f\in H^s(\mathbb R): \partial_x^{-1}f\in L^2(\mathbb R)\}\), \(s>\frac34\), using techniques developed by C. E. Kenig, G. Ponce and L. Vega [J. Am. Math. Soc. 4, No. 2, 323–347 (1991; Zbl 0737.35102)]. For the case \(\beta\cdot\gamma>0\), we also obtain a global result in \(X_1\), using appropriate conservation laws.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
86A05 Hydrology, hydrography, oceanography
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bourgain, J., Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. the KdV equation, Gafa, 3, 209-262, (1993) · Zbl 0787.35098
[2] Chen, G.; Boyd, J.P., Analytical and numerical studies of weakly nonlocal solitary waves of the rotation-modified korteweg – de Vries equation, Phys. D, 155, 201-222, (2001) · Zbl 0985.35077
[3] Iório, R.J.; Nunes, W.V.L., On equations of KP-type, Proc. roy. soc. Edinburgh sect. A, 128, 4, 725-743, (1998) · Zbl 0911.35103
[4] Isaza, P.; Mejia, C.; Stallbohm, V., Regularizing effects for the linearized kadomtsev – petviashvili (KP) equation, Rev. colombiana mat., 31, 37-61, (1997) · Zbl 0887.35140
[5] Kenig, C.E.; Ponce, G.; Vega, L., Oscillatory integrals and regularity of dispersive equations, Indiana univ. math. J., 40, 33-69, (1991) · Zbl 0738.35022
[6] Kenig, C.E.; Ponce, G.; Vega, L., Well-posedness of the initial value problem for the korteweg – de Vries equation, J. amer. math. soc., 4, 323-347, (1991) · Zbl 0737.35102
[7] Kenig, C.E.; Ponce, G.; Vega, L., Well-posedness and scattering results for the generalized korteweg – de Vries equation via contraction principle, Comm. pure appl. math., 46, 527-620, (1993) · Zbl 0808.35128
[8] Kenig, C.E.; Ponce, G.; Vega, L., A bilinear estimate with applications to the KdV equation, J. amer. math. soc., 9, 2, 573-603, (1996) · Zbl 0848.35114
[9] Molinet, L.; Saut, J.C.; Tzvetkov, N., Global well-posedness for the KP-I equation, Math. ann., 324, 2, 255-275, (2002) · Zbl 1008.35060
[10] Ostrovskii, L.A., Nonlinear internal waves in a rotating Ocean, Okeanologiya, 18, 2, 181-191, (1978)
[11] Redekopp, L.G., Nonlinear waves in geophysics: long internal waves, Lectures in appl. math., 20, 59-78, (1983) · Zbl 0543.76032
[12] Varlamov, V.; Liu, Y., Cauchy problem for the Ostrovsky equation, Discrete contin. dyn. syst., 10, 731-753, (2004) · Zbl 1059.35035
[13] Varlamov, V.; Liu, Y., Stability of solitary waves and weak rotation limit for the Ostrovsky equation, J. differential equations, 203, 159-183, (2004) · Zbl 1064.35148
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.