On the cauchy problem for the ostrovsky equation with positive dispersion

Communications in Partial Differential Equations

Volumn 32, Issue 12, 2007, Pages 1895-1916

  Gui, Guilong ^a   Liu, Yue ^b  

^a JIANGSU UNIVERSITY   (China)

^b UNIVERSITY OF TEXAS AT ARLINGTON   (United States)

Author keywords

Cauchy problem; Local and global well posedness; Ostrovsky equation; Regularity; Weak rotation

Indexed keywords

EID: 36949035590     PISSN: 03605302     EISSN: 15324133     Source Type: Journal    
DOI: 10.1080/03605300600987314     Document Type: Article

References (24)

1. Benilov, E. S. (1992). On the surface waves in a shallow channel with an uneven bottom. Stud. Appl. Math. 87:1-14.
2. Bourgain, J. (1993a). Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Part I: Schröinger equation, Part II: the KdV equation. Geom. Funct. Anal. 2:107-156, 209-262.
3. Bourgain, J. (1993b). On the Cauchy problem for the Kadomtsev- Petviashvili equations. Geom. Funct. Anal. 3:315-341.
4. Chen, G. Y., Boyd, J. P. (2001). Analytical and numerical studies of weakly nonlocal solitary waves of the rotation-modified Korteweg-de Vries equation. Physica D. 155:201-222.
5. Choudhury, R., Ivanov, R., Liu, Y. (2007). Integrals of motion, nonintegrability and local bifurcations for the Ostrovsky equation. 34:544-550.
6. Galkin, V. N., Stepanyants, Yu. A. (1991). On the existence of stationary solitary waves in a rotating fluid. J. Appl. Math. Mech. 55(6):939-943.
7. Gilman, O. A., Grimshaw, R., Stepanyants, Yu. A. (1995). Approximate and numerical solutions of the stationary Ostrovsky equation. Stud. Appl. Math. 95:115-126.
8. Ginibre, J., Tsutsumi, Y., Velo, G. (1997). On the Cauchy problem for the Zakharov system. J. Funct. Anal. 151:384-436.
9. Huo, Z., Jia, Y. (2006). Low-regularity solutions for the Ostrovsky equation. Proc. Edin. Math. Soc. 49:87-100.
10. Kenig, C. E., Ponce, G., Vega, L. (1991). Oscillatory integrals and regularity of dispersive equations. Indiana Univ. Math. J. 40:33-69.
11. Kenig, C. E., Ponce, G., Vega, L. (1993). The Cauchy problem for the Korteweg-de vries equation in sobolev spaces of negative indices. Duke Math. J. 71:1-21.
12. Kenig, C. E., Ponce, G., Vega, L. (1996). A bilinear estimate with applications to the KdV equation. J. Amer. Math. Soc. 9:573-603.
13. Levandosky, S., Liu, Y. (2007). Stability and weak rotation limit of solitary waves of the Ostrovsky equation. Discrete Contin. Dynam. Systems - Series B 7:793-806.
14. Linares, F., Milanés, A. (2006). Local and global well-posedness for the Ostrovsky equation. J. Differential Equations 222(2): 325-340.
15. Liu, Y. (2007). On the stability of solitary waves for the Ostrovsky equation. Quart. Appl. Math. 65:571-589.
16. Liu, Y., Varlamov, V. (2004). Stability of solitary waves and weak rotation limit for the Ostrovsky equation. J. Differential Equations 203:159-183.
17. Ostrovsky, L. A. (1978). Nonlinear internal waves in a rotating ocean. Okeanologia 18:181-191.
18. Ostrovsky, L. A., Stepanyants, Yu. A. (1990). Nonlinear Surface and Internal Waves in Rotating Fluids. Research Reports in Physics, Nonlinear wave 3. Berlin, Heidelberg: Springer.
19. Saut, J. C., Tzvetkov, N. (1999). The Cauchy problem for higher-order KP equations. J. Differential Eqns. 153:196-222.
20. Takaoka, H. (2000). Well-posedness for the Kadomtsev-Petviashvili II equation. Adv. Differential Equations 5:1421-1443.
21. Takaoka, H., Tzvetkov, N. (2001). On the local regularity of the Kadomtsev-Petviashvili II equation. International Math. Res. Notices 2:77-114.
22. Tzvetkov, N. (2000). Global low-regularity solutions for Kadomtsev-Petviashvili equations. Differential Integral Eqns. 13:1289-1320.
23. Varlamov, V., Liu, Y. (2004). Cauchy problem for the Ostrovsky equation. Discrete Contin. Dynam. Systems 10:731-753.
24. Xavier, C. (2004). Local well-posedness for a higher order nonlinear Schrödinger equation in Sobolev spaces of negative indices. J. Differential Eqns. 13:1-10.