Global well-posedness of the Benjamin-Ono equation in low-regularity spaces

Journal of the American Mathematical Society

Volumn 20, Issue 3, 2007, Pages 753-798

  Ionescu, Alexandru D ^a   Kenig, Carlos E ^b  

^a UNIVERSITY OF WISCONSIN MADISON   (United States)

^b UNIVERSITY OF CHICAGO   (United States)

Author keywords

Benjamin Ono initial value problem; Gauge transformation; Global well posedness; Local smoothing

Indexed keywords

EID: 34447516606     PISSN: 08940347     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0894-0347-06-00551-0     Document Type: Article

References (25)

1. T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech. 29 (1967), 559-592.
2. 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), 209-262. MR1215780 (95d:35160b)
3. N. Burq and F. Planchon, On well-posedness for the Benjamin-Ono equation, preprint (2005).
4. J. Colliander, C. E. Kenig, and G. Staffilani, Local well-posedness for dispersion-generalized Benjamin-Ono equations, Differential Integral Equations 16 (2003), 1441-1472. MR2029909 (2005a:35240) (Pubitemid 38192158)
5. J. Ginibre and G. Velo, Smoothing properties and existence of solutions for the generalized Benjamin-Ono equation, J. Differential Equations 93 (1991), 150-232. MR1122309 (93b:35116)
6. A. D. Ionescu and C. E. Kenig, Complex-valued solutions of the Benjamin-Ono equation, preprint (2006).
7. R. J. Iorio, On the Cauchy problem for the Benjamin-Ono equation, Comm. Partial Differential Equations 11 (1986), 1031-1081. MR0847994 (88b:35034)
8. C. E. Kenig and K. D. Koenig, On the local well-posedness of the Benjamin-Ono and modified Benjamin-Ono equations, Math. Res. Lett. 10 (2003), 879-895. MR2025062 (2004j:35249) (Pubitemid 38157353)
9. C. E. Kenig, G. Ponce, and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc. 4 (1991), 323-347. MR1086966 (92c:35106)
10. S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (1993), 1221-1268. MR1231427 (94h:35137)
11. S. Klainerman and S. Selberg, Remark on the optimal regularity for equations of wave maps type, Comm. Partial Differential Equations 22 (1997), 901-918. MR1452172 (99c:35163) (Pubitemid 127361558)
12. s (ℝ), Int. Math. Res. Not. 2003 (2003), 1449-1464. MR1976047 (2004b:35284)
13. H. Koch and N. Tzvetkov, Nonlinear wave interactions for the Benjamin-Ono equation, Int. Math. Res. Not. 2005 (2005), 1833-1847. MR2172940 (2006f:35245) (Pubitemid 41228958)
14. L. Molinet, J.-C. Saut, and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal. 33 (2001), 928-988. MR1885293 (2002k:35281)
15. 2 for the periodic Benjamin-Ono equation, preprint (2006).
16. H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan 39 (1975), 10821091. MR0398275 (53:2129)
17. G. Ponce, On the global well-posedness of the Benjamin-Ono equation, Differential Integral Equations 4 (1991), 527-542. MR1097916 (92e:35137)
18. J.-C. Saut, Sur quelques généralisations de l'équation de Korteweg-de Vries, J. Math. Pures Appl. 58 (1979), 21-61. MR0533234 (82m:35133)
19. T. Tao, Global regularity of wave maps I. Small critical Sobolev norm in high dimension, Int. Math. Res. Not. 2001 (2001), 299-328. MR1820329 (2001m:35200)
20. T. Tao, Global regularity of wave maps II. Small energy in two dimensions, Comm. Math. Phys. 224 (2001), 443-544. MR1869874 (2002h:58052) (Pubitemid 33374002)
21. 2 functions, and applications to nonlinear dispersive equations, Amer. J. Math. 123 (2001), 839-908. MR1854113 (2002k:35283) (Pubitemid 33404895)
22. 1 (ℝ), J. Hyperbolic Differ. Equ. 1 (2004), 27-49. MR2052470 (2005f:35273)
23. D. Tataru, Local and global results for wave maps I, Comm. Partial Differential Equations 23 (1998), 1781-1793. MR1641721 (99j:58209) (Pubitemid 128364866)
24. D. Tataru, On global existence and scattering for the wave maps equation, Amer. J. Math. 123 (2001), 37-77. MR1827277 (2002c:58045) (Pubitemid 33749070)
25. M. Tom, Smoothing properties of some weak solutions to the Benjamin-Ono equation, Differential Integral Equations 3 (1990), 683-694. MR1044213 (91e:35191)