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

Ionescu, Alexandru; Kenig, Carlos

doi:10.1090/S0894-0347-06-00551-0

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

Authors: Alexandru D. Ionescu and Carlos E. Kenig
Journal: J. Amer. Math. Soc. 20 (2007), 753-798
MSC (2000): Primary 35Q53
Posted: October 24, 2006
MathSciNet review: 2291918
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Abstract: We prove that the Benjamin-Ono initial-value problem is globally well-posed in the Banach spaces $H^\sigma_r(\mathbb{R})$ , $\sigma\geq 0$ , of real-valued Sobolev functions.

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), no. 3, 209–262. MR 1215780 (95d:35160b), http://dx.doi.org/10.1007/BF01895688
3. N. Burq and F. Planchon, On well-posedness for the Benjamin-Ono equation, preprint (2005).
4. J. Colliander, C. Kenig, and G. Staffilani, Local well-posedness for dispersion-generalized Benjamin-Ono equations, Differential Integral Equations 16 (2003), no. 12, 1441–1472. MR 2029909 (2005a:35240)
5. J. Ginibre and G. Velo, Smoothing properties and existence of solutions for the generalized Benjamin-Ono equation, J. Differential Equations 93 (1991), no. 1, 150–212. MR 1122309 (93b:35116), http://dx.doi.org/10.1016/0022-0396(91)90025-5
6. A. D. Ionescu and C. E. Kenig, Complex-valued solutions of the Benjamin-Ono equation, preprint (2006).
7. Rafael José Iório Jr., On the Cauchy problem for the Benjamin-Ono equation, Comm. Partial Differential Equations 11 (1986), no. 10, 1031–1081. MR 847994 (88b:35034), http://dx.doi.org/10.1080/03605308608820456
8. Carlos E. Kenig and Kenneth D. Koenig, On the local well-posedness of the Benjamin-Ono and modified Benjamin-Ono equations, Math. Res. Lett. 10 (2003), no. 5-6, 879–895. MR 2025062 (2004j:35249)
9. Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc. 4 (1991), no. 2, 323–347. MR 1086966 (92c:35106), http://dx.doi.org/10.1090/S0894-0347-1991-1086966-0
10. S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (1993), no. 9, 1221–1268. MR 1231427 (94h:35137), http://dx.doi.org/10.1002/cpa.3160460902
11. Sergiu Klainerman and Sigmund Selberg, Remark on the optimal regularity for equations of wave maps type, Comm. Partial Differential Equations 22 (1997), no. 5-6, 901–918. MR 1452172 (99c:35163), http://dx.doi.org/10.1080/03605309708821288
12. H. Koch and N. Tzvetkov, On the local well-posedness of the Benjamin-Ono equation in 𝐻^{𝑠}(ℝ), Int. Math. Res. Not. 26 (2003), 1449–1464. MR 1976047 (2004b:35284), http://dx.doi.org/10.1155/S1073792803211260
13. H. Koch and N. Tzvetkov, Nonlinear wave interactions for the Benjamin-Ono equation, Int. Math. Res. Not. 30 (2005), 1833–1847. MR 2172940 (2006f:35245), http://dx.doi.org/10.1155/IMRN.2005.1833
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), no. 4, 982–988 (electronic). MR 1885293 (2002k:35281), http://dx.doi.org/10.1137/S0036141001385307
15. L. Molinet, Global well-posedness in for the periodic Benjamin-Ono equation, preprint (2006).
16. Hiroaki Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan 39 (1975), no. 4, 1082–1091. MR 0398275 (53 #2129)
17. Gustavo Ponce, On the global well-posedness of the Benjamin-Ono equation, Differential Integral Equations 4 (1991), no. 3, 527–542. MR 1097916 (92e:35137)
18. J.-C. Saut, Sur quelques généralisations de l’équation de Korteweg-de Vries, J. Math. Pures Appl. (9) 58 (1979), no. 1, 21–61 (French). MR 533234 (82m:35133)
19. Terence Tao, Global regularity of wave maps. I. Small critical Sobolev norm in high dimension, Internat. Math. Res. Notices 6 (2001), 299–328. MR 1820329 (2001m:35200), http://dx.doi.org/10.1155/S1073792801000150
20. Terence Tao, Global regularity of wave maps. II. Small energy in two dimensions, Comm. Math. Phys. 224 (2001), no. 2, 443–544. MR 1869874 (2002h:58052), http://dx.doi.org/10.1007/PL00005588
21. Terence Tao, Multilinear weighted convolution of 𝐿²-functions, and applications to nonlinear dispersive equations, Amer. J. Math. 123 (2001), no. 5, 839–908. MR 1854113 (2002k:35283)
22. Terence Tao, Global well-posedness of the Benjamin-Ono equation in 𝐻¹(𝑅), J. Hyperbolic Differ. Equ. 1 (2004), no. 1, 27–49. MR 2052470 (2005f:35273), http://dx.doi.org/10.1142/S0219891604000032
23. Daniel Tataru, Local and global results for wave maps. I, Comm. Partial Differential Equations 23 (1998), no. 9-10, 1781–1793. MR 1641721 (99j:58209), http://dx.doi.org/10.1080/03605309808821400
24. Daniel Tataru, On global existence and scattering for the wave maps equation, Amer. J. Math. 123 (2001), no. 1, 37–77. MR 1827277 (2002c:58045)
25. Michael Mudi Tom, Smoothing properties of some weak solutions of the Benjamin-Ono equation, Differential Integral Equations 3 (1990), no. 4, 683–694. MR 1044213 (91e:35191)

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