×

Global well-posedness and inviscid limit for the generalized Benjamin-Ono-Burgers equation. (English) Zbl 1460.35312

Summary: This paper deals with the Cauchy problem for the generalized Benjamin-Ono-Burgers equation \(\partial_tu+\mathcal{H}\partial_x^2u-vu_{xx}+\partial_x(u^{k+1}/(k+1))=0,k\geq 4\), where \(\mathcal{H}\) denotes Hilbert transform. We obtain its global well-posedness results in Besov Spaces if \(k\geq 4\) and the initial data in \(\dot B^{s_k}_{2,1}\) are sufficiently small, where \(s_k:=1/2-1/k\) corresponds to the critical scaling regularity index. Furthermore, we prove its global well-posedness and inviscid limit behavior in Sobolev spaces.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35Q35 PDEs in connection with fluid mechanics
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
44A15 Special integral transforms (Legendre, Hilbert, etc.)
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Benjamin, TB., Internal waves of permanent form in fluids of great depth, J Fluid Mech, 29, 559-592 (1967) · Zbl 0147.46502 · doi:10.1017/S002211206700103X
[2] Ono, H., Algebraic solitary waves in stratified fluids, J Phys Soc Japan, 39, 1082-1091 (1975) · Zbl 1334.76027 · doi:10.1143/JPSJ.39.1082
[3] Edwin, PM; Roberts, B., The Benjamin-Ono-Burgers equation: an application in solar physics, Wave Motion, 8, 151-158 (1986) · Zbl 0587.76193 · doi:10.1016/0165-2125(86)90021-1
[4] Ionescu, AD; Kenig, CE., Global well-posedness of the Benjamin-Ono equation in low-regularity spaces, J Amer Math Soc, 20, 753-798 (2007) · Zbl 1123.35055 · doi:10.1090/S0894-0347-06-00551-0
[5] Otani, M., Bilinear estimates with applications to the generalized Benjamin-Ono-Burgers equations, Differ Integral Equ, 18, 12, 1397-1426 (2005) · Zbl 1212.35326
[6] Vento, S., Well-posedness and ill-posedness results for dissipative Benjamin-Ono equations, Osaka J Math, 48, 4, 933-958 (2011) · Zbl 1232.35148
[7] Abdelouhab, L.; Bona, JL; Felland, M., Nonlocal models for nonlinear dispersive waves, Physica D, 40, 360-392 (1989) · Zbl 0699.35227 · doi:10.1016/0167-2789(89)90050-X
[8] Iorio, RJ., On the Cauchy problem for the Benjamin-Ono equation, Commun Partial Differ Equ, 11, 1031-1081 (1986) · Zbl 0608.35030 · doi:10.1080/03605308608820456
[9] Kenig, CE; Koenig, KD., On the local well-posedness of the Benjamin-Ono and modified Benjamin-Ono equations, Math Res Lett, 10, 879-895 (2003) · Zbl 1044.35072 · doi:10.4310/MRL.2003.v10.n6.a13
[10] Koch, H.; Tzvetkov, N., On the local well-posedness of the Benjamin-Ono equation in \(####\), Int Math Res Not, 2003, 1449-1464 (2003) · Zbl 1039.35106 · doi:10.1155/S1073792803211260
[11] Koch, H.; Tzvetkov, N., Nonlinear wave interactions for the Benjamin-Ono equation, Int Math Res Not, 2005, 30, 1833-1847 (2005) · Zbl 1156.35460 · doi:10.1155/IMRN.2005.1833
[12] Molinet, L.; Saut, JC; Tzvetkov, N., Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J Math Anal, 33, 982-988 (2001) · Zbl 0999.35085 · doi:10.1137/S0036141001385307
[13] Ponce, G., On the global well-posedness of the Benjamin-Ono equation, Differ Integral Equ, 4, 527-542 (1991) · Zbl 0732.35038
[14] Tao, T., Global well-posedness of the Benjamin-Ono equation in \(####\), J Hyperbolic Differ Equ, 1, 27-49 (2004) · Zbl 1055.35104 · doi:10.1142/S0219891604000032
[15] Guo, Z.; Peng, L.; Wang, B., Uniform well-posedness and inviscid limit for the Benjamin-Ono-Burgers equation, Adv Math, 228, 647-677 (2011) · Zbl 1234.35072 · doi:10.1016/j.aim.2011.03.017
[16] Molinet, L., A note on the inviscid limit of the Benjamin-Ono-Burgers equation in the energy space, Proc Amer Math Soc, 141, 8, 2793-2798 (2013) · Zbl 1284.35377 · doi:10.1090/S0002-9939-2013-11693-X
[17] Fokas, AS; Luo, L., Global solutions and their asymptotic behavior for Benjamin-Ono-Burgers type equations, Differ Integral Equ, 13, 1-3, 115-124 (2000) · Zbl 0987.35135
[18] Molinet, L.; Ribaud, F., Well-posedness results for the generalized Benjamin-Ono equation with small initial data, J Math Pures Appl, 83, 277-311 (2004) · Zbl 1084.35094 · doi:10.1016/j.matpur.2003.11.005
[19] Binir, B.; Kenig, CE; Ponce, G., On the ill-posedness of the IVP for the generalized Korteweg de Vries and nonlinear Schrödinger equation, J London Math Soc, 53, 551-559 (1996) · Zbl 0855.35112 · doi:10.1112/jlms/53.3.551
[20] Biagioni, HA; Linares, F., Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations, Trans Amer Math Soc, 353, 9, 3649-3659 (2001) · Zbl 0970.35154 · doi:10.1090/S0002-9947-01-02754-4
[21] Molinet, L.; Ribaud, F., On the Cauchy problem for the generalized Benjamin-Ono equation with small initial data, C R Acad Sci Paris Ser I, 337, 523-526 (2003) · Zbl 1045.35068 · doi:10.1016/j.crma.2003.09.012
[22] Molinet, L.; Ribaud, F., Well-posedness results for the generalized Benjamin-Ono equation with arbitrary large initial data, Int Math Res Not, 2004, 70, 3757-3795 (2004) · Zbl 1064.35149 · doi:10.1155/S107379280414083X
[23] Han, L.; Wang, B., Global wellposedness and limit behavior for the generalized finite-depth-fluid equation with small data in critical Besov spaces \(####\), J Differ Equ, 245, 2103-2144 (2008) · Zbl 1178.35302 · doi:10.1016/j.jde.2008.07.008
[24] Kenig, CE; Ponce, G.; Vega, L., Oscillatory integrals and regularity of dispersive equations, Indiana Univ Math J, 40, 33-69 (1991) · Zbl 0738.35022 · doi:10.1512/iumj.1991.40.40003
[25] Stein, EM., Harmonic analysis (1993), Princeton (NJ): Princeton University Press, Princeton (NJ) · Zbl 0821.42001
[26] Stein, EM; Weiss, G., Introduction to Fourier analysis on euclidean spaces (1971), Princeton (NJ): Princeton University Press, Princeton (NJ) · Zbl 0232.42007
[27] Kenig, CE; Ponce, G.; Vega, L., On the generalized Benjamin-Ono equation, Trans Amer Math Soc, 342, 155-172 (1994) · Zbl 0804.35105 · doi:10.1090/S0002-9947-1994-1153015-4
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.