We study the well-posedness issue of the generalized Benjamin-Ono-Burgers (gBO-B) equations. We solve the initial-value problem (IVP) of the gBO-B equations with data below $L^2 (\mathbf{R})$. Our proof is based on the method of L. Molinet and F. Ribaud, which is analogous to that of J. Bourgain, and C.E. Kenig, G. Ponce, and L. Vega. It is known that such a method cannot be applied to the Benjamin-Ono equation.
References
D. Bekiranov, T. Ogawa and G. Ponce: Interaction equations for short and long dispersive waves, J. Funct. Anal. 158 (1998), 357--388. MR1648479 0909.35123 10.1006/jfan.1998.3257 D. Bekiranov, T. Ogawa and G. Ponce: Interaction equations for short and long dispersive waves, J. Funct. Anal. 158 (1998), 357--388. MR1648479 0909.35123 10.1006/jfan.1998.3257
H.A. Biagioni and F. Linares: Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations, Trans. Amer. Math. Soc. 353 (2001), 3649--3659. MR1837253 10.1090/S0002-9947-01-02754-4 0002-9947%28200109%29353%3A9%3C3649%3AIFTDSA%3E2.0.CO%3B2-R H.A. Biagioni and F. Linares: Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations, Trans. Amer. Math. Soc. 353 (2001), 3649--3659. MR1837253 10.1090/S0002-9947-01-02754-4 0002-9947%28200109%29353%3A9%3C3649%3AIFTDSA%3E2.0.CO%3B2-R
J. Bourgain: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part II The KDV-equation, Geom. Funct. Anal. 3 (1993), 209--262. MR1215780 0787.35098 10.1007/BF01895688 J. Bourgain: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Part II The KDV-equation, Geom. Funct. Anal. 3 (1993), 209--262. MR1215780 0787.35098 10.1007/BF01895688
N. Burq and F. Planchon: On the well-posedness for the Benjamin-Ono equation, arXiv: math.AP/0509096. MR2357995 1148.35074 10.1007/s00208-007-0150-y N. Burq and F. Planchon: On the well-posedness for the Benjamin-Ono equation, arXiv: math.AP/0509096. MR2357995 1148.35074 10.1007/s00208-007-0150-y
M. Christ, J. Colliander and T. Tao: Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocsing equations, Amer. J. Math. 125 (2003), 1235--1293. MR2018661 1048.35101 10.1353/ajm.2003.0040 M. Christ, J. Colliander and T. Tao: Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocsing equations, Amer. J. Math. 125 (2003), 1235--1293. MR2018661 1048.35101 10.1353/ajm.2003.0040
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao: Sharp global well-posedness for KdV and modified KdV on $\mathbb{R}$ and $\mathbb{T}$, J. Amer. Math. Soc. 16 (2003), 705--749. MR1969209 1025.35025 10.1090/S0894-0347-03-00421-1 J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao: Sharp global well-posedness for KdV and modified KdV on $\mathbb{R}$ and $\mathbb{T}$, J. Amer. Math. Soc. 16 (2003), 705--749. MR1969209 1025.35025 10.1090/S0894-0347-03-00421-1
J. Colliander, C. Kenig and G. Staffilani: Local well-posedness for dispersion-generalized Benjamin-Ono equations, Diff. Int. Eqns. 16 (2003), 1441--1472. MR2029909 J. Colliander, C. Kenig and G. Staffilani: Local well-posedness for dispersion-generalized Benjamin-Ono equations, Diff. Int. Eqns. 16 (2003), 1441--1472. MR2029909
D.B. Dix: Nonuniqueness and uniqueness in the initial-value problem for Burgers' equation, SIAM J. Math. Anal. 27 (1996), 708--724. MR1382829 0852.35123 10.1137/0527038 D.B. Dix: Nonuniqueness and uniqueness in the initial-value problem for Burgers' equation, SIAM J. Math. Anal. 27 (1996), 708--724. MR1382829 0852.35123 10.1137/0527038
J. Ginibre, Y. Tsutsumi and G. Velo: On the Cauchy problem for the Zakharov System, J. Funct. Anal. 151 (1997), 384--436. MR1491547 0894.35108 10.1006/jfan.1997.3148 J. Ginibre, Y. Tsutsumi and G. Velo: On the Cauchy problem for the Zakharov System, J. Funct. Anal. 151 (1997), 384--436. MR1491547 0894.35108 10.1006/jfan.1997.3148
A. Grünrock: An improved local wellposedness result for the modified KdV-equation, Internat. Math. Res. Not. 2004 (2004), 3287--3308. MR2096258 1072.35161 A. Grünrock: An improved local wellposedness result for the modified KdV-equation, Internat. Math. Res. Not. 2004 (2004), 3287--3308. MR2096258 1072.35161
S. Herr: Well-posedness for equations of Benjamin-Ono type, Illinois J. Math., to appear. MR2379733 05316802 euclid.ijm/1258131113
S. Herr: Well-posedness for equations of Benjamin-Ono type, Illinois J. Math., to appear. MR2379733 05316802 euclid.ijm/1258131113
A.D. Ionescu and C.E. Kenig: Global well-posedness of the Benjamin-Ono equation in low-regularity spaces, arXiv: math.AP/0508632. MR2291918 1123.35055 10.1090/S0894-0347-06-00551-0 A.D. Ionescu and C.E. Kenig: Global well-posedness of the Benjamin-Ono equation in low-regularity spaces, arXiv: math.AP/0508632. MR2291918 1123.35055 10.1090/S0894-0347-06-00551-0
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 1044.35072 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 1044.35072
C.E. Kenig, G. Ponce and L. Vega: A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9 (1996), 573--603. MR1329387 0848.35114 10.1090/S0894-0347-96-00200-7 0894-0347%28199604%299%3A2%3C573%3AABEWAT%3E2.0.CO%3B2-Q C.E. Kenig, G. Ponce and L. Vega: A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9 (1996), 573--603. MR1329387 0848.35114 10.1090/S0894-0347-96-00200-7 0894-0347%28199604%299%3A2%3C573%3AABEWAT%3E2.0.CO%3B2-Q
N. Kita and J. Segata: Time local well-posedness for the Benjamin-Ono equation with large initial data, Publ. Res. Inst. Math. Sci. 42 (2006), 143--171. MR2215439 10.2977/prims/1166642062 N. Kita and J. Segata: Time local well-posedness for the Benjamin-Ono equation with large initial data, Publ. Res. Inst. Math. Sci. 42 (2006), 143--171. MR2215439 10.2977/prims/1166642062
H. Koch and N. Tzvetkov: Nonlinear wave interactions for the Benjamin-Ono equation, Internat. Math. Res. Not. 2005 (2005), 1833--1847. MR2172940 1156.35460 10.1155/IMRN.2005.1833 H. Koch and N. Tzvetkov: Nonlinear wave interactions for the Benjamin-Ono equation, Internat. Math. Res. Not. 2005 (2005), 1833--1847. MR2172940 1156.35460 10.1155/IMRN.2005.1833
L. Molinet and F. Ribaud: The Cauchy problem for dissipative Korteweg-de Vries equations in Sobolev spaces of negative order, Indiana Univ. Math. J. 50 (2001), 1745--1776. MR1889080 1101.35370 10.1512/iumj.2001.50.2135 L. Molinet and F. Ribaud: The Cauchy problem for dissipative Korteweg-de Vries equations in Sobolev spaces of negative order, Indiana Univ. Math. J. 50 (2001), 1745--1776. MR1889080 1101.35370 10.1512/iumj.2001.50.2135
L. Molinet and F. Ribaud: The global Cauchy problem in Bourgain's-type spaces for a dispersive dissipative semilinear equation, SIAM J. Math. Anal. 33 (2002), 1269--1296. MR1920630 1011.35118 10.1137/S0036141000374634 L. Molinet and F. Ribaud: The global Cauchy problem in Bourgain's-type spaces for a dispersive dissipative semilinear equation, SIAM J. Math. Anal. 33 (2002), 1269--1296. MR1920630 1011.35118 10.1137/S0036141000374634
L. Molinet and F. Ribaud: On the low regularity of the Korteweg-de Vries-Burgers equation, Internat. Math. Res. Not. 2002 (2002), 1979--2005. MR1918236 1031.35126 10.1155/S1073792802112104 L. Molinet and F. Ribaud: On the low regularity of the Korteweg-de Vries-Burgers equation, Internat. Math. Res. Not. 2002 (2002), 1979--2005. MR1918236 1031.35126 10.1155/S1073792802112104
L. Molinet, J.-C. Saut and N. Tzvetkov: Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal. 33 (2001), 982--988. MR1885293 0999.35085 10.1137/S0036141001385307 L. Molinet, J.-C. Saut and N. Tzvetkov: Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal. 33 (2001), 982--988. MR1885293 0999.35085 10.1137/S0036141001385307
T. Tao: Global well-posedness of the Benjamin-Ono equation in $H^1(\mathbf{R})$, J. Hyperbolic Diff. Eqns. 1 (2004), 27--49. MR2052470 1055.35104 10.1142/S0219891604000032 T. Tao: Global well-posedness of the Benjamin-Ono equation in $H^1(\mathbf{R})$, J. Hyperbolic Diff. Eqns. 1 (2004), 27--49. MR2052470 1055.35104 10.1142/S0219891604000032