February  2011, 4(1): 15-50. doi: 10.3934/dcdss.2011.4.15

Large-time asymptotics of the generalized Benjamin-Ono-Burgers equation

1. 

Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607, United States

2. 

Department of Mathematics, New York Institute of Technology, 1855 Broadway, New York, NY 10023, United States

Received  March 2009 Revised  November 2009 Published  October 2010

In this paper, attention is given to pure initial-value problems for the generalized Benjamin-Ono-Burgers (BOB) equation

$ u_t + u_x +(P(u))_{x}-\nu $uxx$ - H$uxx=0,

where $H$ is the Hilbert transform, $\nu > 0$ and $P\ : R \to R$ is a smooth function. We study questions of global existence and of the large-time asymptotics of solutions of the initial-value problem. If $\Lambda (s)$ is defined by $\Lambda '(s) = P(s), \Lambda (0) = 0,$ then solutions of the initial-value problem corresponding to reasonable initial data maintain their integrity for all $t \geq 0$ provided that $\Lambda$ and $P'$ satisfy certain growth restrictions. In case a solution corresponding to initial data that is square integrable is global, it is straightforward to conclude it must decay to zero when $t$ becomes unboundedly large. We investigate the detailed asymptotics of this decay. For generic initial data and weak nonlinearity, it is demonstrated that the final decay is that of the linearized equation in which $P \equiv 0.$ However, if the initial data is drawn from more restricted classes that involve something akin to a condition of zero mean, then enhanced decay rates are established. These results extend the earlier work of Dix who considered the case where $P$ is a quadratic polynomial.

Citation: Jerry L. Bona, Laihan Luo. Large-time asymptotics of the generalized Benjamin-Ono-Burgers equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 15-50. doi: 10.3934/dcdss.2011.4.15
References:
[1]

M. J. Ablowitz and A. S. Fokas, The inverse scattering transform for the Benjamin-Ono equation, A pivot to multidimensional problems,, Stud. Appl. Math., 68 (1983), 1. Google Scholar

[2]

L. Abdelouhab, J. L. Bona, M. Felland and J.-C. Saut, Nonlocal models for nonlinear dispersive waves,, Phys. D, 40 (1989), 360. doi: doi:10.1016/0167-2789(89)90050-X. Google Scholar

[3]

C. J. Amick, J. L. Bona and M. E. Schonbek, Decay of solutions of some nonlinear wave equations,, J. Differential Equations, 81 (1989), 1. doi: doi:10.1016/0022-0396(89)90176-9. Google Scholar

[4]

T. B. Benjamin, Internal waves of permanent form in fluids of great depth,, J. Fluid Mech., 29 (1967), 559. doi: doi:10.1017/S002211206700103X. Google Scholar

[5]

T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems,, Philos. Trans. Royal Soc. London Ser. A, 272 (1972), 47. doi: doi:10.1098/rsta.1972.0032. Google Scholar

[6]

P. Biler, Asymptotic behavior in time of some equations generalizing Korteweg-de Vries-Burgers,, Bull. Polish Acad. Sci., 32 (1984), 275. Google Scholar

[7]

P. Biler, Large-time behavior of periodic solutions of two dispersive equations of Korteweg-de Vries-Burgers type,, Bull. Polish Acad. Sci., 32 (1984), 401. Google Scholar

[8]

J. L. Bona, On solitary waves and their role in the evolution of long waves,, in, (1981), 183. Google Scholar

[9]

J. L. Bona, V. A. Dougalis, O. A. Karakashian and W. R. McKinney, Computations of blow-up and decay for periodic solutions of the generalized Korteweg-de Vries-Burgers equation,, Applied Numerical Math., 10 (1992), 335. doi: doi:10.1016/0168-9274(92)90049-J. Google Scholar

[10]

J. L. Bona, V. A. Dougalis, O. A. Karakashian and W. R. McKinney, Conservative high-order numerical schemes for the generalized Korteweg-de Vries equation,, Philos. Trans. Royal Soc. London, 351 (1995), 107. Google Scholar

[11]

J. L. Bona and H. Kalisch, Singularity formation in the generalized Benjamin-Ono equation,, Discrete Cont. Dyn. Systems Series A, 11 (2004), 27. Google Scholar

[12]

J. L. Bona and L. Luo, Decay of the solutions to nonlinear, dispersive wave equations,, Diff. & Int. Equations, 6 (1993), 961. Google Scholar

[13]

J. L. Bona and L. Luo, More results on the decay of solutions to nonlinear, dispersive wave equations,, Discrete and Continuous Dynamical Systems, 1 (1995), 151. doi: doi:10.3934/dcds.1995.1.151. Google Scholar

[14]

J. L. Bona, W. G. Pritchard and L. R. Scott, An evaluation of a model equation for water waves,, Philos. Trans. Royal Soc. London Ser. A, 302 (1981), 457. doi: doi:10.1098/rsta.1981.0178. Google Scholar

[15]

J. L. Bona, S. Rajopadhye and M. E. Schonbek, Models for propagation of bores I. Two-dimensional theory,, Differential & Int. Equations, 7 (1994), 699. Google Scholar

[16]

J. L. Bona and M. E. Schonbek, Travelling-wave solutions of the Korteweg-de Vries-Burgers equation,, Proc. Royal Soc. Edinburgh A, 101 (1985), 207. Google Scholar

[17]

J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation,, Philos. Trans. Royal Soc. London Ser A, 278 (1975), 555. doi: doi:10.1098/rsta.1975.0035. Google Scholar

[18]

H. Brezis and T. Gallouët, Nonlinear Schrödinger evolution equations,, Nonlinear Anal., 4 (1980), 677. doi: doi:10.1016/0362-546X(80)90068-1. Google Scholar

[19]

H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities,, Comm. P.D.E, 5 (1980), 773. doi: doi:10.1080/03605308008820154. Google Scholar

[20]

D. Derks, "Coherent Structures in the Dynamics of Perturbed Hamiltonian Systems,", Ph.D. Thesis, (1992). Google Scholar

[21]

D. B. Dix, The dissipation of nonlinear dispersive waves: The case of asymptotically weak nonlinearity,, Comm. P.D.E, 17 (1992), 1665. doi: doi:10.1080/03605309208820899. Google Scholar

[22]

D. B. Dix, Temporal asymptotic behavior of solutions of the Benjamin-Ono -Burgers equation,, J. Differential Equations, 90 (1991), 238. doi: doi:10.1016/0022-0396(91)90148-3. Google Scholar

[23]

P. M. Edwin and B. Roberts, The Benjamin-Ono-Burgers equation: An application in solar physics,, Wave Motion, 8 (1986), 151. doi: doi:10.1016/0165-2125(86)90021-1. Google Scholar

[24]

J. Ginibre and G. Velo, Smoothing properties and existence of solutions for the generalized Benjamin-Ono equation,, J. Differential Equations, 93 (1991), 150. doi: doi:10.1016/0022-0396(91)90025-5. Google Scholar

[25]

N. Hayashi and P. I. Naumkin, Large time asymptotics of solutions to the generalized Benjamin-Ono equation,, Trans. American Math. Soc., 351 (1999), 109. doi: doi:10.1090/S0002-9947-99-02285-0. Google Scholar

[26]

R. J. Iório, On the Cauchy problem for the Benjamin-Ono equation,, Comm. P.D.E., 11 (1986), 1031. doi: doi:10.1080/03605308608820456. Google Scholar

[27]

R. S. Johnson, A nonlinear equation incorporating damping and dispersion,, J. Fluid Mech., 42 (1970), 49. doi: doi:10.1017/S0022112070001064. Google Scholar

[28]

R. S. Johnson, Shallow water waves on a viscous fluid--The undular bore,, Phys. of Fluids, 15 (1972), 1693. doi: doi:10.1063/1.1693764. Google Scholar

[29]

H. Kalisch and J. L. Bona, Models for internal waves in deep water,, Discrete Cont. Dyn. Systems, 6 (2000), 1. doi: doi:10.3934/dcds.2000.6.1. Google Scholar

[30]

T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation,, Studies in Applied Math., 8 (1983), 93. Google Scholar

[31]

C. E. Kenig, G. Ponce and L. Vega, On the generalized Benjamin-Ono equation,, Trans. Amer. Math. Soc., 342 (1994), 155. doi: doi:10.2307/2154688. Google Scholar

[32]

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves,, Philos. Mag., 39 (1895), 422. Google Scholar

[33]

J. J. Mahony and W. G. Pritchard, Wave reflexion from beaches,, J. Fluid Mech., 101 (1980), 809. doi: doi:10.1017/S0022112080001942. Google Scholar

[34]

Y. Mammeri, On the decay in time of solutions of some generalized regularized long wave equations,, Comm. Pure Appl. Anal., 7 (2008), 513. doi: doi:10.3934/cpaa.2008.7.513. Google Scholar

[35]

C. C. Mei and L. F. Liu, The damping of surface gravity waves in a bounded liquid,, J. Fluid Mech., 59 (1973), 239. doi: doi:10.1017/S0022112073001540. Google Scholar

[36]

J. W. Miles, Surface-wave damping in a closed basin,, Proc. Royal Soc. London Ser. A, 297 (1967), 459. doi: doi:10.1098/rspa.1967.0081. Google Scholar

[37]

P. I. Naumkin and I. A. Shishmarev, "Nonlinear Nonlocal Equations in the Theory of Waves,", in, 133 (1994). Google Scholar

[38]

H. Ono, Algebraic solitary waves in stratified fluids,, J. Phys. Soc. Japan, 39 (1975), 1082. doi: doi:10.1143/JPSJ.39.1082. Google Scholar

[39]

E. Ott and R. N. Sudan, Damping of solitary waves,, Phys. of Fluids, 13 (1970), 1432. doi: doi:10.1063/1.1693097. Google Scholar

[40]

T. Ozawa, On critical cases of Sobolev's inequality,, J. Functional Anal., 127 (1995), 259. doi: doi:10.1006/jfan.1995.1012. Google Scholar

[41]

D. H. Peregrine, Calculations of the development of an undular bore,, J. Fluid Mech., 25 (1966), 321. doi: doi:10.1017/S0022112066001678. Google Scholar

[42]

E. Schechter, "Well-Behaved Evolutions and the Trotter Product Formulas,", Ph.D. Thesis, (1978). Google Scholar

[43]

M. M. Tom, Smoothing properties of some weak solutions of the Benjamin-Ono equation,, Differential & Int. Equations, 3 (1990), 683. Google Scholar

[44]

S. Vento, Well-posedness for the generalized Benjamin-Ono equation with arbitrary large initial data in the critical space,, Inter. Math. Res. Notices, (2009). doi: doi:10.1093/imrn/rnp133. Google Scholar

[45]

L. Zhang, Decay of solutions to generalized Benjamin-Bona-Mahony-Burgers equations in n-space dimensions,, Nonlinear Analysis, 25 (1995), 1343. doi: doi:10.1016/0362-546X(94)00252-D. Google Scholar

[46]

L. Zhang, Initial value problem for a nonlinear parabolic equation with singular integral-differential term,, ACTA Math. Appl. Sinica, 8 (1992), 367. doi: doi:10.1007/BF02006745. Google Scholar

[47]

Y. Zhou and B. Guo, Initial value problems for a nonlinear singular integral-differential equation of deep water,, Lecture notes in Mathematics, 1306 (1986), 278. Google Scholar

show all references

References:
[1]

M. J. Ablowitz and A. S. Fokas, The inverse scattering transform for the Benjamin-Ono equation, A pivot to multidimensional problems,, Stud. Appl. Math., 68 (1983), 1. Google Scholar

[2]

L. Abdelouhab, J. L. Bona, M. Felland and J.-C. Saut, Nonlocal models for nonlinear dispersive waves,, Phys. D, 40 (1989), 360. doi: doi:10.1016/0167-2789(89)90050-X. Google Scholar

[3]

C. J. Amick, J. L. Bona and M. E. Schonbek, Decay of solutions of some nonlinear wave equations,, J. Differential Equations, 81 (1989), 1. doi: doi:10.1016/0022-0396(89)90176-9. Google Scholar

[4]

T. B. Benjamin, Internal waves of permanent form in fluids of great depth,, J. Fluid Mech., 29 (1967), 559. doi: doi:10.1017/S002211206700103X. Google Scholar

[5]

T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems,, Philos. Trans. Royal Soc. London Ser. A, 272 (1972), 47. doi: doi:10.1098/rsta.1972.0032. Google Scholar

[6]

P. Biler, Asymptotic behavior in time of some equations generalizing Korteweg-de Vries-Burgers,, Bull. Polish Acad. Sci., 32 (1984), 275. Google Scholar

[7]

P. Biler, Large-time behavior of periodic solutions of two dispersive equations of Korteweg-de Vries-Burgers type,, Bull. Polish Acad. Sci., 32 (1984), 401. Google Scholar

[8]

J. L. Bona, On solitary waves and their role in the evolution of long waves,, in, (1981), 183. Google Scholar

[9]

J. L. Bona, V. A. Dougalis, O. A. Karakashian and W. R. McKinney, Computations of blow-up and decay for periodic solutions of the generalized Korteweg-de Vries-Burgers equation,, Applied Numerical Math., 10 (1992), 335. doi: doi:10.1016/0168-9274(92)90049-J. Google Scholar

[10]

J. L. Bona, V. A. Dougalis, O. A. Karakashian and W. R. McKinney, Conservative high-order numerical schemes for the generalized Korteweg-de Vries equation,, Philos. Trans. Royal Soc. London, 351 (1995), 107. Google Scholar

[11]

J. L. Bona and H. Kalisch, Singularity formation in the generalized Benjamin-Ono equation,, Discrete Cont. Dyn. Systems Series A, 11 (2004), 27. Google Scholar

[12]

J. L. Bona and L. Luo, Decay of the solutions to nonlinear, dispersive wave equations,, Diff. & Int. Equations, 6 (1993), 961. Google Scholar

[13]

J. L. Bona and L. Luo, More results on the decay of solutions to nonlinear, dispersive wave equations,, Discrete and Continuous Dynamical Systems, 1 (1995), 151. doi: doi:10.3934/dcds.1995.1.151. Google Scholar

[14]

J. L. Bona, W. G. Pritchard and L. R. Scott, An evaluation of a model equation for water waves,, Philos. Trans. Royal Soc. London Ser. A, 302 (1981), 457. doi: doi:10.1098/rsta.1981.0178. Google Scholar

[15]

J. L. Bona, S. Rajopadhye and M. E. Schonbek, Models for propagation of bores I. Two-dimensional theory,, Differential & Int. Equations, 7 (1994), 699. Google Scholar

[16]

J. L. Bona and M. E. Schonbek, Travelling-wave solutions of the Korteweg-de Vries-Burgers equation,, Proc. Royal Soc. Edinburgh A, 101 (1985), 207. Google Scholar

[17]

J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation,, Philos. Trans. Royal Soc. London Ser A, 278 (1975), 555. doi: doi:10.1098/rsta.1975.0035. Google Scholar

[18]

H. Brezis and T. Gallouët, Nonlinear Schrödinger evolution equations,, Nonlinear Anal., 4 (1980), 677. doi: doi:10.1016/0362-546X(80)90068-1. Google Scholar

[19]

H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities,, Comm. P.D.E, 5 (1980), 773. doi: doi:10.1080/03605308008820154. Google Scholar

[20]

D. Derks, "Coherent Structures in the Dynamics of Perturbed Hamiltonian Systems,", Ph.D. Thesis, (1992). Google Scholar

[21]

D. B. Dix, The dissipation of nonlinear dispersive waves: The case of asymptotically weak nonlinearity,, Comm. P.D.E, 17 (1992), 1665. doi: doi:10.1080/03605309208820899. Google Scholar

[22]

D. B. Dix, Temporal asymptotic behavior of solutions of the Benjamin-Ono -Burgers equation,, J. Differential Equations, 90 (1991), 238. doi: doi:10.1016/0022-0396(91)90148-3. Google Scholar

[23]

P. M. Edwin and B. Roberts, The Benjamin-Ono-Burgers equation: An application in solar physics,, Wave Motion, 8 (1986), 151. doi: doi:10.1016/0165-2125(86)90021-1. Google Scholar

[24]

J. Ginibre and G. Velo, Smoothing properties and existence of solutions for the generalized Benjamin-Ono equation,, J. Differential Equations, 93 (1991), 150. doi: doi:10.1016/0022-0396(91)90025-5. Google Scholar

[25]

N. Hayashi and P. I. Naumkin, Large time asymptotics of solutions to the generalized Benjamin-Ono equation,, Trans. American Math. Soc., 351 (1999), 109. doi: doi:10.1090/S0002-9947-99-02285-0. Google Scholar

[26]

R. J. Iório, On the Cauchy problem for the Benjamin-Ono equation,, Comm. P.D.E., 11 (1986), 1031. doi: doi:10.1080/03605308608820456. Google Scholar

[27]

R. S. Johnson, A nonlinear equation incorporating damping and dispersion,, J. Fluid Mech., 42 (1970), 49. doi: doi:10.1017/S0022112070001064. Google Scholar

[28]

R. S. Johnson, Shallow water waves on a viscous fluid--The undular bore,, Phys. of Fluids, 15 (1972), 1693. doi: doi:10.1063/1.1693764. Google Scholar

[29]

H. Kalisch and J. L. Bona, Models for internal waves in deep water,, Discrete Cont. Dyn. Systems, 6 (2000), 1. doi: doi:10.3934/dcds.2000.6.1. Google Scholar

[30]

T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation,, Studies in Applied Math., 8 (1983), 93. Google Scholar

[31]

C. E. Kenig, G. Ponce and L. Vega, On the generalized Benjamin-Ono equation,, Trans. Amer. Math. Soc., 342 (1994), 155. doi: doi:10.2307/2154688. Google Scholar

[32]

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves,, Philos. Mag., 39 (1895), 422. Google Scholar

[33]

J. J. Mahony and W. G. Pritchard, Wave reflexion from beaches,, J. Fluid Mech., 101 (1980), 809. doi: doi:10.1017/S0022112080001942. Google Scholar

[34]

Y. Mammeri, On the decay in time of solutions of some generalized regularized long wave equations,, Comm. Pure Appl. Anal., 7 (2008), 513. doi: doi:10.3934/cpaa.2008.7.513. Google Scholar

[35]

C. C. Mei and L. F. Liu, The damping of surface gravity waves in a bounded liquid,, J. Fluid Mech., 59 (1973), 239. doi: doi:10.1017/S0022112073001540. Google Scholar

[36]

J. W. Miles, Surface-wave damping in a closed basin,, Proc. Royal Soc. London Ser. A, 297 (1967), 459. doi: doi:10.1098/rspa.1967.0081. Google Scholar

[37]

P. I. Naumkin and I. A. Shishmarev, "Nonlinear Nonlocal Equations in the Theory of Waves,", in, 133 (1994). Google Scholar

[38]

H. Ono, Algebraic solitary waves in stratified fluids,, J. Phys. Soc. Japan, 39 (1975), 1082. doi: doi:10.1143/JPSJ.39.1082. Google Scholar

[39]

E. Ott and R. N. Sudan, Damping of solitary waves,, Phys. of Fluids, 13 (1970), 1432. doi: doi:10.1063/1.1693097. Google Scholar

[40]

T. Ozawa, On critical cases of Sobolev's inequality,, J. Functional Anal., 127 (1995), 259. doi: doi:10.1006/jfan.1995.1012. Google Scholar

[41]

D. H. Peregrine, Calculations of the development of an undular bore,, J. Fluid Mech., 25 (1966), 321. doi: doi:10.1017/S0022112066001678. Google Scholar

[42]

E. Schechter, "Well-Behaved Evolutions and the Trotter Product Formulas,", Ph.D. Thesis, (1978). Google Scholar

[43]

M. M. Tom, Smoothing properties of some weak solutions of the Benjamin-Ono equation,, Differential & Int. Equations, 3 (1990), 683. Google Scholar

[44]

S. Vento, Well-posedness for the generalized Benjamin-Ono equation with arbitrary large initial data in the critical space,, Inter. Math. Res. Notices, (2009). doi: doi:10.1093/imrn/rnp133. Google Scholar

[45]

L. Zhang, Decay of solutions to generalized Benjamin-Bona-Mahony-Burgers equations in n-space dimensions,, Nonlinear Analysis, 25 (1995), 1343. doi: doi:10.1016/0362-546X(94)00252-D. Google Scholar

[46]

L. Zhang, Initial value problem for a nonlinear parabolic equation with singular integral-differential term,, ACTA Math. Appl. Sinica, 8 (1992), 367. doi: doi:10.1007/BF02006745. Google Scholar

[47]

Y. Zhou and B. Guo, Initial value problems for a nonlinear singular integral-differential equation of deep water,, Lecture notes in Mathematics, 1306 (1986), 278. Google Scholar

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