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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[8]

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

[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.

[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.

[11]

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

[12]

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

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[20]

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

[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.

[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.

[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.

[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.

[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.

[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.

[27]

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

[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.

[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.

[30]

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

[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.

[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.

[33]

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

[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.

[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.

[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.

[37]

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

[38]

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

[39]

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

[40]

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

[41]

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

[42]

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

[43]

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

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[8]

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

[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.

[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.

[11]

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

[12]

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

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[20]

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

[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.

[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.

[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.

[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.

[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.

[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.

[27]

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

[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.

[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.

[30]

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

[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.

[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.

[33]

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

[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.

[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.

[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.

[37]

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

[38]

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

[39]

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

[40]

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

[41]

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

[42]

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

[43]

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

[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.

[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.

[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.

[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.

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