Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative

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November 2014, 19(9): 2837-2863. doi: 10.3934/dcdsb.2014.19.2837

Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative

Imen Manoubi ^1,

1.	Unité de recherche : Multifractales et Ondelettes, Faculté des Sciences de Monastir, Av. de l'environnement 5019 Monastir, Tunisia

Received June 2013 Revised March 2014 Published September 2014

Abstract
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In this paper, we study the water wave model with a nonlocal viscous term \begin{equation*} u_t + u_x + \beta u_{x x x} + \frac{\sqrt \nu}{\sqrt \pi}\frac{\partial}{\partial t} \int_0^t\frac{u(s)}{\sqrt{t-s}} ds + u u_x = v u_{xx}, \end{equation*} where $\frac{1}{\sqrt \pi}\frac{\partial}{\partial t} \int_0^t\frac{u(s)}{\sqrt{t-s}} ds $ is the Riemann-Liouville half derivative. We prove the well-posedness of the equation and we investigate theoretically and numerically the asymptotical behavior of the solutions. Also, we compare our theoretical and numerical results with those given in [4] for a similar equation.

Keywords: decay rate, nonlocal viscous model, Waterwaves, Gear scheme., fractional derivatives.

Mathematics Subject Classification: Primary: 35Q35; Secondary: 35Q5.

Citation: Imen Manoubi. Theoretical and numerical analysis of the decay rate of solutions to a water wave model with a nonlocal viscous dispersive term with Riemann-Liouville half derivative. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2837-2863. doi: 10.3934/dcdsb.2014.19.2837

References:

[1]	C. J. Amick, J. L. Bona and M. E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differential Equations, 81 (1989), 1-49. doi: 10.1016/0022-0396(89)90176-9.
[2]	J. L. Bona, M. Chen and J.-C Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media I: Derivation and the linear theory, J. Nonlinear Sci., 12 (2002), 283-318. doi: 10.1007/s00332-002-0466-4.
[3]	J. L. Bona, F. Demengel and K. Promislow, Fourier splitting and dissipation of nonlinear dispersive waves, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 477-502. doi: 10.1017/S0308210500021478.
[4]	M. Chen, S. Dumont, L. Dupaigne and O. Goubet, Decay of solutions to a water wave model with a nonlocal viscous dispersive term, Discrete Contin. Dyn. Syst., 27 (2010), 1473-1492. doi: 10.3934/dcds.2010.27.1473.
[5]	M. Chen and O. Goubet, Long-time asymptotic behavior of 2D dissipative boussinesq system, Discrete Contin. Dyn. Syst., 17 (2007), 509-528. doi: 10.3934/dcds.2007.17.509.
[6]	S. Dumont and J.-B Duval, Numerical investigation of asymptotical properties of solutions to models for waterwaves with non local viscosity, International Journal of Numerical Analysis and Modeling, 10 (2013), 333-349.
[7]	D. Dutykh, Viscous-potential free-surface flows and long wave modelling, Eur. J. Mech. B Fluids, 28 (2009), 430-443. doi: 10.1016/j.euromechflu.2008.11.003.
[8]	D. Dutykh and F. Dias, Viscous potential free surface flows in a fluid layer of finite depth, C.R.A.S, Série I, 345 (2007), 113-118. doi: 10.1016/j.crma.2007.06.007.
[9]	, tm., ().
[10]	A. C Galucio, J.-F Deü and F. Dubois, The $G^\alpha$-scheme for approximation of fractional derivatives: Application to the dynamics of dissipative systems, J. Vib. Control, 14 (2008), 1597-1605. doi: 10.1177/1077546307087427.
[11]	A. C Galucio, J.-F Deü, S. Mengué and F. Dubois, An adaptation of the Gear scheme for fractional derivatives, Comput. Methods Appl. Mech. Engrg., 195 (2006), 6073-6085. doi: 10.1016/j.cma.2005.10.013.
[12]	O. Goubet and G. Warnault, Decay of solutions to a linear viscous asymptotic model for waterwaves, Chinese Ann. Math. Ser. B, 31 (2010), 841-854. doi: 10.1007/s11401-010-0615-2.
[13]	N. Hayashi, E. I. Kaikina, P. I. Naumkin and I. A. Shishmarev, Asymptotics for Dissipative Nonlinear Equations, Lecture Notes in Mathematics, 1884, Springer-Verlag, Berlin, 2006. doi: 10.1007/b133345.
[14]	T. Kakutani and M. Matsuuchi, Effect of viscosity on long gravity waves, J. Phys. Soc. Japan, 39 (1975), 237-246. doi: 10.1143/JPSJ.39.237.
[15]	P. Liu and A. Orfila, Viscous effects on transient long wave propagation, J. Fluid Mech., 520 (2004), 83-92. doi: 10.1017/S0022112004001806.

show all references

References:

[1]	C. J. Amick, J. L. Bona and M. E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differential Equations, 81 (1989), 1-49. doi: 10.1016/0022-0396(89)90176-9.
[2]	J. L. Bona, M. Chen and J.-C Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media I: Derivation and the linear theory, J. Nonlinear Sci., 12 (2002), 283-318. doi: 10.1007/s00332-002-0466-4.
[3]	J. L. Bona, F. Demengel and K. Promislow, Fourier splitting and dissipation of nonlinear dispersive waves, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 477-502. doi: 10.1017/S0308210500021478.
[4]	M. Chen, S. Dumont, L. Dupaigne and O. Goubet, Decay of solutions to a water wave model with a nonlocal viscous dispersive term, Discrete Contin. Dyn. Syst., 27 (2010), 1473-1492. doi: 10.3934/dcds.2010.27.1473.
[5]	M. Chen and O. Goubet, Long-time asymptotic behavior of 2D dissipative boussinesq system, Discrete Contin. Dyn. Syst., 17 (2007), 509-528. doi: 10.3934/dcds.2007.17.509.
[6]	S. Dumont and J.-B Duval, Numerical investigation of asymptotical properties of solutions to models for waterwaves with non local viscosity, International Journal of Numerical Analysis and Modeling, 10 (2013), 333-349.
[7]	D. Dutykh, Viscous-potential free-surface flows and long wave modelling, Eur. J. Mech. B Fluids, 28 (2009), 430-443. doi: 10.1016/j.euromechflu.2008.11.003.
[8]	D. Dutykh and F. Dias, Viscous potential free surface flows in a fluid layer of finite depth, C.R.A.S, Série I, 345 (2007), 113-118. doi: 10.1016/j.crma.2007.06.007.
[9]	, tm., ().
[10]	A. C Galucio, J.-F Deü and F. Dubois, The $G^\alpha$-scheme for approximation of fractional derivatives: Application to the dynamics of dissipative systems, J. Vib. Control, 14 (2008), 1597-1605. doi: 10.1177/1077546307087427.
[11]	A. C Galucio, J.-F Deü, S. Mengué and F. Dubois, An adaptation of the Gear scheme for fractional derivatives, Comput. Methods Appl. Mech. Engrg., 195 (2006), 6073-6085. doi: 10.1016/j.cma.2005.10.013.
[12]	O. Goubet and G. Warnault, Decay of solutions to a linear viscous asymptotic model for waterwaves, Chinese Ann. Math. Ser. B, 31 (2010), 841-854. doi: 10.1007/s11401-010-0615-2.
[13]	N. Hayashi, E. I. Kaikina, P. I. Naumkin and I. A. Shishmarev, Asymptotics for Dissipative Nonlinear Equations, Lecture Notes in Mathematics, 1884, Springer-Verlag, Berlin, 2006. doi: 10.1007/b133345.
[14]	T. Kakutani and M. Matsuuchi, Effect of viscosity on long gravity waves, J. Phys. Soc. Japan, 39 (1975), 237-246. doi: 10.1143/JPSJ.39.237.
[15]	P. Liu and A. Orfila, Viscous effects on transient long wave propagation, J. Fluid Mech., 520 (2004), 83-92. doi: 10.1017/S0022112004001806.

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