# American Institute of Mathematical Sciences

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

 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

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.
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.

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##### 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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