June  2012, 5(3): 419-426. doi: 10.3934/dcdss.2012.5.419

On the instability of a nonlocal conservation law

1. 

Institut de Mathématiques et de Modélisation de Monptellier, Université Montpellier II, 34 095 Montpellier

Received  August 2010 Revised  January 2011 Published  October 2011

We are interested in a nonlocal conservation law which describes the morphodynamics of sand dunes sheared by a fluid flow, recently proposed by Andrew C. Fowler and studied by [1,2]. We prove that constant solutions of Fowler's equation are non-linearly unstable. We also illustrate this fact using a finite difference scheme.
Citation: Afaf Bouharguane. On the instability of a nonlocal conservation law. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 419-426. doi: 10.3934/dcdss.2012.5.419
References:
[1]

N. Alibaud, P. Azerad and D. Isèbe, A non-monotone nonlocal conservation law for dune morphodynamics, Differential and Integral Equations, 23 (2010), 155-188.

[2]

B. Alvarez-Samaniego and P. Azerad, Existence of travelling-wave and local well-posedness of the Fowler equation, Disc. Cont. Dyn. Syst. Ser. B, 12 (2009), 671-692. doi: 10.3934/dcdsb.2009.12.671.

[3]

P. Azerad, A. Bouharguane and J.-F. Crouzet, Simultaneous denoising and enhancement of signals by a fractal conservation law, Communications in Nonlinear Science and Numerical Simulation, 17(2) (2012), pp. 867-881. doi: 10.1016/j.cnsns.2011.07.001.

[4]

A. Bouharguane, Global existence of solutions to the Fowler equation in a neighbourhood of travelling-waves, to appear in International Journal of Differential Equations. Archived at http://arxiv.org/abs/1107.0152.

[5]

P. Azerad and A. Bouharguane, Finite difference approximations for a fractional diffusion/anti-diffusion equation, preprint: http://arxiv.org/abs/1104.4861.

[6]

A. De Bouard, Instability of stationary bubbles, SIAM J .Math. Anal., 26 (1995), 566-582. doi: 10.1137/S0036141092237029.

[7]

A. C. Fowler, Dunes and drumlins, in "Geomorphological Fluid Mechanics" (eds. A. Provenzale and N. Balmforth), 211, Springer-Verlag, Berlin, (2001), 430-454.

[8]

A. C. Fowler, Evolution equations for dunes and drumlins, Mathematics and Environment (Paris, 2002), Rev. R. Acad. de Cienc. Exactas Fis. Nat. Serie A. Mat., 96 (2002), 377-387.

[9]

A. C. Fowler, "Mathematics and the Environment," lecture notes. Available from: http://www2.maths.ox.ac.uk/~fowler/courses/mathenvo.html.

[10]

K. K. J. Kouakou and P.-Y. Lagrée, Evolution of a model dune in a shear flow, Eur. J. Mech. B Fluids, 25 (2006), 348-359. doi: 10.1016/j.euromechflu.2005.09.002.

[11]

P.-Y. Lagrée and K. Kouakou, Stability of an erodible bed in various shear flows, European Physical Journal B - Condensed Matter, 47 (2005), 115-125.

[12]

I. Podlubny, "An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications," Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, 1999.

show all references

References:
[1]

N. Alibaud, P. Azerad and D. Isèbe, A non-monotone nonlocal conservation law for dune morphodynamics, Differential and Integral Equations, 23 (2010), 155-188.

[2]

B. Alvarez-Samaniego and P. Azerad, Existence of travelling-wave and local well-posedness of the Fowler equation, Disc. Cont. Dyn. Syst. Ser. B, 12 (2009), 671-692. doi: 10.3934/dcdsb.2009.12.671.

[3]

P. Azerad, A. Bouharguane and J.-F. Crouzet, Simultaneous denoising and enhancement of signals by a fractal conservation law, Communications in Nonlinear Science and Numerical Simulation, 17(2) (2012), pp. 867-881. doi: 10.1016/j.cnsns.2011.07.001.

[4]

A. Bouharguane, Global existence of solutions to the Fowler equation in a neighbourhood of travelling-waves, to appear in International Journal of Differential Equations. Archived at http://arxiv.org/abs/1107.0152.

[5]

P. Azerad and A. Bouharguane, Finite difference approximations for a fractional diffusion/anti-diffusion equation, preprint: http://arxiv.org/abs/1104.4861.

[6]

A. De Bouard, Instability of stationary bubbles, SIAM J .Math. Anal., 26 (1995), 566-582. doi: 10.1137/S0036141092237029.

[7]

A. C. Fowler, Dunes and drumlins, in "Geomorphological Fluid Mechanics" (eds. A. Provenzale and N. Balmforth), 211, Springer-Verlag, Berlin, (2001), 430-454.

[8]

A. C. Fowler, Evolution equations for dunes and drumlins, Mathematics and Environment (Paris, 2002), Rev. R. Acad. de Cienc. Exactas Fis. Nat. Serie A. Mat., 96 (2002), 377-387.

[9]

A. C. Fowler, "Mathematics and the Environment," lecture notes. Available from: http://www2.maths.ox.ac.uk/~fowler/courses/mathenvo.html.

[10]

K. K. J. Kouakou and P.-Y. Lagrée, Evolution of a model dune in a shear flow, Eur. J. Mech. B Fluids, 25 (2006), 348-359. doi: 10.1016/j.euromechflu.2005.09.002.

[11]

P.-Y. Lagrée and K. Kouakou, Stability of an erodible bed in various shear flows, European Physical Journal B - Condensed Matter, 47 (2005), 115-125.

[12]

I. Podlubny, "An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications," Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, 1999.

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