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On the instability of a nonlocal conservation law

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  • 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.
    Mathematics Subject Classification: 35L65, 45K05, 35G25, 35C07, 35B35, 65M06.

    Citation:

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  • [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. BouharguaneGlobal 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. BouharguaneFinite 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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