Differential and Integral Equations

A non-monotone nonlocal conservation law for dune morphodynamics

Nathaël Alibaud, Pascal Azerad, and Damien Isèbe

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We investigate a non-local, non-linear conservation law, first introduced by A.C. Fowler to describe morphodynamics of dunes, see [6, 7]. A remarkable feature is the violation of the maximum principle, which allows for erosion phenomenon. We prove well posedness for initial data in $L^2$ and give an explicit counterexample for the maximum principle. We also provide numerical simulations corroborating our theoretical results.

Article information

Differential Integral Equations, Volume 23, Number 1/2 (2010), 155-188.

First available in Project Euclid: 20 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47J35: Nonlinear evolution equations [See also 34G20, 35K90, 35L90, 35Qxx, 35R20, 37Kxx, 37Lxx, 47H20, 58D25] 47G20: Integro-differential operators [See also 34K30, 35R09, 35R10, 45Jxx, 45Kxx] 35L65: Conservation laws 35B50: Maximum principles 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20] 65M06: Finite difference methods


Alibaud, Nathaël; Azerad, Pascal; Isèbe, Damien. A non-monotone nonlocal conservation law for dune morphodynamics. Differential Integral Equations 23 (2010), no. 1/2, 155--188. https://projecteuclid.org/euclid.die/1356019392

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