Well-Posedness and Stability Analysis of a Landscape Evolution Model

In this paper, we study a system of partial differential equations modeling the evolution of a landscape in order to describe the mechanisms of pattern formations. A ground surface is eroded by the flow of water over it, by either sedimentation or dilution. We consider a model, composed of three evo...

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Bibliographic Details
Published in:Journal of nonlinear science 2024-02, Vol.34 (1), Article 20
Main Authors: Binard, Julie, Degond, Pierre, Noble, Pascal
Format: Article
Language:English
Subjects:
Analysis
Analysis of PDEs
Classical Mechanics
Dilution
Economic Theory/Quantitative Economics/Mathematical Methods
Evolution
Gullies
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical models
Mathematics
Mathematics and Statistics
Partial differential equations
Sediments
Stability analysis
Theoretical
Well posed problems
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Summary:In this paper, we study a system of partial differential equations modeling the evolution of a landscape in order to describe the mechanisms of pattern formations. A ground surface is eroded by the flow of water over it, by either sedimentation or dilution. We consider a model, composed of three evolution equations: one on the elevation of the ground surface, one on the fluid height and one on the concentration of sediments in the fluid layer. We first establish the well-posedness of the system in short time and under the assumption that the initial fluid height does not vanish. Then, we focus on pattern formation in the case of a film flow over an inclined erodible plane. For that purpose, we carry out a spectral stability analysis of constant state solutions in order to determine instability conditions and identify a mechanism for pattern formations. These patterns, which are rills and gullies, are the starting point of the formation of rivers and valleys in landscapes. Finally, we carry out some numerical simulations of the full system in order to validate the spectral instability scenario, and determine the resulting patterns.
ISSN:0938-8974
1432-1467
DOI:10.1007/s00332-023-09997-9