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Mathematical Analysis of a Model of River Channel Formation

机译：河道形成模型的数学分析

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The study of overland flow of water over an erodible sediment leads to a coupled model describing the evolution of the topographic elevation and the depth of the overland water film. The spatially uniform solution of this model is unstable, and this instability corresponds to the formation of rills, which in reality then grow and coalesce to form large-scale river channels. In this paper we consider the deduction and mathematical analysis of a deterministic model describing river channel formation and the evolution of its depth. The model involves a degenerate nonlinear parabolic equation (satisfied on the interior of the support of the solution) with a super-linear source term and a prescribed constant mass. We propose here a global formulation of the problem (formulated in the whole space, beyond the support of the solution) which allows us to show the existence of a solution and leads to a suitable numerical scheme for its approximation. A particular novelty of the model is that the evolving channel self-determines its own width, without the need to pose any extra conditions at the channel margin.
机译：对可蚀性沉积物上的陆上水流的研究导致了一个耦合模型，该模型描述了地形高程的演变和陆上水膜的深度。该模型在空间上的统一解是不稳定的，这种不稳定性对应于小溪的形成，实际上，小溪随后生长并合并形成大型河道。在本文中，我们考虑了确定性模型的推论和数学分析，该模型描述了河道的形成及其深度的演变。该模型涉及具有超线性源项和规定常数的退化的非线性抛物线方程（在解的载体内部满足）。我们在这里提出问题的整体表述（在整个空间中进行表述，超出了解决方案的支持范围），这使我们能够显示解决方案的存在并为其找到合适的数值方案。该模型的一个特别新颖之处在于，不断演变的信道可以自行确定其自身的宽度，而无需在信道边界处设置任何额外条件。

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