Une équation d’ évolution de surface / A surface evolution equation

Physical context : We are interested in the mathematical study of the evolution of surfaces which builds up when some granular matter is dropped on non flat region. If a matter grain falls onto a position where the slope of the pile is steep, it will slide down, and eventually cause grains of the existing pile to slide down as well. Some connections with the Monge-Kantorovich problem of optimal transportation will be studied also.
 
 
Mathematical problem : Since the work of Prigohzin (cf. 5]), this has been well known (see also [1], [2] and the references therein) that on flat region the evolution of the surface of a granular matter when the angle of stability is equal to π/4 can be described by an evolution equation governed by a subdifferential of the indicator function of K  the set of function with gradient less or equal to tan(π/4)=1.  
 
Such a situation may be visualized as a point u(t) moving inside K and being pushed by the boundary of this convex set K when the contact is established. The theoretical and numerical study of this problem is well understood by now, as well as its connection with the Monge-Kantorovich problem of optimal transportation.  
 
1. In the case of non flat region, we need to introduce a function g : Ω → IR+ describing the eight of the support surface. Then, since the free surface is never bellow the support surface, the set K needs to include the condition that u is biger than g.  If g ∈ K, then the standard theory of first order differential inclusions with the theory of the maximal monotone operators enables to handle the problem by taking g as an initial data.  But, if the gradient of g is biger than 1, then the situation is completely different and our purpose in this project is to introduce and study the right theoretical and numerical way to handle this problem. The main difficulty here come from the loose of convexity in the set K.
2. In the previous application, we assume that the angle of stability of the material is equal to π/4. Now, for fluid material (like water) the angle of stability is equal to 0 and the previous  model may give some metaphoric description of how ”water” originating from ”rain-fall” flows along a landscape of varying height, forming ”lakes” and also ”rivers” and ”surface flows”. To this aim, we propose in this project to study the limit, as δ → 0, of the previous model where the constant 1 is replaced by δ > 0.
 
 
References
[1] G. Aronson, L. C. Evans and Y. Wu,                  Fast/Slow diffusion and growing sandpiles. J. Differential Equations, 131:304–335, 1996.
[2] S. Dumont and N. Igbida, Back on a Dual Formulation for the Growing Sandpile Problem, European Journal of Applied Mathematics, vol. 20 (2009), pp. 169-185.
[3] L. C. Evans and F. Rezakhanlou, A stochastic model for sandpiles and its continium limit. Comm. Math. Phys., 197 (1998), no. 2, 325-345.
[4] N. Igbida, Back on Stochastic Model for Sandpile. Recent developments in Nonlinear Analysis, Proceedings of the conference in Mathematics and Mathematical Physics, Morocco 28-30 October 2008.
[5] L. Prigozhin, Variational model of sandpile growth. Euro. J. Appl. Math. , 7, 225-236, 1996

Contact:[Noureddine Igbida

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