We provide a rather complete description of the results obtained so far on the nonlinear diffusion equation ut = ∇⋅ (um−1∇(−Δ)−su), which describes a flow through a porous medium driven by a nonlocal pressure. We consider constant parameters m > 1 and 0 < s < 1, we assume that the solutions are non-negative, and the problem is posed in the whole space. We present a theory of existence of solutions, results on uniqueness, and relation to other models. As new results of this paper, we prove the existence of self-similar solutions in the range when N = 1 and m > 2, and the asymptotic behavior of solutions when N = 1. The cases m = 1 and m = 2 were rather well known.
CITATION STYLE
Stan, D., del Teso, F., & Vázquez, J. L. (2018). Porous medium equation with nonlocal pressure. In Springer Optimization and Its Applications (Vol. 135, pp. 277–308). Springer International Publishing. https://doi.org/10.1007/978-3-319-89800-1_12
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