Finite and infinite speed of propagation for porous medium equations with nonlocal pressure

Abstract

We study a porous medium equation with fractional potential pressure:∂ t u = ∇ · (u m−1 ∇p), p = (−∆) −s u, for m > 1, 0 < s < 1 and u(x, t) ≥ 0. The problem is posed for x ∈ R N , N ≥ 1, and t > 0. The initial data u(x, 0) is assumed to be a bounded function with compact support or fast decay at infinity. We establish existence of a class of weak solutions for which we determine whether the property of compact support is conserved in time depending on the parameter m, starting from the result of finite propagation known for m = 2. We find that when m ∈ [1, 2) the problem has infinite speed of propagation, while for m ∈ [2, 3) it has finite speed of propagation. In other words m = 2 is critical exponent regarding propagation. The main results have been announced in the note [29].Other related models. Equation (CV) with s = 1/2 in dimension N = 1 has been proposed by Head [20] to describe the dynamics of dislocation in crystals. The model is written in the integrated form asThe dislocation density is u = v x . This model has been recently studied by Biler, Karch and Monneau in [4], where they prove that the problem enjoys the properties of uniqueness and comparison of viscosity solutions. The relation between u and v is very interesting and will be used by us in the final sections.Another possible generalization of the (CV) model is ∂ t u = ∇ · (u∇p), p = (−∆) −s (|u| m−2 u), that has been investigated by Biler, Imbert and Karch in [2,3]. They prove the existence of weak solutions and they find explicit self-similar solutions with compact support for all m > 1. The finite speed of propagation for every weak solution has been done in [22].The second nonlocal version of the classical PME is the model u t = −(−∆) s u m , m > 0,

Lie group analysis of 2‐dimensional space‐fractional model for flow in porous media

“…This equation is studied in previous studies. [16][17][18] It is well known that single-phase fluid flow through the porous media is described by the classical diffusion equation for pressure. In the limiting case of = 0, Equation 3 coincides with this classical equation.…”

“…Such types of anomalous diffusion equations are known and studied. [16][17][18][19][20][21][22] Nevertheless, symmetry properties of fractional equations with the Riesz potential are not yet investigated.…”

Porous Medium Equation with Nonlocal Pressure

“…The fact that such a first contact point happens for t > 0 and x = ∞ is justified by regularization. We also exclude the extreme case where the contact is made at the boundary of the support of U given by |x f (t c )| := b +Ct c (see Lemma 7.2 in [56]). Then, there exists…”

On the well-posedness of solutions with finite energy for nonlocal equations of porous medium type

“…The other main application of the L p -energy estimates is as key steps in Sobolev or Simon type compactness arguments. Such arguments are used in [42,10,9,38,39,40] to prove existence of energy solutions through the resolution of a sequence of smooth approximate problems and passing to the limit in view of compactness.…”