141 research outputs found

    Finite speed of propagation for a non-local porous medium equation

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    This note is concerned with proving the finite speed of propagation for some non-local porous medium equation by adapting arguments developed by Caffarelli and V\'azquez (2010).Comment: 10 pages. New version after revision. Several typos removed and more explainations given the contact analysi

    Level set approach for fractional mean curvature flows

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    This paper is concerned with the study of a geometric flow whose law involves a singular integral operator. This operator is used to define a non-local mean curvature of a set. Moreover the associated flow appears in two important applications: dislocation dynamics and phase field theory for fractional reaction-diffusion equations. It is defined by using the level set method. The main results of this paper are: on one hand, the proper level set formulation of the geometric flow; on the other hand, stability and comparison results for the geometric equation associated with the flow

    Self-similar solutions for a fractional thin film equation governing hydraulic fractures

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    In this paper, self-similar solutions for a fractional thin film equation governing hydraulic fractures are constructed. One of the boundary conditions, which accounts for the energy required to break the rock, involves the toughness coefficient K≄0K\geq 0. Mathematically, this condition plays the same role as the contact angle condition in the thin film equation. We consider two situations: The zero toughness (K=0K=0) and the finite toughness K∈(0,∞)K\in(0,\infty) cases. In the first case, we prove the existence of self-similar solutions with constant mass. In the second case, we prove that for all K\textgreater{}0 there exists an injection rate for the fluid such that self-similar solutions exist

    The Schauder estimate in kinetic theory with application to a toy nonlinear model

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    This article is concerned with the Schauder estimate for linear kinetic Fokker-Planck equations with H\"older continuous coefficients. This equation has an hypoelliptic structure. As an application of this Schauder estimate, we prove the global well-posedness of a toy nonlinear model in kinetic theory. This nonlinear model consists in a non-linear kinetic Fokker-Planck equation whose steady states are Maxwellian and whose diffusion in the velocity variable is proportional to the mass of the solution

    Estimates on elliptic equations that hold only where the gradient is large

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    We consider a function which is a viscosity solution of a uniformly elliptic equation only at those points where the gradient is large. We prove that the H{\"o}lder estimates and the Harnack inequality, as in the theory of Krylov and Safonov, apply to these functions.Comment: 18 page

    Some regularity results for anisotropic motion of fronts

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    We study the regularity of propagating fronts whose motion is anisotropic. We prove that there is at most one normal direction at each point of the front; as an application, we prove that convex fronts are C^{1,1}. These results are by-products of some necessary conditions for viscosity solutions of quasilinear elliptic equations. Besides, these conditions imply some regularity for viscosity solutions of nondegenerate quasilinear elliptic equation

    Some regularity results for anisotropic motion of fronts

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    We study the regularity of propagating fronts whose motion is anisotropic. We prove that there is at most one normal direction at each point of the front; as an application, we prove that convex fronts are C^{1,1}. These results are by-products of some necessary conditions for viscosity solutions of quasilinear elliptic equations. Besides, these conditions imply some regularity for viscosity solutions of nondegenerate quasilinear elliptic equation
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