Revista Matemática Iberoamericana


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Volume 3, Issue 3, 1987, pp. 275–310
DOI: 10.4171/RMI/51

The Relation Between the Porous Medium and the Eikonal Equations in Several Space Dimensions

Pierre-Louis Lions[1], Panagiotis E. Souganidis[2] and Juan Luis Vázquez[3]

(1) CEREMADE, Univ de Paris IX Dauphine, Place du Marechal de Lattre de Tassigny, 75775, PARIS CEDEX 05, FRANCE
(2) Division of Applied Mathematics, Brown University, RI 02912, PROVIDENCE, UNITED STATES
(3) Departamento de Matematicas, Universidad Autónoma de Madrid, Ctra. de Colmenar Viejo, Km. 15, 28049, MADRID, SPAIN

We study the relation between the porous medium equation $u_t = \Delta(u^m), \; m>1$, and the eikonal equation $\nu_t = |D\nu|^2$. Under quite general assumptions, we prove that the pressure and the interface of the solution of the Cauchy problem for the porous medium equation converge as $m \downarrow 1$ to the viscosity solution and the interface of the Cauchy problem for the eikonal equation. We also address the same questions for the case of the Dirichlet boundary value problem.

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Lions Pierre-Louis, Souganidis Panagiotis, Vázquez Juan Luis: The Relation Between the Porous Medium and the Eikonal Equations in Several Space Dimensions. Rev. Mat. Iberoamericana 3 (1987), 275-310. doi: 10.4171/RMI/51