JournalsrmiVol. 3, No. 3pp. 275–310

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

  • Pierre-Louis Lions

    Université de Paris-Dauphine, Paris, France
  • Panagiotis E. Souganidis

    University of Chicago, USA
  • Juan Luis Vázquez

    Universidad Autónoma de Madrid, Spain
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Abstract

We study the relation between the porous medium equation ut=Δ(um),  m>1u_t = \Delta(u^m), \; m>1, and the eikonal equation νt=Dν2\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 m1m \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.

Cite this article

Pierre-Louis Lions, Panagiotis E. Souganidis, Juan Luis Vázquez, The Relation Between the Porous Medium and the Eikonal Equations in Several Space Dimensions. Rev. Mat. Iberoam. 3 (1987), no. 3, pp. 275–310

DOI 10.4171/RMI/51