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The Relation Between the Porous Medium and the Eikonal Equations in Several Space Dimensions
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1987
Year
Viscosity SolutionMonge-ampere EquationPore StructureEngineeringSeveral Space DimensionsPhysicsFree Boundary ProblemGeometric Partial Differential EquationPorous Medium EquationsEikonal EquationsParabolic EquationTransport PhenomenaGeneral AssumptionsNonlinear Hyperbolic ProblemPorous MediumIntegrable SystemEikonal EquationPorous Body
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.