Concavity of Solutions of the Porous Medium Equation.

Abstract

The flow of a gas through a porous medium is governed by a degenerate quasilinear parabolic equation. It is known that the nonnegative solutions to this equation possess a lower bound for the second derivative of the pressure in the spatial variables. This bound plays an important role in the mathematical treatment and is related to the entropy of the flow. Since the solutions exhibit interfaces across which v sub x jumps positively, no upper bound is possible globally for v subxx. Nevertheless it is proven that the concavity of v(.,t) in the region where v is positive is preserved in time. This is in itself an interesting geometric property of the solution. It also allows one to obtain precise information about the asymptotic behaviour of the flow.

Open PDF

Document Details

Document Type
Technical Report
Publication Date
Aug 01, 1985
Accession Number
ADA160969

Entities

People

  • Juan L. Vazquez
  • Philippe Benilan

Organizations

  • University of Wisconsin–Madison

Tags

Communities of Interest

  • C4I

DTIC Thesaurus Topics

  • Abstracts
  • Boundaries
  • Cauchy Problem
  • Classification
  • Contracts
  • Convergence
  • Differential Equations
  • Discontinuities
  • Equations
  • Inequalities
  • Intervals
  • Mathematical Analysis
  • Mathematics
  • North Carolina
  • Two Dimensional
  • United States
  • Wisconsin

Fields of Study

  • Mathematics

Readers

  • Finite Element Method (FEM) for solving Partial Differential Equations (PDEs)
  • Fluid Dynamics.
  • Graph Algorithms and Convex Optimization.