An approach by difference to the porous medium equation with convection
type: Item
instance of: scholarly article
title: An approach by difference to the porous medium equation with convection
zbMATH Open document ID: 0847.35074
author: Michiaki Watanabe
published in: Hiroshima Mathematical Journal
publication date: 1996-04-08T00:00:00Z
review text: The author discusses the existence and the uniqueness of the solution to the Cauchy problem for the porous medium equation with convection terms: \[ (1)\quad u_t= \Delta \phi(u)+ \sum^N_{i= 1} {\partial\over \partial x_i} F_i(u),\;x\in \mathbb{R}^n,\;t> 0;\qquad (2)\quad u(0, x)= u_0(x),\;x\in \mathbb{R}^n. \] The functions \(\phi\) is assumed to be strictly increasing, locally Lipschitz continuous on \(\mathbb{R}\) and satisfies \(\phi(0)= 0\). The functions \(F_i\) are defined on \(\mathbb{R}\), \(F_i(0)= 0\), and \(|F_i(r)- F_i(s)|/ |\phi(r)- \phi(s)|\) are bounded for \(r\), \(s\) in every bounded subinterval of \(\mathbb{R}\). The author provides a direct method for solving the problem (1)--(2) via a method of difference approximation which converges to a limit of a semigroup \(\{S(t): t\geq 0\}\) of contractions on \(L^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)\) associated with the problem (1), (2).
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35K45
Mathematics Subject Classification ID: 76S05
zbMATH DE Number: 863655
zbMATH Keywords: difference scheme
zbMATH Keywords: porous medium equation
zbMATH Keywords: convection
reviewed by: Akira Tsutsumi
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 863655