A product formula approach to first order quasilinear equations
type: Item
instance of: scholarly article
title: A product formula approach to first order quasilinear equations
zbMATH Open document ID: 0573.35009
author: Yoshikazu Kobayashi
published in: Hiroshima Mathematical Journal
publication date: 1985-01-01T00:00:00Z
review text: This work is concerned with a new semigroup approach to the conservation law equation: \(u_ t+\sum^{n}_{n=1}(\phi_ i(u))_{x_ i}=0\) in \(R^+\times R^ n\) where \(\phi =(\phi_ 1,...,\phi_ n)\) is a smooth \(R^ n\) valued function on R. Let \(C_ h: L^ 1(R^ n)\to L^ 1(R^ n)\) be the operator defined by \[ (C_ hu)(x)=\int_{R^ n}2^{- 1}(sign(u(x)-h\phi '(\xi)-\xi)+sign \xi)d\xi;\quad h>0. \] The main result of this paper asserts that the semigroup solution T(t)u of the above equation in the sense of Crandall is given by the Chernoff-Brezis-Paty product formula: \(T(t)u=\lim_{h\to 0}C_ h^{[t/h]}u\).
Mathematics Subject Classification ID: 35F25
Mathematics Subject Classification ID: 47H20
Mathematics Subject Classification ID: 35C05
zbMATH DE Number: 3914792
zbMATH Keywords: quasilinear equations
zbMATH Keywords: semigroup
zbMATH Keywords: conservation law
zbMATH Keywords: product formula
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3914792