A product formula approach to first order quasilinear equations (Q1063169)

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: \(T(t)u=\lim_{h\to 0}C_ h^{[t/h]}u\).