## Differential and Integral Equations

### Fractional derivatives and smoothing in nonlinear conservation laws

#### Abstract

It is shown that the solution of the Riemann problem $$\hbox{ \frac {\partial}{\partial t}} \int_0^t k(t-s)(u(s,x)-u_0(x))\,\rm{d}s + (\sigma(u))_x(t,x) = 0,$$ where $u_0 = \chi_{\rminus}$ is continuous when $t> 0$. Here $k$ is locally integrable, nonnegative, and nonincreasing on $\mathbb{R}plus$ with $k(0+)=\infty$.

#### Article information

Source
Differential Integral Equations, Volume 8, Number 8 (1995), 1961-1976.

Dates
First available in Project Euclid: 20 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1369056135

Mathematical Reviews number (MathSciNet)
MR1348960

Zentralblatt MATH identifier
0885.45005

#### Citation

Gripenberg, Gustaf; Londen, Stig-Olof. Fractional derivatives and smoothing in nonlinear conservation laws. Differential Integral Equations 8 (1995), no. 8, 1961--1976. https://projecteuclid.org/euclid.die/1369056135