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 $k(0+)=\infty$.