Fractional derivatives and smoothing in nonlinear conservation laws
type: Item
instance of: scholarly article
title: Fractional derivatives and smoothing in nonlinear conservation laws
zbMATH Open document ID: 0885.45005
author: Gustaf Gripenberg
author: Stig-Olof Londen
published in: Differential and Integral Equations
publication date: 1998-04-20T00:00:00Z
review text: The authors investigate solutions of the Riemann problem \[ \frac{\partial}{\partial t}\int_0^t k(t-s)(u(s,x)- u_0(x))dx+ (\sigma(u))_x (t,x)=0, \qquad t>0, \quad x\in\mathbb{R}, \tag \(*\) \] where \(k\) is locally integrable, nonnegative and nonincreasing on the positive reals, but unbounded near zero. The climatic theorem states that if \(\sigma\) is continuous and satisfies \(\sigma(v)- \sigma(w)\geq c(v-w)\) for some constant \(c>0\) and all \(v,w\) with \(v\geq w\), then the solution of \((*)\) with \(u_0(x)=1\) if \(x<0\) and \(=0\) if \(x\geq 0\) is continuous on \((0,\infty)\times \mathbb{R}\setminus \{(0,0)\}\). Several other properties of these solutions are established such as the absolute continuity of the mapping \(x\mapsto \sigma(u(t,x))\). A limiting case occurs when \(k\) is replaced by the Dirac operator. This yields the familiar law \(u_t+ \sigma(u)_x=0\) which, as the authors point out, has discontinuous solutions (even for smooth initial conditions). A corollary to the main result is that this conservation law has continuous solutions if \(u_t\) is replaced by \(\frac{\partial^\alpha u}{\partial t^\alpha}\) where \(0<\alpha<1\). The proofs are clearly written, although lengthy and detailed; especially the proof of the overall continuity of the solution.
reviewed by: M.Tarabek
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 35L65
zbMATH DE Number: 811755
zbMATH Keywords: fractional derivatives
zbMATH Keywords: smoothing
zbMATH Keywords: nonlinear conservation laws
zbMATH Keywords: convolution
zbMATH Keywords: Riemann problem
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 811755