Conservation laws with continuous flux functions
type: Item
instance of: scholarly article
title: Conservation laws with continuous flux functions
zbMATH Open document ID: 0961.35088
DOI: 10.1007/BF01193828
author: S. N. Kruzhkov
author: Philippe Benilan
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 1999-06-21T00:00:00Z
review text: The authors investigate the uniqueness problem for generalized entropy solutions (in the sense of S. Kruzhkov) of the stationary problem \[ u+ \text{div}_x\varphi(u)=f \quad \text{on } {\mathbb R}^n \tag{E} \] and the corresponding Cauchy problem \[ u_t+ \text{div}_x\varphi(u)=f \quad \text{on } (0,T)\times{\mathbb R}^n, \qquad u(0,x)=u_0(x). \tag{CP} \] The flux vector \(\varphi(u)=(\varphi_1(u),\ldots,\varphi_n(u))\) is assumed to be only continuous. In this case the effect of infinite speed of propagation generally appears which can lead even to nonuniqueness of generalized entropy solutions (g.e.s.). The authors prove the uniqueness of g.e.s. under the following anisotropic condition on the flux functions: \[ \liminf_{r\to 0} r^{1-n}\prod_{i=1}^n \omega_i(r)<\infty, \] where \(\omega_i(r)\), \(i=1,\ldots,n\) are subadditive increasing functions on \([0,\infty)\) such that for all \(u,v\) \(|\varphi_i(u)-\varphi_i(v)|\leq\omega_i(|u-v|)\). Moreover it is proved that under the condition above \[ \int_{{\mathbb R}^n}(u-v)^+ dx\leq \int_{\{u>v\}}(f-g) dx+ \int_{\{u=v\}}(f-g)^+ dx, \] where \(u\) is a generalized entropy sub-solution, \(v\) is a generalized entropy super-solution of the problem (E) with the source functions \(f\), \(g\) respectively. The similar result is proved for the problem (CP) as well. This property is also established without any restrictions on the flux vector if one of the function \(u\) or \(v\) doesn't depend on \(x\). In this case it implies estimates of \(L^p\)-norms \(\|u\|_p\leq \|f\|_p\) and \(\|u(t,\cdot)\|_p\leq \|u_0\|_p+\int_0^t\|f(\tau,\cdot)\|_p d \tau\) for g.e.s. of the problems (E) and (CP) respectively for all \(1\leq p\leq\infty\). The authors also establish uniqueness of g.e.s. for the special case when \(\varphi_i(u)=|u|^{\alpha_i-1}u\), \(\alpha_i>0\), \(i=1,\ldots,n\), and the input functions are integrable with respect to spatial variables.
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35L45
zbMATH DE Number: 952684
zbMATH Keywords: scalar conservation laws
zbMATH Keywords: infinite speed of propagation
zbMATH Keywords: stationary problem
zbMATH Keywords: generalized entropy solutions
zbMATH Keywords: generalized entropy sub- and super-solutions
zbMATH Keywords: comparison principle
zbMATH Keywords: anisotropic uniqueness conditions
reviewed by: Evgeniy Panov
MaRDI profile type: Publication
cites work: Problème d'obstacle pour une équation quasi-linéaire du premier ordre
cites work: The semigroup approach to first order quasilinear equations in several space variables
description: scientific article; zbMATH DE number 952684