Schonbek, Maria Elena Existence of solutions to singular conservation laws. (English) Zbl 0567.35060 SIAM J. Math. Anal. 15, 1125-1139 (1984). The scalar conservation law \(u_ t+f(u)_ x+g(u)/x=0\) is considered with f,g smooth, f strictly convex or strictly concave and with definite sign of ug(u) at infinity (either \(ug(u)\geq 0\) or \(ug(u)<0\) for large u). Existence of the solution to the initial-value problem for such a conservation law \((u(0,x)=u_ 0(x),\quad x>0)\) is proved by regularizing the equation to the form \(u_ t+f(u)_ x+g(u)/(x+\delta)=\epsilon u_{xx}\) and taking a singular limit. Local \(L^{\infty}\) a priori bounds are obtained either by maximum principle for parabolic equations or by estimating the solution along generalized backward characteristics. Then the method of compensated compactness is applied, which allows to pass to the limit without gradient estimates. Reviewer: A.Doktor Cited in 1 ReviewCited in 7 Documents MSC: 35L65 Hyperbolic conservation laws 35L45 Initial value problems for first-order hyperbolic systems 35A05 General existence and uniqueness theorems (PDE) (MSC2000) Keywords:singular conservation laws; scalar conservation law; Existence; initial-value problem; regularizing; a priori bounds; compensated compactness PDF BibTeX XML Cite \textit{M. E. Schonbek}, SIAM J. Math. Anal. 15, 1125--1139 (1984; Zbl 0567.35060) Full Text: DOI