an:03905104
Zbl 0567.35060
Schonbek, Maria Elena
Existence of solutions to singular conservation laws
EN
SIAM J. Math. Anal. 15, 1125-1139 (1984).
00147526
1984
j
35L65 35L45 35A05
singular conservation laws; scalar conservation law; Existence; initial-value problem; regularizing; a priori bounds; compensated compactness
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.
A.Doktor