Open AccessViscosity solutions and uniqueness for systems of inhomogeneous balance laws
Author(s) -
Graziano Crasta,
Benedetto Piccoli
Publication year - 1997
Publication title -
discrete and continuous dynamical systems
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.289
H-Index - 70
eISSN - 1553-5231
pISSN - 1078-0947
DOI - 10.3934/dcds.1997.3.477
Subject(s) - uniqueness , degenerate energy levels , initial value problem , dimension (graph theory) , nonlinear system , viscosity , cauchy problem , mathematics , mathematical analysis , space (punctuation) , viscosity solution , mathematical physics , pure mathematics , physics , quantum mechanics , computer science , operating system
This paper is concerned with the Cauchy problem $$(*)u_t+[F(u)]_x=g(t,x,u),\quad u(0,x)=\bar{u}(x),$$ for a nonlinear $2\times 2$ hyperbolic system of inhomogeneous balance laws in one space dimension. As usual, we assume that the system is strictly hyperbolic and that each characteristic field is either linearly degenerate or genuinely nonlinear. \par Under suitable assumptions on $g$, we prove that there exists $T>0$ such that, for every $\bar{u}$ with sufficiently small total variation, the Cauchy problem ($*$) has a unique ``viscosity solution'', defined for $t\in [0,T]$, depending continuously on the initial data