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.
Viscosity solutions and uniqueness for systems of inhomogeneous balance laws / Crasta, Graziano; Benedetto, Piccoli. - In: DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. - ISSN 1078-0947. - 3:4(1997), pp. 477-502. [10.3934/dcds.1997.3.477]
Viscosity solutions and uniqueness for systems of inhomogeneous balance laws
CRASTA, Graziano;
1997
Abstract
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.