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In this paper we introduce a concept of "regulated function" $v(t,x)$ of twovariables, which reduces to the classical definition when $v$ is independent of$t$. We then consider a scalar conservation law of the form$u_t+F(v(t,x),u)_x=0$, where $F$ is smooth and $v$ is a regulated function,possibly discontinuous w.r.t.both $t$ and $x$. By adding a small viscosity, oneobtains a well posed parabolic equation. As the viscous term goes to zero, theuniqueness of the vanishing viscosity limit is proved, relying on comparisonestimates for solutions to the corresponding Hamilton--Jacobi equation.  As an application, we obtain the existence and uniqueness of solutions for aclass of $2\times2$ triangular systems of conservation laws with hyperbolicdegeneracy.

Vanishing Viscosity Solutions for Conservation Laws with Regulated Flux