Consider a scalar conservation law with discontinuous flux (1):
\begin{equation*}
\quad u_{t}+f(x,u)_{x}=0,
\qquad f(x,u)= \begin{cases}
f_l(u)\ &\text{if}\ x<0,\\
f_r(u)\ & \text{if} \ x>0,
\end{cases}
\quad \quad \quad(1)
\end{equation*}
where u = u(x, t) is the state variable and fl, fr are strictly convex maps. We study the Cauchy problem for (1) from the point of view of control theory regarding the initial datum as a control. Letting
u(x,t)≐StABu-(x) denote the solution of the Cauchy problem for (1), with initial datum u(⋅,0)=u-, that satisfy at x = 0 the interface entropy condition associated to a connection (A, B) (see Adimurthi, S. Mishra and G.D. Veerappa Gowda, J. Hyperbolic Differ. Equ. 2 (2005) 783–837), we analyze the family of profiles that can be attained by (1) at a given time T > 0:
\mathcal{A}^{AB}(T)=\left\{\mathcal{S}_T^{AB} \,\overline u : \ \overline u\in{\bf L}^\infty(\mathbb{R})\right\}.\
We provide a full characterization of AAB(T) as a class of functions in BVloc(ℝ\{0}) that satisfy suitable Oleǐnik-type inequalities, and that admit one-sided limits at x = 0 which satisfy specific conditions related to the interface entropy criterion. Relying on this characterisation, we establish the Lloc1-compactness of the set of attainable profiles when the initial data u- vary in a given class of uniformly bounded functions, taking values in closed convex sets. We also discuss some applicationsof these results to optimization problems arising in traffic flow.
Global structure of solutions toward the rarefaction waves for the Cauchy problem of the scalar conservation law with nonlinear viscosity
