Attainable profiles for conservation laws with flux function spatially discontinuous at a single point

TL;DR

This paper characterizes the set of attainable solutions at a fixed time for a scalar conservation law with a discontinuous flux at a point, using control theory and entropy conditions, with applications to porous media and traffic flow.

Contribution

It provides a complete description of attainable profiles for conservation laws with flux discontinuity, including their BV regularity and entropy conditions, and proves compactness results.

Findings

Attainable profiles are characterized by BV functions satisfying Ole
2fnik-type inequalities.

The set of attainable profiles is compact in local L^1 topology.

Applications include optimization in porous media and traffic flow models.

Abstract

Consider a scalar conservation law with discontinuous flux \begin{equation*}\tag{1} \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} \end{equation*} where $u=u(x,t)$ is the state variable and $f_{l}$, $f_{r}$ 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 $u(x,t)\doteq \mathcal{S}_t^{AB} \overline u(x)$ denote the solution of the Cauchy problem for (1), with initial datum $u(\cdot,0)=\overline u$, that satisfy at $x=0$ the interface entropy condition associated to a connection $(A,B)$ (see~\cite{MR2195983}), we analyze the family of profiles that can be attained by (1) at a given time $T>0$: \begin{equation*} \mathcal{A}^{AB}(T)=\left\{\mathcal{S}_T^{AB} \,\overline u : \ \overline u\in{\bf L}^\infty(\mathbb{R})\right\}. \end{equation*} We provide a full characterization of $\mathcal{A}^{AB}(T)$ as a class of functions in $BV_{loc}(\mathbb{R}\setminus\{0\})$ that satisfy suitable Ole\v{\i}nik-type inequalities, and that admit one-sided limits at $x=0$ which satisfy specific conditions related to the interface entropy criterium. Relying on this characterisation, we establish the ${\bf L^1}_{loc}$-compactness of the set of attainable profiles when the initial data $\overline u$ vary in a given class of uniformly bounded functions, taking values in closed convex sets. We also discuss some applications of these results to optimization problems arising in porous media flow models for oil recovery and in traffic flow.