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

¹ Dipartimento di Matematica “Tullio Levi-Civita”, Università di Padova, Italy.
² Department of Mathematics, Penn State University, University Park, PA, USA.

^* Corresponding author: ancona@math.unipd.it

Received: 3 March 2020
Accepted: 13 July 2020

Abstract

Consider a scalar conservation law with discontinuous flux (1):

$$u_t+f{\left(x,u\right)}_x=0,f\left(x,u\right)=\{f_l\left(u\right)& \text{if}x<0,\\ f_r\left(u\right)& \text{if}x>0,$$

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)≐S_t^ABu^-(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: $$A^{\mathrm{AB}}\left(T\right)=\left\{S_T^{\mathrm{AB}}\bar{u}:\bar{u}\in L^{\infty }\left(R\right)\right\}.$$

We provide a full characterization of A^AB(T) as a class of functions in BV_loc(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 L_loc¹-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.