Initial boundary value problem for 1D scalar balance laws with strictly convex flux

TL;DR

This paper extends a Lax-Oleinik type formula to 1D scalar balance laws with boundary conditions, addressing complexities due to boundary crossing of minimizers and establishing entropy admissibility of solutions.

Contribution

It introduces a boundary functional for the initial boundary value problem, enabling explicit solution construction and boundary condition verification for scalar balance laws.

Findings

Established a Lax-Oleinik type formula for boundary value problems.

Classified boundary points based on minimizer structure.

Proved the entropy admissibility of the explicit solution.

Abstract

A Lax-Oleinik type explicit formula for 1D scalar balance laws has been recently obtained for the pure initial value problem by Adimurthi et al. in [1]. In this article, by introducing a suitable boundary functional, we establish a Lax-Oleinik type formula for the initial boundary value problem. For the pure initial value problem, the solution for the corresponding Hamilton-Jacobi equation turns out to be the minimizer of a functional on the set of curves known as h-curves. In the present situation, part of the h-curve joining any two points in the quarter plane may cross the boundary $x = 0.$ This phenomenon breaks the simplicity of the minimization process through the boundary functional compared to the case of conservation laws. Moreover, this complicates the verification of the boundary condition in the sense of Bardos, le Roux, and Nedelec [2]. To verify the boundary condition, the boundary points are classified into three types depending on the structure of the minimizers at those points. Finally, by introducing characteristic triangles, we construct generalized characteristics and show that the explicit solution is entropy admissible.