Thibault Liard, Enrique Zuazua. Analysis and numerical solvability of backward-forward conservation laws (2022)
Abstract. In this paper, we study the problem of initial data identification for weak-entropy solutions of the one-dimensional Burgers equation. This problem consists in identifying the set of initial data evolving to a given target at a final time. Due to the time-irreversibility of the Burgers equation, some target functions are unattainable from solutions of this equation, making the identification problem under consideration ill-posed. To get around this issue, we introduce a non-smooth optimization problem which consists in minimizing the difference between the predictions of the Burgers equation and the observations of the system at a final time in L2(R) norm. Here, we fully characterize the set of minimizers of the aforementioned non-smooth optimization problem. One of minimizers is the backward entropy solution, constructed using a backward-forward method. Some simulations are given using a wave-front tracking algorithm.