Analysis and numerical solvability of backward-forward conservation laws

In this paper, we study the problem of initial data identification for 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 inverse problem under consideration ill-posed. To get around this issue, we introduce an 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 L^2(R) norm. The two main contributions of this work are as follows. -We fully characterize the set of minimizers of the aforementioned non-smooth optimization problem. -A wave-front tracking method is implemented to construct numerically all of them. One of minimizers is the backward entropy solution, constructed using a backward-forward method.