Initial Data Identification for the One-Dimensional Burgers Equation

Abstract

In this article, we study the problem of identification for the 1-D Burgers equation. This problem consists in identifying the set of initial data evolving to a given target at a final time. Due to the property of nonbackward uniqueness of the Burgers equation, there may exist multiple initial data leading to the same given target. In articles “Initial data identification in conservation laws and Hamilton-Jacobi equations” ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">arXiv:1903.06448</i> , 2019) and “The inverse problem for Hamilton-Jacobi equations and semiconcave envelopes” ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SIAM Journal on Mathematical Analysis</i> , vol. 52, the authors fully characterize the set of initial data leading to a given target using the classical Lax–Hopf formula. In this article, an alternative proof based only on generalized backward characteristics is given. This leads to the hope of investigating systems of conservation laws in 1-D, where the classical Lax–Hopf formula no more holds. Moreover, numerical illustrations are presented using as a target, a function optimized for minimum pressure rise in the context of sonic-boom minimization problems. All of initial data leading to this given target are constructed using a wavefront tracking algorithm.