Initial data identification in conservation laws and Hamilton–Jacobi equations

Abstract

In the scalar 1D case, conservation laws and Hamilton-Jacobi equations are deeply related. For both, we characterize those profiles that can be attained as solutions at a given positive time corresponding to at least one initial datum. Then, for each of the two equations, we precisely identify all those initial data yielding a solution that coincide with a given profile at that positive time. Various topological and geometrical properties of the set of these initial data are then proved.2000 Mathematics Subject Classification: 35L65, 35F21, 93B30, 35R30.

Initial Data Identification for the One-Dimensional Burgers Equation

“…As (2) is time-irreversibility, some conditions on u T need to be imposed for it to be attainable. This is shown in [12,Theorem 3.1,Corollary 3.2], [19,Corollary 1] or [17] where they prove that u T is truly attainable in an exact manner by a solution of (2) if and only if u T satisfies the one-sided Lipschitz condition [6], [18], [23], [15], i.e…”

“…In [19], the authors prove that the set of initial data evolving to an attainable target u T is a convex set. Later on, the aforementioned set was fully characterized in [12], [16] using the classical Lax-Hopf formula [21,Theorem 2.1].…”

“…In our paper, an alternative proof of the characterization of the set of initial data in [12], [16] is given using only backward generalized characteristics. This leads to the hope of investigate systems of conservation laws in one dimension.…”

“…This leads to the hope of investigate systems of conservation laws in one dimension. More precisely, [12,Theorem 4.1] cannot be extend to systems as Euler equations, Saint-Venant equations or Aw-Rascle-Zhang model since Lax-Hopf formula does not hold anymore. Moreover, a wave-front tracking method is implemented to construct numerically all of them.…”

“…Due to the property of non-backward uniqueness of Burgers equation, there may exist multiple initial data leading to the same given target. In [12], [16], the authors fully characterize the set of initial data leading to a given target using the classical Lax-Hopf formula. In this note, an alternative proof based only on generalized backward characteristics is given.…”