[Paper Review] Initial Data Identification in Conservation Laws and Hamilton-Jacobi Equations

Rinaldo M. Colombo, Vincent Perrollaz|arXiv (Cornell University)|Mar 15, 2019

Geometric Analysis and Curvature Flows19 references4 citations

TL;DR

This paper provides a complete characterization of initial data that evolve into a given profile at a fixed positive time for scalar conservation laws and Hamilton–Jacobi equations in one space dimension. It identifies the set of initial data yielding a specific solution profile, proving convexity, extremal structure, and topological properties such as the absence of finite-dimensional extremal faces.

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.

Motivation & Objective

• To characterize all initial data that generate a given solution profile at a fixed positive time T for scalar conservation laws and Hamilton–Jacobi equations.
• To establish the non-emptiness of the set of such initial data through an explicit constructive method.
• To analyze the topological and geometric structure of the set of initial data yielding a given profile, including convexity and extremal properties.
• To prove that the set of initial data is a cone with no finite-dimensional extremal faces, and that BV-regular initial data exist whenever the solution profile is reachable.
• To provide a framework for inverse problems in conservation laws and optimal control by identifying terminal or initial costs from observed dynamics.

Proposed method

• Uses the duality between conservation laws and Hamilton–Jacobi equations via the relation $ u = \partial_x U $, where $ u $ solves (1.1) and $ U $ solves (1.2).
• Applies Oleïnik-type decay estimates and refined analysis of rarefaction waves to reconstruct initial data from a given profile at time $ T $.
• Introduces two disjoint sets $ X_i $ and $ X_{ii} $: values of initial data on $ X_i $ are uniquely determined, while on $ X_{ii} $ they are loosely constrained.
• Employs viscosity and entropy solution theory to ensure well-posedness and to characterize solutions via the Hamilton–Jacobi equation.
• Constructs perturbations of initial data using compactly supported functions $ A_k $ to prove extremality and non-emptiness of the solution set.
• Uses backward construction by prolonging shocks and analyzing the interaction potential to ensure constant potential on $ (0,T) $, which aids in solution uniqueness and regularity.

Experimental results

Research questions

• RQ1 Which initial data for a scalar conservation law or Hamilton–Jacobi equation evolve into a given solution profile at a fixed time $ T > 0 $?
• RQ2 What is the topological and geometric structure of the set of such initial data, particularly regarding convexity and extremal points?
• RQ3 Can one always find a BV-regular initial datum that produces a given solution profile, even if the profile arises from an initial datum of unbounded variation?
• RQ4 Does the set of initial data yielding a given profile admit finite-dimensional extremal faces, or is it a cone with infinite-dimensional structure?
• RQ5 How can one explicitly construct an initial datum that generates a given solution profile, particularly in the presence of shocks and rarefaction waves?

Key findings

• The set $ I^{ ext{CL}}_T(w) $ of initial data yielding a given profile $ w $ at time $ T $ is non-empty if and only if the profile satisfies certain structural conditions derived from rarefaction wave decay and shock propagation.
• The set $ I^{ ext{CL}}_T(w) $ is convex, and its unique extreme point is fully characterized as the initial datum obtained by prolonging all shocks backward in time.
• The set $ I^{ ext{CL}}_T(w) $ is a cone with no finite-dimensional extremal faces, indicating infinite-dimensional freedom in choosing initial data for a given profile.
• For any profile $ w $ in the image of the solution operator, there exists an initial datum in $ \mathbf{BV}({\mathbb{R}};{\mathbb{R}}) $ that generates it, even if the original initial data had unbounded variation.
• The set $ I^{ ext{CL}}_T(w) $ always contains one-sided Lipschitz continuous functions, and more regular initial data may also exist depending on the profile.
• The solution constructed by prolonging shocks backward yields a constant interaction potential on $ (0,T) $, which is a key structural property for inverse design and control problems.