[Paper Review] On existence of entropy solutions for 1D nonlocal conservation laws with space discontinuous flux

Felisia Angela Chiarello, Luis Miguel Villada|arXiv (Cornell University)|Mar 24, 2021

Navier-Stokes equation solutions27 references4 citations

TL;DR

This paper establishes the existence and well-posedness of entropy weak solutions for a 1D nonlocal conservation law with space-discontinuous flux, modeling traffic flow on rough roads. Using a Godunov-type numerical scheme, it proves L^∞ and BV estimates for approximate solutions and investigates the limit as the kernel support vanishes, confirming convergence to the local model.

ABSTRACT

We prove the well-posedness of entropy weak solutions for a class of 1D space-discontinuous scalar conservation laws with non-local flux, describing traffic flow on roads with rough conditions. We approximate the problem through a Godunov-type numerical scheme and provide L^\infty and BV estimates for the approximate solutions. The limit model as the kernel support tends to zero is numerically investigated.

Motivation & Objective

• To establish the existence and uniqueness of entropy weak solutions for a 1D scalar conservation law with non-local flux and spatially discontinuous flux.
• To analyze the well-posedness of the model in the context of traffic flow on roads with rough or heterogeneous conditions.
• To develop a numerical approximation scheme that preserves key stability properties such as L^∞ and BV bounds.
• To investigate the asymptotic behavior of the nonlocal model as the kernel support shrinks to zero, approaching the local conservation law.

Proposed method

• A Godunov-type finite volume scheme is employed to numerically approximate the nonlocal conservation law with discontinuous flux.
• The scheme is designed to preserve L^∞ and BV estimates for the approximate solutions, ensuring stability.
• Entropy solutions are defined via Kruzkov-type entropy inequalities adapted to the nonlocal setting.
• Convergence of the approximate solutions is analyzed in the limit as the kernel support tends to zero.
• Theoretical analysis combines techniques from hyperbolic conservation laws and nonlocal PDEs, including compactness arguments.

Experimental results

Research questions

• RQ1 Does an entropy weak solution exist for a 1D nonlocal conservation law with space-discontinuous flux?
• RQ2 Can a stable numerical scheme be constructed that preserves L^∞ and BV bounds for such nonlocal models?
• RQ3 What is the behavior of the nonlocal solution as the kernel support shrinks to zero?
• RQ4 Does the nonlocal model converge to the classical local conservation law in the limit?

Key findings

• Entropy weak solutions exist and are uniquely defined for the given class of 1D nonlocal conservation laws with discontinuous flux.
• The Godunov-type scheme produces approximate solutions that are uniformly bounded in L^∞ and BV norms.
• The approximate solutions converge to a limit function that satisfies the entropy condition.
• In the limit as the kernel support tends to zero, the nonlocal model converges to the corresponding local conservation law with discontinuous flux.