Front Tracking for Scalar Conservation Laws with Spatially Heterogeneous Flux

TL;DR

This paper introduces a new front tracking method for scalar conservation laws with spatially heterogeneous flux, successfully handling cases where classical theories fail, including finite-time blow-up scenarios.

Contribution

The paper develops a novel front tracking scheme that converges to entropy solutions for conservation laws with complex, spatially varying flux functions, overcoming limitations of classical methods.

Findings

Convergence of the scheme to entropy solutions is proven.

The method handles fluxes where classical theory fails.

Constructs solutions even with finite-time blow-up of the solution.

Abstract

In this article, we propose a novel front tracking scheme for scalar conservation laws with spatially heterogeneous, uniformly convex flux and prove that approximations converge to the unique entropy solution. The main tools are Dafermos' generalised characteristics and Kruzkov's entropies. Crucially, our method handles fluxes where classical theory fails completely. As a concrete demonstration, we construct entropy solutions for a Cauchy problem with flux $f(x,u)=xu^2$, where bounded initial data can become unbounded in finite time, even on compact spatial domains. This finite-time blow-up violates the maximum principle, rendering all classical existence techniques--based on $L^{\infty}$ estimates and compactness--inapplicable. However, the flux $f(x,u(x,t))$ remains bounded despite $u$ blowing up, and our front tracking scheme exploits this to construct approximations that converge to an entropy solution.