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Front Tracking for Scalar Conservation Laws with Spatially Heterogeneous Flux

Parasuram Venkatesh

TL;DR

This work develops a front-tracking scheme for scalar conservation laws with spatially heterogeneous, uniformly convex flux and proves convergence to the unique entropy solution by leveraging Dafermos' generalized characteristics and Kruzhkov entropies. The approach directly handles flux heterogeneity, including smoothly varying, nonuniform fluxes, and demonstrates well-posedness for BV initial data via δ-approximate front-tracking solutions that preserve a TVD structure. A key contribution is the demonstration that entropy solutions exist even when classical approaches fail, as shown by the flux f(x,u)=x u^2, which can exhibit finite-time blow-up while keeping f(x,u) bounded along characteristics. The results extend the front-tracking framework beyond homogeneous flux, provide a practical scheme for constructing entropy solutions in challenging regimes, and offer insights into unbounded and non-principle-fulfilling dynamics in heterogeneous media.

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.

Front Tracking for Scalar Conservation Laws with Spatially Heterogeneous Flux

TL;DR

This work develops a front-tracking scheme for scalar conservation laws with spatially heterogeneous, uniformly convex flux and proves convergence to the unique entropy solution by leveraging Dafermos' generalized characteristics and Kruzhkov entropies. The approach directly handles flux heterogeneity, including smoothly varying, nonuniform fluxes, and demonstrates well-posedness for BV initial data via δ-approximate front-tracking solutions that preserve a TVD structure. A key contribution is the demonstration that entropy solutions exist even when classical approaches fail, as shown by the flux f(x,u)=x u^2, which can exhibit finite-time blow-up while keeping f(x,u) bounded along characteristics. The results extend the front-tracking framework beyond homogeneous flux, provide a practical scheme for constructing entropy solutions in challenging regimes, and offer insights into unbounded and non-principle-fulfilling dynamics in heterogeneous media.

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 , 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 estimates and compactness--inapplicable. However, the flux remains bounded despite blowing up, and our front tracking scheme exploits this to construct approximations that converge to an entropy solution.

Paper Structure

This paper contains 12 sections, 7 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Structural assumptions
  4. Piecewise stationary data
  5. Generalised Riemann problem
  6. Front tracking
  7. The Cauchy problem
  8. A priori estimates
  9. Passing to the limit
  10. Application of the method
  11. Unbounded solutions
  12. Acknowledgements

Key Result

Lemma 2.1

Suppose $f$ satisfies S0-C2-UC-FSP. Then for all $a> 0$, there exist two global-in-time, classical, stationary solutions $u_a^{\pm}$ of claw, i.e. $\partial_tu_a^{\pm}=0$, such that $f(x,u_a^{\pm}(x,t))\equiv a$. If $a=0$, then $u_a\equiv0$.

Theorems & Definitions (16)

  • Lemma 2.1: Existence of stationary solutions
  • proof
  • Lemma 2.2: Uniform inversion bounds
  • proof
  • Theorem 2.3: Solution of the generalised Riemann problem
  • proof
  • Definition 1
  • Definition 2
  • Theorem 2.4: Existence of front tracking approximations
  • proof
  • ...and 6 more