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Multidimensional Scalar Conservation Laws with Non-Aligned Discontinuous Flux and Singularity of Solutions

Ajlan Zajmović

TL;DR

The paper addresses multidimensional scalar conservation laws with discontinuous, non-aligned flux across a sharp interface and zero initial data. By employing a vanishing viscosity approximation with smooth Heaviside regularization, it proves the boundedness and non-positivity of the approximations and demonstrates that a subsequence converges, in the sense of Radon measures, to a limit that is singular and supported on the flux discontinuity surface. The key finding is that the vanishing viscosity limit yields a measure-valued concentration on the interface, with $u(D)<0$, highlighting a delta-like singularity (delta-shock) that cannot be captured by classical $L^1$-solutions. This work connects to shadow-wave and Colombeau-type approaches and suggests a need for generalized solution concepts to fully describe the interface dynamics, potentially guiding future numerical schemes for capturing interface concentrations in multidimensional settings.

Abstract

We prove that the family of solutions to vanishing viscosity approximation for multidimensional scalar conservation laws with discontinuous non-aligned flux and zero initial data in the limit generates a singular measure supported along the discontinuity surface.

Multidimensional Scalar Conservation Laws with Non-Aligned Discontinuous Flux and Singularity of Solutions

TL;DR

The paper addresses multidimensional scalar conservation laws with discontinuous, non-aligned flux across a sharp interface and zero initial data. By employing a vanishing viscosity approximation with smooth Heaviside regularization, it proves the boundedness and non-positivity of the approximations and demonstrates that a subsequence converges, in the sense of Radon measures, to a limit that is singular and supported on the flux discontinuity surface. The key finding is that the vanishing viscosity limit yields a measure-valued concentration on the interface, with , highlighting a delta-like singularity (delta-shock) that cannot be captured by classical -solutions. This work connects to shadow-wave and Colombeau-type approaches and suggests a need for generalized solution concepts to fully describe the interface dynamics, potentially guiding future numerical schemes for capturing interface concentrations in multidimensional settings.

Abstract

We prove that the family of solutions to vanishing viscosity approximation for multidimensional scalar conservation laws with discontinuous non-aligned flux and zero initial data in the limit generates a singular measure supported along the discontinuity surface.

Paper Structure

This paper contains 17 sections, 5 theorems, 84 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation and Context
  3. Overview and Main Contributions
  4. Notions and Notation
  5. Heaviside Function and Mollifications
  6. Sign Functions and Their Variants
  7. Flux Functions with Discontinuity Interfaces
  8. Function Spaces
  9. Distributional Derivatives and Non-Smooth Terms
  10. Regularized Absolute Value
  11. Sign $\lesssim$
  12. Vector notation with $[\cdot]$
  13. Weak Convergence Notation
  14. Generic Constants and weak lower semicontinuity
  15. A vanishing viscosity approximation
  16. ...and 2 more sections

Key Result

Theorem 1.1

The family of solutions $(u_\varepsilon)$ to viscosity, icviscosity is bounded in $L^1(\mathbb{R}^+\times \mathbb{R}^d)$. The weak subsequential limit $u$ of $(u_\varepsilon)$ in the space of Radon measure ${\mathcal{M}}_{loc}(\mathbb{R}^+\times \mathbb{R}^d))$ is singular on the discontinuity inter

Figures (1)

  • Figure 1: Schematic representation of the flux discontinuity surface in 2D case

Theorems & Definitions (8)

  • Theorem 1.1
  • Definition 2.1: Weak Lower Semicontinuity
  • Theorem 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Remark 3.4