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Scalar conservation laws with discontinuous flux: existence and uniqueness

Darko Mitrovic

TL;DR

The article addresses the well-posedness of multidimensional scalar conservation laws with spatially discontinuous flux, proving existence and uniqueness of entropy solutions under a framework where the flux vanishes at the admissible states $a$ and $b$ and discontinuities occur along a finite set of smooth interfaces. The authors develop a vanishing-viscosity strategy with a local flattening of interfaces and a radial extension of the flux to establish local entropy admissibility and then patch these results to obtain global well-posedness; for general flux in $BV$ with spatial discontinuities, they construct an $(\varepsilon_k)$-vanishing viscosity germ that yields a complete, $L^1$-stable family of solutions corresponding to given initial data. The work extends the entropy framework for discontinuous flux to higher dimensions and provides a robust mechanism (the germ) to understand limits of viscous regularizations, clarifying the role of interface conditions in ensuring uniqueness. These results have practical implications for modeling transport in heterogeneous media and networks, and offer a foundation for stable numerical schemes that respect entropy admissibility at interfaces.

Abstract

We study the well-posedness of the Cauchy problem for scalar conservation laws with discontinuous, non-degenerate fluxes. Locally, the fluxes are piecewise smooth across interfaces described by a Heaviside-type discontinuity, with left and right states depending smoothly on both space and the solution variable. The interface is given by a smooth function, and the fluxes vanish at the boundary values of the admissible interval for the solution. In addition, we consider the more general case of heterogeneous flux functions with bounded variation in the spatial variable and smooth dependence on the solution variable, again vanishing at the prescribed boundary states. For this setting, we construct a stable semigroup of solutions, thereby establishing a well-posed solution framework.

Scalar conservation laws with discontinuous flux: existence and uniqueness

TL;DR

The article addresses the well-posedness of multidimensional scalar conservation laws with spatially discontinuous flux, proving existence and uniqueness of entropy solutions under a framework where the flux vanishes at the admissible states and and discontinuities occur along a finite set of smooth interfaces. The authors develop a vanishing-viscosity strategy with a local flattening of interfaces and a radial extension of the flux to establish local entropy admissibility and then patch these results to obtain global well-posedness; for general flux in with spatial discontinuities, they construct an -vanishing viscosity germ that yields a complete, -stable family of solutions corresponding to given initial data. The work extends the entropy framework for discontinuous flux to higher dimensions and provides a robust mechanism (the germ) to understand limits of viscous regularizations, clarifying the role of interface conditions in ensuring uniqueness. These results have practical implications for modeling transport in heterogeneous media and networks, and offer a foundation for stable numerical schemes that respect entropy admissibility at interfaces.

Abstract

We study the well-posedness of the Cauchy problem for scalar conservation laws with discontinuous, non-degenerate fluxes. Locally, the fluxes are piecewise smooth across interfaces described by a Heaviside-type discontinuity, with left and right states depending smoothly on both space and the solution variable. The interface is given by a smooth function, and the fluxes vanish at the boundary values of the admissible interval for the solution. In addition, we consider the more general case of heterogeneous flux functions with bounded variation in the spatial variable and smooth dependence on the solution variable, again vanishing at the prescribed boundary states. For this setting, we construct a stable semigroup of solutions, thereby establishing a well-posed solution framework.

Paper Structure

This paper contains 3 sections, 3 theorems, 32 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Entropy admissible solutions
  3. $(\varepsilon_k)$-vanishing viscosity germ

Key Result

Theorem 2.1

Under the non-degeneracy assumption non-deg, conditions (a), (b), and (c), and local representation of the flux local-flux, there exists an entropy admissible solution $u$ to eqmain in the sense that there exists a bounded function $p_u:\Bbb{R}^+\times\Bbb{R}^d\to \Bbb{R}$, $p_u=u$ almost everywhere

Figures (1)

  • Figure 1: The discontinuity line is the blue ellipse. $B_i$ and $B_j$ are balls in which we have the representation \ref{['local-flux']}. Green balls are the set $Q_1^\epsilon$ i.e. they turn into $Q_1$ as $\epsilon\to 0$. Red balls are the set $Q_2^\epsilon$ i.e. they turn into $Q_2$ as $\epsilon\to 0$.

Theorems & Definitions (6)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • Definition 3.1
  • Theorem 3.2
  • proof