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Failure of uniqueness for scalar conservation laws

Shyam Sundar Ghoshal, Abraham Sylla, Parasuram Venkatesh

TL;DR

This work reveals that scalar conservation laws with spatially heterogeneous flux can exhibit fundamental pathologies absent in the homogeneous theory: bounded initial data can lead to $L^{\infty}$ blow-up and there may be infinitely many entropy solutions, showing that Kruzhkov’s doubling-of-variables argument relies on $L^{\infty}$ bounds. To restore a form of well-posedness, the authors develop a front-tracking framework for fluxes $f(x,u)=g(x)h(u)$ and prove uniqueness under a Lax-type interface condition, while also constructing explicit ill-posedness examples. The results demonstrate that flux structure, zero sets of $g$, and growth assumptions critically influence existence, uniqueness, and stability, and they establish a refined theory for flux-heterogeneous conservation laws that extends beyond the classical homogeneous case. The findings have implications for modeling with spatial heterogeneity and emphasize the need for additional admissibility criteria beyond entropy inequalities. These contributions collectively advance our understanding of well-posedness in heterogeneous scalar conservation laws and provide explicit mechanisms for when and why uniqueness can fail or be recovered.

Abstract

In this article, we develop what are, to the best of our knowledge, the first negative results for scalar conservation laws. We begin with explicit examples where bounded initial data leads to $L^{\infty}$ blow-up despite flux regularity. More strikingly, we demonstrate that Kružkov's entropy equalities alone fail to ensure uniqueness in this regime by constructing infinitely many entropy solutions to a single Cauchy problem with bounded initial datum, each continuous in time with respect to the $L^{1}$ norm. Thus, we demonstrate that the $L^{\infty}$ assumption is essential for the doubling of variables argument, and hence for the uniqueness of entropy solutions to scalar conservation laws. On the positive side, we develop a novel theory for scalar conservation laws with spatial heterogeneity by adapting the front tracking method. We recover uniqueness by imposing a Lax-type condition in addition to the entropy inequality, motivated by the properties of our front tracking approximations. Unbounded Kružkov solutions do not necessarily satisfy the weak formulation; we show that global weak solutions may not even exist in a natural class for some Cauchy problems of this form, even when Kružkov entropy solutions exist. Finally, we construct an explicit example of global ill-posedness with bounded initial datum.

Failure of uniqueness for scalar conservation laws

TL;DR

This work reveals that scalar conservation laws with spatially heterogeneous flux can exhibit fundamental pathologies absent in the homogeneous theory: bounded initial data can lead to blow-up and there may be infinitely many entropy solutions, showing that Kruzhkov’s doubling-of-variables argument relies on bounds. To restore a form of well-posedness, the authors develop a front-tracking framework for fluxes and prove uniqueness under a Lax-type interface condition, while also constructing explicit ill-posedness examples. The results demonstrate that flux structure, zero sets of , and growth assumptions critically influence existence, uniqueness, and stability, and they establish a refined theory for flux-heterogeneous conservation laws that extends beyond the classical homogeneous case. The findings have implications for modeling with spatial heterogeneity and emphasize the need for additional admissibility criteria beyond entropy inequalities. These contributions collectively advance our understanding of well-posedness in heterogeneous scalar conservation laws and provide explicit mechanisms for when and why uniqueness can fail or be recovered.

Abstract

In this article, we develop what are, to the best of our knowledge, the first negative results for scalar conservation laws. We begin with explicit examples where bounded initial data leads to blow-up despite flux regularity. More strikingly, we demonstrate that Kružkov's entropy equalities alone fail to ensure uniqueness in this regime by constructing infinitely many entropy solutions to a single Cauchy problem with bounded initial datum, each continuous in time with respect to the norm. Thus, we demonstrate that the assumption is essential for the doubling of variables argument, and hence for the uniqueness of entropy solutions to scalar conservation laws. On the positive side, we develop a novel theory for scalar conservation laws with spatial heterogeneity by adapting the front tracking method. We recover uniqueness by imposing a Lax-type condition in addition to the entropy inequality, motivated by the properties of our front tracking approximations. Unbounded Kružkov solutions do not necessarily satisfy the weak formulation; we show that global weak solutions may not even exist in a natural class for some Cauchy problems of this form, even when Kružkov entropy solutions exist. Finally, we construct an explicit example of global ill-posedness with bounded initial datum.
Paper Structure (21 sections, 14 theorems, 86 equations, 3 figures)

This paper contains 21 sections, 14 theorems, 86 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Preliminaries
  4. Structural assumptions
  5. Summary of results
  6. Well-posedness
  7. $\mathbf{L}^{\infty}$ blow-up of entropy solutions
  8. Non-uniqueness
  9. Entropy-preserving solution
  10. Non-uniqueness with one discontinuity
  11. Non-uniqueness from bounded initial data
  12. Non-existence of weak solutions
  13. A sufficient condition for blow-up of entropy solutions
  14. A sufficient condition for non-uniqueness of entropy solutions
  15. Front tracking approximations
  16. ...and 6 more sections

Key Result

Lemma 2.1

If $x\in G^c$ and $t>0$, then for any entropy solution $u$, the left and right spatial traces $u(x\pm,t)$ exist and satisfy the inequality $u(x-,t)\geq u(x+,t)$ if $g(x)>0$ and $u(x-,t)\leq u(x+,t)$ if $g(x)<0$.

Figures (3)

  • Figure 1: Characteristics associated with the Cauchy problem for \ref{['claw']} with $f(x,u)=xu^2$ and initial data given by \ref{['blowupdata']}. The $\mathbf{L}^{\infty}$ norm of the entropy solution $u$ blows up along $\{x=0,t\geq1\}$.
  • Figure 2: One of the infinitely many Kružkov entropy solutions to the Cauchy problem for \ref{['claw']} with flux $f(x,u)=xu^2$ and initial data given by \ref{['shocking nonuniqueness']}.
  • Figure 3: The entropy solution to the Cauchy problem for \ref{['claw']} with flux $f(x,u)=xu^2$ and initial data given by \ref{['bounded nonuniqueness']} evolves into the profile \ref{['shocking nonuniqueness']} at $t=2$, beyond which there are infinitely many entropic continuations.

Theorems & Definitions (25)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.3.1
  • Lemma 2.4: Trace at the interface
  • proof
  • Corollary 2.4.1
  • ...and 15 more