Failure of uniqueness for scalar conservation laws

TL;DR

This paper presents new negative results showing failure of uniqueness for scalar conservation laws under certain conditions, and introduces a novel approach with additional conditions to recover uniqueness, highlighting the importance of boundedness assumptions.

Contribution

The paper provides the first explicit examples of non-uniqueness and develops a new theory for scalar conservation laws with spatial heterogeneity using front tracking.

Findings

Existence of bounded initial data leading to $L^{inity}$ blow-up.

Failure of entropy equalities alone to ensure uniqueness.

Recovery of uniqueness with a Lax-type condition and front tracking.

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 Kru\v{z}kov'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 Kru\v{z}kov 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 Kru\v{z}kov entropy solutions exist. Finally, we construct an explicit example of global ill-posedness with bounded initial datum.