Failure of uniqueness for scalar conservation laws
In this article, we develop the first negative results for scalar conservation laws with finite speed of propagation. While the work of Gargyants, Goritsky, and Panov establishes non-uniqueness of unbounded solutions for scalar conservation laws with spatially homogeneous flux, heterogeneity allows entropy solutions to exhibit more pathological behaviours. In particular, even for smooth and bounded initial data, the entropy solution may not be unique in the full half plane. Furthermore, the multiple entropy solutions all satisfy a finite speed of propagation property. 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 inequalities 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 detail examples demonstrating the sharpness of our assumptions and construct an explicit example of global ill-posedness with bounded initial datum.