The unique limit of the Glimm-Lax construction for Sobolev data and obstructions to 1-d convex integration

TL;DR

This paper proves the uniqueness of Sobolev initial data solutions for 1D hyperbolic conservation laws within the Glimm-Lax framework, extending stability results and addressing non-uniqueness in continuous solutions.

Contribution

It establishes the uniqueness of Sobolev space solutions in the Glimm-Lax class and shows non-uniqueness results do not extend beyond certain fractional Sobolev spaces.

Findings

Solutions with Sobolev initial data are unique in the Glimm-Lax class.

Non-uniqueness results for continuous solutions do not apply to $C^eta$ solutions with $eta > 1/2$.

Developed a weighted relative entropy contraction for rarefaction wave perturbations.

Abstract

We consider a genuinely nonlinear $1$-d system of hyperbolic conservation laws with two unknowns. A famous construction of Glimm & Lax shows that global-in-time "Glimm-Lax" weak entropy solutions exist in this setting for any initial data with small $L^\infty$ norm [Mem. Amer. Math. Soc. (1970), no. 101]. Recent work in the $L^1$-stability theory by Bressan, Marconi & Vaidya has given the first partial uniqueness and stability results for these solutions [Arch. Ration. Mech. Anal. (2025), vol. 249]. In this paper, we build on these results by combining them with recent advances in the $L^2$-theory. We show that solutions with initial data in the Sobolev space $H^s$ for $s>0$ are unique in the full class of Glimm--Lax solutions that decay in total variation at a rate of $1/t$. As a secondary result, our techniques are also used to show the recent non-uniqueness result of Chen, Vasseur & Yu for continuous solutions (arxiv:2407.02927) cannot extend to $C^\alpha$ solutions for $\alpha > 1/2$, alongside some appropriate fractional Sobolev spaces $W^{s,p}$. An auxiliary result of independent interest is the development of a weighted relative entropy contraction for perturbations of rarefaction waves.