The Unique Limit of the Glimm-Lax Construction for Sobolev Data and Obstructions to 1-d Convex Integration

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 [preprint (2024)] 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.