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

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 [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.