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

TL;DR

This work proves a sharp well-posedness theory for genuinely nonlinear 1-D 2×2 systems in the Glimm–Lax class with Sobolev initial data. By combining a refined $L^2$ stability framework with recent $L^1$-stability results, the authors show uniqueness of Glimm–Lax solutions for initial data in fractional Sobolev spaces $W^{s,p}_{loc}$, with explicit Hölder-type dependence on initial data and a decay rate in total variation. They also establish obstructions to extending non-uniqueness results to smoother spaces, proving that any periodic $C^$ solution with $> frac12$ that satisfies the entropy inequality must coincide with the GL solution. The analysis relies on a new relative entropy contraction for rarefaction perturbations and a strengthened $L^2$-based stability theory that broadens applicability to non-BV perturbations, offering new insights into the stability and uniqueness landscape for conservation laws beyond BV theory.

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 $C^α$ solutions for $α> 1/2$, 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.

Figures (2)

• Figure 1: A weak entropy solution is defined for short time by taking a limit of small-BV front tracking approximations on each trapezoid.
• Figure 2: In \ref{['fig:v-def-obs']} we cover the information cone in trapezoids. At the base of each trapezoid we define the approximant $v_i$, which has initial data approximating $u(i\tau,\cdot)$ and is also a solution to the system \ref{['cl']}. \ref{['fig:single-obs']} shows our strategy for estimating the norm $\left\lVert v_{i-1}(t_1,\cdot)-v_i(t_1,\cdot)\right\rVert_{L^1}$. For short time (of size $\tau$ or $2\tau$) we use \ref{['lem:trapezoidconstruction']} to estimate $\left\lVert u - v_i\right\rVert_{L^1}$ and $\left\lVert u-v_{i-1}\right\rVert_{L^1}$. From time $(i+1)\tau$ to $t_1$, we apply the $L^1$-Lipschitz bound from \ref{['prop:L1-bressan']}. This results in a much better estimate than simply iterating \ref{['lem:trapezoidconstruction']}.

Theorems & Definitions (19)

• Theorem 1.1
• Corollary 1.1
• Theorem 1.2
• Remark 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.3
• Proposition 2.1: 2025arXiv250500420B
• Proposition 2.2: 2025arXiv250500420B