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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2601.01349 |
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| _version_ | 1866917183107366912 |
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| author | Cheng, Jeffrey Faile, Cooper Krupa, Sam G. |
| author_facet | Cheng, Jeffrey Faile, Cooper Krupa, Sam G. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_01349 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The unique limit of the Glimm-Lax construction for Sobolev data and obstructions to 1-d convex integration Cheng, Jeffrey Faile, Cooper Krupa, Sam G. Analysis of PDEs 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. |
| title | The unique limit of the Glimm-Lax construction for Sobolev data and obstructions to 1-d convex integration |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2601.01349 |