2024/07/03 by Robin Ming Chen, Alexis Vasseur, Chen, Robin Ming +3 · 1 citation
Computer Science · Engineering · Mathematics · #35L45 #35L65 #76N10 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2407.02927
openalex publication_date 2024/07/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we show that a geometrical condition on 2×2 systems of conservation laws leads to non-uniqueness in the class of 1D continuous functions. This demonstrates that the Liu Entropy Condition alone is insufficient to guarantee uniqueness, even within the mono-dimensional setting. We provide examples of systems where this pathology holds, even if they verify stability and uniqueness for small BV solutions. Our proof is based on the convex integration process. Notably, this result represents the first application of convex integration to construct non-unique continuous solutions in one dimension.