2024/10/04 by Thérèse Moerschell, Massimo Sorella, Moerschell, Thérèse +1
Computer Science · Engineering · Mathematics · #35A02 #35Q49 #76F25 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2410.03308
openalex publication_date 2024/10/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present a novel example of a divergence-free velocity field b ∈ L^∞ ((0,1); Lp (\mathbbT2)) for p<2 arbitrary but fixed which leads to non-unique solutions of advection-diffusion in the class L^∞t,x ∩ L2t H1x while satisfying the local energy inequality. This result complements the known uniqueness result of bounded solutions for divergence-free and L2t,x integrable velocity fields. Additionally, we also prove the necessity of time integrability of the velocity field for the uniqueness result. More precisely, we construct another divergence-free velocity field b ∈ Lp ((0,1); L^∞ (\mathbbT2)), for p< 2 fixed, but arbitrary, with non-unique aforementioned solutions. Our contribution closes the gap between the regime of uniqueness and non-uniqueness in this context. Previously, it was shown with the convex integration technique that for d≥ 3 divergence-free velocity fields b ∈ L^∞((0,1);Lp (\mathbbTd)) with p < (2d)/(d+2) could lead to non-unique solutions in the space L^∞t L(2d)/(d-2)x ∩ L2t H1x. Our proof is based on a stochastic Lagrangian approach and does not rely on convex integration.