Sharp Nonuniqueness in the Transport Equation with Sobolev Velocity Field

2024/05/02 by Elia Brué, Bruè, Elia, Maria Colombo +3 · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2405.01670

openalex publication_date 2024/05/02 · openalex created_date 2024/05/10 · openalex updated_date 2026/07/28

Abstract

Given a divergence-free vector field \bf u ∈ L^∞_t W^1,p_x(\mathbb R^d) and a nonnegative initial datum ρ₀ ∈ L^r, the celebrated DiPerna--Lions theory established the uniqueness of the weak solution in the class of L^∞_t L^r_x densities for (1)/(p) + (1)/(r) ≤ 1. This range was later improved in [BCDL21] to (1)/(p) + (d-1)/(dr) ≤ 1. We prove that this range is sharp by providing a counterexample to uniqueness when (1)/(p) + (d-1)/(dr) > 1. To this end, we introduce a novel flow mechanism. It is not based on convex integration, which has provided a non-optimal result in this context, nor on purely self-similar techniques, but shares features of both, such as a local (discrete) self similar nature and an intermittent space-frequency localization.

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