Nonuniqueness of weak solutions for the transport equation at critical space regularity

2020/04/20 by Cheskidov, Alexey, Luo, Xiaoyutao · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2004.09538

Abstract

We consider the linear transport equations driven by an incompressible flow in dimensions d≥ 3. For divergence-free vector fields u ∈ L¹_t W^1,q, the celebrated DiPerna-Lions theory of the renormalized solutions established the uniqueness of the weak solution in the class L^∞_t L^p when (1)/(p) + (1)/(q) ≤ 1. For such vector fields, we show that in the regime (1)/(p) + (1)/(q) > 1, weak solutions are not unique in the class L¹_t L^p. One crucial ingredient in the proof is the use of both temporal intermittency and oscillation in the convex integration scheme.

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