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

Abstract

We consider the linear transport equations driven by an incompressible flow in dimensions $d\ge 3$. For divergence-free vector fields $u\in L_t^1{W}^{1,q}$, the celebrated DiPerna-Lions theory of the renormalized solutions established the uniqueness of the weak solution in the class $L_t^{\infty }{L}^p$ when $\frac{1}{p}+\frac{1}{q}\le 1$. For such vector fields, we show that in the regime $\frac{1}{p}+\frac{1}{q}>1$, weak solutions are not unique in the class $L_t^1{L}^p$. One crucial ingredient in the proof is the use of both temporal intermittency and oscillation in the convex integration scheme.

30 pages; minor corrections per referee comments, to appear in annals of pde

References in corpus (7)

• Convex integration solutions to the transport equation with full dimensional concentration
• Positive solutions of transport equations and classical nonuniqueness of characteristic curves
• Sharp regularity estimates for solutions of the continuity equation drifted by Sobolev vector fields
• Non-unique weak solutions in Leray-Hopf class of the 3D Hall-MHD system
• Stationary and discontinuous weak solutions of the Navier-Stokes equations
• Non-uniqueness of Weak Solutions to the 3D Quasi-Geostrophic Equations
• Anomalous Dissipation in Passive Scalar Transport

Cited by in corpus (1)

• Sharp nonuniqueness for the Navier-Stokes equations