Positive solutions of transport equations and classical nonuniqueness of characteristic curves

Abstract

The seminal work of DiPerna and Lions [Invent. Math., 98, 1989] guarantees the existence and uniqueness of regular Lagrangian flows for Sobolev vector fields. The latter is a suitable selection of trajectories of the related ODE satisfying additional compressibility/semigroup properties. A long-standing open question is whether the uniqueness of the regular Lagrangian flow is a corollary of the uniqueness of the trajectory of the ODE for a.e. initial datum. Using Ambrosio's superposition principle we relate the latter to the uniqueness of positive solutions of the continuity equation and we then provide a negative answer using tools introduced by Modena and Székelyhidi in the recent groundbreaking work [Ann. PDE, 4, 2018]. On the opposite side, we introduce a new class of asymmetric Lusin-Lipschitz inequalities and use them to prove the uniqueness of positive solutions of the continuity equation in an integrability range which goes beyond the DiPerna-Lions theory.

References in corpus (5)

• Dissipative continuous Euler flows
• Non-uniqueness and h-principle for Hölder-continuous weak solutions of the Euler equations
• Lagrangian flows for vector fields with gradient given by a singular integral
• Convex integration solutions to the transport equation with full dimensional concentration
• Sharp regularity estimates for solutions of the continuity equation drifted by Sobolev vector fields

Cited by in corpus (11)

• A surprising formula for Sobolev norms
• Non Uniqueness of power-law flows
• An Intermittent Onsager Theorem
• Weak stablity and closure in turbulence
• Distribution dependent SDEs driven by additive continuous noise
• Solution theory of fractional SDEs in complete subcritical regimes
• Nonuniqueness of weak solutions for the transport equation at critical space regularity
• The $\alpha$-SQG patch problem is illposed in $C^{2,\beta }$ and $W^{2,p}$
• On measure-preserving selection of solutions of ODEs
• Non-conservative $H^{\frac{1}{2}-}$ weak solutions of the incompressible 3D Euler equations
• Nontrivial absolutely continuous part of anomalous dissipation measures in time