Osgood meets Ambrosio-DiPerna-Lions

2026/07/30 by Guido De Philippis, Leonardo Franchi
Mathematics · #math.AP

paper · pdf

18 pages

arxiv created 2026/07/30 · arxiv updated 2026/07/31

Abstract

In this note we extend the Ambrosio-DiPerna-Lions theory on the well-posedness of the transport equation to the case in which the vector field satisfies an L^p Osgood condition. In particular we show that bounded distributional solutions to the transport equation are unique and thus renormalized. As opposed to the classical proof, we do not rely on the vanishing of a commutator, but we show that a weighted energy built out of the Littlewood-Paley decomposition of the solution is conserved.

Citations

• Quantitative Estimates in Passive Scalar Transport: A Unified Approach in W^1,p via Christ-Journé Commutator Estimates
• Propagation of regularity for transport equations. A Littlewood-Paley approach
• Complex interpolation of function spaces with general weights
• On the Sobolev space of functions with derivative of logarithmic order
• A new approach to bounds on mixing
• Multilinear Singular Integral Forms of Christ-Journé Type

Related

• Transport equations for Osgood velocity fields
• Renormalization of active scalar equations
• Quantitative Estimates in Passive Scalar Transport: A Unified Approach in W^1,p via Christ-Journé Commutator Estimates
• Nonuniqueness of weak solutions for the transport equation at critical space regularity
• Sharp Nonuniqueness in the Transport Equation with Sobolev Velocity Field