A uniqueness lemma with applications to regularization and incompressible fluid mechanics

TL;DR

This paper proves a uniqueness property for transport equations with rough coefficients and applies it to establish uniqueness results for Euler and Navier-Stokes equations at Leray regularity, challenging certain weak solutions.

Contribution

It extends previous results to show uniqueness for transport equations with rough coefficients and applies this to fluid mechanics equations at critical regularity levels.

Findings

Proves uniqueness for transport equations with rough coefficients.

Establishes uniqueness for Euler and Navier-Stokes equations at Leray scale.

Provides barriers against certain weak solutions in fluid dynamics.

Abstract

In this paper, we extend our previous result from [16]. We prove that transport equations with rough coefficients do possess a uniqueness property. Our method relies strongly on duality and bears a strong resemblance with the well-known DiPerna-Lions theory first developed in [8]. As an application, we show a uniqueness result for the Euler and Navier-Stokes equations at the Leray regularity scale. In turn, this theorem stands as a barrier against the paradoxical weak solutions constructed in [17], [18], [19] and later reformulated in [6].