Remarks on backward uniqueness of parabolic equations and incompressible Navier-Stokes well-posedness

TL;DR

This paper critically examines the backward uniqueness theory for parabolic equations and its implications for Navier-Stokes equations, identifying errors and misconceptions in prior proofs and analyses.

Contribution

It highlights technical errors in existing proofs of backward uniqueness for parabolic equations and clarifies misconceptions in Navier-Stokes well-posedness related to these results.

Findings

Identification of errors in backward uniqueness proofs

Clarification of misconceptions in Navier-Stokes analysis

Emphasis on correct application of parabolic scalings

Abstract

We explain why the theory of Escauriaza, Seregin, and Sverak (Russian Math. Surveys, 2003) on potential finite time singularity in Navier-Stokes solutions must be largely misapprehended. It is found that the proofs of the backward uniqueness theorem for parabolic equations contain technical errors. The stated validity of a theorem for vorticity is established on ill-informed analyses as the solenoidal constraint is not taken into account. There are many cases where parabolic scalings are erroneously applied. We briefly discuss a number of related issues.