Estimation of non-uniqueness and short-time asymptotic expansions for Navier-Stokes flows

TL;DR

This paper investigates the potential non-uniqueness of solutions to the 3D Navier-Stokes equations, providing estimates on how quickly solutions can diverge and introducing a new short-time asymptotic expansion method.

Contribution

It introduces an algebraic estimate for the divergence rate of non-unique solutions and develops a novel local short-time asymptotic expansion technique.

Findings

Quantifies the separation rate of possibly non-unique solutions.

Provides a new local short-time asymptotic expansion method.

Addresses solution behavior in critical and sub-critical spaces.

Abstract

There is considerable evidence that solutions to the non-forced 3D Navier-Stokes equations in the natural energy space are not unique. Assuming this is the case, it becomes important to quantify how non-uniqueness evolves. In this paper we provide an algebraic estimate for how rapidly two possibly non-unique solutions can separate over a compact spatial region in which the initial data has sub-critical regularity. Outside of this compact region, the data is only assumed to be in the scaling critical weak Lebesgue space and can be large. In order to establish this separation rate, we develop a new spatially local, short-time asymptotic expansion which is of independent interest.