Date of Graduation

5-2023

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics (PhD)

Degree Level

Graduate

Department

Mathematical Sciences

Advisor/Mentor

Zachary Bradshaw

Committee Member

Ariel Barton

Second Committee Member

Andrew Seth Raich

Keywords

Asymptotics, Local Energy, Navier-Stokes, non-uniqueness, Self-Similar, Separation

Abstract

In this dissertation, we investigate asymptotic properties of local energy solutions to the Navier-Stokes equations and develop an application which controls the separation of non-unique solutions in this class. Specifically, we quantify the rate at which two, possibly unique solutions evolving from the same data may separate pointwise away from a singularity. This is motivated by recent results on non-uniqueness for forced and unforced Navier-Stokes and analytical and numerical evidence suggesting non-uniqueness in the Leray class. Our investigation begins with discretely self-similar solutions known to exist globally in time and to be regular outside a space-time paraboloid. We prove decay rates for these solutions with locally sub-critical data away from the origin and show improved decay for the `non-linear part' of the flow. We also lower the H\"older regularity required to obtain our maximal decay rate. To achieve improved decay, we use Picard iterates to approximate solutions. We demonstrate a scale of decay rates for Picard approximations which determine upper bounds for how non-unique, discretely self-similar solutions may separate. In subsequent sections, we replace the self-similar condition with local sub-critical regularity and are able to obtain all but the maximal separation rate for Lorentz solutions, a subclass of local energy solutions.

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