The second iterate for the Navier-Stokes equation

TL;DR

This paper studies the second iterative step in solving the Navier-Stokes equations, revealing new insights into solution properties, uniqueness, and instability in specific function spaces.

Contribution

It introduces a novel analysis of the second iterate map, offering fresh perspectives on Koch-Tataru solutions and establishing instability results.

Findings

Boundedness properties of the bilinear operator are characterized.

Provides a new perspective on Koch-Tataru solutions.

Establishes an instability result in $B^{-1}_{inite,q}$ for q>2.

Abstract

We consider the iterative resolution scheme for the Navier-Stokes equation, and focus on the second iterate, more precisely on the map from the initial data to the second iterate at a given time t. We investigate boundedness properties of this bilinear operator. This new approach yields very interesting results: a new perspective on Koch-Tataru solutions; a first step towards weak strong uniqueness for Koch-Tataru solutions; and finally an instability result in $B^{-1}_{\infty,q}$, for q>2.