Forces for the Navier-Stokes equations and the Koch and Tataru theorem

TL;DR

This paper investigates the existence and interaction of solutions to the Navier-Stokes equations on , focusing on initial data in BMO^{-1} and forces with small norm, extending Koch and Tataru's theorem.

Contribution

It analyzes the interaction between different classes of solutions and establishes conditions for global existence with small forcing and initial data.

Findings

Existence of mild solutions under small forcing and initial data.

Interaction analysis between Koch-Tataru solutions and other solution classes.

Conditions for global solutions in presence of forcing and initial data.

Abstract

We consider the Cauchy problem for the incompressible Navier--Stokes equations on the whole space $\mathbb{R}^3$, with initial value $\vec u_0\in {\rm BMO}^{-1}$ (as in Koch and Tataru's theorem) and with force $\vec f=\Div \mathbb{F}$ where smallness of $\mathbb{F}$ ensures existence of a mild solution in absence of initial value. We study the interaction of the two solutions and discuss the existence of global solution for the complete problem (i.e. in presence of initial value and forcing term) under smallness assumptions. In particular, we discuss the interaction between Koch and Tataru solutions and Lei-Lin's solutions (in $L^2\mathcal{F}^{-1}L^1$) or solutions in the multiplier space $\mathcal{M}(\dot H^{1/2,1}_{t,x}\mapsto L^2_{t,x})$.