On the use of tent spaces for solving PDEs: A proof of the Koch-Tataru theorem

TL;DR

This paper presents a new proof of the Koch-Tataru theorem on Navier-Stokes equations using tent spaces theory, highlighting its potential for broader applications in fluid mechanics PDEs.

Contribution

It offers a novel proof of the Koch-Tataru theorem employing tent spaces, simplifying the understanding of Navier-Stokes well-posedness in critical spaces.

Findings

Proof of the Koch-Tataru theorem using tent spaces

Demonstrates the applicability of tent spaces in fluid mechanics PDEs

Simplifies the existing proof of Navier-Stokes well-posedness

Abstract

In these notes we will present (a part of) the parabolic tent spaces theory and then apply it in solving some PDE's originated from the fluid mechanics. In more details, to our most interest are the incompressible homogeneous Navier-Stokes equations. These equations have been investigated mathematically for almost one century. Yet, the question of proving well-posedness (i.e. existence, uniqueness and regularity of solutions) lacks satisfactory answer. A large part of the known positive results in connection with Navier-Stokes equations are those in which the initial data $u_0$ is supposed to have a small norm in some critical or scaling invariant functional space. All those spaces are embedded in the homogeneous Besov space $\dot B^{-1}_{\infty,\infty}.$. A breakthrough was made in the paper [16] by Koch and Tataru, where the authors showed the existence and the uniqueness of solutions to the Navier-Stokes system in case when the norm $\|u_0\|_{\mathrm{BMO}^{-1}}$ is small enough. The principal goal of these notes is to present a new proof of the theorem by Koch and Tataru on the Navier-Stokes system, namely the one using the tent spaces theory. We also hope that after having read these notes, the reader will be convinced that the theory of tent spaces is highly likely to be useful in the study of other equations in fluid mechanics. These notes are mainly based on the content of the article [1] by P. Auscher and D. Frey. However, in [1] the authors deal with a slightly more general system of parabolic equations of Navier-Stokes type. Here we have chosen to write down a self-contained text treating only the relatively easier case of the classical incompressible homogeneous Navier-Stokes equations.