On Kato's method for Navier--Stokes Equations

TL;DR

This paper analyzes Kato's method for solving Navier-Stokes equations, providing a unified framework, new proofs, and extending existence and uniqueness results to rough initial data and irregular domains.

Contribution

It offers a generalization of Kato's method using real interpolation, unifies existing results, and establishes new existence and uniqueness theorems for complex initial data and domains.

Findings

Unified approach to Navier-Stokes via Kato's method

New existence and uniqueness results for rough data

Generalized proofs using real interpolation

Abstract

We investigate Kato's method for parabolic equations with a quadratic non-linearity in an abstract form. We extract several properties known from linear systems theory which turn out to be the essential ingredients for the method. We give necessary and sufficient conditions for these conditions and provide new and more general proofs, based on real interpolation. In application to the Navier-Stokes equations, our approach unifies several results known in the literature, partly with different proofs. Moreover, we establish new existence and uniqueness results for rough initial data on arbitrary domains in ${\mathbb R}^3$ and irregular domains in ${\mathbb R}^n$.