On well-posedness of parabolic equations of Navier-Stokes type with BMO^{-1}(\R^n) data

TL;DR

This paper establishes well-posedness results for certain parabolic equations of Navier-Stokes type with initial data in BMO^{-1}( ^n), using tent space techniques, including a new proof for the classical Navier-Stokes case.

Contribution

It introduces a novel approach using tent spaces to analyze parabolic equations with quadratic nonlinearities, extending well-posedness results to models lacking kernel bounds or self-adjointness.

Findings

Reproved well-posedness of Navier-Stokes in BMO^{-1}( ^n) with a new proof.

Extended analysis to models without pointwise kernel bounds.

Developed a framework applicable to broader classes of parabolic equations.

Abstract

We develop a strategy making extensive use of tent spaces to study parabolic equa-tions with quadratic nonlinearities as for the Navier-Stokes system. We begin with a new proof of the well-known result of Koch and Tataru on the well-posedness of Navier-Stokes equations in \R^n with small initial data in BMO^{-1}(\R^n). We then study another model where neither pointwise kernel bounds nor self-adjointness are available.