Well-posedness for the heat flow of harmonic maps and the liquid crystals flow with rough initial data

TL;DR

This paper studies the mathematical well-posedness of heat flow of harmonic maps and nematic liquid crystal flow with rough initial data in specific function spaces, extending understanding of these complex PDEs.

Contribution

It establishes well-posedness results for these flows with initial data in BMO and BMO^{-1} spaces, broadening the class of initial conditions for which solutions exist.

Findings

Proves well-posedness for harmonic map heat flow with BMO initial data.

Demonstrates well-posedness for nematic liquid crystal flow with BMO^{-1} initial data.

Extends previous results to rough initial data in non-smooth function spaces.

Abstract

We investigate the well-posedness of (i) the heat flow of harmonic maps from $R^n$ to a compact Riemannian manifold without boundary for initial data in BMO; and (ii) the hydrodynamic flow $(u,d)$ of nematic liquid crystals on $\mathbb R^n$ for initial data in $BMO^{-1}\times BMO$.