On well-posedness for parabolic Cauchy problems of Lions type with rough initial data

TL;DR

This paper characterizes the well-posedness of parabolic Cauchy problems with rough initial data in Hardy--Sobolev and Besov spaces, providing a comprehensive framework for complex coefficients.

Contribution

It establishes well-posedness results for parabolic problems with initial data in Hardy--Sobolev and Besov spaces, extending previous theories to rough data and complex coefficients.

Findings

Identifies p-ranges for well-posedness with Hardy--Sobolev initial data.

Constructs solutions with gradients in weighted tent spaces.

Extends results to initial data in Besov spaces.

Abstract

We establish a complete picture for well-posedness of parabolic Cauchy problems with time-independent, uniformly elliptic, bounded measurable complex coefficients. We exhibit a range of $p$ for which tempered distributions in homogeneous Hardy--Sobolev spaces $\dot{H}^{s,p}$ with regularity index $s \in (-1,1)$ are initial data. Source terms of Lions' type lie in weighted tent spaces, and weak solutions are built with their gradients in weighted tent spaces as well. A similar result can be achieved for initial data in homogeneous Besov spaces $\dot{B}^{s}_{p,p}$.