Tent space well-posedness for parabolic Cauchy problems with rough coefficients

TL;DR

This paper establishes well-posedness of higher order parabolic Cauchy problems with rough coefficients in certain function spaces, introducing a new class based on tent spaces and extending results to higher order systems.

Contribution

It identifies a new well-posedness class for higher order parabolic equations using tent spaces, extending $L^p$ well-posedness results to systems with rough coefficients.

Findings

Well-posedness class characterized by tent spaces for $p eq 2$

Higher order $L^p$ well-posedness established for $m > 1$

Local Hölder continuity of solutions for $p > 2$

Abstract

We study the well-posedness of Cauchy problems on the upper half space $\mathbb{R}^{n+1}_+$ associated to higher order systems $\partial_t u =(-1)^{m+1}\mbox{div}_m A\nabla ^m u$ with bounded measurable and uniformly elliptic coefficients. We address initial data lying in $L^p$ ($1<p<\infty$) and $BMO$ ($p=\infty$) spaces and work with weak solutions. Our main result is the identification of a new well-posedeness class, given for $p\in(1,\infty]$ by distributions satisfying $\nabla^m u \in T^{p,2}_m$, where $T^{p,2}_m$ is a parabolic version of the tent space of Coifman--Meyer--Stein. In the range $p\in [2,\infty]$, this holds without any further constraints on the operator and for $p=\infty$ it provides a Carleson measure characterization of $BMO$ with non-autonomous operators. We also prove higher order $L^p$ well-posedness, previously only known for the case $m = 1$. The uniform $L^p$ boundedness of propagators of energy solutions plays an important role in the well-podesness theory and we discover that such bounds hold for $p$ close to $2$. This is a consequence of local weak solutions being locally H\"older continuous with values in spatial $L^p_{loc}$ for some $p>2$, what is also new for the case $m>1$.