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

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 ${\nabla }^{m}u\in T_m^{p,2}$, where $T_m^{p,2}$ 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 Hölder continuous with values in spatial $L_{loc}^p$ for some $p>2$, what is also new for the case $m>1$.

Accepted in J. Differential Equations (2020); corrected typos, small exposition changes in the introduction, adjusted references

Cited by in corpus (2)

• On well-posedness for parabolic Cauchy problems of Lions type with rough initial data
• On well-posedness and maximal regularity for parabolic Cauchy problems on weighted tent spaces