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

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 ${\dot{B}}_{p,p}^s$.

arXiv comment51 pages, 2 figures. Comments are welcome

References in corpus (2)

• Tent space well-posedness for parabolic Cauchy problems with rough coefficients
• On well-posedness and maximal regularity for parabolic Cauchy problems on weighted tent spaces

Cited by in corpus (1)

• On regularity of solutions to the Navier--Stokes equation with initial data in ${\mathrm{BMO}}^{-1}$