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 R+n+1 associated to higher order systems ∂tu=(−1)m+1divmA∇mu with bounded measurable and uniformly elliptic coefficients. We address initial data lying in Lp (1<p<∞) and BMO (p=∞) spaces and work with weak solutions. Our main result is the identification of a new well-posedness class, given for p∈(1,∞] by distributions satisfying ∇mu∈Tmp,2, where Tmp,2 is a parabolic version of the tent space of Coifman–Meyer–Stein. In the range p∈[2,∞], this holds without any further constraints on the operator and for p=∞ it provides a Carleson measure characterization of BMO with non-autonomous operators. We also prove higher order Lp well-posedness, previously only known for the case m=1. The uniform Lp boundedness of propagators of energy solutions plays an important role in the well-posedness 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 Llocp for some p>2, what is also new for the case m>1.