Approximate Solutions to Second Order Parabolic Equations I: analytic estimates

Abstract

We establish a new type of local asymptotic formula for the Green's function ${\mathcal G}_t(x,y)$ of a uniformly parabolic linear operator $\partial_t - L$ with non-constant coefficients using dilations and Taylor expansions at a point $z=z(x,y)$, for a function $z$ with bounded derivatives such that $z(x,x)=x \in {\mathbb R}^N$. For $z(x,y) =x$, we recover the known, classical expansion obtained via pseudo-differential calculus. Our method is based on dilation at $z$, Dyson and Taylor series expansions, and the Baker-Campbell-Hausdorff commutator formula. Our procedure leads to an elementary, algorithmic construction of approximate solutions to parabolic equations which are accurate to arbitrary prescribed order in the short-time limit. We establish mapping properties and precise error estimates in the exponentially weighted, $L^{p}$-type Sobolev spaces $W^{s,p}_a({\mathbb R}^N)$ that appear in practice.