We establish a new type of local asymptotic formula for the Green's function Gt(x,y) of a uniformly parabolic linear operator ∂t-L with nonconstant 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∈RN. Our method is based on dilation at z, Dyson, and Taylor series expansions. We use the Baker-Campbell-Hausdorff commutator formula to explicitly compute the terms in the Dyson series. Our procedure leads to an explicit, elementary, algorithmic construction of approximate solutions to parabolic equations that are accurate to arbitrarity prescribed order in the short-time limit. We establish mapping properties and precise error estimates in the exponentially weighed, Lp-type Sobolev spaces Was,p(RN) that appear in practice.
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics