Approximate solutions to second order parabolic equations. I: Analytic estimates

Radu Constantinescu, Nick Costanzino, Anna L. Mazzucato, Victor Nistor
2010 Journal of Mathematical Physics  
We establish a new type of local asymptotic formula for the Green's function G_t(x,y) of a uniformly parabolic linear operator ∂_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 ∈ 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
more » ... 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( R^N) that appear in practice.
doi:10.1063/1.3486357 fatcat:5b2p42gxtzetnjbtajdxp72tr4