TY - JOUR

T1 - Approximate solutions to second order parabolic equations. I

T2 - Analytic estimates

AU - Constantinescu, Radu

AU - Costanzino, Nick

AU - Mazzucato, Anna L.

AU - Nistor, Victor

N1 - Funding Information:
We thank Andrew Lesniewski and Michael Taylor for sending us their papers and for useful discussions. We also thank Richard Melrose for carefully reading our paper and for pointing out a possible misunderstanding in an earlier version of this paper. A. L. M. was partially supported by the NSF under Grant No. DMS-0708902. V. N. was partially supported by the NSF under Grant Nos. DMS-0555831, DMS-0713743, and OCI-0749202.

PY - 2010/10

Y1 - 2010/10

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=78149447512&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78149447512&partnerID=8YFLogxK

U2 - 10.1063/1.3486357

DO - 10.1063/1.3486357

M3 - Article

AN - SCOPUS:78149447512

VL - 51

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 10

M1 - 103502

ER -