Fundamental solutions for second order parabolic systems with drift terms

TL;DR

This paper constructs fundamental solutions for second-order parabolic systems with divergence-free drift terms in BMO^{-1}_x, establishing their Gaussian bounds under certain local boundedness conditions, advancing understanding of such PDEs.

Contribution

It introduces a method to construct fundamental solutions for parabolic systems with divergence-free drifts in BMO^{-1}_x and proves Gaussian bounds for these solutions.

Findings

Constructed fundamental solutions for the specified parabolic systems.

Established Gaussian upper bounds for these fundamental solutions.

Provided conditions under which weak solutions satisfy local boundedness estimates.

Abstract

We construct fundamental solutions of second-order parabolic systems of divergence form with bounded and measurable leading coefficients and divergence free first-order coefficients in the class of $BMO^{-1}_x$, under the assumption that weak solutions of the system satisfy a certain local boundedness estimate. We also establish Gaussian upper bound for such fundamental solutions under the same conditions.