Regularity for parabolic equations with singular non-zero divergence vector fields

TL;DR

This paper derives two-sided Gaussian bounds for the heat kernel of divergence-form parabolic equations with singular, time-dependent vector fields under minimal assumptions, advancing understanding of such equations with irregular coefficients.

Contribution

It provides the first two-sided Gaussian bounds for heat kernels associated with parabolic equations having singular, non-zero divergence vector fields.

Findings

Established two-sided Gaussian bounds for heat kernels

Extended analysis to equations with singular, time-inhomogeneous vector fields

Demonstrated minimal assumptions suffice for bounds

Abstract

We establish two-sided Gaussian bounds on the heat kernel of divergence-form parabolic equation with singular time-inhomogeneous vector field satisfying some minimal assumptions.