Heat kernel bounds for parabolic equations with singular (form-bounded) vector fields

TL;DR

This paper establishes Gaussian bounds for the heat kernel of a Kolmogorov operator with measurable elliptic coefficients and singular vector fields, under various minimal assumptions on the divergence and form-boundedness of the vector field.

Contribution

It provides new Gaussian heat kernel bounds for parabolic equations with singular, form-bounded vector fields, extending previous results to minimal regularity assumptions.

Findings

Gaussian lower bounds under non-negative divergence and form-bounded vector fields.

Gaussian upper bounds when the positive divergence part is in the Kato class.

Both upper and lower bounds when divergence is in the Kato class.

Abstract

We consider Kolmogorov operator $-\nabla \cdot a \cdot \nabla + b \cdot \nabla$ with measurable uniformly elliptic matrix $a$ and prove Gaussian lower and upper bounds on its heat kernel under minimal assumptions on the vector field $b$ and its divergence ${\rm div\,}b$. More precisely, we prove: (1) Gaussian lower bound, provided that ${\rm div\,}b \geq 0$, and $b$ is in the class of form-bounded vector fields (containing e.g.\,the class $L^d$, the weak $L^d$ class, as well as some vector fields that are not even in $L_{\rm loc}^{2+\varepsilon}$, $\varepsilon>0$); in these assumptions, the Gaussian upper bound is in general invalid; (2) Gaussian upper bound, provided that $b$ is form-bounded, and the positive part of ${\rm div\,}b$ is in the Kato class; in these assumptions, the Gaussian lower bound is in general invalid; (3) Gaussian upper and lower bounds, provided that $b$ is form-bounded, ${\rm div\,}b$ is in the Kato class; (4) A priori Gaussian upper and lower bounds, provided that $b$ is in a large class containing the class of form-bounded vector fields, ${\rm div\,}b$ is in the Kato class.