# A Local Energy Identity for Parabolic Equations with Divergence-Free Drift

## TL;DR

This paper establishes a local energy identity for certain solutions of parabolic equations with divergence-free drift, advancing understanding of their energy behavior in a distributional framework.

## Contribution

It introduces a local energy identity for distributional solutions of parabolic equations with divergence-free drift, a novel theoretical result.

## Key findings

- Proves a local energy identity for solutions in $L_{2,
abla} imes W^{1,0}_2$.
- Provides a new tool for analyzing parabolic equations with divergence-free drift.
- Enhances theoretical understanding of energy distribution in such PDEs.

## Abstract

We prove a local energy identity for a class of distributional solutions, in $L_{2,\infty} \cap W^{1,0}_2$, of parabolic equations with divergence-free drift.