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

Abstract

We obtain two-sided Gaussian heat kernel bounds for divergence-form parabolic equation with singular (form-bounded type) time-inhomogeneous vector field that can have singular divergence. In the case of the heat equation (in absence of assumptions on the divergence of the vector field), we establish the existence of a unique weak solution, unifying existing solution theories of this equation.