The second-order parabolic PDEs with singular coefficients and applications.

The goal of this paper is to establish the Lipschitz and W 2 , ∞ estimates for a second-order parabolic PDE ∂ t u (t , x) = 1 2 Δ u (t , x) + f (t , x) on R d with zero initial data and f satisfying a Ladyzhenskaya–Prodi–Serrin type condition. Following the theoretic result, we then give two applications. The first is to discuss the regularity of the stochastic heat equations, and the second is to discuss the Sobolev differentiability of strong solutions to a class of SDEs with singular drift coefficients. [ABSTRACT FROM AUTHOR]