The second-order parabolic PDEs with singular coefficients and applications

Abstract

The goal of this paper is to establish the Lipschitz and $W^{2,\infty }$ estimates for a second-order parabolic PDE ${\partial }_{t}u(t,x)=\frac{1}{2}\Delta u(t,x)+f(t,x)$ on ${\mathbb{R}}^d$ 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.

Keywords:

• Second-order parabolic PDE
• stochastic heat equation
• stochastic flow
• stochastic differential equation
• Sobolev differentiability

AMS SUBJECT CLASSIFICATION:

• 60H15
• 35K08

Acknowledgment

The authors would like to thank the anonymous referees for their careful reading of the text and their subsequent corrections and useful suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

R.R. Tian is supported by the National Natural Science Foundation of China grant 11901442. J.L. Wei is supported by the National Natural Science Foundation of China grant 11501577. Y.B. Tang is supported by the National Natural Science Foundation of China grants 11971188, 11471129.