Parabolic equations with singular divergence‐free drift vector fields

Author: 

Qian, Z
Xi, G

Publication Date: 

6 December 2018

Journal: 

Journal of the London Mathematical Society

Last Updated: 

2020-02-18T04:31:31.117+00:00

Issue: 

1

Volume: 

100

DOI: 

10.1112/jlms.12202

page: 

17-40

abstract: 

In this paper, we study an elliptic operator in divergence form but not necessarily symmetric. In particular, our results can be applied to elliptic operator L=νΔ+u(x,t)·∇, where u(·,t) is a time‐dependent vector field in ℝn, which is divergence‐free in the distributional sense, that is ∇·u=0. Suppose u∈L∞(0,∞;BMO−1(ℝn)). We show the existence of the fundamental solution Γ(x,t;ξ,τ) of the parabolic operator L−∂t and show that Γ satisfies the Aronson estimate with a constant depending only on the dimension n, the elliptic constant λ and the norm ∥u∥L∞t(BMO−1x). Therefore, the existence and uniqueness of solutions to the parabolic equation (L−∂t)v=0 are established for initial data in L2‐space, and their regularity theory is obtained too. In fact, we establish these results for a general non‐symmetric elliptic operator in divergence form.

Symplectic id: 

944246

Submitted to ORA: 

Submitted

Publication Type: 

Journal Article