Weak Solutions of McKean–Vlasov SDEs with Supercritical Drifts

Xicheng Zhang

Communications in Mathematics and Statistics ›› 2023, Vol. 12 ›› Issue (1) : 1-14. DOI: 10.1007/s40304-021-00277-0
Article

Weak Solutions of McKean–Vlasov SDEs with Supercritical Drifts

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Abstract

Consider the following McKean–Vlasov SDE:

d X t = 2 d W t + R d K ( t , X t - y ) μ X t ( d y ) d t , X 0 = x ,
where
μ X t
stands for the distribution of
X t
and
K ( t , x ) : R + × R d R d
is a time-dependent divergence free vector field. Under the assumption
K L t q ( L ~ x p )
with
d p + 2 q < 2
, where
L ~ x p
stands for the localized
L p
-space, we show the existence of weak solutions to the above SDE. As an application, we provide a new proof for the existence of weak solutions to 2D Navier–Stokes equations with measure as initial vorticity.

Keywords

McKean–Vlasov system / Supercritical drift / 2D Navier–Stokes equation / Krylov’s estimate

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Xicheng Zhang. Weak Solutions of McKean–Vlasov SDEs with Supercritical Drifts. Communications in Mathematics and Statistics, 2023, 12(1): 1‒14 https://doi.org/10.1007/s40304-021-00277-0

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