Weak Solutions of McKean–Vlasov SDEs with Supercritical Drifts

Xicheng Zhang

doi:10.1007/s40304-021-00277-0

Weak Solutions of McKean–Vlasov SDEs with Supercritical Drifts

Xicheng Zhang^*

^*此作品的通讯作者

Wuhan University

科研成果: 期刊稿件 › 文章 › 同行评审

4 引用（Scopus）

摘要

Consider the following McKean–Vlasov SDE: (Formula presented.) where μX_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 dp+2q<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.

源语言	英语
页（从-至）	1-14
页数	14
期刊	Communications in Mathematics and Statistics
卷	12
期	1
DOI	https://doi.org/10.1007/s40304-021-00277-0
出版状态	已出版 - 3月 2024
已对外发布	是

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title = "Weak Solutions of McKean{\textendash}Vlasov SDEs with Supercritical Drifts",

abstract = "Consider the following McKean{\textendash}Vlasov SDE: (Formula presented.) where μXt stands for the distribution of Xt and K(t,x):R+{\texttimes}Rd→Rd is a time-dependent divergence free vector field. Under the assumption K∈Ltq(L\textasciitilde{}xp) with dp+2q<2, where L\textasciitilde{}xp stands for the localized Lp-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{\textendash}Stokes equations with measure as initial vorticity.",

keywords = "2D Navier{\textendash}Stokes equation, 35K55, 60H10, Krylov{\textquoteright}s estimate, McKean{\textendash}Vlasov system, Supercritical drift",

author = "Xicheng Zhang",

note = "Publisher Copyright: {\textcopyright} School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature 2023.",

year = "2024",

month = mar,

doi = "10.1007/s40304-021-00277-0",

language = "English",

volume = "12",

pages = "1--14",

journal = "Communications in Mathematics and Statistics",

issn = "2194-6701",

publisher = "Springer Verlag",

number = "1",

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TY - JOUR

T1 - Weak Solutions of McKean–Vlasov SDEs with Supercritical Drifts

AU - Zhang, Xicheng

N1 - Publisher Copyright: © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature 2023.

PY - 2024/3

Y1 - 2024/3

N2 - Consider the following McKean–Vlasov SDE: (Formula presented.) where μXt stands for the distribution of Xt and K(t,x):R+×Rd→Rd is a time-dependent divergence free vector field. Under the assumption K∈Ltq(L~xp) with dp+2q<2, where L~xp stands for the localized Lp-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.

AB - Consider the following McKean–Vlasov SDE: (Formula presented.) where μXt stands for the distribution of Xt and K(t,x):R+×Rd→Rd is a time-dependent divergence free vector field. Under the assumption K∈Ltq(L~xp) with dp+2q<2, where L~xp stands for the localized Lp-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.

KW - 2D Navier–Stokes equation

KW - 35K55

KW - 60H10

KW - Krylov’s estimate

KW - McKean–Vlasov system

KW - Supercritical drift

UR - https://www.scopus.com/pages/publications/85146153418

U2 - 10.1007/s40304-021-00277-0

DO - 10.1007/s40304-021-00277-0

M3 - Article

AN - SCOPUS:85146153418

SN - 2194-6701

VL - 12

SP - 1

EP - 14

JO - Communications in Mathematics and Statistics

JF - Communications in Mathematics and Statistics

IS - 1

ER -

Weak Solutions of McKean–Vlasov SDEs with Supercritical Drifts

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