A remark on global solutions to random 3D vorticity equations for small initial data

Michael Röckner; Rongchan Zhu; Xiangchan Zhu

doi:10.3934/dcdsb.2019048

Article Contents

2019, Volume 24, Issue 8: 4021-4030. Doi: 10.3934/dcdsb.2019048

This issue Previous Article Trajectory and global attractors for generalized processes Next Article Nonlocal time-porous medium equation: Weak solutions and finite speed of propagation

A remark on global solutions to random 3D vorticity equations for small initial data

a.
Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China
b.
School of Science, Beijing Jiaotong University, Beijing 100044, China
c.
Department of Mathematics, University of Bielefeld, D-33615 Bielefeld, Germany
d.
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

^* Corresponding author
^* Corresponding author

Received: March 2018

Revised: May 2018

Early access: February 19, 2019

Published online: February 19, 2019

Supported in part by NSFC (11671035, 11771037). Financial support by the DFG through the CRC 1283"Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications" is acknowledged.

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Abstract

In this paper, we prove that the solution constructed in [2] satisfies the stochastic vorticity equations with the stochastic integration being understood in the sense of the integration of controlled rough path introduced in [8]. As a result, we obtain the existence and uniqueness of the global solutions to the stochastic vorticity equations in 3D case for the small initial data independent of time, which can be viewed as a stochastic version of the Kato-Fujita result (see [10]).

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Mathematics Subject Classification: 60H15, 82C28.

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References

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