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Three-dimensional stochastic Navier–Stokes equations with Markov switching

Po-Han Hsu

https://orcid.org/0000-0003-3010-6853

Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung City 80424, Taiwan

E-mail Address: phsu2@math.nsysu.edu.tw

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and

Padmanabhan Sundar

https://orcid.org/0000-0003-1388-7555

316 Lockett Hall, Louisiana State University, Baton Rouge, LA 70803-4918, USA

E-mail Address: psundar@lsu.edu

Corresponding author.

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https://doi.org/10.1142/S0219493723500570Cited by:0 (Source: Crossref)

Abstract

A finite-state Markov chain is introduced in the noise terms of the three-dimensional stochastic Navier–Stokes equations in order to allow for transitions between two types of multiplicative noises. We call such systems as stochastic Navier–Stokes equations with Markov switching. To solve such a system, a family of regularized stochastic systems is introduced. For each such regularized system, the existence of a unique strong solution (in the sense of stochastic analysis) is established by the method of martingale problems and pathwise uniqueness. The regularization is removed in the limit by obtaining a weakly convergent sequence from the family of regularized solutions, and identifying the limit as a solution of the three-dimensional stochastic Navier–Stokes equation with Markov switching.

Keywords:

AMSC: 60H15, 76D05

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