Open Access
2023 Stochastic primitive equations with horizontal viscosity and diffusivity
Martin Saal, Jakub Slavík
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Electron. J. Probab. 28: 1-56 (2023). DOI: 10.1214/23-EJP940


We establish the existence and uniqueness of pathwise strong solutions to the stochastic 3D primitive equations with only horizontal viscosity and diffusivity driven by transport noise on a cylindrical domain M=(h,0)×G, GR2 bounded and smooth, with the physical Dirichlet boundary conditions on the lateral part of the boundary. Compared to the deterministic case where the uniqueness of z-weak solutions holds in L2, more regular initial data are necessary to establish uniqueness in the anisotropic space Hz1Lxy2 so that the existence of local pathwise solutions can be deduced from the Gyöngy-Krylov theorem. Global existence is established using the logarithmic Sobolev embedding, the stochastic Gronwall lemma and an iterated stopping time argument.

Funding Statement

The first author gratefully acknowledges the financial support of the Deutsche Forschungsgemeinschaft (DFG) through the research fellowship SA 3887/1-1.


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Martin Saal. Jakub Slavík. "Stochastic primitive equations with horizontal viscosity and diffusivity." Electron. J. Probab. 28 1 - 56, 2023.


Received: 18 October 2021; Accepted: 24 March 2023; Published: 2023
First available in Project Euclid: 12 April 2023

MathSciNet: MR4574481
zbMATH: 1517.35183
arXiv: 2109.14568
Digital Object Identifier: 10.1214/23-EJP940

Primary: 35Q35
Secondary: 35A01 , 35K65 , 35M10 , 35Q86 , 35R60 , 60H15 , 76D03 , 86A05 , 86A10

Keywords: horizontal viscosity , Multiplicative noise , Nonlinear stochastic PDE , primitive equations

Vol.28 • 2023
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