Stochastic Primitive Equations with Horizontal Viscosity and Diffusivity

2021/09/29 by Saal, Martin, Slavík, Jakub
#35A01 #35K65 #35M10 #35Q35 (primary) #35Q86 #35R60 #60H15 #76D03 #86A05 #86A10 (secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2109.14568

Abstract

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, G⊂ ℝ² 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 H¹_z L²_xy 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.

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