Pathwise Solutions for Stochastic Hydrostatic Euler Equations under the Local Rayleigh Condition

Abstract

The hydrostatic Euler equations are important in the study of atmospheric and oceanic dynamics in the planetary scale. While its deterministic version has been widely studied in the literature, its stochastic version is far less understood. In this paper, we consider the two-dimensional stochastic hydrostatic Euler equations with initial data that are random variables in a suitable Sobolev space satisfying the local Rayleigh condition. We establish local-in-time existence and uniqueness of maximal pathwise solutions. Our work provides the first result on existence and uniqueness in Sobolev spaces, and establishes the first existence of pathwise solutions to the stochastic hydrostatic Euler equations.

References in corpus (8)

• Local Martingale and Pathwise Solutions for an Abstract Fluids Model
• Solution properties of a 3D stochastic Euler fluid equation
• Global Strong Well-posedness of the Three Dimensional Primitive equations in $L^p$-spaces
• Global Existence and Regularity for the 3D Stochastic Primitive Equations of the Ocean and Atmosphere with Multiplicative White Noise
• Ill-posedness of the hydrostatic Euler and singular Vlasov equations
• On the effect of fast rotation and vertical viscosity on the lifespan of the $3D$ primitive equations
• On the Well-posedness of Reduced $3D$ Primitive Geostrophic Adjustment Model with Weak Dissipation
• Local Martingale Solutions and Pathwise Uniqueness for the Three-dimensional Stochastic Inviscid Primitive Equations