Global Existence and Regularity for the 3D Stochastic Primitive Equations of the Ocean and Atmosphere with Multiplicative White Noise

Abstract

The Primitive Equations are a basic model in the study of large scale Oceanic and Atmospheric dynamics. These systems form the analytical core of the most advanced General Circulation Models. For this reason and due to their challenging nonlinear and anisotropic structure the Primitive Equations have recently received considerable attention from the mathematical community. In view of the complex multi-scale nature of the earth's climate system, many uncertainties appear that should be accounted for in the basic dynamical models of atmospheric and oceanic processes. In the climate community stochastic methods have come into extensive use in this connection. For this reason there has appeared a need to further develop the foundations of nonlinear stochastic partial differential equations in connection with the Primitive Equations and more generally. In this work we study a stochastic version of the Primitive Equations. We establish the global existence of strong, pathwise solutions for these equations in dimension 3 for the case of a nonlinear multiplicative noise. The proof makes use of anisotropic estimates, $L_t^p{L}_x^q$ estimates on the pressure and stopping time arguments.

To appear in Nonlinearity

References in corpus (2)

• Global L_2-solutions of stochastic Navier-Stokes equations
• Local Martingale and Pathwise Solutions for an Abstract Fluids Model

Cited by in corpus (25)

• Stochastic Parameterization: Towards a new view of Weather and Climate Models
• Local Martingale and Pathwise Solutions for an Abstract Fluids Model
• Existence and Regularity of Invariant Measures for the Three Dimensional Stochastic Primitive Equations
• On Unique Ergodicity in Nonlinear Stochastic Partial Differential Equations
• Exponential mixing for a class of dissipative PDEs with bounded degenerate noise
• On the small time asymptotics of 3D stochastic primitive equations
• Markov Selection and $W$-strong Feller for 3D Stochastic Primitive Equations
• Well-Posedness of the 3D Stochastic Primitive Equations with Transport Noise
• The primitive equations with stochastic wind driven boundary conditions
• Local Martingale Solutions and Pathwise Uniqueness for the Three-dimensional Stochastic Inviscid Primitive Equations
• Exponential Mixing for 3D Stochastic Primitive Equations
• A squeezing property and its applications to a description of long time behaviour in the 3D viscous primitive equations
• Gaussian invariant measures and stationary solutions of 2D Primitive Equations
• Exponential stability of 3D stochastic primitive equations driven by fractional noise
• Time Discrete Approximation of Weak Solutions for Stochastic Equations of Geophysical Fluid Dynamics and Applications
• Large deviation principles for 3D stochastic primitive equations
• Asymptotic behavior of 3-D stochastic primitive equations of large-scale moist atmosphere with additive noise
• On weak-strong uniqueness and singular limit for the compressible Primitive Equations
• Global well-posedness and regularity of 3D Burgers equation with multiplicative noise
• Pathwise Solutions for Stochastic Hydrostatic Euler Equations under the Local Rayleigh Condition
• A comparison of data-driven approaches to build low-dimensional ocean models
• A moderate deviation principle for 2D stochastic primitive equations
• Some Results for the Primitive Equations with Physical Boundary Conditions
• The asymptotic behavior of primitive equations with multiplicative noise
• Splitting up method for 2D stochastic primitive equations with multiplicative noise