Global existence and regularity for the 3D stochastic primitive equations of the ocean and atmosphere with multiplicative white noise

• Published in: Nonlinearity Vol. 25; no. 7; pp. 2093 - 2118
• Main Authors: DEBUSSCHE, A, GLATT-HOLTZ, N, TEMAM, R, ZIANE, M
• Format: Journal Article
• Language: English
• Published: Bristol Institute of Physics 01.07.2012; IOP Publishing
• Subjects: Analysis of PDEs; Anisotropy; Atmospherics; Dynamical systems; Exact sciences and technology; Global analysis, analysis on manifolds; Mathematical methods in physics; Mathematical models; Mathematics; Nonlinear dynamics; Nonlinearity; Other topics in mathematical methods in physics; Physics; Primitive equations; Probability; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications); Stochastic analysis; Stochasticity; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Solution uniqueness; Uncertainty; Three-dimensional calculations; Stopping time; Nonlinear differential equations; Nonlinear analysis; Primitive equation; Model study; Stochastic method; Multiplicative noise; Atmospheric dynamics; Foundations; Anisotropic structure; Strong uniqueness; Atmosphere; White noise; Ocean; Large scale; Dynamic model; Regularity; Mathematical physics; Stochastic equation; system; well-posedness; Dynamics equations; backscatter; martingale; Pathwise solutions; Large-scale ocean; space; Navier-Stokes equations
• ISSN: 0951-7715; 1361-6544
• DOI: 10.1088/0951-7715/25/7/2093

The primitive equations (PEs) 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 PEs have recently received considerable attention from the mathematical community. On the other hand, 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 PEs and more generally. In this work we study a stochastic version of the PEs. We establish the global existence and uniqueness 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 super(p)dt super(q) sub(x)estimates on the pressure and stopping time arguments.