The stochastic primitive equations with transport noise and turbulent pressure

• Published in: Stochastic partial differential equations : analysis and computations Vol. 12; no. 1; pp. 53 - 133
• Main Authors: Agresti, Antonio, Hieber, Matthias, Hussein, Amru, Saal, Martin
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.03.2024
• Subjects: Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Mathematics; Mathematics and Statistics; Numerical Analysis; Partial Differential Equations; Probability Theory and Stochastic Processes; Statistical Theory and Methods; 76M35; 76U60; Stochastic partial differential equations; Primitive equations; Kraichnan’s turbulence; Primary 35Q86; Global existence; Turbulence flows; Secondary 35R60; 60H15; Transport noise; Stochastic maximal regularity
• ISSN: 2194-0401; 2194-041X
• DOI: 10.1007/s40072-022-00277-3

In this paper we consider the stochastic primitive equation for geophysical flows subject to transport noise and turbulent pressure. Admitting very rough noise terms, the global existence and uniqueness of solutions to this stochastic partial differential equation are proven using stochastic maximal L 2 -regularity, the theory of critical spaces for stochastic evolution equations, and global a priori bounds. Compared to other results in this direction, we do not need any smallness assumption on the transport noise which acts directly on the velocity field and we also allow rougher noise terms. The adaptation to Stratonovich type noise and, more generally, to variable viscosity and/or conductivity are discussed as well.