Well-posedness for the three dimensional stochastic planetary geostrophic equations of large-scale ocean circulation

Bo You

doi:10.3934/dcds.2020332

Article Contents

2021, Volume 41, Issue 4: 1579-1604. Doi: 10.3934/dcds.2020332

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Well-posedness for the three dimensional stochastic planetary geostrophic equations of large-scale ocean circulation

Bo You

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, P. R. China

Received: March 2020

Revised: August 2020

Early access: September 2020

Published: April 2021

This work was supported by the National Science Foundation of China Grant (11401459, 11871389), the Natural Science Foundation of Shaanxi Province (2018JM1012) and the Fundamental Research Funds for the Central Universities (xjj2018088)

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Abstract

The objective of this paper is to study the well-posedness of solutions for the three dimensional planetary geostrophic equations of large-scale ocean circulation with additive noise. Since strong coupling terms and the noise term create some difficulties in the process of showing the existence of weak solutions, we will first show the existence of weak solutions by the monotonicity methods when the initial data satisfies some "regular" condition. For the case of general initial data, we will establish the existence of weak solutions by taking a sequence of "regular" initial data and proving the convergence in probability as well as some weak convergence of the corresponding solution sequences. Finally, we establish the existence of weak $ \mathcal{D} $-pullback mean random attractors in the framework developed in [11,25].

Keywords:

Well-posedness,
Planetary geostrophic equations,
Additive noise,
Weak $ \mathcal{D} $-pullback mean random attractors.

Mathematics Subject Classification: Primary: 35R60, 37L55, 60H30; Secondary: 35Q86.

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