The Primitive Equations are a basic model in the study of large scale Oceanicand Atmospheric dynamics. These systems form the analytical core of the mostadvanced General Circulation Models. For this reason and due to theirchallenging nonlinear and anisotropic structure the Primitive Equations haverecently received considerable attention from the mathematical community. In view of the complex multi-scale nature of the earth's climate system, manyuncertainties appear that should be accounted for in the basic dynamical modelsof atmospheric and oceanic processes. In the climate community stochasticmethods have come into extensive use in this connection. For this reason therehas appeared a need to further develop the foundations of nonlinear stochasticpartial differential equations in connection with the Primitive Equations andmore generally. In this work we study a stochastic version of the Primitive Equations. Weestablish the global existence of strong, pathwise solutions for theseequations in dimension 3 for the case of a nonlinear multiplicative noise. Theproof makes use of anisotropic estimates, $L^{p}_{t}L^{q}_{x}$ estimates on thepressure and stopping time arguments.
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机译：原始方程是研究大规模海洋和大气动力学的基本模型。这些系统构成了最先进的一般流通模型的分析核心。由于这个原因，并且由于它们具有挑战性的非线性和各向异性结构，本原方程最近受到了数学界的广泛关注。鉴于地球气候系统的复杂的多尺度性质，出现了许多不确定性，这些不确定性应在大气和海洋过程的基本动力学模型中加以考虑。在气候界，随机方法已广泛用于此方面。因此，有必要结合原始方程，更一般地进一步发展非线性随机偏微分方程的基础。在这项工作中，我们研究了本原方程的随机形式。对于非线性乘性噪声的情况，我们针对维度3中的这些方程式建立了强大的路径解的全局存在。该证明利用各向异性估计，即压力和停止时间参数的$ L ^ {p} _ {t} L ^ {q} _ {x} $估计。
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