Stochastic Partial Differential Equations with applications in Modelling of Oceans and Atmosphere

人类对海洋和大气层的影响是一个紧迫的问题。这是一个迅速演变的问题，它不仅表明需要采取行动，而且表明我们面临的不确定程度的现实。例如，我们可以看到我们的二氧化碳排放和海洋中的塑料废物的影响。特别是上层海洋感受到这种影响，因为海洋生物的比例很高。上层海洋通过与空气相互作用对这些影响作出反应，从而影响天气和气候。该项目旨在为流体动力学的建模方面做出贡献。对海洋上层可用的各种模型进行严格的数学分析，将为应对气候变化而对海平面上升、热量吸收、pH值变化等进行任何进一步严格的预测方法奠定基础。特别是，该项目将分析在流体动力学模型中使用随机性。这是必要的，因为高分辨率数据在确定性模拟中并不总是可预测的。然后，将通过数值工作验证随机数学模型的推导和分析，以便它们科普观测和模拟中的不确定性。作为一个特定的目标，我们将研究具有边界的随机偏微分方程的适定性结果（海洋有海岸！）。一类新的随机方程被认为是可行的流体动力学模型，增加了不确定性，在运输的流体包裹，以反映未解决的尺度。这些模型提供了适当的能量循环和守恒动力学，并保持其确定性对应的基本属性。他们的分析仍然是一个悬而未决的问题，是目前项目的重点。具体的例子包括欧拉和旋转浅水方程。该计划是遵循一个程序，涉及一个近似序列的解决方案，通过截断方程，并恢复一个解决方案，通过相对紧凑性的论点。在这个新的随机框架中的边界处理是未知的领域，取决于长期建立的确定性结果和这种新的随机方法之间的相互作用。具有光滑边界的随机二维欧拉方程的适定性是该项目令人兴奋的第一个目标，随后的目标是研究更微妙和相关的场景。与此协同，我们可以探索一个关于增加随机性可能产生的潜在正则化效应的一般结果，这将构成一个巨大的发展，在处理新的一类方程的范围内，他们的确定性对应物已被研究。这是一个已经被理论化和尝试的概念，并取得了不同的成功。概述的计划与EPSRC的数学科学战略主题和数学分析研究领域保持一致。

The human impact on oceans and atmosphere is a pressing concern. It is a problem that has been rapidly evolving which makes clear not only the need for action, but the reality of the level of uncertainty that we face. For example, one sees the effects of our CO2 emissions and plastic waste in the oceans. In particular the upper ocean feels that effect with the high proportion of marine life towards the surface. The upper ocean responds to these effects by interacting with the air and thus influencing the weather and climate. The project aims to contribute to the modelling side of fluid dynamics. A rigorous mathematical analysis of the various models available for the upper layer of the ocean will underpin any further rigorous prediction methods for rising sea levels, heat uptake, changes in pH and more in order to combat climate change. In particular, the project will analyse the use of stochasticity in fluid dynamics models. This is needed as high-resolution data will is not be always predictable in deterministic simulations. The derivation and analysis of stochastic mathematical models will then be validated through the numerical work and so that they cope with uncertainty in both observation and simulation. As a specific objective we will look at well-posedness results for stochastic partial differential equation with boundaries (the ocean has shores!). The new class of stochastic equations considered as viable fluid dynamics models has added uncertainty in the transport of fluid parcels to reflect the unresolved scales. These models afford proper energy circulation and conservation dynamics and preserve fundamental properties of their deterministic counterparts. Their analysis is still an open problem and is the focus of the current project. Specific examples included the Euler and Rotating Shallow Water Equations. The plan is to follow a procedure involving an approximating sequence of solutions via a truncated equation and recover a solution through relative-compactness arguments. A treatment of the boundary in this new stochastic framework is uncharted territory and hinges on the interplay between long established deterministic results and this novel stochastic methodology. Well-posedness of the stochastic 2D Euler equation with a smooth boundary represents an exciting first aim for the project, with subsequent aims to examine more delicate and relevant scenarios. Synergistically with this we could explore a general result about the potential regularising effect adding stochasticity could have, which will constitute a massive development in tackling the new class of equations given the extent to which their deterministic counterparts have been studied. It is a concept which has been theorised and attempted with varying success sofar. The outlined plan firmly aligns with the EPSRC's Strategic Theme of Mathematical Sciences, and Research Area of Mathematical Analysis.