Analysis of Partial Differential Equations with Cross-Diffusion and Stochastic Driving

含有交叉扩散项的二阶发展偏微分方程在数学生物学中经常被使用，对数学分析提出了挑战。熵方法是证明时间上全局解存在的一种有效的技术工具。在这种情况下，一个合适的功能提供了全球先验界，使存在性可以通过一个固定点的论点。此外，泛函可以用来显示在某些制度的整体衰减到平衡。从数学生物学的角度来看，交叉扩散确定性偏微分方程是一个平均场模型，它是非常可取的，包括由于有限尺寸效应或随机外部强迫的随机效应。这自然导致人们考虑随机强迫偏微分方程（SPDE）。对于交叉扩散SPDE，基本上不存在数学分析，本项目的一个主要目标是填补我们知识中的这一重大空白。从技术角度来看，SPDE具有挑战性，因为在许多情况下必须处理非常低的规则性噪声项。在这里，我们提出了一个多方面的方法来解决这个问题。对于局部时间存在理论，我们的目标是研究三种不同的解决方案的概念交叉扩散SPDE：温和的解决方案，弱解决方案（在概率意义上），和重整化正则结构的解决方案。一个关键的问题将是识别噪声规律性的障碍，其中每种不同的技术都适用于交叉扩散SPDE。此外，我们的目标是通过正则化（噪声，扩散，并通过有限维近似）对解决方案的近似方法做出贡献。对于时间上的全局存在性，我们建议将熵方法技术从偏微分方程转移到随机偏微分方程的分析中。在这方面的一个主要主题将是熵变量与全球先验界限相结合的使用。总之，我们的目标是合并和显着扩展理论从几个不同的数学领域的交叉扩散偏微分方程的框架内。

Second-order evolution partial differential equations (PDEs) involving cross-diffusion terms are frequently used in mathematical biology and present a challenge for mathematical analysis. An efficient technical tool to prove the existence of global-in-time solutions are entropy methods. In this context, a suitable functional provides global a-priori bounds so that existence can be obtained via a fixed-point argument. In addition, the functional can be used to show global decay to equilibrium in certain regimes. From the viewpoint of mathematical biology, cross-diffusion deterministic PDEs are a mean-field model and it is highly desirable to include stochastic effects due to finite-size effects or random external forcing. This naturally leads one to consider stochastically-forced partial differential equations (SPDEs). For cross-diffusion SPDEs, basically no mathematical analysis exists and one major goal of this project is to fill this substantial gap in our knowledge. From a technical point of view, SPDEs are challenging as one has to deal with very low-regularity noise terms in many cases. Here we propose a multi-faceted approach towards this problem. For local-in-time existence theory, we aim to study three different solution concepts for cross-diffusion SPDEs: mild solutions, weak solutions (in the probabilistic sense), and renormalized regularity-structure solutions. One key question will be to identify the barrier of noise regularity, where each different technique works for cross-diffusions SPDEs. Furthermore, we aim to contribute towards approximation methods for solutions by regularization (of the noise, of the diffusion, and via finite-dimensional approximations). For global-in-time-existence, we propose to transfer entropy method techniques from PDEs to the analysis of SPDEs. A main theme in this context will be the use of entropy variables in combination with global a-priori bounds. In summary, we aim to merge and significantly extend theories from several different mathematical areas within the framework of cross-diffusion PDEs.