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

点击翻译按钮获取中文摘要

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.