Agresti Antonio, Veraar Mark
Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box , 5031 2600 GA Delft, The Netherlands.
Present Address: Department of Mathematics Guido Castelnuovo, Sapienza University of Rome, P.le Aldo Moro 5, 00185 Rome, Italy.
Nonlinear Differ Equ Appl. 2025;32(6):123. doi: 10.1007/s00030-025-01090-2. Epub 2025 Sep 1.
In this survey, we provide an in-depth exposition of our recent results on the well-posedness theory for stochastic evolution equations, employing maximal regularity techniques. The core of our approach is an abstract notion of critical spaces, which, when applied to nonlinear SPDEs, coincides with the concept of scaling-invariant spaces. This framework leads to several sharp blow-up criteria and enables one to obtain instantaneous regularization results. Additionally, we refine and unify our previous results, while also presenting several new contributions. In the second part of the survey, we apply the abstract results to several concrete SPDEs. In particular, we give applications to stochastic perturbations of quasi-geostrophic equations, Navier-Stokes equations, and reaction-diffusion systems (including Allen-Cahn, Cahn-Hilliard and Lotka-Volterra models). Moreover, for the Navier-Stokes equations, we establish new Serrin-type blow-up criteria. While some applications are addressed using -theory, many require a more general -framework. In the final section, we outline several open problems, covering both abstract aspects of stochastic evolution equations, and concrete questions in the study of linear and nonlinear SPDEs.
在本次综述中，我们运用极大正则性技术，深入阐述了我们近期在随机发展方程适定性理论方面的成果。我们方法的核心是临界空间的抽象概念，当将其应用于非线性随机偏微分方程时，它与尺度不变空间的概念一致。该框架导出了几个精确的爆破准则，并使我们能够得到即时正则化结果。此外，我们完善并统一了之前的结果，同时还给出了一些新的贡献。在综述的第二部分，我们将抽象结果应用于几个具体的随机偏微分方程。特别地，我们给出了对准地转方程、纳维 - 斯托克斯方程以及反应扩散系统（包括艾伦 - 卡恩、卡恩 - 希利亚德和洛特卡 - 沃尔泰拉模型）的随机扰动的应用。此外，对于纳维 - 斯托克斯方程，我们建立了新的塞林型爆破准则。虽然一些应用是使用(L^p)理论处理的，但许多应用需要更一般的(L^p)框架。在最后一节，我们概述了几个开放问题，涵盖了随机发展方程的抽象方面以及线性和非线性随机偏微分方程研究中的具体问题。
