论文标题
奇异SDE的局部时间理论，其应用于具有传输噪声的流体模型
A local-in-time theory for singular SDEs with applications to fluid models with transport noise
论文作者
论文摘要
在本文中，我们为在某些希尔伯特空间中的一个奇异SDE家族而建立了当地理论，即存在，独特性和爆炸标准。关键要求是一个近似属性，它使我们能够将单数漂移和扩散映射嵌入到常规映射的层次结构中，这些映射相对于希尔伯特空间不变，并享受取消属性。 具有传输噪声的流体动力学中的各种非线性模型属于这种类型的单数SDE。通过对广义谎言导数运算符的取消估计，我们可以为涉及Lie衍生物操作员的案件构建此类常规近似值，或者更一般地，是一个具有合适系数的订单的差分运算符。特别是，我们将抽象理论应用于随机两组分子Camassa-Holm（CH）系统以及随机Córdoba-Córdoba-Fontelos（CCF）模型的新型本地结果。
In this paper, we establish a local theory, i.e., existence, uniqueness and blow-up criterion, for a general family of singular SDEs in some Hilbert space. The key requirement is an approximation property that allows us to embed the singular drift and diffusion mappings into a hierarchy of regular mappings that are invariant with respect to the Hilbert space and enjoy a cancellation property. Various nonlinear models in fluid dynamics with transport noise belong to this type of singular SDEs. With a cancellation estimate for generalized Lie derivative operators, we can construct such regular approximations for cases involving the Lie derivative operators, or more generally, differential operators of order one with suitable coefficients. In particular, we apply the abstract theory to achieve novel local-in-time results for the stochastic two-component Camassa--Holm (CH) system and for the stochastic Córdoba-Córdoba-Fontelos (CCF) model.
