Abstract

By using a regularity approximation argument, the global existence and uniqueness are derived for a class of nonlinear SPDEs depending on both the whole history and the distribution under strong enough noise. As applications, the global existence and uniqueness are proved for distribution-path dependent stochastic transport type equations, which are arising from stochastic fluid mechanics with forces depending on the history and the environment. In particular, the distribution-path dependent stochastic Camassa-Holm equation with or without Coriolis effect has a unique global solution when the noise is strong enough, whereas for the deterministic model wave-breaking may occur. This indicates that the noise may prevent blow-up almost surely.

中文翻译：

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分布路径相关的非线性 SPDE 及其在随机传输类型方程中的应用

摘要

通过使用正则近似论证，根据整个历史和足够强噪声下的分布，推导了一类非线性 SPDE 的全局存在性和唯一性。作为应用，证明了分布路径相关的随机输运型方程的全局存在性和唯一性，该方程是由力取决于历史和环境的随机流体力学产生的。特别是，当噪声足够强时，无论有或没有科里奥利效应，分布路径相关的随机卡马萨-霍尔姆方程都具有唯一的全局解，而对于确定性模型，可能会发生破波。这表明噪声几乎肯定可以防止爆炸。