随机强迫对二维有界区域上不可压切片模型强解的影响
The influence of stochastic forcing on strong solutions to the Incompressible Slice Model in 2D bounded domain
摘要：
Cotter-Holm切片模式（CHSM）是一种用于研究大气锋形成的模式。本文考虑的是在光滑的二维有界区域中的不可压缩切片模型（ISM）的随机强迫的影响，这可以通过调整拉格朗日函数的Hamilton原理CHSM的欧拉-Boussinesq Eady不可压缩的情况下。首先，我们在巴拿赫空间$W^{k,p}(D)$中用$k>1+1/p$和$p\geq 2$建立了非线性乘性随机强迫扰动下ISM的局部路径解（概率强解）的存在唯一性.通过引入合适的截断算子应用于速度场和温度场的$W^{1,\infty}$ -范数，使用随机紧性方法和基于依概率收敛的Gyöngy-Krylov特征的Yamada-Watanabe型参数，得到了该解。然后，当ISM受线性乘性随机强迫扰动，且位温在$y$方向上不线性变化时，证明了在初始数据足够小或扩散参数足够大的条件下，相应的Cauchy问题存在唯一的高概率全局路径解.这些结果部分回答了Alonso-Or{á}n等人（Physica D 392：99- 118，2019，pp. 117）。
英文摘要：
The Cotter-Holm Slice Model (CHSM) was introduced to study the behavior of whether and specifically the formulation of atmospheric fronts, whose prediction is fundamental in meteorology. Considered herein is the influence of stochastic forcing on the Incompressible Slice Model (ISM) in a smooth 2D bounded domain, which can be derived by adapting the Lagrangian function in Hamilton's principle for CHSM to the Euler-Boussinesq Eady incompressible case. First, we establish the existence and uniqueness of local pathwise solution (probability strong solution) to the ISM perturbed by nonlinear multiplicative stochastic forcing in Banach spaces $W^{k,p}(D)$ with $k>1+1/p$ and $p\geq 2$. The solution is obtained by introducing suitable cut-off operators applied to the $W^{1,\infty}$-norm of the velocity and temperature fields, using the stochastic compactness method and the Yamada-Watanabe type argument based on the Gyöngy-Krylov characterization of convergence in probability. Then, when the ISM is perturbed by linear multiplicative stochastic forcing and the potential temperature does not vary linearly on the $y$-direction, we prove that the associated Cauchy problem admits a unique global-in-time pathwise solution with high probability, provided that the initial data is sufficiently small or the diffusion parameter is large enough. The results partially answer the problems left open in Alonso-Or{á}n et al. (Physica D 392:99--118, 2019, pp. 117).
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