带非等温湍流压强的随机原始方程
The stochastic primitive equations with non-isothermal turbulent pressure
AI总结：
该研究引入并分析带非等温湍流压强与输运噪声的随机原始方程，推导自纳维-斯托克斯方程，证明了其H¹空间全局适定性，得到初值连续依赖与能量估计等新结果。
摘要：
本文引入并研究了带非等温湍流压强与输运噪声的原始方程。这类方程通过采用随机版本的Boussinesq（布西内斯克）近似与静水压近似，从Navier-Stokes（纳维-斯托克斯）方程推导而来。湍流压强的温度依赖性可视为作用于小尺度垂直动力学的加性噪声导致的结果。针对该模型，我们证明了其在$H^1$空间中的全局适定性，其中噪声同时采用Itô（伊藤）与Stratonovich（斯特拉托诺维奇）两种表述形式。与此前的原始方程变体相比，本文研究的模型在速度场与温度之间呈现出更复杂的耦合关系，相应的分析也比确定性情形要复杂得多。最后，本文证明的初值连续依赖性与能量估计均为新结果，即便在等温湍流压强的情形下也同样成立。
英文摘要：
In this paper, we introduce and study the primitive equations with $\textit{non}$-isothermal turbulent pressure and transport noise. They are derived from the Navier-Stokes equations by employing stochastic versions of the Boussinesq and the hydrostatic approximations. The temperature dependence of the turbulent pressure can be seen as a consequence of an additive noise acting on the small vertical dynamics. For such a model we prove global well-posedness in $H^1$ where the noise is considered in both the Itô and Stratonovich formulations. Compared to previous variants of the primitive equations, the one considered here presents a more intricate coupling between the velocity field and the temperature. The corresponding analysis is seriously more involved than in the deterministic setting. Finally, the continuous dependence on the initial data and the energy estimates proven here are new, even in the case of isothermal turbulent pressure.
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