论文标题
由有限维力驱动的3D原始方程的可控性和登山性
Controllability and ergodicity of 3D primitive equations driven by a finite-dimensional force
论文作者
论文摘要
我们研究了建模大规模海洋和大气运动的3D原始方程系统的可控性和成真性的问题。该系统是由仅作用于温度方程中有限数量的傅立叶模式的加性力驱动的。我们首先表明方程的速度和温度成分可以同时近似控制到相位空间中的任意位置。该证明基于Agrachev-Sarychev型几何控制方法。 接下来，我们研究围绕随机强制系统的非平稳轨迹的原始方程线性化的可控性。假设强迫的概率定律是可分解和可观察的，我们几乎可以通过使用与非线性设置相同的傅立叶模式来确定近似可控性。最后，将线性化系统的可控性与Arxiv：1802.03250的标准相结合，我们用随机力为非线性原始方程建立了指数混合。
We study the problems of controllability and ergodicity of the system of 3D primitive equations modeling large-scale oceanic and atmospheric motions. The system is driven by an additive force acting only on a finite number of Fourier modes in the temperature equation. We first show that the velocity and temperature components of the equations can be simultaneously approximately controlled to arbitrary position in the phase space. The proof is based on Agrachev-Sarychev type geometric control approach. Next, we study the controllability of the linearisation of primitive equations around a non-stationary trajectory of randomly forced system. Assuming that the probability law of the forcing is decomposable and observable, we prove almost sure approximate controllability by using the same Fourier modes as in the nonlinear setting. Finally, combining the controllability of the linearised system with a criterion from arXiv:1802.03250, we establish exponential mixing for the nonlinear primitive equations with a random force.
