Title: Data Assimilation to the Primitive Equations in $H^2$ Show Chinese title

Title: 数据同化到原始方程在$H^2$

Abstract: In this paper we prove that the solution to the primitive equations is predicted by the corresponding data assimilation(DA) equations in $H^2$. Although, the DA equation does not include the direct information about the base solution and its initial conditions, the solution to the DA equation exponentially convergence to the base(original) solution when the external forces are known even before they are observed. Additionally, when the external force is not completely known but its spatially dense observations are available, then the DA is stable, $i.e.$ the DA solution lies in a sufficiently small neighborhood of the base solution. Show Chinese abstract

Abstract: 在本文中，我们证明了原始方程的解由相应的数据同化(DA)方程在$H^2$中预测。 尽管DA方程不包含关于基础解及其初始条件的直接信息，但当外部力已知时，即使在它们被观测之前，DA方程的解也会指数收敛到基础（原始）解。 此外，当外部力不完全已知但存在空间密集的观测值时，DA是稳定的，$i.e.$DA解位于基础解的一个足够小的邻域内。