Sensitivity Analysis for the 2D Navier–Stokes Equations with Applications to Continuous Data Assimilation
Sensitivity Analysis for the 2D Navier–Stokes Equations with Applications to Continuous Data Assimilation
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DOI：
10.1007/s00332-021-09739-9
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发表时间：
2020-07
影响因子：
3
通讯作者：
Elizabeth Carlson;Adam Larios
中科院分区：
文献类型：
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作者：
Elizabeth Carlson;Adam Larios
严格证明了二维不可压Navier-Stokes方程形式灵敏度方程关于粘性的适定性。此外，我们这样做，通过显示一系列的差的导数收敛到唯一的解决方案的灵敏度方程的二维Navier-Stokes方程和相关的数据同化方程，利用连续的数据同化算法提出的Azouani，奥尔森和Titi。因此，这种证明方法提供了不同的参数的统一界限，证明参数恢复算法，随着系统的发展而改变参数将表现良好。此外，我们的分析可以扩展到分析的敏感性的二维欧拉方程的粘性正则化。我们还注意到，这似乎是第一个这样严格的证明的强或弱的灵敏度方程的2D Navier-Stokes方程（在自然情况下的零初始数据）的整体存在性和唯一性的强或弱的解决方案，他们可以获得作为一个限制的差异相对于粘度的一致性。
We rigorously prove the well-posedness of the formal sensitivity equations with respect to the viscosity corresponding to the 2D incompressible Navier–Stokes equations. Moreover, we do so by showing a sequence of difference quotients converges to the unique solution of the sensitivity equations for both the 2D Navier–Stokes equations and the related data assimilation equations, which utilize the continuous data assimilation algorithm proposed by Azouani, Olson, and Titi. As a result, this method of proof provides uniform bounds on difference quotients, demonstrating parameter recovery algorithms that change parameters as the system evolves will be well behaved. Furthermore, our analysis can be extended to analyze the sensitivity of the 2D Euler equations to a viscous regularization. We also note that this appears to be the first such rigorous proof of global existence and uniqueness to strong or weak solutions to the sensitivity equations for the 2D Navier–Stokes equations (in the natural case of zero initial data), and that they can be obtained as a limit of difference quotients with respect to the viscosity.