AI 中文总结
研究二维纳维 - 斯托克斯方程在部分切向边界观测下的连续数据同化，通过构建边界反馈产生强制谱隙，结合非线性误差估计得到指数同步准则，并在多种情况下验证了该准则。
AI 中文摘要
我们研究在具有纳维 - 滑移边界条件的光滑、有界、连通区域上的二维纳维 - 斯托克斯方程的连续数据同化，不使用内部观测。可用数据仅为由非空相对开子集\(\Gamma\subset\partial\Omega\)上切向速度的有限维测量组成。我们证明，由足够精细的观测构建的足够强的边界反馈会为同化误差产生一个强制谱隙。极限间隙与通过在\(\Gamma\)上施加齐次狄利克雷条件得到的混合边界问题的间隙相同，且显示为\(\nu\)阶。将这种反馈诱导的强制性与根据参考解的长时间平均对称梯度能量对非线性误差产生的估计相结合，我们得到了指数同步的充分准则。我们在无外力情况、当未受扰动的纳维 - 滑移形式具有\(\mathrm{L}^2\)谱隙时足够小的外力、在没有切向刚性运动的区域中足够大的粘性以及在存在正边界摩擦时足够大的粘性等情况下验证了该准则。我们还处理了允许切向刚性运动的区域上的完全滑移情况，其中在对力的非刚性螺线管分量的一个小条件下实现同步。
英文摘要
We study continuous data assimilation for the two-dimensional Navier--Stokes equations on a smooth, bounded, connected domain with Navier-slip boundary conditions, using no interior observations. The available data consist only of finite-dimensional measurements of the tangential velocity on a non-empty relatively open subset $Γ\subset\partialΩ$. We prove that sufficiently strong boundary feedback, constructed from sufficiently fine observations, generates a coercive spectral gap for the assimilation error. The limiting gap is identified with that of a mixed-boundary problem obtained by imposing a homogeneous Dirichlet condition on $Γ$, and is shown to be of order $ν$. Combining this feedback-induced coercivity with an estimate of the non-linear error production in terms of the long-time averaged symmetric-gradient energy of the reference solution, we obtain a sufficient criterion for exponential synchronisation. We verify this criterion in the unforced case, for sufficiently small forcing when the unnudged Navier-slip form has an $\mathrm{L}^2$ spectral gap, for sufficiently large viscosity on domains without tangential rigid motions, and for sufficiently large viscosity in the presence of positive boundary friction. We also treat perfect slip on domains admitting tangential rigid motions, where synchronisation follows under a smallness condition on the non-rigid solenoidal component of the forcing.
