Inspired by the numerical immersed boundary method, we introduce regularized Stokes immersed boundary problems in two dimensions to describe regularized motion of a 1-D closed elastic string in a 2-D Stokes flow, in which a regularized $\delta$-function is used to mollify the flow field and singular forcing. We establish global well-posedness of the regularized problems, and prove that as the regularization parameter diminishes, string dynamics in the regularized problems converge to that in the Stokes immersed boundary problem with no regularization. Viewing the un-regularized problem as a benchmark, we derive error estimates under various norms for the string dynamics. Our rigorous analysis shows that the regularized problems achieve improved accuracy if the regularized $\delta$-function is suitably chosen. This may imply potential improvement in the numerical method, which is worth further investigation.

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二维正则化斯托克斯浸入边界问题：适定性、奇异极限和误差估计

受数值浸入边界法的启发，我们引入了二维正则化斯托克斯浸入边界问题来描述二维斯托克斯流中一维闭合弹性弦的正则化运动，其中使用正则化的$\delta$函数缓和流场和奇异强迫。我们建立正则化问题的全局适定性，并证明随着正则化参数的减小，正则化问题中的弦动力学收敛到没有正则化的斯托克斯浸入边界问题中的弦动力学。将非正则化问题视为基准，我们得出字符串动力学在各种规范下的误差估计。我们严格的分析表明，如果适当地选择正则化的 $\delta$ 函数，正则化问题可以提高准确性。