AI 中文总结
研究三维佩斯金问题，利用最优正则性空间\(W^{1,\infty}(\mathbb{S}^2)\)中的初始数据，通过抛物平滑效应去奇异化，结合非线性估计与结构解耦，证明解在\(C^1\)拓扑中指数收敛到共形球面，实现全局渐近稳定性。
AI 中文摘要
我们证明了三维佩斯金问题的全局适定性和渐近稳定性，该问题模拟了浸入不可压缩斯托克斯流体中的封闭弹性膜。我们使用最优正则性空间\(W^{1,\infty}(\mathbb{S}^2)\)中的初始数据，其可能包含无限多个角点。这些初始构型通过流动的抛物平滑效应立即去奇异化，对于所有\(t > 0\)都变得光滑。然后我们证明解在\(C^1\)拓扑中指数收敛到一个平移和伸缩的共形球面。通过将我们的非线性估计与共形稳态的10维流形的精确结构解耦相结合来实现稳定性，表明无限维耗散扰动受到严格控制。我们分析的核心是在球面\(\mathbb{S}^2\)上的一个泛函框架，它使用谱利特尔伍德 - 佩利投影来控制由流体非线性产生的高度奇异的多线性算子。
英文摘要
We prove global well-posedness and asymptotic stability for the three-dimensional Peskin problem, which models a closed, elastic membrane immersed in an incompressible Stokes fluid. We work with initial data in the optimal regularity space $W^{1,\infty}(\mathbb{S}^2)$, which may contain infinitely many corners. These initial configurations are instantly desingularized by the flow's parabolic smoothing effect, becoming smooth for all $t > 0$. Then we establish that the solutions converge exponentially in the $C^1$ topology to a translated and dilated conformal sphere. The stability is achieved by combining our nonlinear estimates with an exact structural decoupling of the 10-dimensional manifold of conformal steady states, demonstrating that the infinite-dimensional dissipative perturbation is strictly controlled. The core of our analysis is a functional framework on the sphere $\mathbb{S}^2$ that uses spectral Littlewood-Paley projections to control the highly singular multilinear operators arising from the fluid nonlinearity
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