Well-posedness and global behavior of the Peskin problem of an immersed elastic filament in Stokes flow

Mori, Yoichiro; Rodenberg, Analise; Spirn, Daniel

doi:10.48550/arxiv.1704.08392

Cited by 1 publication

(2 citation statements)

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“…Regularity of u recovered from X(s, t) is carefully studied in [11]. In a parallel work, Mori et al [13] establish similar local and global well-posedness results for (6) in C 1,α -spaces. They also show improved regularity of X(s, t) for positive time and a blowup criterion.…”

Section: The 2-d Stokes Immersed Boundary Problemmentioning

confidence: 96%

“…Recall that in the analysis of the original problem with the Hookean elasticity [2,13], (1)-( 4) is first reduced (under some assumptions) to the contour dynamic equation (6), and it is then rewritten as ∂ t X = LX + g X . Here LX = − 1 4 (−∆) 1/2 X captures the principal singular part in U X that is derived by linearizing the integrand of (6) around s ′ = s, while g X is a nonlinear nonlocal term collecting all the remaining terms.…”

Section: Scheme Of the Proofs And Organization Of The Papermentioning

confidence: 99%

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Regularized Stokes Immersed Boundary Problems in Two Dimensions: Well-posedness, Singular Limit, and Error Estimates

Tong

2019

Preprint

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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 δ-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 δ-function is suitably chosen. This may imply potential improvement in the numerical method, which is worth further investigation.

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Section: The 2-d Stokes Immersed Boundary Problemmentioning

confidence: 96%

Section: Scheme Of the Proofs And Organization Of The Papermentioning

confidence: 99%