We study dynamics of immersed interface problems in Stokes flow. One of the simplest of such problems is the 2D Peskin problem, in which a 1D closed elastic structure is immersed in a 2D Stokes fluid. This has been studied computationally and analytically. We extend the 2D Peskin problem into two different cases: (1) 2D inextensible interface problem. (2) 3D Peskin problem.

In the 2D inextensible interface problem, we assume that the interface is inextensible instead of extensible.
Through the boundary integral method, we reformulate the problem into two contour equations, an evolution equation and a tension determination equation.
The tension determination equation is for determining the tension on the filament that ensures the filament inextensibility.
We first study the well-posedness and the regularity of the generalized tension determination problem in H\"older spaces.
We use the small scale decomposition to split the equation into a principal part and a remainder part.
Then, we use Fredholm alternative theorem to obtain the well-posedness.
Next, we use a suitable time-weighted H\"older spaces to study the well-posedness and the regularity of the dynamic problem.
We take the principal part of the evolution with the small scale decomposition and reformulate it into the Duhamel's form.
Then, we use the fixed point arument to prove the well-posedness and use the standard parabolic bootstrap argument for the regularity.     We also study the Peskin problem in the 3D case.    With the boundary integral method, the 3D Peskin may be reformulated to an evolution equation on an unit sphere $\mathbb{S}^2$ for the elastic interface.
We use more than one local charts to prove that the problem is well-posed in low-regularity H\"older spaces.
Moreover, we prove that the elastic membrane becomes smooth instantly in time.