The Immersed Boundary Problem in 2-D: the Navier-Stokes Case

We study the immersed boundary problem in 2-D. It models a 1-D elastic closed string immersed and moving in a fluid that fills the entire plane, where the fluid motion is governed by the 2-D incompressible Navier-Stokes equation with a positive Reynolds number subject to a singular forcing exerted by the string. We introduce the notion of mild solutions to this system, and prove its existence, uniqueness, and optimal regularity estimates when the initial string configuration is $C^1$ and satisfies the well-stretched condition and when the initial flow field $u_0$ lies in $L^p(\mathbb{R}^2)$ with $p\in (2,\infty)$. A blow-up criterion is also established. When the Reynolds number is sent to zero, we show convergence in short time of the solution to that of the Stokes case of 2-D immersed boundary problem, with the optimal error estimates derived. We prove the energy law of the system when $u_0$ additionally belongs to $L^2(\mathbb{R}^2)$. Lastly, we show that the solution is global when the initial data is sufficiently close to an equilibrium state.