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

TL;DR

This paper analyzes the 2-D immersed boundary problem with Navier-Stokes equations, establishing existence, uniqueness, regularity, blow-up criteria, convergence to Stokes flow, energy laws, and conditions for global solutions.

Contribution

It introduces the notion of mild solutions for the 2-D immersed boundary problem and proves key properties including existence, uniqueness, regularity, and convergence results.

Findings

Existence and uniqueness of mild solutions under specified conditions.

Convergence of Navier-Stokes solutions to Stokes solutions as Reynolds number approaches zero.

Energy law and global existence for solutions near equilibrium.

Abstract

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.