An $$L^2$$ approach to viscous flow in the half space with free elastic surface

Abstract

We consider a linearized fluid-structure interaction problem, namely the flow of an incompressible viscous fluid in the half space $${\mathbb {R}}^n_+$$ R + n such that on the lower boundary a function h satisfying an undamped Kirchhoff-type plate equation is coupled to the flow field. Originally, h describes the height of an underlying nonlinear free surface problem. Since the plate equation contains no damping term, this article uses $$L^2$$ L 2 -theory yielding the existence of strong solutions on finite time intervals in the framework of homogeneous Bessel potential spaces. The proof is based on $$L^2$$ L 2 -Fourier analysis and also deals with inhomogeneous boundary data of the velocity field on the “free boundary” $$x_n=0$$ x n = 0 .