The Navier-Stokes Equations with Elastic Boundary and Boundary Conditions of Friction Type

Abstract

In this work we consider the incompressbile Navier-Stokes equations from fluid mechanics in combination with two different types of boundary conditions. One problem that we consider is a free boundary problem. Here, the upper, elastic boundary part of a cylindric container, that contains a viscous fluid, is set in motion due to an interior force, induced by the velocity field of the fluid, and due to a given exterior force. In Chapter 2, we consider this problem formulated on the fixed, upper half space. Via partial Fourier transform we show the existence of solutions which belong to homogeneous Sobolev spaces. Then, we identify admissible spaces for the initial values. The second boundary condition models friction of a viscous fluid with the boundary of the domain. In Chapter 3, we show the existence of global, weak solutions that are strong solutions with respect to time in two dimensions. In the three-dimensional case, we show short time existence of strong solutions and global existence if the given data are small enough. Furthermore, strong-weak uniqueness is shown, as well as a connection of this boundary condition to the Dirichlet- and perfect slip boundary condition. In Chapter 4, we combine the above boundary conditions in order to quantify the elasticity of an upper, free boundary of a container, that is filled with a viscous fluid. We show the existence of weak solutions for this hybrid model.