2021, Hintermüller, Michael, Kröner, Axel

In this paper we consider a fluid-structure interaction problem given by the steady Navier Stokes equations coupled with linear elasticity taken from [Lasiecka, Szulc, and Zochoswki, Nonl. Anal.: Real World Appl., 44, 2018]. An elastic body surrounded by a liquid in a rectangular domain is deformed by the flow which can be controlled by the Dirichlet boundary condition at the inlet. On the walls along the channel homogeneous Dirichlet boundary conditions and on the outflow boundary do-nothing conditions are prescribed. We recall existence results for the nonlinear system from that reference and analyze the control to state mapping generaziling the results of [Wollner and Wick, J. Math. Fluid Mech., 21, 2019] to the setting of the nonlinear Navier-Stokes equation for the fluid and the situation of mixed boundary conditions in a domain with corners.

2008, Anoschenko, Olga, Khruslov, Evgeni, Stephan, Holger

We consider the Navier-Stokes-Vlasov-Poisson system of partial differential equations, describing the motion of a viscous incompressible fluid with small solid charged particles therein. We prove the existence of a weak global solution of the initial boundary value problem for this system.

2015, Mielke, Alexander, Naumann, Joachim

We consider Kolmogorov's model for the turbulent motion of an incompressible fluid in 3. This model consists in a Navier-Stokes type system for the mean flow u and two further partial differential equations: an equation for the frequency and for the kinetic energy k each. We investigate this system of partial differential equations in a cylinder x ]0,T[ ( 3 cube, 0 < T < +∞) under spatial periodic boundary conditions on x ]0,T[ and initial conditions in x {0}. We present an existence result for a weak solution {u, , k} to the problem under consideration, with , k obeying the inequalities formula1 and formula2.

2018, Mielke, Alexander, Naumann, Joachim

This paper is concerned with Kolmogorov's two-equation model for free turbulence in space dimension 3, involving the mean velocity u, the pressure p, an average frequency omega, and a mean turbulent kinetic energy k. We first discuss scaling laws for a slightly more general two-equation models to highlight the special role of the model devised by Kolmogorov in 1942. The main part of the paper consists in proving the existence of weak solutions of Kolmogorov's two-equation model under space-periodic boundary conditions in cubes with positive side length l. To this end, we provide new a priori estimates and invoke existence result for pseudo-monotone operators.