On the existence of weak solutions to a coupled system of two turbulent flows

Joachim Naumann , Jörg Wolf

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 12, 39

MSC 2000

35J55 Boundary value problems for elliptic systems

35Q30 Stokes and Navier-Stokes equations

76F25 Turbulent transport, mixing

Abstract
In this paper, we study a model problem for the stationary turbulent motion of two fluids in disjoint bounded domains $\Omega_1$ and $\Omega_2$ such that $\Gamma:=\bar{\Omega}_1 \cap \bar{\Omega}_2 \not= \varnothing$. The specific difficulty of this problem arises from the boundary condition which characterizes the interaction of the fluid motions along $\Gamma$.\par We prove the existence of a weak solution to the problem under consideration which is more regular than the solution obtained in [3]. Moreover, we establish some regularity results for any weak solution. Our discussion is heavily based on the results in appendices 1 and 2 which seem to be of independent interest.