Existence of a weak solution to the Navier-Stokes equation in a general time-varying domain by the Rothe method

We assume that Omega(t) is a domain in R-3, arbitrarily (but continuously) varying for 0 <= t <= T. We impose no conditions on smoothness or shape of Omega(t). We prove the global in time existence of a weak solution of the Navier-Stokes equation with Dirichlet's homogeneous or inhomogeneous boundary condition in Q([0,T)):={(x,t);0 <= t <= T, x epsilon Omega(t)). The solution satisfies the energy-type inequality and is weakly continuous in dependence of time in a certain sense. As particular examples, we consider flows around rotating bodies and around a body striking a rigid wall. Copyright (C) 2008 John Wiley & Sons, Ltd.