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Existence of a weak solution to the Navier-Stokes equation in a general time-varying domain by the Rothe method. (English) Zbl 1160.35494
Summary: We assume that $$\Omega ^t$$ is a domain in $$\mathbb R^{3}$$, arbitrarily (but continuously) varying for $$0 \leqslant t \leqslant T$$. 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 $$Q_{[0,T)} := \{(x,t);0 \leqslant t \leqslant T, x \in \Omega ^t\}$$ 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.

##### MSC:
 35Q30 Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D05 Navier-Stokes equations for incompressible viscous fluids
##### Keywords:
Navier-Stokes equations; weak solutions
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##### References:
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