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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
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