Global Weak Solutions of the Navier–Stokes Equations with Nonhomogeneous Boundary Data and Divergence

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Consider a smooth bounded domain \Omega\subseteq\mathbb R^3 with boundary \partial\Omega , a time interval [0,T) , with T\in(0,\infty] , and the Navier–Stokes system in [0,T) \times \Omega , with initial value u_0 \in L^2_{\sigma} (\Omega) and external force f= {\mathrm{div}}\,F , F \in L^2 (0,T;L^2(\Omega)) . Our aim is to extend the well-known class of Leray-Hopf weak solutions u satisfying u_{\vert{\partial \Omega}}=0 , {\mathrm{div}}\,u=0 to the more general class of Leray-Hopf type weak solutions u with general data u_{\vert{\partial \Omega}} =g , {\mathrm{div}}\,u=k satisfying a certain energy inequality. Our method rests on a perturbation argument writing u in the form u=v+E with some vector field E in [0,T)\times \Omega satisfying the (linear) Stokes system with f=0 and nonhomogeneous data. This reduces the general system to a perturbed Navier–Stokes system with homogeneous data, containing an additional perturbation term. Using arguments as for the usual Navier–Stokes system we get the existence of global weak solutions for the more general system.