Consider a smooth bounded domain with boundary , a time interval , with , and the Navier-Stokes system in , with initial value and external force , . Our aim is to extend the well-known class of Leray-Hopf weak solutions satisfying , to the more general class of Leray-Hopf type weak solutions with general data , satisfying a certain energy inequality. Our method rests on a perturbation argument writing in the form with some vector field in satisfying the (linear) Stokes system with 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.
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Reinhard Farwig, H. Kozono, H. Sohr, Global Weak Solutions of the Navier-Stokes Equations with Nonhomogeneous Boundary Data and Divergence. Rend. Sem. Mat. Univ. Padova 125 (2011), pp. 51–70DOI 10.4171/RSMUP/125-4