Global Weak Solutions of the Navier–Stokes Equations with Nonhomogeneous Boundary Data and Divergence
Reinhard Farwig
Technische Hochschule Darmstadt, GermanyH. Kozono
Tohoku University, Sendai, JapanH. Sohr
Universität Paderborn, Germany
Abstract
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.
Cite this article
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. 51–70
DOI 10.4171/RSMUP/125-4