Global weak solutions of the Navier-Stokes equations with nonhomogeneous boundary data and divergence
Summary: Consider a smooth bounded domain \(\Omega\subseteq\mathbb{R}^3\) with boundary \(\partial\Omega\), a time interval \([0,T)\), \(0<T\leq\infty\), and the Navier-Stokes system in \([0,T)\times\Omega\), with initial value \(u_0\in L^2_\sigma(\Omega)\) and external force \(f=\text{div}\,F\), \(F \in L^2(0,T;L^2(\Omega))\). Our aim is to extend the well-known class of Leray-Hopf weak solutions \(u\) satisfying \(u|_{\partial\Omega}=0\), \(\text{div}\,u=0\) to the more general class of Leray-Hopf type weak solutions \(u\) with general data \(u|_{\partial\Omega}=g\), \(\text{div}\,u=k\) 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.
scientific article; zbMATH DE number 5961388
scientific article