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

Abstract

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 $u_0\in L_{\sigma }^2(\Omega )$ 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 $u_{|\partial \Omega }=0$, $\mathrm{div}u=0$ to the more general class of Leray-Hopf type weak solutions $u$ with general data $u_{|\partial \Omega }=g$, $\mathrm{div}u=k$ 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.