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.