Rendiconti del Seminario Matematico della Università di Padova


Full-Text PDF (139 KB) | Metadata | Table of Contents | RSMUP summary
Volume 125, 2011, pp. 51–70
DOI: 10.4171/RSMUP/125-4

Published online: 2011-06-30

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

Reinhard Farwig[1], H. Kozono[2] and H. Sohr[3]

(1) Technische Hochschule Darmstadt, Germany
(2) Tohoku University, Sendai, Japan
(3) Universität Paderborn, Germany

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

No keywords available for this article.

Farwig Reinhard, Kozono H., Sohr H.: Global Weak Solutions of the Navier-Stokes Equations with Nonhomogeneous Boundary Data and Divergence. Rend. Sem. Mat. Univ. Padova 125 (2011), 51-70. doi: 10.4171/RSMUP/125-4