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European Mathematical Society Publishing House
2016-09-19 17:05:48
Rendiconti del Seminario Matematico della Università di Padova
Rend. Sem. Mat. Univ. Padova
RSMUP
0041-8994
2240-2926
General
10.4171/RSMUP
http://www.ems-ph.org/doi/10.4171/RSMUP
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2013)
125
2011
0
Global Weak Solutions of the Navier-Stokes Equations with Nonhomogeneous Boundary Data and Divergence
Reinhard
Farwig
Technische Hochschule Darmstadt, DARMSTADT, GERMANY
H.
Kozono
Tohoku University, SENDAI, JAPAN
H.
Sohr
Universität Paderborn, 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.
General
51
70
10.4171/RSMUP/125-4
http://www.ems-ph.org/doi/10.4171/RSMUP/125-4