J Glob Optim (2008) 40:277–288
DOI 10.1007/s10898-007-9197-2
On the interior regularity of weak solutions
to the non-stationary Stokes system
Joachim Naumann · Jörg Wolf
Received: 30 May 2007 / Accepted: 9 June 2007 / Published online: 26 July 2007
© Springer Science+Business Media, LLC 2007
Abstract In this paper, we prove that any weak solution to the non-stationary Stokes system
in 3D with right hand side -div f satisfying (1.4) below, belongs to C ( ]0, T [; C
α
()). The
proof is based on Campanato-type inequalities and the existence of a local pressure introduced
in Wolf [13].
Keywords Non-stationary Stokes system · Interior regularity
Mathematics Subject Classification (2000) 35Q30 · MSC 35D10
1 Introduction. Statement of main result
Let be a bounded domain in R
3
, let 0 < T < ∞ and define Q := ×]0, T [. We consider
the Stokes system
∂ u
∂ t
- u +∇ p =-div f in Q, (1)
div u = 0 in Q, (2)
where u = (u
1
, u
2
, u
3
) denotes the velocity vector, p the pressure and -div f an external
force.
Proc. Conference “Variational analysis and PDE’s”. Intern. Centre “E. Majorana”, Erice, July 5–14, 2006.
J. Naumann · J. Wolf (B )
Humboldt-Universität zu Berlin, Institut für Mathematik, Unter den Linden 6, 10099 Berlin, Germany
e-mail: jwolf@mathematik.hu-berlin.de
J. Naumann
e-mail: jnaumann@mathematik.hu-berlin.de
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