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On the theory of nonstationary hydrodynamic potentials. (English) Zbl 0995.35044
Salvi, Rodolfo (ed.), The Navier-Stokes equations: theory and numerical methods. Proceedings of the international conference, Varenna, Lecco, Italy, 2000. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 223, 113-129 (2002).
The author considers the initial-boundary value problem for the Stokes equations \[ \vec{v}_t-\Delta \vec{v}+\nabla p=0,\;\nabla \cdot \vec{v}=0,\;x\in \Omega ,\;t\in ( 0,T) , \] \[ \vec{v}\mid _{t=0}=\vec{v}_0( x) ,\;\vec{v}\mid _S=\vec{a}( x',t) , \] in a bounded convex domain \(\Omega \subset {\mathbb R}^n,n\geq 2,\) with a smooth boundary \(S.\) The main result is the following : Assume that \(S\in C^{2+\alpha },\alpha \in ( 0,1) \). For arbitrary \(\vec{a}( x,t) \) and \(\vec{v}_0( x) \) which are continuous and satisfy the compatibility conditions \(\vec{a}( x,0) =\) \( \vec{v}_0( x) \mid _S,\;\nabla \cdot \vec{v} _0( x) =0,\;\vec{a}( x,0) \cdot \vec{n}( x) \mid _S=0,\) the problem has a continuous solution satisfying the inequality \[ \sup _{x\in \Omega } \sup _{t<T}|\vec{v}( x,t) |\leq c( t) \left( \sup _{x\in S} \sup _{t<T}|\vec{a}( x,t) |+\sup _{x\in \Omega } |\vec{v}_0( x) |\right). \]
For the entire collection see [Zbl 0972.00046].

MSC:
35Q30 Navier-Stokes equations
76D07 Stokes and related (Oseen, etc.) flows
35B35 Stability in context of PDEs
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