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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