On the theory of nonstationary hydrodynamic potentials (Q2771086)

The author considers the initial-boundary value problem for the Stokes equations NEWLINE\[NEWLINE \vec{v}_t-\Delta \vec{v}+\nabla p=0,\;\nabla \cdot \vec{v}=0,\;x\in \Omega ,\;t\in ( 0,T) , NEWLINE\]NEWLINE NEWLINE\[NEWLINE \vec{v}\mid _{t=0}=\vec{v}_0( x) ,\;\vec{v}\mid _S=\vec{a}( x',t) , NEWLINE\]NEWLINE 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 NEWLINE\[NEWLINE \sup _{x\in \Omega } \sup _{t<T}|\vec{v}( x,t) |\leq c( t) \left( \sup _{x\in S} \sup _{t<T}|\vec{a}( x,t) |+\sup _{x\in \Omega } |\vec{v}_0( x) |\right).NEWLINE\]NEWLINENEWLINENEWLINEFor the entire collection see [Zbl 0972.00046].