Maximum modulus estimate for the solution of the nonstationary Stokes equations

A maximum modulus estimate for the nonstationary Stokes equations in $C^2$ domain is found. The singular part and regular part of Poisson kernel are analyzed. The singular part consists of the gradient of single layer potential and the gradient of composite potential defined on only normal component of the boundary data. Furthermore, the normal velocity near the boundary is bounded if the boundary data is bounded. If the normal component of the boundary data is Dini-continuous and the tangential component of the boundary data is bounded, then the maximum modulus of velocity is bounded in whole domain.

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