Nonstationary Stokes system in Sobolev–Slobodetski spaces

Abstract

We consider an initial-boundary value problem for the nonstationary Stokes system in a bounded domain $$\Omega \subset \mathbb R ^3$$ with slip boundary conditions. We prove the existence in the Hilbert–Sobolev–Slobodetski spaces with fractional derivatives. The proof is divided into two main steps. In the first step by applying the compatibility conditions an extension of initial data transforms the considered problem to a problem with vanishing initial data such that the right-hand sides data functions can be extended by zero on the negative half-axis of time in the above mentioned spaces. The problem with vanishing initial data is transformed to a functional equation by applying an appropriate partition of unity. The existence of solutions of the equation is proved by a fixed point theorem. We prove the existence of such solutions that $$v\in H^{l+2,l/2+1}(\Omega \times (0,T)),\,\nabla p\in H^{l,l/2}(\Omega \times (0,T)),\,v$$ —velocity, $$p$$ —pressure, $$l\in \mathbb R _+\cup \{0\},\,l \ne [l]+\frac{1}{2}$$ and the spaces are introduced by Slobodetski and used extensively by Lions–Magenes. We should underline that to show solvability of the Stokes system we need only solvability of the heat and the Poisson equations in $$\mathbb R ^3$$ and $$\mathbb R _+^3$$ . This is possible because the slip boundary conditions are considered.