Existence of solutions to the nonstationary Stokes system in H-μ2,1, μ∈(0,1), in a domain with a distinguished axis. Part 2. Estimate in the 3d case

W. M. Zajączkowski
2007 Applicationes Mathematicae  
EXISTENCE OF SOLUTIONS TO THE NONSTATIONARY STOKES SYSTEM IN H 2,1 −µ , µ ∈ (0, 1), IN A DOMAIN WITH A DISTINGUISHED AXIS. PART 2. ESTIMATE IN THE 3d CASE Abstract. We examine the regularity of solutions to the Stokes system in a neighbourhood of the distinguished axis under the assumptions that the initial velocity v 0 and the external force f belong to some weighted Sobolev spaces. It is assumed that the weight is the (−µ)th power of the distance to the axis. Let f ∈ L 2,−µ , v 0 ∈ H 1 −µ , µ
more » ... , v 0 ∈ H 1 −µ , µ ∈ (0, 1). We prove an estimate of the velocity in the H 2,1 −µ norm and of the gradient of the pressure in the norm of L 2,−µ . We apply the Fourier transform with respect to the variable along the axis and the Laplace transform with respect to time. Then we obtain two-dimensional problems with parameters. Deriving an appropriate estimate with a constant independent of the parameters and using estimates in the two-dimensional case yields the result. The existence and regularity in a bounded domain will be shown in another paper.
doi:10.4064/am34-2-2 fatcat:tuksbqfxxbbonh4u7ehow2p3qq