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On the \(L_{p}\)-\(L_{q}\) maximal regularity of the Neumann problem for the Stokes equations in a bounded domain. (English) Zbl 1145.35053

This paper is concerning with the \(L_p\)-\(L_q\) maximal regularity of the Neumann problem for the Stokes equations in a bounded domain. The proof of the main results is divided into three steps. The first step shows the \(L_p\)-\(L_q\) maximal regularity of solutions to the model problems in the whole space and in the half-space by using the operator valued Fourier multiplier theorem. In the second step the authors consider the Stokes equation with zero Neumann condition in a general bounded domain and, using the localization procedure, reduce the problem to the model problems in the whole space and half-space. In the third step the authors use a solution to the Laplace equation with the zero Dirichlet boundary condition, to reduce the model problem to the divergence free case.

MSC:

35B45 A priori estimates in context of PDEs
35Q30 Navier-Stokes equations
35B65 Smoothness and regularity of solutions to PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D07 Stokes and related (Oseen, etc.) flows
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