On the ‐boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer |
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| Authors: | Hirokazu Saito |
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| Affiliation: | Department of Pure and Applied Mathematics, Graduate School of Fundamental Science and Engineering, Waseda University, Shinjuku‐ku, Tokyo 169‐8555, Japan |
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| Abstract: | In this paper, we prove the  ‐boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer with resolvent parameter  , where  , and our boundary conditions are nonhomogeneous Neumann on upper boundary and Dirichlet on lower boundary. We want to emphasize that we can choose 0 < ? < π ∕ 2 and γ0 > 0 arbitrarily, although usual parabolic theorem tells us that we must choose a large γ0 > 0 for given 0 < ? < π ∕ 2. We also prove the maximal Lp ? Lq regularity theorem of the nonstationary Stokes problem as an application of the  ‐boundedness. The key of our approach is to apply several technical lemmas to the exact solution formulas of a resolvent problem. The formulas are obtained through the solutions of the ODEs, in the Fourier space, driven by the partial Fourier transform with respect to tangential space variable  . Copyright © 2014 John Wiley & Sons, Ltd. |
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| Keywords: | Stokes equations infinite layer ‐boundedness maximal regularity |
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