On the ‐boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer

• Published in: Mathematical methods in the applied sciences Vol. 38; no. 9; pp. 1888 - 1925
• Main Author: Saito, Hirokazu
• Format: Journal Article
• Language: English; Japanese
• Published: 01.06.2015
• Subjects: Boundaries; Dirichlet problem; Fluid flow; Fourier analysis; Fourier transforms; Mathematical analysis; Operators; Theorems
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.3201

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 <  ϵ  <  π  ∕ 2 and γ 0  > 0 arbitrarily, although usual parabolic theorem tells us that we must choose a large γ 0  > 0 for given 0 <  ϵ  <  π  ∕ 2. We also prove the maximal L p  −  L q 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.