Abstract
The time‐periodic Stokes problem in a half‐space with fully inhomogeneous right‐hand side is investigated. Maximal regularity in a time‐periodic Lp setting is established. A method based on Fourier multipliers is employed that leads to a decomposition of the solution into a steady‐state and a purely oscillatory part in order to identify the suitable function spaces.
Highlights
We investigate the T -time-periodic Stokes problem∂tu − ∆u + ∇p = f div u = g u=h u(T + t, ·) = u(t, ·), in R × Rn+, in R × Rn+, on R × ∂Rn+, (1.1)in a half-space Rn+ of dimension n ≥ 2
A method based on Fourier-multipliers is employed that leads to a decomposition of the solution into a steady-state and a purely oscillatory part in order to identify the optimal function spaces
It is convenient to formulate T -time-periodic problems in a setting of function spaces where the torus T := R/T Z is used as a time-axis
Summary
In a half-space Rn+ of dimension n ≥ 2. We shall establish maximal regularity Lp estimates that include the case h = 0. It is convenient to formulate T -time-periodic problems in a setting of function spaces where the torus T := R/T Z is used as a time-axis. Via lifting with the quotient map π : R → T, T -time-periodic functions are canonically identified as functions defined on T and vice versa For such functions, we introduce the simple decomposition. The two separated estimates (1.6) and (1.7) of different regularity type for the steadystate Pu and the purely oscillatory part P⊥u of the solution, respectively, demonstrate the necessity of the decomposition. Denote the space of smooth T -time-periodic E-valued functions.
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