Time-Periodic Linearized Navier–Stokes Equations: An Approach Based on Fourier Multipliers

Abstract

The Stokes and Oseen linearizations of the time-periodic Navier–Stokes equations in the n-dimensional whole space for n ≥ 2 are investigated. An approach based on Fourier multipliers is introduced to establish $\mathrm{L}^{q}$ estimates and to identify function spaces of maximal regularity for the corresponding operators. Moreover, the representation of a solution in terms of a Fourier multiplier is used to introduce the concept of a time-periodic fundamental solution. The main idea is to replace the time axis by a torus group and to study the system in a setting of functions defined on a locally compact abelian group G. For this purpose, we develop the required formalism. More specifically, we introduce the space $\mathcal{S}(G)$ of Schwartz-Bruhat functions and investigate the Stokes and Oseen systems in the corresponding space of tempered distributions $\mathcal{S^{{\prime}}}(G)$. Moreover, we give a detailed proof of the so-called Transference Principle, which enables us to employ Fourier multipliers in a group setting in order to establish $\mathrm{L}^{q}$ estimates.