L^q-theory of a singular “winding” integral operator arising from fluid dynamics

Abstract

We analyze in classical L q (R n )-spaces, n = 2 or n = 3, 1 < q < ∞, a singular integral operator arising from the linearization of a hydrodynamical problem with a rotating obstacle. The corresponding system of partial differential equations of second order involves an angular derivative which is not subordinate to the Laplacian. The main tools are LittlewoodPaley theory and a decomposition of the singular kernel in Fourier space.

Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle

“…The spaceŴ k,q (Ω) consists of equivalence classes of L 1 loc -functions being unique only up to elements from Π k−1 and is equipped with the norm |α|=k ∂ α u q , where · q denotes the L q -norm. However, sometimes being less careful, we will consider v ∈Ŵ k,q (Ω) as a function (representative) rather than an equivalence class of functions, i.e., v ∈ L 1 loc (Ω) such that ∂ α v ∈ L q (Ω) for every multi-index α with |α| = k. For further results on similar problems we refer to [4], [8], [9], [10], [11], [12] and [13].…”

Time-Periodic Solutions to the Navier-Stokes Equations

“…In particular, we shall give simple proofs of certain L q -estimates originally established by Farwig, Hishida, and Müller in [9] and by Farwig in [6].…”

“…Maximal regularity in an L q -setting of the system (3.88) was established for the first time in the Stokes case (λ = 0) by Farwig, Hishida, and Müller in [9], and in the Oseen case (λ = 0) by Farwig in [6]. More specifically, in [9, Theorem 1.1] the following result was obtained:…”

“…The proofs of Theorem 3.7.1 and Theorem 3.7.2 presented in [9] and [6], respectively, are very technical and rely on a non-trivial application of the Littlewood-Paley decomposition. Here, simple proofs based on the maximal regularity results for the time-periodic problem obtained in the previous section will be presented.…”

“…Similar estimates for weak solutions in the Oseen case (λ = 0) were established in [37] and [38] by Kračmar, Nečasová, and Penel. The results in [31] and [37], [38] were obtained using the same very technical approach that was used in [9] and [6] based on the Littlewood-Paley decomposition. Simple proofs of these estimates were recently made available by Galdi and Kyed in [22].…”