Time-Periodic Solutions to the Equations of Magnetohydrodynamics with Background Magnetic Field

Abstract

Contents p 2 (1 − 3 q) ∈ (0, 1). Note that their result does not include the solutions constructed by Akiyama. The focus of this thesis lies on the existence of time-periodic solution to (MHDT), i.e., we want to find solutions (u, H, p) such that u(t+T , x) = u(t, x), H(t+T , x) = H(t, x) and p(t+T , x) = p(t, x) n 2 T 0 R n f (t, x) e −ix•ξ−i 2π T kt dxdt. The resulting function F Gn [f ] is defined on Z × R n , and the purely periodic part satisfies F Gn [P ⊥ f ](0, ξ) = 0. To construct time-periodic functions, a combination of classical Fourier multiplier results and a transference principle can be applied to yield existence of solutions on T×R n. Afterwards, classical methods of reflection and localisation can be used to construct solutions in sufficiently smooth domains. Applications of this technique can for example be found in Celik and Kyed [20,21], Eiter and Kyed [30] or Kyed and Sauer [61]. The advantage of this approach is clear: One directly constructs time-periodic solutions and therefore avoids considerations of initial value problems or the concept of R-boundedness. But since v 2,(2,1) (q,p),2 (T × R n). As a next step we come back to the trace problem and we see the advantage of working with Triebel-Lizorkin spaces in this context. By the Paley-Wiener-Schwartz theorem it is well-known that the vi ∞ (Ω) ≤ κ. This type of behaviour of the constant is to be expected, see for example Galdi and Kyed [38, Lemma 2.4]. Using Banach's fixed-point theorem together with the stated estimate, we show existence of time-periodic solutions to (MHDE) without general smallness assumptions on B 1. Note that this includes all constant magnetic fields H 0 , which is analogue to the results for the Oseen equations from [38].