Characterization of Three-Dimensional Euler Flows Supported on Finitely Many Fourier Modes

• Published in: Journal of mathematical fluid mechanics Vol. 24; no. 3
• Main Authors: Kishimoto, Nobu, Yoneda, Tsuyoshi
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.08.2022; Springer Nature B.V
• Subjects: Beltrami flow; Classical and Continuum Physics; Coriolis effect; Euler-Lagrange equation; Fluid flow; Fluid mechanics; Fluid- and Aerodynamics; Mathematical analysis; Mathematical Methods in Physics; Physics; Physics and Astronomy; Stretching; Theoretical mathematics; Three dimensional flow; Turbulence; Two dimensional flow; Three-dimensional incompressible Euler flow; Characterization; 35Q31; Finitely many Fourier modes; Beltrami flow; 76B03
• ISSN: 1422-6928; 1422-6952
• DOI: 10.1007/s00021-022-00703-5

Recently, the Nash-style convex integration has been becoming the main scheme for the mathematical study of turbulence, and the main building block of it has been either Beltrami flow (finite mode) or Mikado flow (compactly supported in the physical side). On the other hand, in physics, it is observed that turbulence is composed of a hierarchy of scale-by-scale vortex stretching. Thus our mathematical motivation in this study is to find another type of building blocks accompanied by vortex stretching and scale locality (possibly finitely many Fourier modes). In this paper, we give a complete list of solutions to the 3D Euler equations with finitely many Fourier modes, which is an extension of the corresponding 2D result by Elgindi et al. (Comm Math Phys 355(1): 145–159, 2017). In particular, we show that there are no 3D Euler flows with finitely many Fourier modes, except for stationary 2D-like flows and Beltrami flows. We also discuss the case when viscosity and Coriolis effect are present.