2023/10/31 by Dallas Albritton, Wojciech Ożański, Albritton, Dallas +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.2310.20674
openalex publication_date 2023/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider vortex column solutions v = V(r) eθ+ W(r) ez to the 3D Euler equations. We give a mathematically rigorous construction of the countable family of unstable modes discovered by Liebovich and Stewartson (J. Fluid Mech. 126, 1983) via formal asymptotic analysis. The unstable modes exhibit O(1) growth rates and concentrate on a ring r= r0 asymptotically as the azimuthal and axial wavenumbers n, α→ ∞ with a fixed ratio. We construct these so-called ring modes with an inner-outer gluing procedure. Finally, we prove that each linear instability implies nonlinear instability for vortex columns. In particular, our analysis yields nonlinear instability for the Batchelor trailing line vortex V(r) :=(q)/(r) (1-e-r2) and W(r) :=e-r2 when 0