Eulerian and Lagrangian Stability in Zeitlin’s Model of Hydrodynamics
Artikel i vetenskaplig tidskrift, 2024

The two-dimensional (2-D) Euler equations of a perfect fluid possess a beautiful geometric description: they are reduced geodesic equations on the infinite-dimensional Lie group of symplectomorphims with respect to a right-invariant Riemannian metric. This structure enables insights to Eulerian and Lagrangian stability via sectional curvature and Jacobi equations. The Zeitlin model is a finite-dimensional analogue of the 2-D Euler equations; the only known discretization that preserves the rich geometric structure. Theoretical and numerical studies indicate that Zeitlin’s model provides consistent long-time behaviour on large scales, but to which extent it truly reflects the Euler equations is mainly open. Towards progress, we give here two results. First, convergence of the sectional curvature in the Euler–Zeitlin equations on the Lie algebra su(N) to that of the Euler equations on the sphere. Second, L2-convergence of the corresponding Jacobi equations for Lagrangian and Eulerian stability. The results allow geometric conclusions about Zeitlin’s model to be transferred to Euler’s equations and vice versa, which could expedite the ultimate aim: to characterize the generic long-time behaviour of perfect 2-D fluids.

Författare

Klas Modin

Chalmers, Matematiska vetenskaper, Tillämpad matematik och statistik

Göteborgs universitet

Manolis Perrot

Université Grenoble Alpes

Communications in Mathematical Physics

0010-3616 (ISSN) 1432-0916 (eISSN)

Vol. 405 8 177

Långtidsbeteende för tvådimensionella inkompressibla flöden via kvantisering

Vetenskapsrådet (VR) (2022-03453), 2023-01-01 -- 2026-12-31.

Ämneskategorier

Teknisk mekanik

Beräkningsmatematik

DOI

10.1007/s00220-024-05047-x

Mer information

Senast uppdaterat

2024-08-09