2010/12/13 by Antoine Choffrut, Vladimír Šverák · 43 citations
Mathematics · Physics and Astronomy · #Backward Euler method #Compressibility #Degeneracy (biology) #Degenerate energy levels #Euler equations #Euler's formula #Geodesic #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Invariant (physics) #Lie group #Mathematical analysis #Mathematical physics #Mathematics #Metric (unit) #Nonlinear Waves and Solitons #Pure mathematics #Semi-implicit Euler method #math.AP
paper · pdf · doi:10.1007/s00039-012-0149-8
published in Geometric and Functional Analysis 22(1), 136-201 (Birkhäuser) · 81 pages
arxiv created 2010/12/13 · openalex publication_date 2012/01/24 · arxiv updated 2013/04/05 · openalex created_date 2020/11/23 · openalex updated_date 2026/08/05
It is well known that the incompressible Euler equations can be formulated in a very geometric language. The geometric structures provide very valuable insights into the properties of the solutions. Analogies with the finite-dimensional model of geodesics on a Lie group with left-invariant metric can be very instructive, but it is often difficult to prove analogues of finite-dimensional results in the infinite-dimensional setting of Euler's equations. In this paper we establish a result in this direction in the simple case of steady-state solutions in two dimensions, under some non-degeneracy assumptions. In particular, we establish, in a non-degenerate situation, a local one-to-one correspondence between steady-states and co-adjoint orbits.