2019/01/01 by Holger R. Dullin, Joachim Worthington · 1 citation
Engineering · Mathematics · #Euler equations #Euler method #Euler's formula #Fluid Dynamics and Thin Films #Fluid Dynamics and Turbulent Flows #Fourier transform #Navier-Stokes equation solutions #Nilpotent #Operator (biology) #Shear flow #Stability (learning theory) #Vorticity #math.DS
paper · pdf · doi:10.1137/19m1252260
published in SIAM Journal on Applied Mathematics 79(5), 2168-2191 (Society for Industrial and Applied Mathematics)
openalex publication_date 2019/01/01 · arxiv created 2019/03/24 · openalex created_date 2019/04/01 · arxiv updated 2020/09/07 · openalex updated_date 2026/08/05
The Euler equations on a three-dimensional periodic domain have a family of shear flow steady states. We derive a formulation of the dynamics of the vorticity Fourier modes on a periodic domain and linearize around the shear flows. The linearized system has a decomposition into infinitely many classes of modes, each of which decouples from all other modes. By projecting to the divergence-free subspace, and with the help of a Lax pair, the dynamics of each class are significantly simplified. Most classes can be reduced to corresponding classes from the two-dimensional problem, and thus many results from the two-dimensional case can be applied in the three-dimensional case, leading to results about linear stability and spectral instability. The most interesting and novel case is unique to the three-dimensional problem: for a dense set of parameter values, the linearized operator has a nilpotent part, leading to linear instability, despite spectral stability. This is connected to the nonnormality of the linearized dynamics. We show that all shear flows in the family considered (even the linearly stable ones) are parametrically unstable, because the nilpotent classes, even though exceptional, are dense.