2004/10/17 by Fabian Waleffe, Waleffe, Fabian
Engineering · Mathematics · Physics and Astronomy · #35Q30 #35Q35 #76B03 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Mathematical Physics (math-ph) #Navier-Stokes equation solutions #math-ph #math.AP #math.MP #msc:35Q30 #msc:35Q35 #msc:76B03 #physics.flu-dyn
paper · pdf · doi:10.48550/arxiv.math/0410380
10 pages, submitted to AMS Proc. v2: slight generalization of formula (31) and Theorem 1 (wavenumber mu=lambda>1 and mu=2, instead of mu=2 only). Clarification of a remark on Katz and Pavlovic's work on top of page 9. First 6 pages identical to v1
openalex publication_date 2004/10/17 · arxiv created 2004/10/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Katz and Pavlovic recently proposed a dyadic model of the Euler equations for which they proved finite time blow-up in the H3/2+ε Sobolev norm. It is shown that their model can be reduced to the dyadic inviscid Burgers equation where nonlinear interactions are restricted to dyadic wavenumbers. The inviscid Burgers equation exhibits finite time blow-up in Hα, for α≥ 1/2, but its dyadic restriction is even more singular, exhibiting blow-up for any α> 0. Friedlander and Pavlovic developed a closely related model for which they also prove finite time blow-up in H3/2+ε. Some inconsistent assumptions in the construction of their model are outlined. Finite time blow-up in the Hα norm, with α> 0, is proven for a class of models that includes all those models. An alternative shell model of the Navier-Stokes equations is discussed.