On Some Dyadic Models of the Euler Equations
نویسنده
چکیده
Katz and Pavlovic recently proposed a dyadic model of the Euler equations for which they proved finite time blow-up in the H3/2+ǫ 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+ǫ. Some inconsistent assumptions in the construction of their model are outlined. Finite time blowup 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.
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