Blow-up in finite time for the dyadic model of the Navier-Stokes equations

By A. Cheskidov. 2006

84 citations recorded by the source corpus.

We study the dyadic model of the Navier-Stokes equations introduced by Katz and Pavlovic. They showed a finite time blow-up in the ca se where the dissipation degreeis less than 1/4. In this paper we prove the existence of weak solutions for all �, energy inequality for every weak solution with nonnegative initial data starting from any time, local regularity for � > 1/3, and global regularity for � � 1/2. In addition, we prove a finite time blow-up in the case where � < 1/3. It is remarkable that the model with � = 1/3 enjoys the same estimates on the nonlinear term as the 4D Navier-Stokes equations. Finally, we discuss a weak global attractor, which coincides with a maximal bounded invariant set for alland becomes a strong global attractor for � � 1/2.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bc56026598972c18973375c0d27c3be76ea3f757

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In Applied Mathematics

In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

Global Attractors of Evolutionary Systems (2006, methodology): “See also Cheskidov [7] for an application ofthis theorem to a viscous dyadic model.”
Finite time blowup for an averaged three-dimensional Navier-Stokes equation (2014, methodology): “Note that the blowup here is available for all values of the dissipation parameter up to the critical value of 1/2, in contrast to the results in [26] and [10] for the untruncated equation (1.”
On global attractors of the 3D Navier-Stokes equations (2006, result): “Moreover, a similar analysis of the dyadic models also results in a finite time blow-up (see [ 5 ]).”
Lectures on the Onsager conjecture (2010, result): “We also consider the inviscid variant of (47) as a model for the Euler equation: (48) d dt aq = λ c q−1a 2 q−1 − λcqaqaq+1 + fq, q = 0, 1, ... Long after the origination analytical studies of the models resumed in recent works of Katz and Pavlovic [31], Cheskidov et al [13, 10], Kiselev and Zlatoš ”