International Journal of Mathematics and Computational Science
Articles Information
International Journal of Mathematics and Computational Science, Vol.1, No.5, Oct. 2015, Pub. Date: Aug. 13, 2015
"Restricted Euler Equations" Model Is Not Very Suitable for Revealing Properties Inherent to Euler Equations
Pages: 332-338 Views: 2093 Downloads: 762
[01] Vladimir A. Dobrynskii, Institute for Metal Physics of N.A.S.U., Kiev, Ukraine.
For the restricted Euler equations, we prove the following: 1) If there is a time instant such that the perfect vector alignment between the vorticity vector and the strain matrix eigenvector happens at some point inside the 3D incompressible restricted Euler flow, it continues then permanently and keeps forever in the sense that one happens successively at all points which belong to the given point trajectory generated by the flow. 2) If the aforementioned trajectory exists, then, depending on initial data, one can either blow up for a finite time or not. What is interesting in doing so is that the aforesaid finite-time blow ups can be various and very different. In particular, we found the blow up such that the strain matrix eigenvalues all go at infinity (which can be both positive and negative) whereas the vorticity remains bounded at the same time. Such kind the vorticity behavior should be considered, generally speaking, as slightly unexpected if to take into account the well-known Beale-Kato-Majda (abbr. BKM) criterion for solutions of the genuine Euler equations to blow up at a finite-time. On the other hand, there are solutions to the restricted Euler equations which blow up at a finite time by a scenario similar to that is described in the BKM criterion. Summarizing all what is the aforesaid we see that the restricted Euler equations are not suitable enough for revealing properties inherent to the Euler equations.
Finite-Time Blow up, the Euler Equations, the Restricted Euler Equations, Alignment, Vorticity, Strain, Vortex Stretching
[01] Viellefosse P.: Local interaction between vorticity and shear in a perfect incompressible fluid. J. Phys. Paris. vol.43 (1982), 837-842.
[02] Viellefosse P.: Internal motion of a small element of fluid in an inviscid flow. Physica A. vol.125 (1984), 150-162.
[03] Novikov E. A.: Internal dynamics of flows and formation of singularities. Fluid Dynam. Res. vol.6 (1990), 79-89.
[04] Cantwell B. J.: Exact solution of a restricted Euler equation for the velocity gradient tensor. Phys. Fluids. vol.4, no. 4 (1992), 782-793.
[05] Liu H. and Tadmor E.: Spectral dynamics of the velocity gradient field in restricted flows. Commun. Math. Phys. 228 (2002), 435-466.
[06] Liu H. and Tadmor E.: Critical thresholds in 2D restricted Euler-Poisson equations. SIAM J. Appl. Math., vol. 63, no. 6 (2003), 1889-1910.
[07] Liu H., Tadmor E. and Wei D.: Global regularity of the 4D restricted Euler equations. Physica D (2009), doi: 10.1016/j.phesd.2009.07.009.
[08] Gibbon J. D.: The three-dimensional Euler equations: Where do we stand? Physica D. vol. 237 (2008), 1894-1904.
[09] Gantmakher F. R., Theory of Matrices, 3th edition, "Nauka" Publishing House, Moscow, 1967, 575 p. (in Russian).
[10] Beale J. T., Kato T. and Majda A.: Remarks on the breakdown of smooth solutions for 3-D Euler equtions. Commun. Math. Phys. 94 (1984), 61-66.
[11] Dobrynskii V. A.: Perfect vector alignment of the vorticity and the vortex stretching is one forever. Forum Mathematicum (2011, september), DOI~10.1515~/FORM.2011.149, 13 pp.
[12] Ponce G.: Remarks on a paper by J.T.Beale, T.Kato and A.Majda. Commun. Math. Phys. 98 (1985), 349-353.
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