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Comment on “Application of the extended Lie group analysis to the Hopf functional formulation of the Burgers equation” [J. Math. Phys. 54, 072901 (2013)]

### Abstract

The quest to find new statistical symmetries in the theory of turbulence is an ongoing research endeavor which is still in its beginning and exploratory stage. In our comment we show that the recently performed study of Wacławczyk and Oberlack [J. Math. Phys. 54, 072901 (2013)] failed to present such new statistical symmetries. Despite their existence within a functional Fourier space of the statistical Burgers equation, they all can be reduced to the classical and well-known symmetries of the underlying deterministic Burgers equation itself, except for one symmetry, but which, as we will demonstrate, is only a mathematical artefact without any physical meaning. Moreover, we show that the proposed connection between the translation invariance of the multi-point moments and a symmetry transformation associated to a certain invariant solution of the inviscid functional Burgers equation is invalid. In general, their study constructs and discusses new particular solutions of the functional Burgers equation without referring them to the well-established general solution. Finally, we also see a shortcoming in the presented methodology as being too restricted to construct a complete set of Lie point symmetries for functional equations. In particular, for the considered Burgers equation essential symmetries are not captured.

© 2016 AIP Publishing LLC

Received 06 August 2015
Accepted 27 November 2015
Published online 22 March 2016

Article outline:

I. THE NEW STATISTICAL SYMMETRIES AND THEIR DETERMINISTIC ORIGIN
II. ON S_{1} AND ITS NON-CONNECTEDNESS TO THE TRANSLATION INVARIANCE OF THE MULTI-POINT MOMENTS
III. FINAL REMARKS

/content/aip/journal/jmp/57/3/10.1063/1.4940357

1.

1.M. Wacławczyk and M. Oberlack, “Application of the extended Lie group analysis to the Hopf functional formulation of the Burgers equation,” J. Math. Phys. 54, 072901 (2013).

http://dx.doi.org/10.1063/1.4812803
3.

3. In Ref. 2 the Fourier transform is defined oppositely than in Ref. 1. As a result, the respective equations in Ref. 1 and Ref. 2, corresponding to (5) and (10), only differ by their sign in the imaginary unit.

4.

4. Note that in Ref. 1 the sampled values for u are denoted by v; here, however, we stay with the notation from Ref. 2.

5.

5. For the Euler and Navier-Stokes equations, the fields y(x) and z(k) must be additionally solenoidal, i.e., ∇ ⋅ y = 0 and k ⋅ z = 0, in order to eliminate the pressure field in each case; for the 1D Burgers equation, however, this restriction is of course irrelevant.

6.

6.P. J. Olver, Applications of Lie Groups to Differential Equations, 2nd ed. (Springer, Verlag, 1993).

9.

9.M. Frewer,

G. Khujadze, and

H. Foysi, “

On the physical inconsistency of a new statistical scaling symmetry in incompressible Navier-Stokes turbulence,” e-print

arXiv:1412.3061 (

2014).

10.

10.M. Frewer,

G. Khujadze, and

H. Foysi, “

A critical examination of the statistical symmetries admitted by the Lundgren-Monin-Novikov hierarchy of unconfined turbulence,” e-print

arXiv:1412.6949 (

2014).

11.

11. The error in Eq. [114] in Ref. 1 also has an effect on the subsequent solution given by Eqs. [129-131].

12.

12. Under the additional assumption that z(0) = z(k)|_{k=0} = 0.

13.

13. However, note that ultimately the Galilean invariance (22) is not compatible with the underlying constraint of a time-independent z-field, which basically is a defining condition of the functional Burgers equation (8), and which, of course, must hold in the transformed domain as well in order to warrant an overall consistent symmetry analysis of this equation. That means, since the Galilean symmetry (22) does not induce the necessary constraint in the transformed domain from ∂_{t}z = 0, and vice versa, the presence of this constraint ultimately breaks the Galilean symmetry (22). Unfortunately, this symmetry breaking mechanism of the underlying constraints ∂_{t}y(x) = 0 and ∂_{t}z(k) = 0 of the corresponding functional Burgers equations (7) and (8), has not been taken into account or discussed in Ref. 1.

14.

14.J. Kevorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods (Springer, Verlag, 1996).

16.

16.I. Hosokawa and K. Yamamoto, “Numerical study of the Burgers’ model of turbulence based on the characteristic functional formalism,” Phys. Fluids 13, 1683–1692 (1970).

http://dx.doi.org/10.1063/1.1693141
17.

17.M. Oberlack and M. Wacławczyk, “On the extension of Lie group analysis to functional differential equations,” Arch. Mech. 58, 597–618 (2006).

18.

18. Note that: (i) For scaling symmetry S_{5} the functional derivative δ/δz = ∂/(∂z dk) transforms invariantly. (ii) The projective symmetry S_{6} only exists in physical space as its underlying deterministic symmetry is ill-defined in Fourier space.

19.

19. However, note again that ultimately also symmetry S_{6} gets broken by the underlying constraint ∂_{t}y(x) = 0 of the functional Burgers equation (7), in the same way as explained in Ref. 13.

http://aip.metastore.ingenta.com/content/aip/journal/jmp/57/3/10.1063/1.4940357

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2016-03-22

2016-10-27

### Abstract

The quest to find new statistical symmetries in the theory of turbulence is an ongoing research endeavor which is still in its beginning and exploratory stage. In our comment we show that the recently performed study of Wacławczyk and Oberlack [J. Math. Phys. 54, 072901 (2013)] failed to present such new statistical symmetries. Despite their existence within a functional Fourier space of the statistical Burgers equation, they all can be reduced to the classical and well-known symmetries of the underlying deterministic Burgers equation itself, except for one symmetry, but which, as we will demonstrate, is only a mathematical artefact without any physical meaning. Moreover, we show that the proposed connection between the translation invariance of the multi-point moments and a symmetry transformation associated to a certain invariant solution of the inviscid functional Burgers equation is invalid. In general, their study constructs and discusses new particular solutions of the functional Burgers equation without referring them to the well-established general solution. Finally, we also see a shortcoming in the presented methodology as being too restricted to construct a complete set of Lie point symmetries for functional equations. In particular, for the considered Burgers equation essential symmetries are not captured.

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