On the crest factor and its relevance in detecting turbulent behaviour in solutions of partial differential equations

Michele V Bartuccelli; Guido Gentile; Michele V Bartuccelli; Guido Gentile

doi:10.3934/mbe.2022385

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 8: 8273-8287. doi: 10.3934/mbe.2022385

Previous Article Next Article

Research article Special Issues

On the crest factor and its relevance in detecting turbulent behaviour in solutions of partial differential equations

Michele V Bartuccelli ^{1
,
,},
Guido Gentile ^{2
,
,}

1.
Department of Mathematics, University of Surrey, Guildford, GU2 7XH, UK
2.
Dipartimento di Matematica e Fisica, Università Roma Tre, Roma, I-00146, Italy

Academic Editor: Rongsong Liu

Received: 30 January 2022 Revised: 25 April 2022 Accepted: 11 May 2022 Published: 08 June 2022

In this work we investigate the connection between two fundamental features of solutions of partial differential equations (PDEs), namely the crest factor and the length scale associated to each solution. We illustrate how the crest factor of solutions of some linear and non-linear PDEs, including the incompressible two-dimensional Navier-Stokes equations, has the capability for detecting turbulent and non-turbulent behaviour.
- partial differential equations,
- analysis of solutions,
- crest factor,
- length scales,
- turbulence
Citation: Michele V Bartuccelli, Guido Gentile. On the crest factor and its relevance in detecting turbulent behaviour in solutions of partial differential equations[J]. Mathematical Biosciences and Engineering, 2022, 19(8): 8273-8287. doi: 10.3934/mbe.2022385

Related Papers:

Abstract

In this work we investigate the connection between two fundamental features of solutions of partial differential equations (PDEs), namely the crest factor and the length scale associated to each solution. We illustrate how the crest factor of solutions of some linear and non-linear PDEs, including the incompressible two-dimensional Navier-Stokes equations, has the capability for detecting turbulent and non-turbulent behaviour.

References

[1]	P. Voke, L. Kleiser, J. P. Chollet, Direct and Large-Eddy Simulation I: Selected papers from the First ERCOFTAC Workshop on Direct and Large-Eddy Simulations, The University of Surrey, Guildford, UK, 27-30 March 1994, Springer Science and Business Media, Springer, Dordrecht, 1994. https://doi.org/10.1007/978-94-011-1000-6
[2]	R. R. Singh, Basic Electrical Engineering, McGraw Hill, Chennai, 2018.
[3]	M. P. Norton, D. G. Karczub, Fundamentals of Noise and Vibration Analysis for Engineers, Cambridge University Press, Cambridge, 2003. https://doi.org/10.1017/CBO9781139163927
[4]	M. Chasin, Musicians and Hearing Aids. A Clinical Approach, Plural Publishing, San Diego, 2022.
[5]	M. J. Griffin, Handbook of Human Vibration, Academic Press, London, 1990.
[6]	C. Jannot, Analysis and Management of Sleep Data, in Biomedical Signal Processing and Artificial Intelligence in Healthcare, pp. 207–240, Ed. W. A. Zgallai, Elsevier, London, 2020. https://doi.org/10.1016/B978-0-12-818946-7.00008-1
[7]	R. A. Adams, Sobolev spaces, Academic Press, New York, 1975.
[8]	R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Second edition, Appl. Math. Sci., 68, Springer, New York, 1997. https://doi.org/10.1007/978-1-4612-0645-3
[9]	J. C. Robinson, Infinite-dimensional dynamical systems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001.
[10]	A. V. Babin, M. I. Vishik, Attractors for evolution equations, Nauka, Moscow, 1989.
[11]	M. V. Bartuccelli, J. D. Gibbon, Sharp constants in the Sobolev embedding theorem and a derivation of the Brezis-Gallouet interpolation inequality, J. Math. Phys., 52 (2011), 093706. https://doi.org/10.1063/1.3638056 doi: 10.1063/1.3638056
[12]	M. V. Bartuccelli, Sharp constants for the $L^{\infty}$-norm on the torus and applications to dissipative partial differential equations, Differ. Integral Equ., 27 (2014), 59–80.
[13]	M. V. Bartuccelli, C. R. Doering, J. D. Gibbon, S. A. Malham, Length Scales in Solutions of the Navier-Stokes equations, Nonlinearity, 6 (1993), 549–568. https://doi.org/10.1088/0951-7715/6/4/003 doi: 10.1088/0951-7715/6/4/003
[14]	C. R. Doering, J. D. Gibbon, Applied Analysis of the Navier-Stokes Equations, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1995.
[15]	S. A. Gourley, M. V. Bartuccelli, Length scales in solutions of a scalar reaction-diffusion equation with delay, Phys. Lett. A, 202 (1995), 79–87. https://doi.org/10.1016/0375-9601(95)00334-Y doi: 10.1016/0375-9601(95)00334-Y
[16]	R. Dascaliuc, C. Foias, M. S. Jolly, Relations between the Energy and enstrophy on the global attractor of the 2-D Navier-Stokes equations, J. Dyn. Differ. Equ., 17 (2005), 643–736. https://doi.org/10.1007/s10884-005-8269-6 doi: 10.1007/s10884-005-8269-6
[17]	R. A. Fisher, The wave of advance of advantageous gene, Ann. Eugenics, 7 (1937), 335–369. https://doi.org/10.1111/j.1469-1809.1937.tb02153.x doi: 10.1111/j.1469-1809.1937.tb02153.x
[18]	A. Kolmogorov, I. Petrovsky, N. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bull. Univ. Moscou, Ser. Int. Sec., 1 (1937), 1–25.
[19]	M. V. Bartuccelli, On the crest factor for dissipative partial differential equations, Proc. Math. Phys. Eng. Sci., 475 (2019), 20190322. https://doi.org/10.1098/rspa.2019.0322 doi: 10.1098/rspa.2019.0322
[20]	L. C. Berselli, T. Iliescu, W. J. Layton, Mathematics of Large Eddy Simulation of Turbulent Flows, Springer series in Scientific Computation, Springer, Berlin, 2006.
[21]	U. Frisch, Turbulence: The legacy of A. N. Kolmogorov, Cambridge University Press, Cambridge, 1995.
[22]	M. V. Bartuccelli, On the nature of space fluctuations of solutions of dissipative partial differential, Appl. Math. Lett., 96 (2019), 14–19. https://doi.org/10.1016/j.aml.2019.04.011 doi: 10.1016/j.aml.2019.04.011
[23]	M. V. Bartuccelli, J. H. B. Deane, G. Gentile, Explicit estimates on the torus for the sup-norm and the Crest Factor of solutions of the modified Kuramoto-Sivashinsky Equation in one and two space dimensions, J. Dyn. Differ. Equ., 32 (2020), 791–807. https://doi.org/10.1007/s10884-019-09762-1 doi: 10.1007/s10884-019-09762-1
[24]	F. Calogero, A. Degasperis, The Spectral Transform and Solitons, North-Holland, Amsterdam, 1982.
[25]	R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, H. C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, London, 1982.
[26]	P. Constantin, C. Foias, Navier–Stokes Equations, University of Chicago Press, Chicago, 1988.
[27]	A. A. Ilyin, Best constants in multiplicative inequalities for sup-norms, J. Lond. Math. Soc., 58 (1998), 84–96. https://doi.org/10.1112/S002461079800653X doi: 10.1112/S002461079800653X
[28]	A. A. Ilyin, E. S. Titi, Sharp estimates for the number of degrees of freedom for the damped-driven 2D Navier-Stokes equations, J. Nonlinear Sci., 16 (2006), 233–253. https://doi.org/10.1007/s00332-005-0720-7 doi: 10.1007/s00332-005-0720-7
[29]	C. Marchioro, M. Pulvirenti, Mathematical Theory of Incompressible Nonviscous Fluids, Appl. Math. Sci., 94, Springer-Verlag, New York, 1994. https://doi.org/10.1007/978-1-4612-4284-0
[30]	F. Fantuzzi, D. Goluskin, D. Huang, S. Chernyshenko, Bounds for Deterministic and Stochastic Dynamical Systems using Sum of Squares Optimization, SIAM J. Appl. Dyn. Syst., 16 (2016), 1962–1988. https://doi.org/10.1137/15M1053347 doi: 10.1137/15M1053347
[31]	D. Goluskin, F. Fantuzzi, Bounds on mean energy in the Kuramoto–Sivashinsky equation computed using semidefinite programming, Nonlinearity, 32 (2019), 1705–1730. https://doi.org/10.1088/1361-6544/ab018b doi: 10.1088/1361-6544/ab018b
[32]	M. V. Bartuccelli, Explicit estimates on the torus for the sup-norm and the dissipative length scale of solutions of the Swift-Hohenberg equation in one and two space dimensions, J. Math. Anal. Appl., 411 (2014), 166–176. https://doi.org/10.1016/j.jmaa.2013.09.027 doi: 10.1016/j.jmaa.2013.09.027
[33]	M. V. Bartuccelli, S. A. Gourley, A. A. Ilyin, Positivity and the attractor dimension in a fourth-order reaction diffusion equation, Proc. Roy. Soc. London A, 458 (2002), 1431–1446. https://doi.org/10.1098/rspa.2001.0931 doi: 10.1098/rspa.2001.0931

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)