Speaker: John D. Gibbon, Dept of Mathematics, Imperial College London

Time : 15:00 - 16.00 CEST (Rome/Paris)

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy

Zoom :  A zoom meeitng link will appear here, one hour before the talk

Organizers : Pavan Pranjivan Mehta* (pavan.mehta@sissa.it) and Arran Fernandez** (arran.fernandez@emu.edu.tr)

* SISSA, International School of Advanced Studies, Italy

** Eastern Mediterranean University, Northern Cyprus

Keywords:  Multifractal model; Navier-Stokes equations; Toner-Tu equations

Abstract: The multifractal model (MFM) of Parisi and Frisch (1985) has been fundamental to our understanding and interpretation of homogeneous isotropic turbulence [1,2]. How it corresponds to the Navier-Stokes equations (NSEs) is the topic of this talk. By studying the NS energy dissipation in higher L𝑝-norms we look at the properties of Leray-Hopf solutions which can be formally reworked so that these appear to display a spread of dimensions. When compared to the MFM it is found that the range of h, the MFM invariance parameter, lies in the range -2/3 ≤ h ≤ 1/3. It also recovers the inverse Paladin-Vulpiani scale Lη^{-1} ~ Re^{1/(1+h)}. We then briefly discuss the fractional NSEs in a similar context. If time permits, I will also perform a brief survey of the incompressible Toner-Tu equations (ITT) which govern flocking phenomena. These are the NSEs on the LHS with terms α u - β u |u|^{2} on the RHS and share many similar properties of the NSEs themselves.

Biography: John Gibbon joined the Mathematics Department at Imperial College London in 1980 as a 31-year-old lecturer. He had been an undergraduate in Mathematics at the University of Birmingham (1967-70) and studied for his PhD at the University of Manchester Science & Technology (1970-73). For five years he had had postdoc positions and then held a lectureship at UCD Dublin for 2 years. His early work had been on nonlinear waves and solitons before he moved on to the study of Navier-Stokes and Euler turbulence.

Bibliography

[1] G. Parisi, U. Frisch, in: M. Ghil, R. Benzi, G. Parisi (Eds.), Turbulence and Predictability in Geophysical Fluid Dynamics, in: Proc. Int. School of Physics E. Fermi 1983 Varenna, Italy, North-Holland, Amsterdam, 1985, pp. 84–87.

[2] U. Frisch, Turbulence : The Legacy of A. N. Kolmogorov, CUP, Cambridge, 1995.

[3] J. D. Gibbon, Identifying the multifractal set on which energy dissipates in a turbulent Navier–Stokes fluid, Physica D 445 (2023) 133654.