Date: Friday, 21 June, 2024 - 15:00 to 16:00
Title : Phase transitions in the fractional three-dimensional Navier-Stokes equations
Speaker: Daniel W. Boutros, University of Cambridge

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy
Organizers : Pavan Pranjivan Mehta* and Arran Fernandez**
SISSA, International School of Advanced Studies, Italy
** Eastern Mediterranean University, Northern Cyprus

Keywords: fractional Navier–Stokes equations, regularity criteria, energy balance, phase transition, hypodissipation

Abstract

The fractional Navier-Stokes equations are a generalisation of the Navier-Stokes equations, in which the Laplacian in the viscous term is replaced by a fractional Laplacian. We study the functional properties of solutions to these equations as the exponent of the fractional Laplacian (which we will refer to as 𝑠) goes to zero (while keeping the viscosity fixed), instead of studying the conventional inviscid limit in which the viscosity is sent to zero. We find four critical values of the exponent, at which a qualitative change in behaviour of the solutions occurs. Such a change is referred to as a phase transition. In particular, we establish three results: i) a continuation criterion for strong solutions for 𝑠 greater than 1/3 ii) an equation of local energy balance if 𝑠 greater than or equal to 3/4 iii) an infinite hierarchy of regularity estimates on higher order derivatives for Leray-Hopf solutions if 𝑠 greater than 5/6. This paper [1] is joint work with John D. Gibbon (Imperial College London).

Biography

Daniel W. Boutros is a PhD student at the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge, under the supervision of Professor Edriss S. Titi. He received his Bachelor’s degrees from the University of Groningen, and his Master’s degree from the University of Cambridge. His research interests lie in the analysis of partial differential equations, in particular in connection with hydrodynamic turbulence theory and geophysical fluid mechanics. In particular, he has worked on the mathematical analysis of large-scale oceanic dynamics, subgrid-scale turbulence modelling and the study of turbulent wall-bounded flows.

Bibliography

[1] Daniel W. Boutros and John D. Gibbon. “Phase transitions in the fractional three-dimensional Navier–Stokes equations”. In: Nonlinearity 37.4 (2024), pp. 1-27