Centre de Physique Théorique


Wednesday 13 January 2021

14h00 – 15h00, online-for link write to Annalisa Panati

Simon Thalabard (IMPA, Brésil)

Simon Thalabard (IMPA, Brésil)

In this talk, I will outline outgoing efforts to analyse self-similar
solutions to non-linear diffusions. In the context of hydrodynamic turbulence,
non-linear diffusions arise as closures for the kinetic energy dynamics and are
historically referred to as ’’Leith models’’ In the context of wave
turbulence, they arise from an approximation of local interactions within the
relevant kinetic equations. Phenomenology of this class of non-linear
diffusions can be described from the interplay between scaling solutions and
equilibrium solutions. Among salient features, a subclass of models with
physical interest feature explosive behavior leading to finite-time singularity
. In those cases, careful numerics reveal systematic anomalous transients,
whose scalings deviate slightly from Kolmogorov-Zakharov constant-flux
exponent. Our efforts aim at characterizing those anomalies in terms of an
eigen-value problem involving self-similar solutions to the equations.
For models involving second-order derivatives only, which usually feature
direct cascade of energy, such an approach was shown to be successful and the
anomalous exponent relates to well-defined heteroclinic bifurcation.
Our recent results show that the approach can be extended to more elaborated
fourth-order models, which appear in the study of gravitational wave
turbulence or formation of Bose-Einstein condensates, and for which one
expects a dual cascade scenario. In this case, the anomalous exponent relates
to the presence of an infinitely large limit-cycle in a suitably defined 4-th
order autonomous system of ordinary differential equations.
If time permits, I will finally mention recent insights from such non-linear
diffusions on the problem of strong hydro-dynamical turbulence, and in
particular on Gallavotti’s conjecture, which states that in the limit of
vanishing viscosity, replacing viscous damping by a reversible thermostat
yield statistical features of turbulence unaltered.