# Agenda

# Wednesday 13 January 2021

### 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.