14:00 David Villringer, Imperial College London
Enhanced dissipation by transport type noise
A fundamental model in mathematical fluid dynamics is that of a passive scalar subject to both molecular diffusion, as well as advection due to the motion of some fluid. In particular, one expects the interplay of the advection and diffusion terms to result in a rate of convergence to the equilibrium for the passive scalar that is far quicker than the natural dissipative one, a phenomenon known as enhanced dissipation. This effect is particularly pronounced when the fluid behaves in a disorderly and turbulent way. A particularly simple model for such fluid motion is given by stochastic transport noise, which manages to model the chaotic and seemingly random nature of turbulence very effectively. In this talk, we will discuss the heuristics behind why certain random flows can be seen to enhance dissipation, and derive a spectral criterion for when a given SPDE driven by transport noise achieves enhanced dissipation. Furthermore, we will obtain explicit rates for enhanced dissipation by transport noise, as well as the precise hypoelliptic regularisation it enjoys, in the special case when this noise generates a shear flow.
14:35 Shyam Popat, University of Oxford
Working towards a central limit theorem for the Dean--Kawaski equation
In this short talk, I'll begin by introducing the Dean--Kawasaki equation and motivate why kinetic solution theory is needed to study it. Then I will state the well-posedness results of [Popat '24]. Finally, I will state some preliminary results for a central limit theorem and state some conjectures which are currently work in progress. I will try to keep the talk as accessible as possible. The results are based on joint work with Benjamin Fehrman.
15:10 Dr Giuseppe Cannizzaro, the University of Warwick
Superdiffusive Central Limit Theorem for the critical Stochastic Burgers Equation
The Stochastic Burgers Equation (SBE) was introduced in the eighties by van Beijren, Kutner and Spohn as a way to encode the fluctuations of driven diffusive systems with one conserved quantity. In the subcritical dimension d=1, it coincides with the derivative of the KPZ equation whose large-scale behaviour is polynomially superdiffusive and given by the KPZ Fixed Point, and in the super-critical dimensions d>2, it was recently shown to be diffusive and rescale to a biased Stochastic Heat equation. At the critical dimension d=2, the SBE was conjectured to be logarithmically superdiffusive with a precise exponent but this has only been shown in the discrete setting and up to lower order corrections. In the present talk, we pin down the logarithmic superdiffusivity exactly by identifying the limit of the so-called diffusion coefficient and show that, once the logarithmic corrections to the scaling are taken into account, the solution of the SBE satisfies a central limit theorem.
This is joint work with Q. Moulard and F. Toninelli.
