Lisbon Webinar in Analysis and Differential Equations

Kolmogorov theory of turbulence and a rigorous theory of one-dimensional turbulence

Sala 6.2.33, Ciências ULisboa (com transmissão via Zoom)

Por Sergei Kuksin (Univ. Paris VII).

My talk is a review of the results on turbulence in the 1d Burgers equation, presented in my book One-dimensional turbulence and stochastic Burgers equation (2021), written jointly with A. Boritchev. Namely, I will talk about the viscous Burgers equation on a circle, perturbed by a random force which is smooth in $x$ and white in time $t$, and explain that Sobolev norms of its solutions admit upper and lower estimates, which are asymptotically sharp as the viscosity goes to zero. This assertion allows to derive for solutions of the equation results, which are rigorous analogies of the main predictions of the Kolmogorov theory of turbulence. Namely, of the Kolmogorov 2/3-law for increments of the turbulent velocity-fields and of the Kolmogorov-Obukhov 5/3-law for the energy spectrum of turbulence (I will explain these laws). The results were non-rigorously obtained by physicists in 1990s (and earlier by J. Burgers in 1948, even more heuristically).

Transmissão via Zoom.

The Lisbon Webinar in Analysis in Differential Equations is a joint iniciative of CAMGSD, CMAFcIO and GFM, three research centers of the University of Lisbon. It is aimed at filling the absence of face-to-face seminars and wishes to be a meeting point of mathematicians working in the field.