Abstract:
My talk is a review of the results on turbulence in the 1d viscous Burgers equation on a circle, obtained by myself and my former PhD students, Andrey Biryuk and Alexandre Boritchev; now the results are presented in a MS of my joint book with A.Boritchev. Namely, I will talk about the 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 laws for the increments of the turbulent vector-fields and of the Kolmogorov-Obukhov law. The results were non-rigorously obtained by physicists Aurell-Frisch-Lutsko-Vergassola in 1992 (and by Burgers in 1948, even more heuristically).
Biography:
Sergei Borisovich Kuksin is a Russian mathematician, specializing in partial differential equations (PDEs). Kuksin received his doctorate under the supervision of Mark Vishik at Moscow State University in 1981. He was at the Steklov Institute in Moscow and at the Heriot-Watt University and is a directeur de recherché (senior researcher) at the Institut Mathématiques de Jussieu of the Paris Diderot University (Paris VII). His research deals with KAM theory in partial differential equations (i.e. infinite dimensional Hamiltonian systems);[2] partial differential equations involved with random perturbations, turbulence and statistical hydrodynamics; and elliptic PDEs for functions between compact manifolds.
In 1992 he was an invited speaker with talk KAM theory for partial differential equations at the European Congress of European Mathematicians in Paris. In 1998 he was an invited speaker at International Congress of Mathematicians in Berlin. In 2016 he received the Lyapunov Prize from the Russian Academy of Sciences.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai