Rigorous theory of 1d turbulence
发布人: 曹思圆   发布时间: 2019-11-12   浏览次数: 35

报告人: Sergei Kuksin

邀请人:张静 副教授

报告时间:2019-11-18 上午9:30—10:30

报告地址: 数学楼 102


S. B. Kuksin教授,University Paris-Diderot  and Shandong University,爱丁堡皇家学会成员(Royal Society of Edinburgh)。主要研究方向包括:无穷维哈密顿系统(包括哈密顿偏微分方程);KAM理论(KdV方程的拟周期解的存在性);偏微分方程和随机流体动力学系统的随机扰动;非线性偏微分方程的湍流问题;紧流型上的椭圆方程。Kuksin教授曾在很多具有影响力的会议上做过报告(如:1998年ICM会议上报告Hamiltonian PDE,2009 ICMP会议上报告Averaging in Nonlinear PDE,等)。现任多本数学杂志的编委: GAFA,Proc. A Royal Soc. Edinburgh,Discrete Continuous Dynamical Systems Series A,Stoch. PDE Anal Appl.等.


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 for the energy spectrum of turbulence (I will explain these notions). The results were non-rigorously obtained by physicists Aurell-Frisch-Lutsko-Vergassola in 1992 (and by Burgers in 1948, even more heuristically).