Lectures at IMPA and UFRJ by Prof. Uriel Frisch

Laboratoire Lagrange, Observatoire and Universite Cote d'Azur Nice, France




Uriel Frisch is the author of a 1995 book on turbulence and of over 200 research publications [Google Scholar]. One of his most cited works, published in 1986, concerns the lattice gas automaton method of simulating fluid dynamics using a cellular automaton. Uriel is also known for his work with Giorgio Parisi on the analysis of the fine structure of turbulent flows, for his early advocacy of multifractal systems in modeling physical processes, and for his research on using transportation theory to reconstruct the distribution of matter in the early universe. See more in wikipedia.




Lectures at IMPA

Friday, May 11: Mini-course, class 1:  10:30 - Room 228

Monday, May 14: Mini-course, class 2:  10:30 - Room 228

Tuesday, May 15: Special lecture:  14:00 - Auditorium 1

Wednesday, May 16: Mini-course, class 3:  10:30 - Room 236

Thursday, May 17: Seminar Lecture:  14:00 - Room 333


Lectures at UFRJ

2a. feira, 21/maio, 14h00, IM/UFRJ: Mini-curso, aula 1, sala C-116

3a. feira, 22/maio, 14h00, IM/UFRJ: Mini-curso, aula 2, sala C-116

4a. feira, 23/maio, 17h00, IM/UFRJ: palestra no Seminário de História da Matemática e da Física. Título:  "Differential Geometry and  Hydrodynamics: more than two Centuries of Interaction”. Palestra ministrada por Uriel Frisch, em inglês. Sala C-116.    LECTURE NOTES

5a. feira, 24/maio, 14h00, IM/UFRJ: Mini-curso, aula 3, sala C-116.


Special lecture

Introduction to turbulence and blow up

Tuesday, May 15th 2018, 14:00 - Auditorium 1


This lecture is intended to give a rough idea of some of questions arising in turbulence: Reynolds number and instabilities, chaos and the butterfly effect, fully developed turbulence as a spatial variant of Brownian motion, fractals in Nature and in turbulence, simple models leading to finite time explosion (blow-up), nonlinear depletion and possible avoidance of blow-up.

The lecture will be elementary and accessible to students whose background is in mathematics, or physics, or engineering or numerical analysis.



Cauchy invariants and Cauchy-Lagrangian numerical methods

1st class: Friday, May 11th, 10:30 - Room 228
2nd class: Monday, May 14th, 10:30 - Room 228
3rd class: Wednesday, May 16th, 10:30 - Sala 236


Cauchy's invariants for 3D incompressible Euler flow, recently rediscovered after 200 years, give us a powerful tool for investigating the Lagrangian structure of such flow. Among the topics we shall discuss: how the Cauchy invariants relate to the Lie-transport (pullback) invariance of the vorticity 2-form; how they generate recursion relations - resembling those of the Lagrangian perturbation theory in cosmology - giving a constructive hold on time-analyticity of the Lagrangian map and thereby allowing the development of Cauchy-Lagrange numerical schemes that can be orders of magnitude faster than the usual Eulerian schemes.

The first lecture will be  devoted to historical background: how the invariants were discovered by Cauchy in 1815 and then almost completely forgotten (except by Hermann Hankel), rediscovered many times in the late 20th Century and finally reattributed to Cauchy by the Russian school.

The second lecture will deal with the mathematics of time-analyticity with and without solid boundaries.

The third lecture will present the new Cauchy-Lagrange numerical method.


Some useful references:

  • Frisch, U. and Villone, B. 2014. Cauchy's almost forgotten Lagrangian formulation of the Euler equation for 3D incompressible flow, Europ. Phys. J. H 39, 325--351. arXiv:1402.4957 [math.HO] 
  • Frisch, U. and Zheligovsky, V. 2014. A very smooth ride in a rough sea, Commun.  Math. Phys., 326, 499--505. arXiv:1212.4333 [math.AP] 
  • Zheligovsky, V. and Frisch, U. 2014. Time-analyticity of Lagrangian particle trajectories in ideal fluid flow, J. Fluid Mech., 749, 404--430.  arXiv:1312.6320 [math.AP]
  • Rampf, C., Villone, B and Frisch, U. 2015. How smooth are  particle trajectories   in a Lambda CDM Universe?, Mon. Not. R. Astron. Soc., 452, 1421--1436. arXiv:1504.00032 [astro-ph.CO]
  • Podvigina, O., Zheligovsky, V. and Frisch, U. 2015. The Cauchy-Lagrangian method for numerical analysis of Euler flow. J. Comput. Phys.,  306, 320--342. 
  • arXiv:1504.05030v1 [math.NA]
  • Besse, N. and Frisch, U.  2017 A constructive approach to regularity of Lagrangian  trajectories for incompressible Euler flow in a bounded domain, Comm. Math.  Phys. 351, 689-707. arXiv:1603.09219v1 [math.AP].
  • Besse, N. and Frisch, U. 2017 Geometric formulation of the Cauchy invariants for  incompressible Euler flow in flat and curved spaces.  J. Fluid Mech. 825, 412--478.  arXiv:1701.01592 [physics.flu-dyn].


Lecture at the Seminar for Applied and Computational Mathematics

Burgers turbulence and the spurious -3 energy spectrum debunked

Thursday, May 17th, 14:00 - Room 333


Formal manipulations of the one-dimensional inviscid and unforced Burgers equation suggest that the energy spectrum E(k)  might follow a k-3 rather than the  k-2, generally associated with shocks. The use of the Fourier-Lagrangian solution, introduced by Fournier and Frisch (L'equation de Burgers deterministe et statistique. 1983. J. Mec. Theor. Appl, vol. 2, pp. 699-750.) shows that the spurious k-3 solutions stem from an improper use of the multi-stream behaviour associated with the  Burgers-Vlasov equation.

This will be a blackboard lecture intended for advanced students and specialists in PDEs, field theory and/or catastrophy theory.