Mathematics of Turbulent Flows: A Million Dollar Problem! by Edriss S Titi

URL: https://www.icts.res.in/lecture/1/details/1661/

Turbulence is a classical physical phenomenon that has been a great challenge to mathematicians, physicists, engineers and computational scientists. Chaos theory has been developed in the end of the last century to address similar phenomena that occur in a wide range of applied sciences, but the eyes have always been on the big ball – Turbulence.Controlling the identifying the onset of turbulence has a great economic and industrial impact ranging from reducing the drag on cars and commercial airplanes to better design of fuel engins, weather and climate predictions. It is widely accepted by the scientific community that turbulent flows are governed by the Navier-Stokes equations, for large values Reynolds numbers, i.e. when the nonlinear effects are dominating the viscous linear effects (internal friction within the fluids) in the Navier-Stokes equations. As such, the Navier-Stokes equations form the main building block in any fluid model, in particular in global climate models. Whether the solutions to the three-dimensional Navier-Stokes equations remain smooth, indefinitely in time, is one of the most challenging mathematical problems. Therefore, it was identified by the Clay Institute of Mathematics as one of the seven most outstanding Millennium Problems in mathematics, and it has set one million US dollars prize for solving it. Notably, reliable computer simulations of turbulent flows is way out of reach even for the most powerful state-of-the art supercomputers. In this talk I will describe, using layman language, the main challenges that the different scientific communities facing while attempting to attack this problem. In particular, I will emphasize the mathematical point of view of turbulence.

Table of Contents (powered by https://videoken.com)
0:00:00 Introduction
0:04:25 Introduction to Speaker
0:07:13 Mathematics of Turbulent Flows: A Million Dollar Problem!
0:11:45 What is:
0:14:29 This is a very complex phenomenon since it involves a wide range of dynamically
0:14:40 Can one develop a mathematical framework to understand thiscomplex phenomenon?
0:16:39 Why do we want to understand turbulence?
0:19:20 The Navier-Stokes Equations
0:23:07 Rayleigh Bernard Convection Boussinesq Approximation
0:25:18 What is the difference between Ordinary and Evolutionary Partial Differential Equations?
0:26:51 ODE: The unknown is a function of one variable,
0:27:04 A major difference between finite and infinitedimensional space is:
0:30:59 Sobolev Spaces
0:32:03 The Navier-Stokes Equations
0:33:11 Navier-Stokes Equations Estimates
0:34:15 By Poincare inequality
0:35:14 Theorem (Leray 1932-34)
0:37:59 Strong Solutions of Navier-Stokes
0:38:25 Formal Enstrophy Estimates
0:38:40 Nonlinear Estimates
0:39:41 Calculus/Interpolation (Ladyzhenskaya) Inequalities
0:42:16 The Two-dimensional Case
0:43:05 The Three-dimensional Case
0:43:30 The Question Is Again Whether:
0:44:52 Foias-Ladyzhenskaya-Prodi-Serrin Conditions
0:46:51 Navier-Stokes Equations
0:48:21 Vorticity Formulation
0:51:06 The Three dimensional Case
0:51:10 Euler Equations
0:54:26 Beale-Kato-Majda
0:54:54 Weak Solutions for 3D Euler
0:57:14 The present proof is not a traditional PDE proof.
0:59:53 lll-posedness of 3D Euler
1:00:10 Special Results of Global Existence for the three-dimensional Navier-Stokes
1:01:25 Let us move to Cylindrical coordinates
1:01:56 Theorem (Leiboviz, mahalov and E.S.T.)
1:02:50 Remarks
1:04:49 Does 2D Flow Remain 2D?
1:08:03 Theorem [Cannone, Meyer & Planchon] [Bondarevsky] 1996
1:09:32 Raugel and Sell (Thin Domains)
1:09:34 Stability of Strong Solutions
1:09:39 The Effect of Rotation
1:11:04 An Illustrative Example The Effect of the Rotation
1:11:16 The Effect of the Rotation
1:11:42 Fast Rotation = Averaging
1:14:14 How can the computer help in solving the3D Navier-Stokesequations and turbulent flows?
1:14:50 Weather Prediction
1:16:02 Flow Around the Car
1:16:12 How long does it take to compute the flow around the car for a short time?
1:17:30 Experimental data from Wind Tunnel
1:18:13 Histogram for the experimental data
1:19:15 Statistical Solutions of the Navier-Stokes Equations
1:20:54 Thank You!
1:21:01 Q&A

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