SoSe 19: The Navier-Stokes Equations and Applications Toward Data Assimilation
Titi Edriss
Comments
In these lectures I will introduce the basics of the mathematical analysis of the Navier-Stokes equations describing the flow of viscous incompressible fluids. These equations appear in a wide range of physical and biological applications, varying from oceanic and atmospheric dynamics, to combustion theory and body-fluid transport. From the mathematical point of view, these equations have been identified among the most challenging problems in Applied Analysis. At the same time, from the computational point of view they are prohibitively expensive to simulate in many real-life applications and out of reach even for most powerful state-of-the-art computers.
In this course we will focus on the analytical properties of the solutions of these equations. In particular, on establishing the fact that solutions feature finitely many determining parameters. We will capitalize on this property to introduce a nudging downscaling data assimilation algorithm based on coarse mesh spatial measurements, i.e., on a conceptually simple but efficient means of informing a flow simulation of observational data in such a way that the numerical solution provably approaches the observed flow in suitable limits.
Students who are interested in attending the course are expected to have a good background in Real Analysis (and basics of Functional Analysis). The rest of the course will be, to a large extend, self-contained.
Proposed topics to be covered
- Introducing the Navier-Stokes equations from basic physical principles. Introducing the appropriate Functional Spaces.
- Steady state solutions to the Navier-Stokes equations and their regularity.
- Time dependent weak solutions.
- Global regularity of strong solutions for the two-dimensional case.
- Short time existence of strong solutions in the three-dimensional case.
- Time Analyticity and Gevrey regularity.
- Long-time behavior of solutions, global attractors, determining modes, nodes and other degrees of freedom.
- A downscaling data assimilation algorithm based on spatial coarse mesh measurements.
Suggested reading
Primary References:
P. Constantin and C. Foias, Navier-Stokes Equations, University of Chicago Press, 1988.
C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001.
R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holand. New print published by the AMS 2001.
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd Ed, Applied Math Sci. 68, Springer-Verlag,
1997.
Various classical and recent papers on these topics will be pointed out in the course of the semester.
Additional References:
A. Chorin and J. Marsden, A Mathematical Introduction to Fluid Mechanics, Springer-Verlag.
C. Doering and J. Gibbon, Applied Analysis of the Navier-Stokes Equations, Cambridge University Press.
J. C. Robinson, Infinite-dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts in Applied Mathematics.
R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conference Series in Applied Math 66, SIAM, 2nd Ed, 1995.
close14 Class schedule
Regular appointments