Topics in Analysis: Fluid Mechanics¶
Contact info¶
 Website
 Office
3.120, OskarMorgensternPlatz 1
 Office hours
By appointment
 Class time
Mondays 13:15–14:45 and 15:00–15:45
 Class location
Seminarraum 12, OskarMorgensternPlatz 1, 2.Stock
Course description¶
Fluid mechanics has long been intertwined with both pure and applied mathematics, from the early history of complex analysis to more recent developments in areas such as harmonic analysis, integrable systems, dynamical systems, and the calculus of variations. After deriving the Euler and Navier–Stokes equations from first principles, we will cover several topics of mathematical interest, motivated in part by the interests of the class. No prior knowledge of fluid mechanics is assumed.
Prerequisites: Real and complex analysis, ordinary differential equations. Familiarity with functional analysis and/or partial differential equations may be helpful but is not required. Absolutely no prior knowledge of fluid mechanics (or physics more generally) is assumed.
Schedule and lecture notes¶
The dates below link to (rough, handwritten) notes for the lectures. For the presentations there are links to the references.
Week 
Date 
Topics 

1 
Introduction, Lagrangian and Eulerian points of view 

2 
Conservation of mass, incompressible Euler equations, vorticity 

3 
Circulation, steady potential flow in two dimensions 

4 
Classical solutions for 2D Euler [Ka67] 

5 
Classical solutions for 2D Euler continued 

6 
The Navier–Stokes equations in three dimensions 

7 
Existence of Leray–Hopf weak solutions (Part II of [Ch06]) 

8 
Existence of Leray–Hopf weak solutions continued 

9 
Boundary layers and singular perturbation theory ([CM79]) 

10 
Water waves ([St57]) 

11 
10.12 
Week 1 of presentations: People who volunteered to go first 
12 
07.01 
Week 2 of presentations: Vorticity and rotation 
13 
14.01 
Week 3 of presentations: Fluids and geometry 
14 
21.01 
Week 4 of presentations: Famous calculations 
15 
28.01 
Week 5 of presentations: Other effects 
References¶
For the lectures¶
Our derivation of the incompressible Euler and Navier–Stokes equations is standard, and versions of it can be found in almost any good textbook on fluid mechanics. I have been interpolating between [CM79] and [Ba00], with a little bit of [Ar62], but unfortunately none of these is freely available online through the university library.
The freely available lecture notes [MW16] provide brief derivations of the incompressible Euler and NavierStokes equations, and also include a proof of the existence of LerayHopf weak solutions in 3D for bounded domains.
We used [Ka67] as a reference for the existence of classical solutions to the Euler equations in 2D. For weaker solutions, see for instance [CP12]; this is also a potential presentation topic.
For the existence of weak solutions to the Navier–Stokes equations we followed “Part II” of [Ch06], which is available online through the university library (you should be able to save all 70 pages of Part II to a pdf in one go).
The freely available textbook [Te18] contains most if not all of the functional analysis background we will need. The Schauder fixedpoint theorem (Theorem 18.7) is proved using (the beautiful and extremely useful) Leray–Schauder degree theory. Those uninterested in the degree theory may want to look at Schauder’s shorter original proof, or at the still shorter proofs in other textbooks (e.g. Theorem 11.1 in [GT98]).
Our brief discussion of boundary layers followed [CM79]. Much more information (from a mathematical point of view) can be found in the online lecture notes [Ng15].
Our brief discussion of water waves was somewhat similar to “Part I” of [St57].
 Ar62
R. Aris. Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Dover Publications, New York, 1962.
 Ba00
G. Batchelor. An Introduction to Fluid Dynamics (Cambridge Mathematical Library). Cambridge: Cambridge University Press, 2000.
 Ch06(1,2)
JeanYves Chemin, Mathematical geophysics: an introduction to rotating fluids and the NavierStokes equations. Oxford University Press, 2006.
 CM79(1,2,3,4)
A. J. Chorin and J. E. Marsden, A mathematical introduction to fluid mechanics. SpringerVerlag, New YorkHeidelberg, 1979.
 CP12
C. Marchioro and M. Pulvirenti, Mathematical theory of incompressible nonviscous fluids. Springer, 2012.
 GT98
David Gilbarg and Neil S. Trudinger. Elliptic partial differential equations of second order. Springer, 2015.
 Ka67(1,2)
Tosio Kato, On classical solutions of the twodimensional nonstationary Euler equation. Arch. Rational Mech. Anal. 25: 188–200, 1967.
 MW16
S. Mishra and F. Weber, Topics in Mathematical and Computational Fluid Dynamics, https://www2.math.ethz.ch/education/bachelor/lectures/fs2016/math/macfd.html
 Ng15
Toan Nguyen. Lectures notes for Math 597F, https://nttoan81.wordpress.com/category/math597ftopicsonboundarylayers/, 2015.
 St57(1,2,3)
J. J. Stoker, Water Waves: The Mathematical Theory with Applications. Interscience Publishers, Inc., New York, 1957.
 Te18(1,2)
Gerald Teschl, Topics in Real and Functional Analysis, Graduate Studies in Mathematics, Volume XXX, Amer. Math. Soc., Providence, (to appear).
For the presentations¶
 ADS09
Marino Arroyo and Antonio DeSimone. Relaxation dynamics of fluid membranes. Physical Review E 79.3, 031915 (2009).
 Ar79
Hassan Aref. Motion of three vortices. The Physics of Fluids 22, 393 (1979).
 Ar10
Hassan Aref. Selfsimilar motion of three point vortices. Physics of Fluids 22, 057104 (2010).
 Be85
T. Brooke Benjamin. Note on added mass and drift. Journal of Fluid Mechanics 169, 251256 (1985).
 Br09
Alberto Bressan. Lecture Notes on Hyperbolic Conservation Laws. (2009).
 BKM84
J. T. Beale, T. Kato, and A. Majda. Remarks on the breakdown of smooth solutions for the 3D Euler equations. Comm. Math. Phys., 94:1, 6166 (1984).
 CJ15
Constantin, A., Johnson, R.S.: The dynamics of waves interacting with the Equatorial Undercurrent. Geophys. Astrophys. Fluid Dyn. 109, 311–358 (2015)
 EM69
Ebin, David G.; Marsden, Jerrold E. Groups of diffeomorphisms and the solution of the classical Euler equations for a perfect fluid. Bull. Amer. Math. Soc. 75:5, 962967 (1969).
 EM70
David G. Ebin and Jerrold Marsden. Groups of Diffeomorphisms and the Motion of an Incompressible Fluid. Annals of Mathematics Second Series 92:1, 102163 (1970).
 Ev10
Lawrence C. Evans. Partial Differential Equations. Graduate Studies in Mathematics, Volume 19, Amer. Math. Soc., Providence (2010).
 Fi16
Richard Fitzpatrick. Plasma Physics. (2016)
 Ha18
SV Haziot. Explicit twodimensional solutions for the ocean flow in arctic gyres. Monatshefte für Mathematik, 2018
 HM18
Susanna V. Haziot and Kateryna Marynets. Applying the Stereographic Projection to Modeling of the Flow of the Antarctic Circumpolar Current. Oceanography 31:3, 6875 (2018)
 Ir14
Fridtjov Irgens. Rheology and NonNewtonian Fluids. Springer International Publishing Switzerland, 2014.
 KC08
Kundu, Pijush K., and Ira M. Cohen. Fluid Mechanics. 4th ed. Amsterdam ; Boston: Academic Press, 2008.
 Ja13
Jardetzky, W. S. Theories of figures of celestial bodies. Courier Corporation, 2013. Harvard
 JOR18
Thomas Jankuhn, Maxim A. Olshanskii, Arnold Reusken. Incompressible fluid problems on embedded surfaces: Modeling and variational formulations. arXiv preprint, (2018).
 MR05
J. Málek and K. R. Rajagopal. Mathematical issues concerning the NavierStokes equations and some of their generalizations. (2019)
 Ne14
Paul K Newton. Point vortex dynamics in the postAref era. Fluid Dynamics Research 46:3 (2014).
 No75
E. A. Novikov. Dynamics and statistics of a system of vortices. Zh. Eksp. Teor. Fiz 68, 1868–1882 (1975).
 Sa67
P. G. Saffman. The selfpropulsion of a deformable body in a perfect fluid. Journal of Fluid Dynamics 28:2, 385389 (1967).
 Sa14
Anders Sandberg. TorusEarth. Andart (blog), 2014.
 Sc60
L.E. Scriven. Dynamics of a fluid interface equation of motion for Newtonian surface fluids. Chemical Engineering Science 12.2 (1960): 98108.
 ST18
Maryam Samavaki, Jukka Tuomela. NavierStokes equations on Riemannian manifolds. arXiv preprint (2018).
 Sy49
J. L. Synge. On the motion of three vortices. Canadian J. Math. 1, (1949). 257–270
 Ta10
Terence Tau. The EulerArnold equation. terrytao.wordpress.com (2014)
 Th87(1,2)
William Thomson. On Ship Waves. Proceedings of the Institution of Mechanical Engineers 38:1, 409434 (1887).
 Ta14
Terence Tao. Noether’s theorem and the conservation laws for the Euler equations. terrytao.wordpress.com (2010)
 Ur53
F. Ursell. Mass transport in gravity waves. Mathematical Proceedings of the Cambridge Philosophical Society, 49:1, 145150 (1953).
 Wo74
CY. Wong. Toroidal figures of equilibrium. The Astrophysical Journal 190, 675694 (1974).
 Yi84
ChiaShun Yih. New derivations of Darwin’s theorem. Journal of Fluid Mechanics 152, 163172 (1985)
Presentation topics¶
The assessment for the course is based on inclass presentations. See the schedule above for the 13 topics that were ultimately picked by students, as well as links to the references they used. Below is a list of some other suggested topics which were not chosen:
Applications of Leray–Schauder degree theory (e.g. [Te18])
Unique weak solutions to 2D Euler (Yudovich’s theorem)
Contour dynamics methods for vortex patches
Arnold stability theorem for shear flows
Vortex filaments
Gerstner and/or Crapper waves (explicit solutions in Lagrangian coordinates)
Weak solutions to the Euler equations and isometric embeddings