Introduction to Fluid Mechanics
from 12:00h to 13:30h.
From February 14th
(about 15 weeks).
Aula T1, Edifici històric,
Universitat de Barcelona
(Gran Via de les Corts Catalanes, 585, 08007 Barcelona).
The course will also be entirely accessible online.
Course Description
This course will introduce the Euler and NavierStokes equations, the fundamental equations in Fluid Mechanics, as well as other relevant models in physics and engineering. It will cover a variety of mathematical concepts and techniques that are ubiquitous in Partial Differential Equations of fluid interface problems. For instance, we will study harmonic analysis, singular integrals, maximal regularity of the heat equation, calculus of variations, bifurcation theory and computer assistedproofs, among others.
No prerequisites are assumed, other than some Real analysis and some basic PDE theory. The course is planned to be adequate for Masters students or higher.
Organizer: Javier GómezSerrano (UB)
Contents

 Euler and NavierStokes
 Introduction to the Euler and Navier Stokes equations
 Weak solutions: Global regularity for the Vortex Patch problem
 LittlewoodPaley and Besov spaces
 Maximal regularity for the heat equation
 The Boussinesq equation and the temperature patch problem
 Inhomogeneous NavierStokes density patch problem
 Euler and NavierStokes

 Bifurcation theory in Fluid Mechanics
 Local Bifurcation Theory
 Global Bifurcation Theory
 Singular Integrals
 Applications to Rotating solutions of the Vortex Patch equation (VStates)
 Vortex points and vortex sheets
 Differentiable NashMoser Scheme
 Bifurcation theory in Fluid Mechanics

 Rigidity and flexibility results for active scalar equations
 General active scalar equations
 Rigidity of active scalars and other relevant models
 Rigidity and flexibility results for active scalar equations

 Other models and numerical aspects
 Numerical simulations, computerassisted proofs and applications to fluids
 Other dispersive models: water waves, the Whitham equations…
 Other models and numerical aspects
The four content blocks are selfcontained and everyone can join/leave at any point.
Speakers
References
 J. Majda and A. L. Bertozzi. Vorticity and incompressible flow, Vol. 27 of Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, (2002).
 C. Robinson, J. L. Rodrigo, W. Sadowski. The threedimensional NavierStokes equations. Cambridge Studies in Advanced Mathematics, 157, Cambridge University Press, Cambridge, England, (2016).
 P.L. Lions. Mathematical topics in fluid mechanics. Vol. 1, volume 3 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York, 1996. Incompressible models, Oxford Science Publications, (1996).
 Bahouri, J.Y. Chemin, R. Danchin. Fourier analysis and nonlinear partial differential equations, volume 343. Springer Science & Business Media, (2011).Burbea, Motions of vortex patches. Lett. Math. Phys. 6 (1982), 116.
 M.G. Crandall, P. H. Rabinowitz, Bifurcation from simple eigenvalues. J. Func. Anal. 8 (1971), 321340.
 L. L. Helms, Potential theory, SpringerVerlag London, 2009.
 H. Kielhofer, Bifurcation Theory: An Introduction with Applications to PDEs. SpringerVerlag, Berlin Heidelberg New York, 2004.
 J. GomezSerrano, J. Park, J. Shi, Y. Yao. Symmetry in stationary and uniformly rotating solutions of active scalar equations. Duke Mathematical Journal 1, no. 1 (2021): 182.
 E. H. Lieb, M. Loss, Analysis. Second edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, (2021).
 M. Berti, Lecture notes on NashMoser theory and Hamiltonian PDEs, http://indico.ictp.it/event/a05229/session/81/contribution/58/material/0/0.pdf.
 J. GomezSerrano, Computerassisted proofs in PDE: a survey. SeMA Journal, 76, no. 3, 459484 (2019).
 W. Tucker. Validated numerics. Princeton University Press, Princeton, NJ, 2011.