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Introduction to Fluid Mechanics

Monday and Wednesday
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 Navier-Stokes 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 assisted-proofs, 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 Gómez-Serrano (UB)

    1. Euler and Navier-Stokes
      • Introduction to the Euler and Navier Stokes equations
      • Weak solutions: Global regularity for the Vortex Patch problem
      • Littlewood-Paley and Besov spaces
      • Maximal regularity for the heat equation
      • The Boussinesq equation and the temperature patch problem
      • Inhomogeneous Navier-Stokes density patch problem
    1. Bifurcation theory in Fluid Mechanics
      • Local Bifurcation Theory
      • Global Bifurcation Theory
      • Singular Integrals
      • Applications to Rotating solutions of the Vortex Patch equation (V-States)
      • Vortex points and vortex sheets
      • Differentiable Nash-Moser Scheme
    1. Rigidity and flexibility results for active scalar equations
      • General active scalar equations
      • Rigidity of active scalars and other relevant models
    1. Other models and numerical aspects
      • Numerical simulations, computer-assisted proofs and applications to fluids
      • Other dispersive models: water waves, the Whitham equations…

The four content blocks are self-contained and everyone can join/leave at any point.

  1. J. Majda and A. L. Bertozzi. Vorticity and incompressible flow, Vol. 27 of Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, (2002).
  2. C. Robinson, J. L. Rodrigo, W. Sadowski. The three-dimensional Navier-Stokes equations. Cambridge Studies in Advanced Mathematics, 157, Cambridge University Press, Cambridge, England, (2016).
  3. 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).
  4. 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), 1-16.
  5. M.G. Crandall, P. H. Rabinowitz, Bifurcation from simple eigenvalues. J. Func. Anal. 8 (1971), 321-340.
  6. L. L. Helms, Potential theory, Springer-Verlag London, 2009.
  7. H. Kielhofer, Bifurcation Theory: An Introduction with Applications to PDEs. Springer-Verlag, Berlin- Heidelberg- New York, 2004.
  8. J. Gomez-Serrano, J. Park, J. Shi, Y. Yao. Symmetry in stationary and uniformly rotating solutions of active scalar equations. Duke Mathematical Journal 1, no. 1 (2021): 1-82.
  9. E. H. Lieb, M. Loss, Analysis. Second edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, (2021).
  10. M. Berti, Lecture notes on Nash-Moser theory and Hamiltonian PDEs,
  11. J. Gomez-Serrano, Computer-assisted proofs in PDE: a survey. SeMA Journal, 76, no. 3, 459-484 (2019).
  12. W. Tucker. Validated numerics. Princeton University Press, Princeton, NJ, 2011.