- BCAM - Severo Ochoa course:
"Introduction to the mathematical theory of incompressible fluids" (February - April 2024)
In collaboration with Renato Lucà.
Course addressed to second year Master students and Ph.D. students.
Part 1: ideal flows
- First lecture (26/02)
• Generalities on the Euler equations: derivation of the equations, notion of solutions, the pressure and the Leray-Helmholtz projector, conservation of the kinetic energy.
• Elements of Littlewood-Paley theory: basic definitions, Littlewood-Paley decomposition of tempered distributions, Bernstein's inequalities.
- Second lecture (28/02)
• Elements of Littlewood-Paley theory: dyadic characterisation of Besov spaces, tame estimates.
• Strong solutions theory for the Euler equations: a priori estimates in Hs, study of the pressure; equivalence with the projected system (by using the Leray-Helmholtz projector).
- Third lecture (04/03)
• Strong solutions theory for the Euler equations: proof of existence (construction of approximate solutions via Friedrichs method; uniform bounds; convergence to a solution by a weak compactness argument) and lower bound for the lifespan.
• Strong solutions theory for the Euler equations: stability estimates in L2 and proof of uniqueness.
• Strong solutions theory for the Euler equations: proof of the blow-up criterion / continuation criterion.
- Fourth lecture (06/03)
• Strong solutions theory for the Euler equations: summary and remarks.
• Vorticity: definition; vorticity formulation of the Euler equations and Biot-Savart law.
• The Beale-Kato-Majda continuation criterion.
• Around the finite energy condition for two-dimensional flows: radial energy decomposition.
- Fifth lecture (11/03)
• Around the finite energy condition for two-dimensional flows: radial energy decomposition.
• The Yudovich theorem: the statement; static estimates, log-Lipschitz regularity of the velocity field.
• The Yudovich theorem: proof of existence.
- Sixth lecture (13/03)
• The Yudovich theorem: proof of existence (end).
• The Yudovich theorem: proof of uniqueness.
• Vortex patches: definition and basic properties.
• Persistence of the boundary regularity: Chemin's theorem; example of the flat case.
- Seventh lecture (18/03)
• Chemin's theorem about persistence of boundary regularity of vortex patches: proof (only a priori estimates).
• Concluding remarks about the incompressible Euler equations and vortex patches structures.
- Eighth lecture (20/03)
• The ideal MHD equations: Hs theory; Elsässer formulation and L∞ theory; improved lifespan in 2D (with a sketch of the proof).
Didactic material
- Video recordings of the lectures
- Lecture notes
 ( updated on 19/03/2024 )
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