- BCAM - Severo Ochoa course:
"Selected topics in the mathematical theory of incompressible fluids" (February - March 2026)
Course addressed to second year Master students and Ph.D. students.
Contents of the lectures
- First lecture (09/02)
• 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 (13/02)
• 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 (16/02)
• 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.
- Fourth lecture (18/02)
• Strong solutions theory for the Euler equations: end of the proof of existence; more on the lower bound for the lifespan of the solution; stability estimates in L2 and proof of uniqueness; proof of the blow-up criterion / continuation criterion.
• Vorticity: definition; vorticity formulation of the Euler equations and Biot-Savart law.
• The Beale-Kato-Majda continuation criterion: statement; logarithmic interpolation inequality (statement and proof).
- Fifth lecture (23/02)
• The Beale-Kato-Majda continuation criterion: proof.
• Fast rotating fluids: introduction about geophysical flows, generalities; scales of motion; the Euler-Coriolis system. Statement of the Taylor-Proudman theorem.
- Sixth lecture (25/02)
• Fast rotating fluids: proof of the Taylor-Proudman theorem. Decomposition of the problem into a 2D part and a 3D perturbation part.
• Study of the linear problem: diagonalisation of the matrix, eigenvalues and eigenvectors, definition of the integral kernels.
- Seventh lecture (02/03)
• Study of the linear problem: dispersive estimates; Strichartz estimates (statement and first part of the proof).
- Eighth lecture (04/03)
• Study of the linear problem: proof of the Strichartz estimates; corollary on global bounds for the solution.
• Study of the non-linear problem: statement of the main theorem ( (improved lifespan of the solution)); local well-posedness theory for the perturbed Euler-Coriolis system.
- Ninth lecture (09/03)
• Study of the non-linear problem: proof of the main theorem (improved lifespan of the solution).
• Stabilisation by transport: setting of the problem; generalities about mixing;
the case of the linearisation around a Couette flow (equations and remarks).
- Tenth lecture (13/03)
• Linear inviscid damping for Couette flow: statement of the theorem; proof via explicit computations in Fourier variables.
• The general case of a monotone shear flow: statement of the main theorem.
- Eleventh lecture (16/03)
• Linear inviscid damping for a monotone shear flow: statement of the main theorem; Step 1 of the proof (damping assuming regularity; scattering assuming regularity); Step 2 of the proof (introducing the linearised Euler equations in the scattering formulation).
- Twelfth lecture (18/03)
• Linear inviscid damping for a monotone shear flow: Step 2 of the proof (propagation of regularity for Euler equations in the scattering formulation: L2 regularity, higher regularity); Step 3 ofthe proof (conclusion).
Didactic material
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