PROGRAM
The goal of the present course is to present two give an overview of recent advances in the mathematical study of models from incompressible fluid mechanics. We will focus on two main topics: on the one hand, mixing, inviscid damping and enhanced dissipation, and related phenomena based on transport by special vector fields; on the other hand, topics related to geophysical flows, with a special interest in dispersion and stabilisation effects due to (fast) rotation.
Dates: Mon-Wed, From February 9th to March 27th * + Fridays February 13th and March 27th.
On Wednesday, February 11th, there will be no course.
DETAILED (TENTATIVE) PROGRAM
1) PART 1: REMINDERS ON THE EULER AND NAVIER-STOKES EQUATIONS (6 hours)
- The equations of fluid mechanics: derivation and basic properties
- Strong solutions theory, weak solutions theory.
- Beale-Kato-Majda continuation criterion and applications: global existence in 2-D
2) PART 2: STABILISATION BY TRANSPORT (14 hours)
- Transport equations: reminders of the classical theory
- Inviscid damping and mixing via incompressible flows: the passive scalar case.
- Enhanced dissipation for transport-diffusion equations.
- The non-linear theory: the case of the Euler equations.
3) PART 3: FAST ROTATING FLUIDS (10 hours)
- Introduction to geophysical flows
- The asymptotic dynamics: convergence via weak compactness methods.
- Dispersion and stabilisation for fast rotating fluids.
- Applications: improved lifespan in presence of fast rotation.