Mathematical Fluid Mechanics, spring 2022

Lecture 1 (January 17)
Derivation of the fundamental equations. Integral quantity advected by the flow.
Continuity equation and momentum balance. The compressible Navier-Stokes system.

Lecture 2 (January 24)
Boundary conditions, energy balance, linearization at a uniform steady state.
The incompressible Euler equation in the whole space, existence of classical solutions.
Step 1: Elimination of the pressure. Step 2: Regularization of the system.

Lecture 3 (February 3, 14h-16h, room 15 : modified schedule!)
Step 3: A oriori estimates. Step 4: taking the limit as ε tends to zero.
Step 5: additional properties of the solution.

Lecture 4 (February 7)
Global existence of solutions to the Euler equations, the Beale-Kato-Majda criterion.
Properties of the Biot-Savart law. The Yudovich theorem in the two-dimensional case.

Lecture 5 (February 10, 14h-16h, room 15 : modified schedule!)
Kelvin's circulation theorem. Potential flow past an obstacle, the d'Alembert paradox.
Stability of shear flows, Rayleigh's inflexion point criterion.

Lecture 6 (February 28)
The Navier-Stokes equations in R^d. Elimination of the pressure, integral equation
Local well posedness in L^p(R^d) when p > d, the fixed point argument.

Lecture 7 (March 7)
The critical case p = d. Local existence for arbitrary data, global existence for small data.
The energy balance and the H¹ energy estimate. The two-dimensional case.

Lecture 8 (March 14)
Function spaces in a bounded domain. The normal trace of L² vector fields with L² divergence.
The Leray-Hopf projection into divergence-free vector fields with vanishing normal trace.
The Stokes operator (proof of the representation theorem).

Lecture 9 (March 21)
Spectrum and fractional powers of the Stokes operator, the Stokes semigroup.
Local well-posedness of the Navier-Stokes equations in the 3D case with initial data in D(A^1/4).
The Stokes flow past a sphere.

Lecture 10 (March 28)
The cylindrical Poiseuille flow. Generalities on the inviscid limit. Derivation of the Prandtl equation.
The stationary case (von Mise transform) and the time-dependent case (Crocco transform).

Lecture 11 (April 4)
The Kato convergence criterion, statement and proof.
Inviscid vortices in the plane, conserved quantities.

Lecture 12 (April 11)
The point vortex system, derivation and general properties, the cases n = 2 and n = 3.
The viscous vorticity equation, Oseen vortices and long-time asymptotics.