In this talk, I will discuss some recent results on Euler and Navier Stokes equations concerning the construction of quasi-periodic solutions and the problem of the inviscid limit for the Navier Stokes equation. I will discuss the following results:
- Construction of quasi-periodic solutions for the Euler equation with a time quasi-periodic external force, bifurcating from a constant, diophantine velocity field;
- I shall discuss the inviscid limit problem from the perspective of KAM theory, namely, I shall prove the existence of quasi-periodic solutions of the Navier Stokes equation converging to the one of the Euler equation constructed in 1). The main difficulty is that this is a singular limit problem. We overcome this difficulty by implementing a normal form methods which allow to prove sharp estimates (global in time) w.r. to the viscosity parameter;
- (time permitting) I also discuss the construction of quasi-periodic traveling waves bifurcating from the Couette flow for the autonomous 2D Euler equation.
These works are based on collaborations with Luca Franzoi and Nader Masmoudi.