Séminaire EDP-Analyse ICJ

On the inviscid limit of the Navier-Stokes equations with Dirichlet boundary conditions

par Vlad Vicol (Princeton University)

Europe/Paris
Fokko Du Cloux (Université Claude Bernard Lyon 1 - Campus de la Doua, Bâtiment Braconnier)

Fokko Du Cloux

Université Claude Bernard Lyon 1 - Campus de la Doua, Bâtiment Braconnier

Description
We consider the vanishing viscosity limit of the Navier-Stokes equations in a half space, with Dirichlet boundary conditions. We prove that the inviscid limit holds in the energy norm if the Navier-Stokes solutions remain bounded in $L^2_t L^\infty_x$ independently of the kinematic viscosity, and if they are equicontinuous at $x_2 = 0$. These conditions imply that there is no boundary layer separation: the Lagrangian paths originating in a boundary layer, stay in a proportional boundary layer during the time interval considered. We then give a proof of the (numerical) conjecture of vanDommelen and Shen (1980) which predicts the finite time blowup of the displacement thickness in the Prandtl boundary layer equations. This shows that the Prandtl layer exhibits separation in finite time.