The inviscid limit and Prandtl's asymptotic expansion for incompressible flows in the half space

The validity of the inviscid limit for the incompressible Navier-Stokes equations is one of the most important and challenging problems in the mathematical theory of fluid dynamics: the motion of inviscid fluids is described by the Euler equations, so, when the viscosity goes to zero, one would expect the convergence of NS solutions to the Euler solutions. However, NS equations are a singular perturbation of the Euler equations: the change of order of the equation implies that fewer boundary conditions can be imposed on the inviscid flows. Therefore, the no-slip boundary conditions, imposed on the NS solutions, are not satisfied by the Euler flow, for which a tangential slip is allowed. This mismatch between the behaviour at the boundary of the NS solutions and the same behaviour of their supposed limit creates a boundary layer, with large gradients of velocity in the normal direction, which make the diffusive effects comparable to the inertial ones: this situation is classically described by Prandtl's equation. The ill posedness of Prandtl's equation in Sobolev settings require the use of more regular functional spaces: in a holomorphic setting, Prandtl's equation is well posed, and the inviscid limit holds. In this thesis, we extend this result for incompatible initial data, which satisfy the no-penetration boundary condition, but allow a tangential slip, a kind of data of both numerical and theoretical interest: this extension is not trivial, since the singularity formed by this kind of initial data forces us to use different function spaces, where some of the properties used in the proof of the compatible case do not hold. In Sobolev settings, we see that, for the linearization around an inviscid flow of the NS equations, the inviscid limit actually holds: in this case, Prandtl's asymptotic expansion is not necessary, and convergence can be proved through energy methods in conormal Sobolev spaces. The linearization can be limited to the tangential part of the flow: indeed, this is enough to avoid the interaction between the diffusive effects on the boundary and strong inertial terms, which is believed to cause boundary layer separation.