T.Alazard,N.Burq,and C.Zuily:Low Regularity Cauchy Theory for the Water-waves Problem:Canals and Wave Pools
Jean-Yves Chemin:Navier-Stokes System
Isabelle Gallagher:Semi-classical Analysis of Oceanic Flows
Patrick Gérard:An Introduction to the Cubic Szeg(o) Equation
David Gérard-Varet:Some Recent Mathematical Results on Fluid-Solid Interaction
Hideo Kozono and Taku Yanagisawa:Lr-Helmholtz Decomposition and Its Application to the Navier-Stokes Equations
F.Rousset and N.Tzvetkov:Lectures on Transverse Instability of Solitary Water-waves
J.-C.Saut:Lectures on the Mathematical Theory of Viscoelastic Fluids
3.2 Domains with boundaries: the boundary layer
When the Navier-Stokes solutions uε and the Euler solution u are defined in a domain Ω with boundaries,the convergence issue gets considerably harder.The difficulty lies in the boundary conditions that are added at δΩ.For ε=Ω(Euler equation),only the tangency condition
is satisfied.But for ε≠0,that is when the viscosity is turned on,the fluid must stick at the boundary,which translates into the Dirichlet condition
In order to satisfy this no-slip condition,the tangential momentum at the boundary is somehow diffused into the domain.As the viscosity is very small,this change of momentum is concentrated near the boundary,in a thin zone called a boundary layer.Mathematically,this boundary layer corresponds to a singular dependence of uε with respect to ε.Hence,the whole point is to understand this boundary layer and its impact on the asymptotics ε→0.
One can even be more specific: the whole point is to determine wether or not the velocity field uε concentrates in an e-neighborhood of the boundary.