On the Production of Dissipation by Interaction of Forced Oscillating Waves in Fluid Dynamics

Abstract

In the context of some bidimensionnal Navier-Stokes model, we exhibit a family of exact oscillating solutions $\{u_{\epsilon }{\}}_{\epsilon }$ defined on some strip $[0,T]\times {\mathbb{R}}^2$ which does not depend on $\epsilon \in ]0,1]$. The exact solutions is described thanks to a complete expansions which reveal a boundary layer in time $t=0$. The interactions of the various scales (1, $1/\epsilon$ and $1/{\epsilon }^2$) produce a macroscopic effect given by the addition of a diffusion. To justify the existence of $\{u_{\epsilon }{\}}_{\epsilon }$, we need to perform various Sobolev estimates that rely on a refined balance between the informations coming from the hyperbolic and parabolic parts of the equations.