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

TL;DR

This paper constructs exact oscillating solutions in a 2D Navier-Stokes model, revealing how wave interactions produce macroscopic diffusion effects through boundary layers and scale interactions.

Contribution

It introduces a family of explicit oscillating solutions demonstrating wave interactions leading to diffusion in fluid dynamics, with detailed Sobolev estimates for justification.

Findings

Identification of boundary layers at t=0

Demonstration of scale interactions producing diffusion

Explicit construction of oscillating solutions

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 $[0,T]\times\R^2$ 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.