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

Aurélien Klak (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

In the context of some bidimensionnal Navier-Stokes model, we exhibit a family of exact oscillating solutions $\{u_{\varepsilon}\}_{\varepsilon}$ defined on some strip $[0,T]\times\R^2$ which does not depend on $\varepsilon\in]0,1]$. 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 ($1$, $1/\varepsilon$ and $1/\varepsilon^2$) produce a macroscopic effect given by the addition of a diffusion. To justify the existence of $\{u_\varepsilon\}_{\varepsilon}$, 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.

Mots clés: Equations of mixed hyperbolic and parabolic type; oscillations; BKW Calculus; stability

Domaines: Equations aux dérivées partielles [math.AP]

Aurélien Klak. On the Production of Dissipation by Interaction of Forced Oscillating Waves in Fluid Dynamics. 2011. ⟨hal-00605895⟩