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

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

2011-07-05

arxiv.org/abs/1107.0787v1

Mathematics

Analysis of PDEs

Scientific paper

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.

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