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We use layer potentials for the Stokes operator and an energetic formulation for the problem. Romain Nguyen - Institut für Mathematik - Freie Universität Berlin Non-stationary boundary layers and energy dissipation in incompressible flows In fully turbulent incompressible flows, the two presumed culprits for energy dissipation are thin boundary layers on the one hand, and three-dimensional stretching of vortex tubes on the other hand. Although both suspects seem equally important, much more theoretical effort has been invested to study the latter, due to the emphasis that has been put on homogeneous turbulence since the s. We argue that the same effort should be dedicated to interrogate the former, especially since it abides the simpler two-dimensional 2D setting. Indeed, while the shortcomings of the perfect fluid model, which lead to the d'Alembert paradox, are well understood, the failure of the Prandtl viscous boundary layer theory to predict the right scaling of energy dissipation still lacks a complete explanation. Several possibilities have been explored in recent years, like finite time singularities in the Prandtl equations, ill-posedness of the Prandtl equations, or even earlier breakdown of the asymptotic expansion itself. What happens beyond the Prandtl regime is even more mysterious. According to a criterion proven by Kato, energy should then be dissipated at scales as fine as the inverse of the Reynolds number, but the process by which this could happen remains elusive. We will review the main issues at hand, using a 2D dipole-wall interaction as illustrative test case.

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