For equations of order two with the Dirichlet boundary condition, as the Laplace problem, the Stokes and the Navier-Stokes systems, perforated domains were only studied when the distance between the holes dε is equal to or much larger than the size of the holes ε. Such a diluted porous medium is interesting because it contains some cases where we have a non-negligible effect on the solution when (ε, dε) → (0, 0). Smaller distances were avoided for mathematical reasons and for these large distances, the geometry of the holes does not affect or rarely affect the asymptotic result. Very recently, it was shown for the 2D-Euler equations that a porous medium is non-negligible only for inter-holes distances much smaller than the size of the holes. For this result, the boundary regularity of holes plays a crucial role, and the permeability criterion depends on the geometry of the lateral boundary. In this paper, we relax slightly the regularity condition, allowing a corner, and we note that a line of irregular obstacles cannot slow down a perfect fluid in any regime such that ε ln dε → 0.
ideal fluids homogenization in perforated domains shrinking obstacles and the porous medium domains with corners
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The first author was supported by the CNRS (program Tellus), the Agence Nationale de la Recherche: Project IFSMACS (Grant No. ANR-15-CE40-0010) and Project SINGFLOWS (Grant No. ANR-18-CE40-0027-01). The second author was supported by National Natural Science Foundation of China (Grant No. 11701016). This work has been supported by the Sino-French Research Program in Mathematics (SFRPM), which made several visits between the authors possible. The first author would like to acknowledge the hospitality and financial support of Peking University to conduct part of this work.
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