Impermeability Through a Perforated Domain for the Incompressible two dimensional Euler Equations
We study the asymptotic behavior of the motion of an ideal incompressible fluid in a perforated domain. The porous medium is composed of inclusions of size \varepsilon}$e separated by distances {d_{\varepsilon}$de and the fluid fills the exterior. If the inclusions are distributed on the unit square, the asymptotic behavior depends on the limit of \frac{d_{\varepsilon}\varepsilon}$dee when \varepsilon}$e goes to zero. If \frac{d_{\varepsilon}\varepsilon \to \infty}$dee then the limit motion is not perturbed by the porous medium, namely, we recover the Euler solution in the whole space. If, on the contrary, \frac{d_{\varepsilon}\varepsilon \to 0}$dee 0, then the fluid cannot penetrate the porous region, namely, the limit velocity verifies the Euler equations in the exterior of an impermeable square. If the inclusions are distributed on the unit segment then the behavior depends on the geometry of the inclusion: it is determined by the limit of \frac{d_{\varepsilon}\varepsilon^{2+\frac1\gamma}$dee2+1g where \gamma \in (0,\infty]$g(0, is related to the geometry of the lateral boundaries of the obstacles. If \frac{d_{\varepsilon}\varepsilon^{2+\frac1\gamma} \to \infty}$dee2+1g then the presence of holes is not felt at the limit, whereas an impermeable wall appears if this limit is zero. Therefore, for a distribution in one direction, the critical distance depends on the shape of the inclusions; in particular, it is equal to \varepsilon^{3}$e3 for balls.