ASYMPTOTIC ANALYSIS OF A NEUMANN PROBLEM IN A DOMAIN WITH CUSP APPLICATION TO THE COLLISION PROBLEM OF RIGID BODIES IN A PERFECT FLUID

We study a two dimensional collision problem for a rigid solid immersed in a cavity filled with a perfect fluid. We are led to investigate the asymptotic behavior of the Dirichlet energy associated with the solution of a Laplace Neumann problem as the distance epsilon > 0 between the solid and the cavity's bottom tends to zero. Denoting by alpha > 0 the tangency exponent at the contact point, we prove that the solid always reaches the cavity in finite time, but with a nonzero velocity for alpha < 2 (real shock case), and with null velocity for alpha >= 2 (smooth landing case). Our proof is based on a suitable change of variables sending to infinity the cusp singularity at the contact. More precisely, for every epsilon >= 0, we transform the Laplace-Neumann problem into a generalized Neumann problem set on a domain containing a horizontal strip ]0, l(epsilon)[x]0, 1[, where l(epsilon) -> +infinity.