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 $\varepsilon>0$ 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 $\alpha \geqslant 2$ (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 $\varepsilon\geqslant 0$, we transform the Laplace--Neumann problem into a generalized Neumann problem set on a domain containing a horizontal strip $]0,\ell_\varepsilon[\times ]0,1[$, where $\ell_\varepsilon\to +\infty$.