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Asymptotic analysis of a Neumann problem in a domain with cusp. Application to the collision problem of rigid bodies in a perfect fluid. (English) Zbl 1333.35208

The authors consider a contact-collision problem between a solid immersed in a cavity filled with a perfect fluid and the bottom of the cavity. The problem reduces to a singular limit problem with a small parameter \(\varepsilon\) modelling the solid-cavity distance when \(\varepsilon\to 0\), for an elliptic problem posed in a domain \(\Omega_{\varepsilon}\) with mixed Dirichlet-Neumann boundary conditions. The domain converges to a singular domain with cusps and the problem is supposed to be axisymmetric. If \(\alpha>0\) denotes the tangency exponent (a parameter characterizing the contact point), the authors prove that for \(\alpha<2\) the limit solution has finite energy. Instead, for \(\alpha\geq 2\) the limit energy is unbounded and the type of blow-up is different if \(\alpha=2\) or if \(\alpha>2\).

MSC:

35Q35 PDEs in connection with fluid mechanics
35B40 Asymptotic behavior of solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
35J20 Variational methods for second-order elliptic equations
35C20 Asymptotic expansions of solutions to PDEs
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
35B44 Blow-up in context of PDEs
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References:

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