Partial Differential Equations Seminar

We consider the evolution of a small rigid body in an incompressible viscous fluid filling the whole space R^3 . When the small rigid body shrinks to a point in the sense that its density is constant, we prove that the solution of the fluid-rigid body system converges to a solution of the Navier–Stokes equations in the full space. Based on some L^p − L^q estimates of the fluid–structure semigroup and a fixed point argument, we obtain a uniform estimate of velocity of the rigid body. This allows us to construct admissible test functions which plays a key role in the procedure of passing to the limit. This is a joint work with Pei Su (Nantes).

Allen-Cahn equations are popular phenomenological models for phase transitions that describe materials on a mesoscopic scale. The macroscopic scale observed by us humans corresponds to a singular scaling limit, the so-called sharp-interface limit, in which transition layers collapse onto hypersurfaces. Since Ilmanen’s pioneering work [J. Differential Geom. 38, 417-461, (1993)] it has been a major open problem to understand this sharp-interface limit of systems of coupled Allen-Cahn equations. In this talk, I will present the first global, unconditional convergence result for such a coupled system and show that the limit is a multiphase mean curvature flow in the sense of Brakke. 

This is joint work with Keisuke Takasao (Kyoto University).
 

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