## Seminar, November 6th 2014

__Thursday 6th__

*ReaDi-EHESS, 190-198 av. de France, 75013 Paris, Room 466*

15h : Mi-Ho Giga (Graduate School of Mathematical Sciences, The University of Tokyo, Japan)

*Small growing crystals in the plane become fully faceted by crystalline energy*We consider crystalline curvature flow equations with a spatially homogeneous driving force for a planar curve. If initial shape is large so that it includes the equilibrium shape, the solution curve grows and eventually sweeps out the whole plane. A level set method for a crystalline curvature flow has developed by Yoshikazu Giga and the author around a decade ago. It enables us to start with arbitrary initial data not necessarily a polygon but a general curve.

Based on this theory we shall prove that if the initial shape is convex and larger but sufficiently close to the equilibrium shape, then the solution curve becomes fully faceted under some assumptions of interfacial energy and mobility. Our results lead the first step for a mathematical explanation to the fact that a snow crystal becomes a hexagonal prism when it is very small.

This is a joint work with Yoshikazu Giga.

15h30 : Yoshikazu Giga (Graduate School of Mathematical Sciences, The University of Tokyo, Japan)

*On a non-blow up criterion involving vorticity direction under the non-slip boundary condition for the three-dimensional Navier-Stokes flow*We give a geometric non-blow up criterion on the direction of the vorticity for the three-dimensional half-space Navier-Stokes flow under the non-slip boundary condition whose initial data is just bounded and may have infinite energy. We prove that under a restriction on behavior in time (type I condition) the solution does not blow up if the vorticity direction is uniformly continuous at the place where the vorticity is large even if we impose the Dirichlet boundary conditions. A similar geometric regularity criterion for non-blow up has been proved by P. Constantin and C. Fefferman (1993) under Lipschitz regularity condition for the whole space has been established by H. Miura and the author (2011), and for the half space with the slip boundary condition. Their argument does not directly apply to the non-slip boundary condition since a key Liouville result for the two dimensional flow does not directly extend to the case of non-slip boundary.

We apply a representation formula for the vorticity (Y. Maekawa (2012)) and establish a Liouville type result under the non-slip boundary condition for type I blow-up. This enables us to prove that a continuous alignment condition for the vorticity prevents the blow-up even under the non-slip boundary condition which may produce a lot of vorticity near the boundary.

This is a joint work with P.-Y. Hsu and Y. Maekawa.

16h - 18h :
Presentations by researchers of the ReaDi team