Name	Naoto Nakano
	Department	Department of Mathematics, School of Fundamental Science and Technology
	Research Fields	Partial Differential Equations, Mathematical Physics

"A Mathematical Analysis of the Behaviour of Granular Materials with the method of Fluid Mechanics"
The recent topic of my study is the behaviour of Granular Materials. Granular Materials, as the nomenclature would suggest, consists of grains, and these grains are invariably filled with a fluid. Granular Materials include not only sands and powder but soil, a mass of snow, polymers, etc. The object of a study of them has a wide range.
Currently I am working on analysing the motion of Granular Materials in mathematical physics. Natural phenomena concerning Granular Materials are quite remarkable, that is, for instance, pyramids stand still on the sands over the centuries even though motion of quicksands can be seen like that of fluid. Those materials seem to have both solid and fluid phases, therefore their dynamic and static behaviour can be greatly interesting. Since investigation of the mechanism of these phenomena has not been completed yet, I devote to the research on Granular Materials.
If one considers the interaction between particles to analyse motion of Granular Materials, one should solve the so-called n-body problem. In fact, the impossibility of solving the general n-body problem is well known since the work by H. Poincare. Then, in the same way as that used for fluid mechanics, we look Granular Materials as continua to describe the behaviour of them in virtue of their features of response. My main work is to solve a system of partial differential equations (as conservation laws of mass, linear momentum, angular momentum and energy) governing motion of Granular Materials obtained by the continuum approximation. I have already solved a problem for the model of "inhomogeneous incompressible fluid-like bodies".
Furthermore, when one consider phenomena of Granular Materials, the characteristic of them on the surface of the body is also important. It is said that the occurrence of slips is essential to the motion of Granular Materials. Namely, we should adopt boundary conditions which include the slip condition to solve the equations mentioned above. Although there are not many results of solvability theorems of partial differential equations with the slip boundary condition because of its difficulty, we should try to solve these slip boundary problems in order to obtain the deeper investigation into granular phenomena.