Mathematical concepts and their physical foundation in the nonstandard analysis theory of turbulence

Main mathematical concepts and their physical foundation in thenonstandard analysis theory of turbulence are presented anddiscussed. The underlying fact is that there does not exist theabsolute zero fluid-volume. Therefore, the physical objectcorresponding to the absolute point is just the uniformfluid-particle. The fluid-particle, in general, corresponds to themonad. The uniform fluid-particle corresponds to the uniform monad,while the nonuniform fluid-particle to the nonuniform monad. Thereare two kinds of the differentiations, one is based on the absolutepoint, and the other based on the monad. The former is adopted inthe Navier--Stokes equations, and the latter in the fundamentalequations presented in this paper for the nonstandard analysistheory of turbulence. The continuity of fluid is elucidated byvirtue of the concepts of the fluid-particle and fluid-particle at alower level. Furthermore, the characters of the continuity in twocases, i.e. in the standard and nonstandard analyses, are presentedin this paper. And the difference in discretization between theNavier--Stokes equations and the fundamental equations given hereinis also pointed out.