Leírás
Abstract:
Consider the flow of the ideal incompressible fluid in a bounded domain $M\subset{\mathbb R}^n$; in the classical fluid dynamics it is described by the Euler Eqiation
$$\frac{\partial u}{\partial t}+P(\nabla u\cdot u)=0$$
where $P$ is the orthogonal projector in $L^2(M,{\mathbb R}^n)$ onto the subspace of divergence-free vector fields tangent to $\partial M$. However, for the irregular (turbulent) flows the Euler equation makes no direct sense, and should be substituted by some intergral relations called Weak Euler Equations (corr. weak Euler solutions). Currently we know many (even too many) such solutions which are mostly devoid of physical sense.
This talk is devoted to a different way to describe the irregular flows which is orthogonal to the previously known ones (hence the title). It is based on the Generalized D'Alembert Principle (GD'AP). The classical D'Alembert Principle (D'AP) states that if a material point is forced to move without friction along a smooth manifold $D$ embedded into Euclidean space $E$, then its acceleration is orthogonal to $E$: $\ddot x(t)\perp T_{x(t)}D$. In case of the fluid, the configuration space is the set $D=SDiff(M)$ of volume-preserving diffeomorphisms of the flow domail $M$ naturally embedded into the Euclidean space $X=L^2(M, {\mathbb R}^n)$. However, the "manifold" $D$ is quite irregular, and the D'AP has only formal sense, and holds only for sufficiently smooth flows which (almost) surely do exist only on a bounded time interval. So, for the generic turbulent flows we need to introduce a GD'AP.
In fact, we need a formulation which is applicable to a wide class of nonsmooth manifolds in Euclidean space, namely the manifolds having a reasonable tangent plane at every point. The formulation is given in the language of Non-Standard Analysis (and it can hardly be given without it). We consider first the simplest pertinent case, namely, a nonsmooth planar curve. Even in this case the result is nontrivial: the kinetic energy of material point moving along such curve monotonically decreases. The same can happen for the motion along a nonsmooth surface of any dimension. At last, we consider the fluid motion as a motion of a material point along the "surface" $D$. We prove the existence of weak solution for all the time (this is the theorem which holds automatically for all the nonsmooth manifords).
The talk is intended to the general mathematical people, and will be kept as elementary as possible.
Online access:
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