From daisymae@math.mit.edu Tue Feb 1 09:38:44 2000
Date: Fri, 28 Jan 2000 18:53:23 -0500 (EST)
From: Shirley Entzminger-Merritt
To: fluids@math.mit.edu
Subject: Physical Mathematics Seminar -- Tuesday, February 1, 2000
PHYSICAL MATHEMATICS SEMINAR
TOPIC: ON THE NONLINEAR STABILITY OF FLUID FLOWS:
ARNOLD'S APPROACH
SPEAKER: VLADIMIR A. VLADIMIROV
Department of Mathematics
Hong Kong University of Science and Technology
ABSTRACT:
Energy variational principles represent legitimate children of more
general laws of the Nature, such as the principle of the least action.
But often they have broader areas of living, and can be applicable, say,
to some dissipative systems. In fluid mechanics they compose the core of
the stability theory and have a long history. More than a century ago (in
1887) Lord Kelvin wrote: "The condition for steady motion of an
incompressible inviscid fluid filling a finite fixed portion of space ...
is that, {\it with a given vorticity}, the energy is a thorough maximum,
or a thorough minimum, or a minimax." These words represent the first
attempt to formulate an important conjecture that was developed in
mathematical terms only in 1965 by Vladimir Arnold. He established a
variational principle which states that {\it on the set of all
`isovortical flows' of an ideal fluid the kinetic energy attains its
stationary values at steady flows}. Discovery of the {\it `isovorticity
conditions'} caused several significant changes in theoretical fluid
dynamics. Namely: (i) it disclosed a hidden unity of hydrodynamic
stability theory; different stability criteria appeared to be of the same
variational nature; (ii) it led to proving of stability for number of new
classes of flows; main achievements here are the natural definitions of
stability; (iii) it caused rapid developments of Hamiltonian approaches
for fluid flows.
In this lecture we present Arnold's principle in a simplest possible way
and give its generalizations for more sophisticated systems such as
stratified fluid, magneto-hydrodynamics and dynamical systems
'solid+fluid'. All considerations are based on a new form of the {\it
'isovorticity conditions'} which are formulated in the spirit of early
Arnold's papers. Several practical results of applications to obtaining
of stability criteria are given. The lecture is based on papers.
DATE: TUESDAY, FEBRUARY 1, 2000
TIME: 2:30 PM
LOCATION: Room 2-338
Refreshments will be served at 3:30 PM in Room 2-349.
Massachusetts Institute of Technology
Department of Mathematics
Cambridge, MA 02139