Some Open Problems and Research Directions in the Mathematical Study of Fluid Dynamics
In an editorial of the Notices of the AMS (vol. 47, Number 3, March 2000), Felix Browder, President of the AMS, refers to “ ... some of the major classical problems: the Riemann Hypothesis, the Poincaré Conjecture, and the regularity of three-dimensional
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In an editorial of the Notices of the AMS (vol. 47, Number 3, March 2000), Felix Browder, President of the AMS, refers to" . . . some of the major classical problems: the Riemann Hypothesis, the Poincare Conjecture, and the regularity of three-dimensional fluid flows". I imagine that many beginning graduate students in Mathematics have heard of the first two of these problems, but maybe not so many know about the third. I would like to describe here this third problem in a broader context, involving not only POE questions of existence, uniqueness and regularity of solutions, but also dynamical issues concerning stability and statistical questions raised by instability. Ordinary incompressible Newtonian fluids are described by the NavierStokes equations. These equations have been used by engineers and physicists with a great deal of success. The range of their validity and applicability is well established. Together with other fundamental systems like the Schrodinger and Maxwell equations, these equations are among the most important equations of mathematical physics.
l.PDE There are two ways of describing fluids. The Eulerian description is concerned with the fluid velocity u(x, t), density p(x, t), and pressure p(x, t) recorded at fixed positions x E Rn, n = 3 as functions of timet. The Navier-Stokes equations relating these quantities to each other is an expression of the balance of forces according to Newton's second law,
F=ma. I will take the density p to be constant in order to write the simplest form of the equations. There are n equations
aui n aui ap -+"'uj--+-=v.dui, at ~ ax·J ax·l j=l
i=l, ... ,n,
representing the actual balance of forces and one more
L n
i=l
aui
-=0
ax·l
representing the constraint of incompressibility. The positive coefficient v is the kinematic viscosity and it is a fixed, given parameter, describing a quality of the B. Engquist et al. (eds.), Mathematics Unlimited — 2001 and Beyond © Springer-Verlag Berlin Heidelberg 2001
354
P.
CONSTANTIN
fluid that is not changing in time under the conditions discussed here. L1 = V 2 is the Laplacian. The equations need boundary conditions. If one considers fluids inside some domain Q c R 3 , then the fluid particles stick to the walls ()Q of the domain u(x,t)=O, xEo!J. The equations are nonlinear and non-local. The term non-local refers to therelationship between velocity and pressure: the pressure is computed by applying linear singular integral operators to quadratic expressions involving the velocity components. The total kinetic energy of the fluid is
There is no external source of energy in the situation depicted above; therefore the kinetic energy dissipates
Solutions with finite kinetic energy and with a finite average rate of dissipation of kinetic energy should, in principle, exist forever and decay to 0. Unfortunately, the dissipation of kinetic energy is the strongest quantitative information about the Navier-Stokes equations that is presently known for general solutions. In his classical work ([1]) Leray used this diss
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