Tiểu Luận Di usion and mixing in uid ow

Diusion and mixing in
uid
ow
By P. Constantin, A. Kiselev, L. Ryzhik, and A. Zlato s
Abstract
We study enhancement of diusive mixing on a compact Riemannian man-ifold by a fast incompressible
ow. Our main result is a sharp description of
the class of
ows that make the deviation of the solution from its average arbi-trarily small in an arbitrarily short time, provided that the
ow amplitude is
large enough. The necessary and sucient condition on such
ows is expressed
naturally in terms of the spectral properties of the dynamical system associated
with the
ow. In particular, we nd that weakly mixing
ows always enhance
dissipation in this sense. The proofs are based on a general criterion for the
decay of the semigroup generated by an operator of the form + iAL with
a negative unbounded self-adjoint operator , a self-adjoint operator L, and
parameter A  1. In particular, they employ the RAGE theorem describing
evolution of a quantum state belonging to the continuous spectral subspace
of the hamiltonian (related to a classical theorem of Wiener on Fourier trans-forms of measures). Applications to quenching in reaction-diusion equations
are also considered.
1. Introduction
Let M be a smooth compact d-dimensional Riemannian manifold. The
main objective of this paper is the study of the eect of a strong incompressible


ow on diusion on M: Namely, we consider solutions of the passive scalar
equation
(1.1) 
A
t
(x; t) + Au
r
A
(x; t)

A
(x; t) = 0; 
A
(x; 0) = 
0
(x):
Here
is the Laplace-Beltrami operator on M; u is a divergence free vector
eld, r is the covariant derivative, and A 2 R is a parameter regulating the
strength of the
ow. We are interested in the behavior of solutions of (1.1) for
A  1 at a xed time  > 0.
644 P. CONSTANTIN, A. KISELEV, L. RYZHIK, AND A. ZLATO

S
It is well known that as time tends to innity, the solution 
A
(x; t) will
tend to its average,
 
1
jM j
Z
M

A
(x; t) d =
1
jM j
Z
M

0
(x) d;
with jM j being the volume of M . We would like to understand how the speed of
convergence to the average depends on the properties of the
ow and determine
which
ows are ecient in enhancing the relaxation process.
The question of the in
uence of advection on diusion is very natural and
physically relevant, and the subject has a long history. The passive scalar
model is one of the most studied PDEs in both mathematical and physical
literature. One important direction of research focused on homogenization,
where in a long time{large propagation distance limit the solution of a passive
advection-diusion equation converges to a solution of an eective diusion
equation. Then one is interested in the dependence of the diusion coecient
on the strength of the
uid
ow. We refer to [29] for more details and references.
The main dierence in the present work is that here we are interested in the


ow eect in a nite time without the long time limit.
On the other hand, the Freidlin-Wentzell theory [16], [17], [18], [19] studies
(1.1) in R
2
and, for a class of Hamiltonian
ows, proves the convergence of
solutions as A ! 1 to solutions of an eective diusion equation on the Reeb
graph of the hamiltonian. The graph, essentially, is obtained by identifying all
points on any streamline. The conditions on the
ows for which the procedure
can be carried out are given in terms of certain non-degeneracy and growth
assumptions on the stream function. The Freidlin-Wentzell method does not
apply, in particular, to ergodic
ows or in odd dimensions.
Perhaps the closest to our setting is the work of Kifer and more recently a
result of Berestycki, Hamel and Nadirashvili. Kifer's work (see [21], [22], [23],
[24] where further references can be found) employs probabilistic methods and
is focused, in particular, on the estimates of the principal eigenvalue (and, in
some special situations, other eigenvalues) of the operator 
+ u
r when 
is small, mainly in the case of the Dirichlet boundary conditions. In particular,
the asymptotic behavior of the principal eigenvalue 

0
and the corresponding
positive eigenfunction 

0
for small  has been described in the case where the
operator u
r has a discrete spectrum and suciently smooth eigenfunctions.
It is well known that the principal eigenvalue determines the asymptotic rate
of decay of the solutions of the initial value problem, namely
(1.2) lim
t!1
t
1
log k

(x; t)k
L
2 = 

0
(see e.g. [22]). In a related recent work [2], Berestycki, Hamel and Nadirashvili
utilize PDE methods to prove a sharp result on the behavior of the principal