Đề tài " Di_usion and mixing in uid ow " pot

Our main result is a sharp description ofthe class of flows that make the deviation of the solution from its average arbi-trarily small in an arbitrarily short time, provided that the fl

Diffusion and mixing in fluid flow

By P Constantin, A Kiselev, L Ryzhik, and A Zlatoˇs

Abstract

We study enhancement of diffusive mixing on a compact Riemannian ifold by a fast incompressible flow Our main result is a sharp description ofthe class of flows that make the deviation of the solution from its average arbi-trarily small in an arbitrarily short time, provided that the flow amplitude islarge enough The necessary and sufficient condition on such flows is expressednaturally in terms of the spectral properties of the dynamical system associatedwith the flow In particular, we find that weakly mixing flows always enhancedissipation in this sense The proofs are based on a general criterion for thedecay of the semigroup generated by an operator of the form Γ + iAL with

man-a negman-ative unbounded self-man-adjoint operman-ator Γ, man-a self-man-adjoint operman-ator L, man-andparameter A  1 In particular, they employ the RAGE theorem describingevolution of a quantum state belonging to the continuous spectral subspace

of the hamiltonian (related to a classical theorem of Wiener on Fourier forms of measures) Applications to quenching in reaction-diffusion equationsare also considered

trans-1 IntroductionLet M be a smooth compact d-dimensional Riemannian manifold Themain objective of this paper is the study of the effect of a strong incompressibleflow on diffusion on M Namely, we consider solutions of the passive scalarequation

(1.1) φAt(x, t) + Au · ∇φA(x, t) − ∆φA(x, t) = 0, φA(x, 0) = φ0(x)

Here ∆ is the Laplace-Beltrami operator on M, u is a divergence free vectorfield, ∇ is the covariant derivative, and A ∈ R is a parameter regulating thestrength of the flow We are interested in the behavior of solutions of (1.1) for

A  1 at a fixed time τ > 0

It is well known that as time tends to infinity, the solution φA(x, t) willtend to its average,

|M |Z

M

φA(x, t) dµ = 1

|M |Z

M

φ0(x) dµ,

with |M | being the volume of M We would like to understand how the speed ofconvergence to the average depends on the properties of the flow and determinewhich flows are efficient in enhancing the relaxation process

The question of the influence of advection on diffusion is very natural andphysically relevant, and the subject has a long history The passive scalarmodel is one of the most studied PDEs in both mathematical and physicalliterature One important direction of research focused on homogenization,where in a long time–large propagation distance limit the solution of a passiveadvection-diffusion equation converges to a solution of an effective diffusionequation Then one is interested in the dependence of the diffusion coefficient

on the strength of the fluid flow We refer to [29] for more details and references.The main difference in the present work is that here we are interested in theflow effect in a finite time without the long time limit

On the other hand, the Freidlin-Wentzell theory [16], [17], [18], [19] studies(1.1) in R2 and, for a class of Hamiltonian flows, proves the convergence ofsolutions as A → ∞ to solutions of an effective diffusion equation on the Reebgraph of the hamiltonian The graph, essentially, is obtained by identifying allpoints on any streamline The conditions on the flows for which the procedurecan be carried out are given in terms of certain non-degeneracy and growthassumptions on the stream function The Freidlin-Wentzell method does notapply, in particular, to ergodic flows or in odd dimensions

Perhaps the closest to our setting is the work of Kifer and more recently aresult 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, insome special situations, other eigenvalues) of the operator −∆ + u · ∇ when 

is small, mainly in the case of the Dirichlet boundary conditions In particular,the asymptotic behavior of the principal eigenvalue λ0 and the correspondingpositive eigenfunction φ0 for small  has been described in the case where theoperator u · ∇ has a discrete spectrum and sufficiently 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

t→∞t−1log kφ(x, t)kL2 = −λ0(see e.g [22]) In a related recent work [2], Berestycki, Hamel and Nadirashviliutilize PDE methods to prove a sharp result on the behavior of the principal

eigenvalue λA of the operator −∆ + Au · ∇ defined on a bounded domain

Ω ⊂ Rdwith the Dirichlet boundary conditions

The main conclusion is that λA stays bounded as A → ∞ if and only if uhas a first integral w in H01(Ω) (that is, u · ∇w = 0) An elegant variationalprinciple determining the limit of λAas A → ∞ is also proved In addition, [2]provides a direct link between the behavior of the principal eigenvalue and thedynamics which is more robust than (1.2): it is shown that kφA(·, 1)kL2 (Ω) can

be made arbitrarily small for any initial datum by increasing A if and only if

λA→ ∞ as A → ∞ (and, therefore, if and only if the flow u does not have afirst integral in H01(Ω)) We should mention that there are many earlier worksproviding variational characterization of the principal eigenvalues, and refer to[2], [24] for more references

Many of the studies mentioned above also apply in the case of a compactmanifold without boundary or Neumann boundary conditions, which are theprimary focus of this paper However, in this case the principal eigenvalue

is simply zero and corresponds to the constant eigenfunction Instead one

is interested in the speed of convergence of the solution to its average, therelaxation speed A recent work of Franke [15] provides estimates on the heatkernels corresponding to the incompressible drift and diffusion on manifolds,but these estimates lead to upper bounds on kφA(1) − φk which essentially

do not improve as A → ∞ One way to study the convergence speed is toestimate the spectral gap – the difference between the principal eigenvalue andthe real part of the next eigenvalue To the best of our knowledge, there is verylittle known about such estimates in the context of (1.1); see [22] p 251 for

a discussion Neither probabilistic methods nor PDE methods of [2] seem toapply in this situation, in particular because the eigenfunction corresponding

to the eigenvalue(s) with the second smallest real part is no longer positive andthe eigenvalue itself does not need to be real

Moreover, even if the spectral gap estimate were available, generally itonly yields a limited asymptotic in time dynamical information of type (1.2),and how fast the long time limit is achieved may depend on A Part of ourmotivation for studying the advection-enhanced diffusion comes from the ap-plications to quenching in reaction-diffusion equations (see e.g [4], [12], [27],[34], citeZ), which we discuss in Section 7 For these applications, one needsestimates on the A-dependent L∞ norm decay at a fixed positive time, thetype of information the bound like (1.2) does not provide We are aware ofonly one case where enhanced relaxation estimates of this kind are available It

is the recent work of Fannjiang, Nonnemacher and Wolowski [10], [11], wheresuch estimates are provided in the discrete setting (see also [22] for some re-lated earlier references) In these papers a unitary evolution step (a certainmeasure-preserving map on the torus) alternates with a dissipation step, which,for example, acts simply by multiplying the Fourier coefficients by damping

factors The absence of sufficiently regular eigenfunctions appears as a key forthe lack of enhanced relaxation in this particular class of dynamical systems.

In [10], [11], the authors also provide finer estimates of the dissipation timefor particular classes of toral automorphisms (that is, they estimate how manysteps are needed to reduce the L2 norm of the solution by a factor of two ifthe diffusion strength is )

Our main goal in this paper is to provide a sharp characterization ofincompressible flows that are relaxation enhancing, in a quite general setup

We work directly with dynamical estimates, and do not discuss the spectralgap The following natural definition will be used in this paper as a measure

of the flow efficiency in improving the solution relaxation

Definition 1.1 Let M be a smooth compact Riemannian manifold Theincompressible flow u on M is called relaxation enhancing if for every τ > 0 and

δ > 0, there exist A(τ, δ) such that for any A > A(τ, δ) and any φ0 ∈ L2(M )with kφ0kL2 (M ) = 1,

where φA(x, t) is the solution of (1.1) and φ the average of φ0

Remarks 1 In Theorem 5.5 we show that the choice of the L2 norm

in the definition is not essential and can be replaced by any Lp-norm with

1 ≤ p ≤ ∞

2 It follows from the proofs of our main results that the hancing class is not changed even when we allow the flow strength that ensures(1.3) to depend on φ0, that is, if we require (1.3) to hold for all φ0 ∈ L2(M )with kφ0kL2 (M ) = 1 and all A > A(τ, δ, φ0)

relaxation-en-Our first result is as follows

Lipschitz continuous incompressible flow u ∈ Lip(M ) is relaxation-enhancing

if and only if the operator u · ∇ has no eigenfunctions in H1(M ), other thanthe constant function

Any incompressible flow u ∈ Lip(M ) generates a unitary evolution group

Uton L2(M ), defined by Utf (x) = f (Φ−t(x)) Here Φt(x) is a ing transformation associated with the flow, defined by dtdΦt(x) = u(Φt(x)),

measure-preserv-Φ0(x) = x Recall that a flow u is called weakly mixing if the corresponding erator U has only continuous spectrum The weakly mixing flows are ergodic,but not necessarily mixing (see e.g [5]) There exist fairly explicit examples

op-of weakly mixing flows [1], [13], [14], [28], [35],u [33], some op-of which we willdiscuss in Section 6 A direct consequence of Theorem 1.2 is the followingcorollary

Corollary 1.3 Any weakly mixing incompressible flow u ∈ Lip(M ) isrelaxation enhancing.

Theorem 1.2, as we will see in Section 5, in its turn follows from a quitegeneral abstract criterion, which we are now going to describe Let Γ be

a self-adjoint, positive, unbounded operator with a discrete spectrum on aseparable Hilbert space H Let 0 < λ1 ≤ λ2 ≤ be the eigenvalues of Γ,and ej the corresponding orthonormal eigenvectors forming a basis in H The(homogeneous) Sobolev space Hm(Γ) associated with Γ is formed by all vectors

an example, that this is not the case in general

Consider a solution φA(t) of the Bochner differential equation

dtφ

A(t) = iALφA(t) − ΓφA(t), φA(0) = φ0

Theorem 1.4 Let Γ be a self-adjoint, positive, unbounded operator with

a discrete spectrum and let a self-adjoint operator L satisfy conditions (1.4).Then the following two statements are equivalent :

• For any τ, δ > 0 there exists A(τ, δ) such that for any A > A(τ, δ) andany φ0 ∈ H with kφ0kH = 1, the solution φA(t) of the equation (1.5)satisfies kφA(τ )k2H < δ

• The operator L has no eigenvectors lying in H1(Γ)

Remark Here L corresponds to iu · ∇ (or, to be precise, a self-adjointoperator generating the unitary evolution group Ut which is equal to iu · ∇

on H1(M )), and Γ to −∆ in Theorem 1.2, with H ⊂ L2(M ) the subspace ofmean zero functions

Theorem 1.4 provides a sharp answer to the general question of when acombination of fast unitary evolution and dissipation produces a significantlystronger dissipative effect than dissipation alone It can be useful in any model

describing a physical situation which involves fast unitary dynamics with sipation (or, equivalently, unitary dynamics with weak dissipation) We proveTheorem 1.4 in Section 3 The proof uses ideas from quantum dynamics, inparticularly the RAGE theorem (see e.g., [6]) describing evolution of a quantumstate belonging to the continuous spectral subspace of a self-adjoint operator.

dis-A natural concern is the consistency of the existence of rough tors of L and condition (1.4) which says that the dynamics corresponding to

eigenvec-L preserves H1(Γ) In Section 4 we establish this consistency by providing amples where rough eigenfunctions exist yet (1.4) holds One of them involves

ex-a discrete version of the celebrex-ated Wigner-von Neumex-ann construction of ex-animbedded eigenvalue of a Schr¨odinger operator [32] Moreover, in Section 6

we describe an example of a smooth flow on the two dimensional torus T2with discrete spectrum and rough (not H1(T2)) eigenfunctions – this exampleessentially goes back to Kolmogorov [28] Thus, the result of Theorem 1.4 isprecise

In Section 7, we discuss the application of Theorem 1.2 to quenching forreaction-diffusion equations on compact manifolds and domains This corre-sponds to adding a non-negative reaction term f (T ) on the right-hand side of(1.1), with f (0) = f (1) = 0 Then the long-term dynamics can lead to twooutcomes: φA → 1 at every point (complete combustion), or φA → c < 1(quenching) The latter case is only possible if f is of the ignition type; that

is, there exists θ0 such that f (T ) = 0 for T ≤ θ0, and c ≤ θ0 The question isthen how the presence of strong fluid flow may aid the quenching process Wenote that quenching/front propagation in infinite domains is also of consider-able interest Theorem 1.2 has applications in that setting as well, but theywill be considered elsewhere

2 Preliminaries

In this section we collect some elementary facts and estimates for theequation (1.5) Henceforth we are going to denote the standard norm in theHilbert space H by k · k, the inner product in H by h·, ·i, the Sobolev spaces

Hm(Γ) simply by Hm and norms in these Sobolev spaces by k · km We havethe following existence and uniqueness theorem

Theorem 2.1 Assume that for any ψ ∈ H1,

Then for any T > 0, there exists a unique solution φ(t) of the equation

φ0(t) = (iL − Γ)φ(t), φ(0) = φ0∈ H1.This solution satisfies

(2.2) φ(t) ∈ L2([0, T ], H2) ∩ C([0, T ], H1), φ0(t) ∈ L2([0, T ], H)

Remarks 1 The proof of Theorem 2.1 is standard, and can proceed byconstruction of a weak solution using Galerkin approximations and then estab-lishing uniqueness and regularity We refer, for example, to Evans [8] wherethe construction is carried out for parabolic PDEs but, given the assumption(2.1), can be applied verbatim in the general case.

2 The existence theorem is also valid for initial data φ0 ∈ H, but thesolution has rougher properties at intervals containing t = 0, namely

(2.3) φ(t) ∈ L2([0, T ], H1) ∩ C([0, T ], H), φ0(t) ∈ L2([0, T ], H−1).The existence of a rougher solution can also be derived from the general semi-group theory, by checking that iL−Γ satisfies the conditions of the Hille-Yosidatheorem and thus generates a strongly continuous contraction semigroup in H(see, e.g [7])

Next we establish a few properties that are more specific to our particularproblem It will be more convenient for us, in terms of notation, to work with

an equivalent reformulation of (1.5), by setting  = A−1 and rescaling time bythe factor −1, thus arriving at the equation

An immediate consequence of (2.6) is the following result, that we statehere as a separate lemma for convenience

Lemma 2.3 Suppose that for all times t ∈ (a, b) we have kφ(t)k2

1 ≥

N kφ(t)k2 Then the following decay estimate holds:

kφ(b)k2≤ e−2N (b−a)kφ(a)k2.Next we need an estimate on the growth of the difference between solutionscorresponding to  > 0 and  = 0 in the H-norm

Lemma 2.4 Assume, in addition to (2.1), that for any ψ ∈ H1 and t > 0,

for some B(t) ∈ L2

loc[0, ∞) Let φ0(t), φ(t) be solutions of(φ0)0(t) = iLφ0(t), (φ)0(t) = (iL − Γ)φ(t),satisfying φ0(0) = φ(0) = φ0∈ H1 Then

Proof The regularity guaranteed by conditions (2.1), (2.7) and rem 2.1 allows us to multiply the equation

The following corollary is immediate

Corollary 2.5 Assume that (2.1) and (2.7) are satisfied, and the initialdata φ0∈ H1 Then the solutions φ(t) and φ0(t) defined in Lemma 2.4 satisfy

kφ(t) − φ0(t)k2≤ 1

2kφ0k

2 1

Z τ 0

B2(t) dt

for any time t ≤ τ

Finally, we observe that conditions (2.1) and (2.7) are independent ing L = Γ shows that (2.7) does not imply (2.1), because in this case theevolution eiLtis unitary on H1 but the domain of L is H2

Tak-( H1 On the otherhand, (2.1) does not imply (2.7), as is the case in the following example Let

H ≡ L2(0, 1), define the operator Γ by Γf (x) ≡P

nen2f (n)eˆ 2πinxfor all f ∈ Hsuch that en2f (n) ∈ `ˆ 2(Z), and take Lf (x) ≡ xf (x) Then L is bounded on Hand so (2.1) holds automatically, but

eitLf (x) = f (x)eitx

so that e2πiLe2πinx = e2πi(n+1)x It follows that e2πiL is not bounded on H1(and neither is eiLt for any t 6= 0)

3 The abstract criterionOne direction in the proof of Theorem 1.4 is much easier We start byproving this easy direction: that existence of H1(Γ) eigenvectors of L ensuresexistence of τ, δ > 0 and φ0 with kφ0k = 1 such that kφA(τ )k > δ for all

A – that is, if such eigenvectors exist, then the operator L is not relaxationenhancing

Proof of the first part of Theorem 1.4 Assume that the initial datum

φ0 ∈ H1 for (1.5) is an eigenvector of L corresponding to an eigenvalue E,normalized so that kφ0k = 1 Take the inner product of (1.5) with φ0 Wearrive at

d

dt e

−iAEthφA(t), φ0i

Note also that we have proved that in the presence of an H1 eigenvector

of L, enhanced relaxation does not happen for some φ0 even if we allow A(τ, δ)

to be φ0-dependent as well This explains Remark 2 after Definition 1.1.The proof of the converse is more subtle, and will require some prepara-tion We switch to the equivalent formulation (2.4) We need to show that if Lhas no H1 eigenvectors, then for all τ, δ > 0 there exists 0(τ, δ) > 0 such that

if  < 0, then kφ(τ /)k < δ whenever kφ0k = 1 The main idea of the proofcan be naively described as follows If the operator L has purely continuousspectrum or its eigenfunctions are rough then the H1-norm of the free evolution(with  = 0) is large most of the time However, the mechanism of this effect

is quite different for the continuous and point spectra On the other hand, wewill show that for small  the full evolution is close to the free evolution for asufficiently long time This clearly leads to dissipation enhancement

The first ingredient that we need to recall is the so-called RAGE theorem.Theorem 3.1 (RAGE) Let L be a self-adjoint operator in a Hilbertspace H Let Pc be the spectral projection on its continuous spectral subspace.Let C be any compact operator Then for any φ0∈ H,

lim

T →∞

1T

T

Z

0

kCeiLtPcφ0k2dt = 0

Clearly, the result can be equivalently stated for a unitary operator U ,replacing eiLt with Ut The proof of the RAGE theorem can be found, forexample, in [6].

A direct consequence of the RAGE theorem is the following lemma Recallthat we denote by 0 < λ1 ≤ λ2 ≤ the eigenvalues of the operator Γand by e1, e2, the corresponding orthonormal eigenvectors Let us alsodenote by PN the orthogonal projection on the subspace spanned by the first

N eigenvectors e1, , eN and by S = {φ ∈ H : kφk = 1} the unit sphere

in H The following lemma shows that if the initial data lie in the continuousspectrum of L then the L-evolution will spend most of its time in the highermodes of Γ

Lemma 3.2 Let K ⊂ S be a compact set For any N, σ > 0, there exists

Tc(N, σ, K) such that for all T ≥ Tc(N, σ, K) and any φ ∈ K,

Remark The key observation of Lemma 3.2 is that the time Tc(N, σ, K)

is uniform for all φ ∈ K

Proof Since PN is compact, we see that for any vector φ ∈ S, thereexists a time Tc(N, σ, φ) that depends on the function φ such that (3.1) holdsfor T > Tc(N, σ, φ) – this is assured by Theorem 3.1 To prove the uniformity

in φ, note that the function

Lemma 3.3 Assume that not a single eigenvector of the operator L longs to H1(Γ) Let K ⊂ S be a compact set Consider the set K1 ≡{φ ∈ K | kPpφk ≥ 1/2} Then for any B > 0 we can find Np(B, K) and

be-Tp(B, K) such that for any N ≥ Np(B, K), any T ≥ Tp(B, K) and any φ ∈ K1,

Remark Note that unlike (3.1), we have the H1 norm in (3.2)

Proof The set K1may be empty, in which case there is nothing to prove.Otherwise, denote by Ejthe eigenvalues of L (distinct, without repetitions) and

by Qj the orthogonal projection on the space spanned by the eigenfunctionscorresponding to Ej First, let us show that for any B > 0 there is N (B, K)such that for any φ ∈ K1,

j

kPNQjφk21 ≥ 2B

if N ≥ N (B, K) It is clear that for each fixed φ with Ppφ 6= 0 we can find

N (B, φ) so that (3.3) holds, since by assumption Qjφ does not belong to H1whenever Qjφ 6= 0 Assume that N (B, K) cannot be chosen uniformly for

φ ∈ K1 This means that for any n, there exists φn∈ K1 such that

kPNQjφk˜ 21 ≤ 2B for any N, a contradiction since ˜φ ∈ K1

Next, take φ ∈ K1 and consider

In (3.4), we set (ei(Ej −El)T − 1)/i(Ej− El)T ≡ 1 if j = l Notice that the sumabove converges absolutely Indeed, PNQjφ =PN

i=1hQjφ, eiiei, and hΓei, eki =

jkΓ1/2PNQjφk2=P

=

X

l6=j

ei(Ej −El)T − 1i(Ej− El)T hΓPNQjφ, PNQlφi

... contradiction since ˜φ ∈ K1

Next, take φ ∈ K1 and consider

In (3.4),... uniformity

in φ, note that the function

Lemma 3.3 Assume that not a single eigenvector... Note that since φ0/kφ0k ∈ K by the hypothesis, we can apply

Lemma