On a conjecture of De Giorgi related to homogenization

For a periodic vector field $\bf F$, let ${\bf X}^\epsilon$ solve the dynamical system \begin{equation*} \frac{d{\bf X}^\epsilon}{dt} = {\bf F}\left(\frac {{\bf X}^\epsilon}\epsilon\right) . \end{equation*} In \cite{DeGiorgi} Ennio De Giorgi enquiers whether from the existence of the limit ${\bf X}^0(t):=\lim\limits_{\epsilon\to 0}{\bf X}^\epsilon(t)$ one can conclude that $ \frac{d{\bf X}^0}{dt}= constant$. Our main result settles this conjecture under fairly general assumptions on $\bf F$, which may also depend on $t$-variable. Once the above problem is solved, one can apply the result to the transport equation, in a standard way. This is also touched upon in the text to follow.

1. Introduction 1.1. Problem setting. For each i " 1, . . . , d let F i : R d Ñ R be Z d periodic smooth function and ε ą 0 being small parameter. In this paper we consider the first order system of differential equations with oscillating structure Our primary motivation for studying (1) comes from a conjecture posed by Ennio De Giorgi in [6] (Conjecture 1.1 page 175) concerning the homogenization of the transport equation B t u ε px, tq`F px{ε, t{εq¨∇ x u ε px, tq " 0, u ε px, 0q " u 0 pxq @x P R d , with F " pF 1 , F 2 , . . . , F d q Lipschitz continuous and periodic in px, tq. The Lipschitz continuous initial condition u 0 pxq is specified at the initial time t " 0.
He also conjectured that if (2) is homogenizable then the following property must be true, see [6] page 177: Let X ε ptq be the solution of Then there exists (4) lim εÑ0 X ε ptq " X 0 ptq Key words and phrases. dynamical system, ODE, transport, homogenization, convergence rate. H. Shahgholian was supported by Swedish Research Council. A.Karakhanyan was partly supported by EPSRC grant. We thank Michael Benedicks for his insightful comments on the dynamical system issues of the current note, and Björn Engquist for bringing to our attention the paper [8].
for any t, p. Moreover, there is a vector B P R d such that On the other hand Peirone [14] showed that if F does not depend on t then the asymptotic linearity of X ε ptq as t Ñ 8 implies that (2) is homogenizable, see Remark 3.
1.2. Background litteratures. As mentioned above, the homogenisation of (3) is closely related to the homogenisation of the first order transport equations B t u`F∇ u " 0 describing miscible flow in porous media [19]. One of the central questions concerning (2) is the strong convergence which is not true in general as the example of equation (2) with Fpx 1 , x 2 , tq " p0, sin x 1 q, d " 2 shows, see [6] page 176. It is known, for instance, that if div F " 0 (i.e. the case of unit integral density ρ " 1) then the effective equation has arithmetic averages p´T 2 F 1 pxqdx,´T 2 F 2 pxqdxq as the forcing velocity, whereas the shear field Fpxq " aϕpxq, a " p1, γq P R 2 yields harmonic averages, i.e. in the homogenised equation the forcing velocity is a´T 2 dx ϕpxq . The interested reader can find more on this problem in the works [19], [9] and [4] and the references therein.
The homogenization of more general transport equations with divergence free vector fields under the assumption a ε " apx, x{εq and div x apx, yq " div y apx, yq " 0, is studied in [7]. The case when a ε " apx{εq is studied in [8]. It is also shown that solutions of (6) converge in L 2 and the limit equation is either a constant coefficient linear transport equation (ergodic case) or an infinite dimensional dynamical system, see [7,8].
In [18] Tartar studied some transport equations with memory effects. He addressed the question of importance of considering the limit function rather than the equation it satisfies. The question he raised was whether the limit retains, in some sense, the structure of linear transport equations (say, it is traveling wave solution).
Some of this questions were addressed by Tassa in [19], in particular he showed that for shear flow d " 2 the limit is a travelling wave (Theorems 4.2 and 4.5 in [19])as well as some convergence rates which depend on whether the rotation number is rational or irrational and the smoothness of the forcing vector field. In fact for rational rotation number (Theorem 4.5 in [19]) the limit is determined by some function a η see (3.13) in [19] and the limit function is a traveling wave if a η " const for all η P r0, 1s.
It seems plausible that the techniques here can (partially) be applied to more general context involving random structure, i.e. stochastic differential equations. Similar type of problems, have been studied in recent works of Bardi-Cesaroni-Scotti [2].
A further direction, that our approach might be possible to extend to, is that of multi-scale problems. More exactly, one may consider F that has both slow and fast variable Fpx, x{εq. A particular case of this was studied by G. Menon [13], with Fpx, x{εq " divpKpxq`εApx{εqq.

1.3.
A few standing assumptions. The non-oscillating system (i.e. when ε " 1) enjoys a number of remarkable properties. Suppose that (7) has invariant measure dµ x " ρpxqdx with density ρ ą 0; see Section 1.6 for details. For the two dimensional problem, d " 2, Kolmogorov proved that if Fpxq " pF 1 px 1 , x 2 q, F 2 px 1 , x 2 qq " 0 is Z 2 periodic and analytic in px 1 , x 2 q variables, then there is an analytic transformation of coordinates y " fpxq such that (7) transforms into shear flow system (8) dy i dt " a i Gpy 1 , . . . , y d q with constants a 1 " 1, a 2 P R and G being a periodic scalar function.
In fact, is now the integral invariant of the new system of differential equations [10]. The constant γ " a 2 is called the rotation number and the system (7) is ergodic if γ is irrational [17].
The main goal of this article is to analyse the behaviour of the solution X ε ptq (to equation (3)) as ε Ñ 0 under some conditions imposed on the vector-field F " pF 1 , . . . , F d q.
We make the following standing assumptions about the vectorfield F : There is a bounded function ρ ą 0 such that divpρFq " 0. Here ρ is called the density of invariant measure.
1.4. The approach and methodology. De Giorgi's conjecture has (more or less) been ignored completely. Everyone, that has touched upon the homogenization of the transport problem, have ignored the fact that convergence of the underlying dynamical system would give the convergence of the transport problem. Our result (read observation) should be seen in the light of homogenization of the dynamical system, rather than the transport problem; even though this directly implies the convergence of the transport problem. The approach we have taken here is a combination of a few, already worked out, methods (originating in the work of Kolmogorov [10], and later Bogolyubov [3] ). Indeed, amalgamating Kolmogorov's method (and its refinement due to Tassa [19]) and Bogolyubov's method allows one to apply singular perturbation techniques to homogenisation problems for dynamical systems giving convergence rates. This has not been done to the best of our knowledge, and hence worth noticing. Such a composition of hybrid techniques-of combining singular perturbations, dynamical systems and homogenisation-gives new insights and opens up for the study of convergence rates for similar problems.
We also want to stress that although our result seems to be new, it does not use any new technique, and most probably, if the problem were noticed by others, that have worked with the related transport problem, one should have done a similar observation.
1.5. Main results. In order to specify the rates of convergence it is convenient to introduce the nondecreasing function (10) δpεq " sup where G 0 is the average defined by ℓˆℓ 0 Gps, yqds if the limit above exists uniformly in y. Note that if Gpt, xq is periodic in t then the limit in (11) exists and, moreover, Indeed, for periodic Gpsq we have that G 0 psq "´1 0 Gpsqds " const and thuŝ where rℓs is the integer part of ℓ ą 0. Substituting ℓ " s ε the result follows. We formulate our main results below starting off the one dimensional problem. Theorem 1. (d " 1) Let G : TˆR be positive Lipschitz continuous function in both variables pt, xq P TˆR, Gpt,¨q is 1-periodic in t. Let X ε be the solution to initial value problem and δpεq be as in (10). Then there is a Lipschitz continuous function X 0 ptq such that (14) |X ε ptq´X 0 ptq| δpεq, t P r0, T s.
Furthermore, if Gpt, ηq does not depend on t and is almost periodic in η, then X 0 ptq " p 0`β t.
In the proof of Theorem 1 we will use a simple version of Bogolyubov's method, tailored for the Cauchy problem Now we dwell on the multidimensional problem with periodic scalar function G, such that F satisfies (F.1)-(F.3). Assume further a P R n satisfyyies a Diophantine type condition |xa, my| ě C |m| d`κ , @m P Z d zt0u (15) where C and κ are positive constants. If X ε is the solution to Cauchy where the last inequality follows from (12). (b) If d " 2 and F " FpXq is independent of t and Z 2 periodic in X, (F1)-(F3) hold and X ε solves the Cauchy problem then there is a linear function X 0 ptq " p`Bt such that (17) |X ε ptq´X 0 ptq| ε, t P r0, T s.
In the proof of Theorem 2 we shall employ Kolmogorov's theorem on coordinate transformation as in [10], its refinement for d " 2 as in [19] and generalisations for d ě 3 as done [1], [11].
1.6. Invariant measure. Suppose F P C 1 is given and let ρ be sought as the solution of Liouville's equation divpρFq " 0. Let τ " x d , x 1 " px 1 , x 2 , . . . , x d´1 , 0q and assume that F d ą 0 then Liouville's equation can be rewritten as follows . . , F d´1 , 0q. We can specify initial condition at time τ " 0, i.e. x d " 0 and then by [12] (Proposition II.1 and Remark afterwards) there is L 8 solution of this Cauchy problem in R dˆr 0, 8q, provided that F P C 0,1 and so is the initial data at τ " 0.

Proof of Theorem 1
Lemma to follow is the scaled version of one dimensions Bogolyubov's estimate. Lemma 1. Let X ε ptq be the solution of the Cauchy problem dΘ dt ε ptq " Gˆt ε , Θ ε ptq˙, Θ ε p0q " X 0 . Suppose that the following limit exists uniformly in y P D for some compact D Ă R and the Cauchy problem dΘ 0 dt " G 0 pΘ 0 q, Θ 0 p0q " X 0 has a unique solution on the finite interval r0, T 1 s then (18) |Θ ε´Θ0 | ď δpεq, as ε Ñ 0 on 0 ď t ď T and δpεq is given by (10).
Let us denote F pt, uq " 1 Gpu,tq ą 0 then F satisfies (F.1)-(F.3). Let X ε be the solution to the initial value problem dX ε dt " F pt, X ε {εq, X ε p0q " p. Clearly 0 ă dX ε dt ď sup R F ă 8 and therefore tX ε u is uniformly Lipschitz continuous on any finite interval r0, T s. Furthermore, X ε is strictly monotone because F ą 0. Thus X ε has inverse which we denote by h ε , ξ " X ε phpξqq.

3.
Multi-dimensional problem: Proof of Theorem 2 (a) 3.1. Change of variables for d ě 3. Let dµ " ρdx be the invariant measure of the system dY dt " FpYq and introduce the following numbers where F " pF 1 , . . . , F d q is the vectorfield on the right hand side of the equation (7).

Proposition 1. There exists a change of variables transforming
with invariant measure dµ w " dw Gpw1,...,w d q and some constant C " 0. Furthermore, if a " pa 1 , . . . , a d q is diophantine then there exists a transformation into the Employing Kozlov's theorem one can reduce the system dY dt " FpYq to (26).

Proof. We shall use the coordinate transformation introduced in [11] Theorem 2:
if uptq solves the shear system du dt " q Gpuq with diophantine q then the mapping given by the equations G is the mean value of G and f is determined from the first order differential equation x∇f, qy " G´MpGq.
If fact, this mapping is non degenerate (i.e. has nontrivial Jacobian) and one-to-one.
Taking εu " z we see that du dt " q Gpuq with q " a ε . From Fourier's expansion we have G m e 2πixm,uy which by integration gives G m 2πixm, ay e 2πixm,uy , and the series is convergent, due to the assumption that a is Diophantine (see (15)). Notice that the sum is bounded because 1 G satisfies the assumptions pF.1q´pF.3q.
implying a j t`p j´z ε j ptq " Opεq and the proof follows.
Remark 3. Peirone showed that if F P C 1 pT d q is Z d periodic, u 0 P C 1 and the limit lim tÑ8 S t F pxq t exists for a.e. x P T d then the problem (2), is homogenizable, [14] Lemma 2.2 (b). Here S t F is the semigroup generated by (7). Our result establishes the converse of this statement for homogenizable (2).

Proof of Theorem 2(b)
Our goal here is to apply Kolmogorov's coordinate transformation in order to reduce the general problem to shear flow. Tassa [19] found an explicit formula for this that we will write below. We should point out that Kolmogorov's proof in [10] is not constructive.
It is convenient to introduce some basic facts about the equation dX dt " FpXq with F satisfying the properties (F.1)-(F3). Let dµ " ρdx be the invariant measure corresponding to this system, then by definition divpρFq " 0. Thus the vectorfield b " pb 1 , b 2 q " ρF is divergence free. This yields that the integral´1 0 b 1 px 1 , x 2 qdx 2 is constant since Similarly we have that´1 0 b 2 px 1 , x 2 qdx 1 is constant. Denote b 1 "´1 0 b 1 px 1 , x 2 qdx 2 , b 2 "´1 0 b 2 px 1 , x 2 qdx 1 (which are the mean integrals of b 1 , b 2 over T 2 scone they are constants) and set It is shown in [19] that in the new coordinate system we get the shear flow dy dt " a Gpyq with a " p1, γq, where γ is the rotation number, see [17]. Furthermore, we have thatˇˇˇˇB py 1 , y 2 q Bpx 1 , x 2 qˇˇˇˇ" and the invariant measure density is with g " pg 1 , g 2 q being the inverse of f " pf 1 , f 2 q, see [19], page 1395.
In order to take advantage of (28) we introduce the function z ε ptq " X ε ptq{ε.
Then z ε ptq solves the Cauchy problem dz ε dt " Fpz ε q ε , z ε p0q " p ε . Clearly, the invariant measure now is dµ z " 1 ε ρdz and b ε " p b1 ε , b2 ε q is divergence free. Note and therefore applying the change of variables y " fpzq, with mapping f given by (28) we obtain the shear flow In order to get rid of the ε in the denominator we set w ε ptq " εy ε ptq. Then w ε ptq solves the equation By (28) we have that and similarly in view of the periodicity of b. Consequently if e i , 1 ď i ď 2 is the unit vector in the canonical basis of R 2 then this translates to the inverse of f, namely we have g j pη`e i q " g j pηq`M ij , 1 ď i, j ď 2 where M ij P Z, see [19] equation (2.5). This yields that 1 Gpηq " b2pg1pηq,0q b2 By Lemma 2, there is a linear function w 0 such that |w ε ptq´w 0 ptq| δpεq, t P r0, T s. Then |x ε 1 ptq´w 0 1 ptq| ď Opδpεqq`O´ε x ε 1 ptq¯. Finally for x ε 2 we have

Examples
Example 1: Let F be 1-periodic vector filed such that F 2 " 1 and Let y ε ptq be the solution to the following initial value problem Consider the ε 2 ?
2 -strip of the line p`sp2, 1q, i.e. S ε " tx P R n : |x´rp`sp2, 1qs| ď ε 2 ? 2 , s ě 0u. Thus as ε Ñ 0 the trajectory (i.e. the set of points) converges to the line ℓpsq " p`sp2, 1q, s ě 0 in Hausdorff distance. Hence the trajectory of the limit is the line ℓpsq. As for the speed of the convergence, we note first that by definition y and after differentiationˇˇˇˇd z 0 ptq dtˇˇˇˇ" b 1 2 . The astute reader has probably noticed that we did not use condition F.3 here, but could still obtain a convergence rate. This is due to the one-dimensional character of the problem, since F 2 " 1 here.
Example 2: Another example is given by F with saw-like graph  Figure 1. In this example a " 1, h " 3 periodically extended over R, see Figure 1. Here a ą 0 is the periodicity of F and h " max F is the pick of F . We can solve this equations explicitly: indeed we have that dy ε dt " " 2h a p y ε ε´k aq`σ if y ε ε P ak`r0, a 2 q 2h a papk`1q´y ε ε q`σ if y ε ε P ak`r a 2 , aq After integration one gets if y ε P εak`r0, aε 2 q C`pkqe´2 ht εa`ε pk`1qa´a σε 2h if y ε P εak`r aε 2 , aεq with some constants C˘pkq and k P Z. Clearly this solution y ε is monotone and hence the argument above works. Obviously 1 a´a 0 dτ F pτ q " a h log`h`σ σ˘" β and therefore we infer that y ε converges uniformly to y 0 ptq " p`t β on any finite closed interval r0, T s. Example 3: Theorem 1 is still valid if the periodicity of Gpt,¨q is replaced with almost periodicity in t because all we needed in the proof was the convergence rate for G 0 . In this case one may get weaker error estimates, see [20] Example 11.13. Indeed, the function F pxq " ř 8 k"0 1 p2k`1q 2 sin´x 2k`1¯i s almost periodic. By direct computationˆT 0 F pxqdx "