Weak convergence and averaging for ODE

Nonlinear Analysis

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Nonlinear Analysis www.elsevier.com/locate/na

Weak convergence and averaging for ODE Lawrence C. Evans ∗,1 , Te Zhang Department of Mathematics, University of California, Berkeley, United States

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Article history: Received 3 June 2015 Accepted 8 October 2015 Communicated by Enzo Mitidieri For Juan Luis Vazquez on his 70th birthday Keywords: Weak convergence Stabilization Adiabatic invariance

abstract This mostly expository paper shows how weak convergence methods provide simple, elegant proofs of (i) the stabilization of an inverted pendulum under fast vertical oscillations, (ii) the existence of particle trapsinduced by rapidly varying electric fields and (iii) the adiabatic invariance of p dx for slowing varying planar Γ Hamiltonian dynamics. Under an appropriate, but very restrictive, unique ergodicity assumption, the proof of (iii) extends also to many degrees of freedom. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction The rigorous mathematical analysis of nonlinear differential equations depends primarily upon deriving estimates, but typically also upon using these estimates to justify limiting procedures of various sorts. For the latter, so-called weak convergence methods can be extremely valuable, as illustrated by many examples in the booklet [7]. This paper provides some more examples, concerning averaging effects for singularly perturbed nonlinear ODE. Section 2 shows how some simple “nonlinear resonance” effects (occurring when the weak limit of the product of two sequences of functions is not the product of the individual weak limits) appear for Kapitsa’s inverted pendulum and its generalizations. Section 3 invokes the more sophisticated tools of Young measures to document the adiabatic invariance of the volume within constant energy surfaces for slowly changing Hamiltonian systems, provided an appropriate ergodic type condition holds. Our proofs are perhaps new, at least in the elegant versions we provide, and our presentation is largely expository. We wish also to call attention to Bornemann’s book [3], a very interesting discussion of weak convergence methods applied to singularly perturbed mechanical and quantum systems. His primary interest is explaining how increasingly singular potentials enforce holonomic constraints in the limit. ∗ Corresponding author. E-mail address: [email protected] (L.C. Evans). 1 Class of 1961 Collegium Chair.

http://dx.doi.org/10.1016/j.na.2015.10.011 0362-546X/© 2015 Elsevier Ltd. All rights reserved.

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The results in Section 2 appear in somewhat different form in the second author’s 2014 PhD thesis from UC Berkeley. We thank M. Zworski for explaining to us about ergodicity for Hamiltonian systems. 2. Averaging and stability 2.1. The inverted pendulum The equation of motion for an inverted pendulum over a vertically oscillating pivot is   b t ′′ θϵ − a + cos sin θϵ = 0, ϵ ϵ

(2.1)

where θϵ = θϵ (t) denotes the angle from the vertical and a := gl > 0, l denoting the length. This is Kapitsa’s pendulum: see for example Landau–Lifshitz [8, Section 30], Arnold [1, Section 25.E] and Levi [9]. 2 We provide a simple proof that solutions of (2.1) converge as ϵ → 0 to solutions of θ′′ + b4 sin 2θ−a sin θ = 0. 2 This ODE has the form θtt + F ′ (θ) = 0, for which the solution θ ≡ 0 is stable provided F ′′ (0) = b2 − a > 0; √ that is, if and only if |b| ≥ 2a. This is the well-known stability condition for the inverted pendulum in the high frequency limit. We turn now to a rigorous proof. Consider the following initial-value problem:    b t ′′    θϵ = a + ϵ cos ϵ sin θϵ (t ≥ 0) (2.2) θϵ (0) = α    ′ θϵ (0) = β.

Theorem 2.1. As ϵ → 0, θϵ converges uniformly on each finite time interval [0, T ] to the solution θ of  b2   sin 2θ  θ′′ = a sin θ − 4 (2.3) θ(0) = α    ′ θ (0) = β. The main idea will be to rewrite the ODE (2.2) into the form 

θϵ′

t − b sin sin θϵ ϵ

′

t = a sin θϵ − b sin cos θϵ θϵ′ . ϵ

(2.4)

Proof. 1. First we show that for each T > 0, we have the estimate max |θϵ |, |θϵ′ | ≤ CT ,

0≤t≤T

(2.5)

for a constant CT > 0 that only depends on T , α and β. To confirm this, integrate (2.4), to find  t ′ |θϵ (t)| ≤ C1 + C2 |θϵ′ | ds 0

for 0 ≤ t ≤ T and constants C1 , C2 ≥ 0. According then to Gronwall’s inequality, we have the estimate   |θϵ′ (t)| ≤ C1 1 + C2 teC1 t ≤ CT for each 0 ≤ t ≤ T and a constant CT > 0 that only depends on T .

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2. Using (2.5), we can find a subsequence ϵj → 0 such that  θϵ j → θ uniformly on [0, T ] θϵ′ j ⇀ θ′ weakly in L2 (0, T ). We next claim that θϵ′ j sin

t b ⇀ sin θ ϵj 2

weakly in L2 (0, T )

(2.6)

as ϵ → 0. To see this, observe that for all ψ ∈ C ∞ ([0, T ]) vanishing near t = 0, T , we have ′  T  T  t t ψ sin θϵ′ dt = −ϵ ψ cos θϵ′ dt ϵ ϵ 0 0  T  T t t ′ ′ ψ cos θϵ′′ dt =ϵ ψ cos θϵ dt + ϵ ϵ ϵ 0 0    T t t = O(ϵ) + ϵa + b cos sin θϵ dt ψ cos ϵ ϵ 0  T t = O(ϵ) + bψ cos2 sin θϵ dt ϵ 0  t b → sin θψ dt 2 0 as ϵ = ϵj → 0, since cos2

t ϵ

⇀ 12 . This proves (2.6).

3. Now integrate (2.4): t θϵ′ (t) − b sin sin θϵ (t) = β + ϵ



t

a sin θϵ − b sin 0

s cos θϵ θϵ′ ds ϵ

for 0 < t < T . Let ϵ = ϵj → 0 and pass to weak limits, recalling (2.6):  t  t b2 b2 θ′ (t) = β + a sin θ − a sin θ − cos θ sin θ ds = β + sin 2θ ds. 2 4 0 0 The function θ is therefore smooth on [0, T ] and solves the ODE θ′′ = a sin θ − Since θϵj → θ locally uniformly, θ(0) = α as well.

b2 4

sin 2θ, with θ′ (0) = β.

Since the initial value problem (2.3) has a unique solution θ, we see that in fact the full sequence {θϵ }ϵ>0 converges: θϵ → θ locally uniformly.  2.2. Generalization We next generalize to the system of ODE 1 g ϵ xϵ (0) = α    ′ xϵ (0) = β.    

xϵ′′ =

  t f (xϵ ) ϵ

Here xϵ = xϵ (t) = (x1ϵ (t), . . . , xnϵ (t)) and f : Rn → Rn is a smooth function with sup |Df | < ∞. Rn

(2.7)

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We assume g : R → R is continuous, 1-periodic, and   1 g(t) dt = t 0

)

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1

tg(t) dt = 0.

(2.8)

0

0

Define G(t) :=

(

g(s) ds; then G is 1-periodic, 

1

G(0) = G(1) = 0,

G dt = 0. 0

Write 2



⟨G ⟩ :=

1

G2 dt.

0

Theorem 2.2. As ϵ → 0, we have xϵ → x uniformly on each finite time interval [0, T ], where x is the unique solution of  ′′ 2   x = −⟨G ⟩Df (x)f (x) (2.9) x(0) = α  x′ (0) = β. Remark. In the conservative case that f = Dφ for a scalar potential function φ, the limit dynamics read x′′ = −Dψ(x) for the new potential function ⟨G2 ⟩ |Dφ|2 . 2 We consequently have local stability near any nondegenerate critical point of φ. As explained by M. Levi in [10], this is the principle behind the “Paul trap” in physics. ψ :=

Proof. 1. Rewrite the ODE (2.7) as    ′   t t f (xϵ ) = −G Df (xϵ )xϵ′ . xϵ′ − G ϵ ϵ

(2.10)

Integrating and noting |f (z)| ≤ C + C|z|, we see that |xϵ′ (t)| ≤ C + C|xϵ (t)| + C



t

|xϵ′ (s)| ds

0

for all t ≥ 0. Since |xϵ (t)| ≤ C +

t 0

|xϵ′ (s)| ds, it follows that  t ′ |xϵ (t)| ≤ C + C |xϵ′ (s)| ds 0

for appropriate constants C. Gronwall’s inequality therefore implies that for each T > 0, we have the estimate max |xϵ (t)|, |xϵ′ (t)| ≤ CT ,

0≤t≤T

for a constant CT > 0 that only depends on T , α and β. 2. Hence for some sequence ϵj → 0, xϵj → x

uniformly on [0, T ],

xϵ′ j ⇀ x′

weakly in L2 (0, T ; Rn ).

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We claim that  G



t ϵj

xϵ′ j ⇀ ⟨G2 ⟩f (x)

weakly in L2 (0, T ; Rn ).

(2.11)

To see this, select any ψ ∈ C ∞ ([0, T ]) vanishing near t = 0, T , and observe    ′  T  T t t ′ xϵ′ dt ψG xϵ dt = ϵ ψΓ ϵ ϵ 0 0 t 1 where Γ (t) := 0 G(s) ds. Since 0 G dt = 0, the function Γ is 1-periodic. Therefore      T  T t t ψΓ ψG xϵ′ dt = O(ϵ) − ϵ xϵ′′ dt ϵ ϵ 0 0      T t t = O(ϵ) − ψΓ g f (xϵ ) dt ϵ ϵ 0  T → −⟨Γ g⟩ f (x)ψ dt 0

as ϵ = ϵj → 0, where 

1

⟨Γ g⟩ :=

 Γ g dt =

0

1

Γ G′ dt = −

0



1

Γ ′ G dt = −⟨G2 ⟩.

0

This proves (2.11). 3. Now integrate (2.10): xϵ′ (t) − G

   t   s t f (xϵ (t)) = β − G Df (xϵ )xϵ′ ds, ϵ ϵ 0

and use (2.11) to pass to weak limits as ϵ = ϵj → 0: x′ (t) = β − ⟨G2 ⟩



t

Df (x)f (x) ds. 0

1 Notice that G( ϵt ) ⇀ 0 G dt = 0. It follows that x is smooth when f is, and solves the ODE and second initial condition in (2.9). The first initial condition is also clear, since xϵj → x locally uniformly. As (2.9) has a unique solution, in fact the full sequence converges as ϵ → 0.



3. Averaging and adiabatic invariance 3.1. Slowly varying Hamiltonians Let H : Rn × Rn × R → R, H = H(p, x, t), be a smooth, time-dependent family of Hamiltonians. Fix T > 0 and consider then the system of ODE    T x˙ ϵ = Dp H(pϵ , xϵ , ϵτ ) , (3.1) 0≤τ ≤ ϵ p˙ ϵ = −Dx H(pϵ , xϵ , ϵτ ) d where ˙ = dτ , with given initial conditions xϵ (0) = x0 , pϵ (0) = p0 . In these dynamics the Hamiltonians are varying slowly, but for a long time. An adiabatic invariant for (3.1) is some quantity involving the trajectory (p, x) that is approximately constant for times 0 ≤ τ ≤ Tϵ . Consult Arnold [1] and Arnold–Kozlov–Neishtadt [2] for the theory of adiabatic invariants and Crawford [5] for many examples.

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For n = 1 degrees of freedom, it is standard wisdom in physics that the action  Φ= p dx

(3.2)

Γ

is an adiabatic invariant, where the integral is over a complete cycle Γ of the motion. Arnold and others have given a rigorous interpretation and derivation of this assertion. We provide next a proof using weak convergence tricks, valid even for more degrees of freedom if the Hamiltonian dynamics are appropriately uniquely ergodic on each energy surface. We do not use action–angle variables. 3.2. Rescaling, weak convergence We hereafter assume that H ≥ 0, lim

H = ∞ uniformly on [0, T ],

(3.3)

on Rn × Rn × [0, T ]

(3.4)

(p,x)→∞

and |Ht | ≤ C(1 + H) for some constant C. We now rescale in time, setting xϵ (t) := xϵ

t := ϵτ,

  t , ϵ

pϵ (t) := pϵ

  t . ϵ

Then  1   xϵ′ = Dp H(pϵ , xϵ , t) ϵ  ′ p = − 1 Dx H(pϵ , xϵ , t) ϵ ϵ where

′

=

d dt .

(0 ≤ t ≤ T ),

(3.5)

Equivalently, we write

1 JDz H(zϵ , t) (0 ≤ t ≤ T ) ϵ   for zϵ := (pϵ , xϵ ), Dz H = (Dp H, Dx H), and J := OI −I . O The energy at time t is z′ϵ =

(3.6)

eϵ := H(pϵ , xϵ , t). Lemma 3.1. (i) We have e′ϵ = Ht (pϵ , xϵ , t)

(0 ≤ t ≤ T ).

(3.7)

(ii) There exists a constant C such that sup |pϵ |, |xϵ | ≤ C

(3.8)

0≤t≤T

for each 0 < ϵ ≤ 1. (iii) There exist a sequence ϵj → 0 and a continuous function e such that eϵj → e uniformly on [0, T ].

(3.9)

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Proof. Calculating (3.7) is immediate from (3.5). It follows then from hypothesis (3.4) that e′ϵ ≤ C + Ceϵ ; whence Gronwall’s inequality implies sup0≤t≤T eϵ ≤ C. The estimate (3.8) is now a consequence of the coercivity assumption (3.3). Finally, (3.7) and (3.8) imply (3.9) for an appropriate subsequence.  Notation. We write for 0 ≤ t ≤ T Γ (t) := {(p, x) | H(p, x, t) = e(t)},

Γϵ (t) := {(p, x) | H(p, x, t) = eϵ (t)}

∆(t) := {(p, x) | H(p, x, t) ≤ e(t)},

∆ϵ (t) := {(p, x) | H(p, x, t) ≤ eϵ (t)}.

We hereafter assume also that |Dz H| ≥ γ > 0

on Γ (t), Γϵ (t)

(0 ≤ t ≤ T ).

Consequently, Γ (t), Γϵ (t) are smooth hypersurfaces, with outward unit normal ν := as well that Γ (t), Γϵ (t) are connected. Then for each continuous function F ,   F dH2n−1 → F dH2n−1

Dz H |Dz H| ;

and we suppose

(3.10)

Γ (t)

Γϵj (t)

uniformly on [0, T ], where H2n−1 denotes Hausdorff measure. Lemma 3.2. Passing if necessary to a further subsequence and reindexing, we have for almost every time 0 ≤ t ≤ T a Borel probability measure σ(t) on Rn × Rn such that spt σ(t) ⊆ Γ (t),

(3.11)

{H(·, t), σ(t)} = div(JDz H(·, t)σ(t)) = 0

(3.12)

weakly in Rn × Rn , and F (pϵj , xϵj , t) ⇀ F¯ :=

 F (p, x, t) dσ(t)

(3.13)

Γ (t)

for each continuous function F . Proof. 1. The existence of a (possibly further) subsequence ϵj → 0 and Young measures σ(t) such that  F (pϵj , xϵj , t) ⇀ F (p, x, t) dσ(t) (3.14) Rn ×Rn

for continuous functions F follows as in Tartar [11] or [7]. The assertion (3.11) follows from (3.10), since (pϵj , xϵj ) ∈ Γϵj (t). Consequently (3.13) holds. 2. To prove (3.12), let φ = φ(p, x, t) be smooth, with compact support in Rn × Rn × (0, T ). Put F = JDH · Dφ. Then, since φ vanishes at t = 0, T , we have  T O(ϵ) = −ϵ φt (zϵ , t) dt  =ϵ 

0 T

φ(zϵ , t)′ − φt (zϵ , t) dt

0 T

JDz H · Dφ(zϵ , t) dt.

= 0

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Therefore (3.13) implies  ϵj →0

T



T



JDz H · Dφ(zϵj , t) dt =

0 = lim

0

JDz H · Dφ dσ(t) dt. 0

Rn ×Rn

The validity of this identity for each φ is the weak formulation of (3.12).



We introduce next Liouville measure on Γ (t), defined for Borel sets E by the rule  1 1 µ(t)(E) := dH2n−1 Z(t) Γ (t)∩E |Dz H|  and normalized by Z(t) := Γ (t) |Dz1H| dH2n−1 .

(3.15)

Lemma 3.3. For each time 0 ≤ t ≤ T we have {H(·, t), µ(t)} = 0

(3.16)

weakly in Rn × Rn . Proof. Let φ = φ(p, x) be smooth, with compact support. Then   1 ν · J T Dφ dH2n−1 JDz H · Dφ dµ(t) = Z(t) Γ (t) Rn ×Rn  1 = div(J T Dφ) dz Z(t) ∆(t) = 0, since J T is antisymmetric. Therefore {H(·, t), µ(t)} = div(JDz H(·, t)µ(t)) = 0 in the weak sense.



Now define Φ(t) := |∆(t)|,

Φϵ (t) := |∆ϵ (t)| (0 ≤ t ≤ T )

(3.17)

to be the 2n-dimensional volumes of ∆(t), ∆ϵ (t). Theorem 3.4. Assume for each time 0 ≤ t ≤ T that the Liouville measure µ(t) is the unique Borel probability measure µ supported on Γ (t) solving {H(·, t), µ} = 0

(3.18)

Φϵ′ ⇀ 0

(3.19)

Φ is constant on [0, T ].

(3.20)

weakly in Rn × Rn . Then

as ϵ → 0, and consequently

Remark. The hypothesis that (3.18) has a unique solution supported in Γ (t) is called unique ergodicity and is extremely strong for n > 1. A heuristic derivation of a special case of this assertion, but without the uniqueness hypothesis for (3.18), is in Appendix D of Campisi–Kobe [4], who discuss also the interpretation of S = kB log Φ as the

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microcanonical Gibbs entropy of the classical Hamiltonian system. See also Dunkel–Hilbert [6] for more discussion; they credit Hertz with the observation that Φ, and thus S, are adiabatic invariants. Bornemann [3] uses weak convergence methods to derive quantum adiabatic theorems. Proof. The hypersurface Γϵ (t) is the 0 level set of the function W ϵ = W ϵ (z, t) := H(z, t) − eϵ (t), whose Wtϵ outward normal velocity is therefore − |Dz W ϵ | . Thus  Wtϵ dH2n−1 Φϵ (t)′ = − ϵ Γϵ (t) |Dz W |  e′ϵ (t) − Ht (p, x, t) = dH2n−1 |Dz H| Γϵ (t)  Ht (pϵ , xϵ , t) − Ht (p, x, t) = dH2n−1 , |Dz H| Γϵ (t) according to (3.7). But owing to (3.11)–(3.13) and the assumed uniqueness of probability measures solving (3.18), it follows that the Young measure σ(t) equals the normalized Liouville measure for a.e. time. Thus  1 Ht Ht (pϵj , xϵj , t) ⇀ dH2n−1 . Z(t) Γ (t) |Dz H|     Since Γϵ (t) |DHz tH| dH2n−1 → Γ (t) |DHz tH| dH2n−1 and Γϵ (t) |Dz1H| dH2n−1 → Γ (t) |Dz1H| dH2n−1 = Z(t) j

j

uniformly on [0, T ], we have Φϵ′ j ⇀ 0.

This assertion holds as well for an appropriate subsequence of any given sequence ϵk → 0, and consequently Φϵ′ ⇀ 0.  3.3. One degree of freedom For one degree of freedom, the unique ergodicity hypothesis holds automatically, since the level sets of H(·, t) are diffeomorphic to circles: Theorem 3.5. If n = 1, then Φϵ′ ⇀ 0

(3.21)

Φ is constant on [0, T ].

(3.22)

as ϵ → 0, and consequently

 Since Green’s Theorem implies Γ (t) p dx = ±Φ(t) (depending upon the orientation), we recover the classical assertion about adiabatic invariance for one degree of freedom. Proof. Suppose µ is a Borel probability measure supported in Γ = Γ (t) and satisfying {H, µ} = 0 for H = H(·, t). Then  Jν · Dφ|Dz H| dµ = 0 Γ

for each smooth φ.

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Let {w(s) = (p(s), x(s)) | 0 ≤ s ≤ L} be a unit speed parameterization of Γ , oriented so that w′ = τ := Jν d and ′ = ds . Then   L 0= Jν · Dφ|Dz H| dµ = ψ ′ d˜ ν, Γ

0 −1

where ν˜ is the pushforward of ν = |Dz H|µ under w and ψ = φ(w). The foregoing identity for each ψ satisfying ψ(0) = ψ(L) implies that ν˜ is a constant multiple of one-dimensional Lebesgue measure. This shows that ν = H1 on Γ , times an appropriate normalizing constant. Hence µ is the Liouville measure.  Acknowledgment The first author was supported in part by NSF Grant DMS-1301661. References [1] V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer, 1978, trans by K. Vogtmann and A. Weinstein. [2] V.I. Arnold, V.V. Kozlov, A.I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, in: Encyclopedia of Mathematical Sciences, vol. 3, Springer, 1988, trans by A. Iacob. [3] F. Bornemann, Homogenization in Time for Singularly Perturbed Mechanical Systems, in: Lecture Notes in Mathematics, vol. 1687, Springer, 1998. [4] M. Campisi, D.H. Kobe, Derivation of the Boltzmann principle, Amer. J. Phys. 78 (2010) 608–615. [5] F. Crawford, Elementary examples of adiabatic invariance, Amer. J. Phys. 58 (1990) 337–344. [6] J. Dunkel, S. Hilbert, Consistent thermostatics forbids negative temperatures, Nat. Phys. 10 (2014) 67–72. [7] L.C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, in: American Math Society, CBMS, vol. 74, American Mathematical Society, 1990, Third printing, 2002. [8] L.D. Landau, E.M. Lifshitz, Mechanics, third ed., in: Course in Theoretical Physics, vol. 1, Pergammon, 1976, trans by J.B. Sykes and J.S. Bell. [9] M. Levi, Geometry of Kapitsa’s potentials, Nonlinearity 11 (1998) 1365–1368. [10] M. Levi, Geometry and physics of averaging with applications, Physica D 132 (1999) 150–164. [11] L. Tartar, Compensated compactness and applications to partial differential equations, in: Nonlinear Analysis and Mechanics, Heriot-Watt Symposium, Vol. IV, Pitman, 1979, pp. 136–212.