On 2{d Euler equations with partial damping and some...

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On 2–d Euler equations with partial damping and some related model problems. Wenqing Hu. 1 1. Department of Mathematics and Statistics, Missouri S&T.

Transcript of On 2{d Euler equations with partial damping and some...

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On 2–d Euler equations with partialdamping and some related model problems.

Wenqing Hu. 1

1. Department of Mathematics and Statistics, Missouri S&T.

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The 2–d Navier–Stokes equation on a torus.

I ut + u∇u +∇p − ν∆u = f ,

divu = 0 ,u|t=0 = u0 .

I u = u(x , t) = (u1(x , t), u2(x , t)),

I p = p(x , t),

I x ∈ T2 = R2/2πZ2.

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The 2–d Navier–Stokes equation on a torus.

I Solution u(x , t) can be written as a Fourier series

u(x , t) =1

∑k∈Z2

u(k , t)e ikx .

I The Laplace operator ∆ is a “damping” term if we view itfrom the Fourier space

(∆u)(k) = −|k |2u(k) for all k ∈ Z2 .

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Modification of the 2–d Navier–Stokes equation on a torus.

I Introduce the “partial damping” operator Y :

(Yu)(k) =

{−|k |2u(k) , k 6∈ K ,0 , k ∈ K .

I We will assume that K is symmetric (i.e. invariant underk → −k) to keep the solution real–valued.

I Ultimate goal is to studyut + u∇u +∇p − νYu = f ,

divu = 0 ,u|t=0 = u0 .

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Background.

I Canonical (conjectural) picture due to Kraichnan 2 : theenergy and enstrophy cascades which spread the excitations toother Fourier modes through the nonlinearity.

I Nonlinear interactions should tend to distribute energyuniformly between all degrees of freedom, and hence a systemfor which some of the degrees of freedom are forced whilesome other degrees of freedom are damped should still reachsome kind of dynamical equilibrium. 3

I Kraichnan’s downward cascade of energy : if in the caseK = {k0,−k0} where k0 is one of the lowest non–trivialfrequencies, the solution will presumably not stay boundedgenerically, even if the forcing acts “far away” in the Fourierspace.

2. Kraichnan, R.H., Inertial ranges in two dimensional turbulence, Physics ofFluids, 10(7), pp. 1417–1423, 1967.

3. At least if there is enough interaction between the damped and forcedparts of the system.

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Background.

I The simplest way to establish “dynamical equilibrium” is toassume that f is random, and validate the fact that thesystem admits a unique invariant measure. 4

I “Asymptotic Strong Feller” property.

I It does not work in the case of partial damping.

4. Hairer, M., Mattingly, J.C., Ergodicity of the 2D Navier–Stokes equationswith degenerate stochastic forcing, Annals of Mathematics, (2) 164 (2006), no.3., pp. 993–1032.

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Our problem.

I We study the modest caseut + u∇u +∇p − νYu = 0 ,

divu = 0 ,u|t=0 = u0 .

I Case 1 : K is finite (remove damping from finitely manymodes) ;

I Case 2 : K is co–finite (i.e. Z2\K is finite, leave damping atfinitely many modes) ;

I Case 3 : Both K and Z2\K are infinite.

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General Result.

I Write

(Zu)(k) =

{−|k|2u(k) , k ∈ K ,0 , k 6∈ K .

I Y + Z = ∆.

I “Regularization”.

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General Result.

I Theorem. Let K be any symmetric subset of Z2. For eachdivergence–free vector field u0 ∈ L2 on the torus T2 theinitial–value problem

(uε)t + uε∇uε +∇pε − νYuε − εZuε = 0 ,divuε = 0 ,

uε|t=0 = u0

has a unique solution uε ∈ C ([0,∞), L2x) ∩ L2

t H2x . The solution

uε is smooth in T2 × (0,∞) and satisfies the energy identity∫T2

1

2|uε(x , t)|2dx +

∫ t

0

∫T2

(−ν(Yuε)uε − ε(Zuε)uε)dxdt ′

=

∫T2

1

2|u0(x)|2dx

for each t ≥ 0.

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General Result.

I Theorem. (continued) Moreover, if ω0 = curlu0 ∈ L2, thenuε ∈ L2

t H2x and satisfies∫

T2

1

2ω2ε(x , t)dx +

∫ t

0

∫T2

(−ν(Yuε)uε − ε(Zuε)uε)dxdt ′

=

∫T2

1

2ω2

0(x)dx

for each t ≥ 0.

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General Result.

I Weak Solution.

u ∈ C ([0,∞), L2) ∩ L∞t H1x ∩ {v ,Yv ∈ L2

t L2x} .

I Corollary. For any symmetric K ⊂ Z2 and any initial datumu0 ∈ H1 our problem has at least one weak solution.

I Leray–Hopf argument.

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Case 1. Finitely many undamped frequencies.

I Let κ = maxk∈K|k|.

I ‖∇v‖2 ≤ −(Yv , v) + κ2‖v‖2L2x.

I Uniform control of the solutions uε in L∞t L2x ∩ L2

t H1x .

I Standard embedding argument gives existence and uniquenessof solutions to the initial–value problem with u0 ∈ L2 in theclass C ([0,∞), L2) ∩ L2

t H1x .

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Case 1. Finitely many undamped frequencies.

I Long–time behavior :

EK ,E ,I =

v : T2 → R2,

v solves the 2d incompressibleEuler equation and, in addition

v(k) = 0for each k 6∈ K ,∫T2 |v |2 = 2E ,

∫T2 |curlv |2 = I .

.

I Krasovskii–LaSalle principle.

I Question : What is the structure of the solution of 2–d Euleron T2 that is supported on finitely many Fourier modes ?

I Answer : Independent of time and is supported either on a linepassing through the origin or on a circle centered at theorigin. (Will come back to this later.)

I So in the case finitely many undamped modes solutionsconverge to steady state !

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Case 2. Finitely many damped frequencies.

I Vorticity formulation : ωt + (u · ∇)ω = νYω .

I Bound ‖Yω‖L∞ ≤ cK‖ω‖L2 due to the finiteness of Z2\K .

I ‖ω(t)‖L2 is not increasing.

I Thusd

dt‖ω(t)‖L∞ ≤ cK‖ω0‖L2 .

I We arrive at the a–priori estimate

‖ω(t)‖L∞ ≤ ‖ω0‖L∞ + cK‖ω0‖L2t , t > 0 .

I If ω0 ∈ L∞, uniqueness follows from Yudovich theory.

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The structure of Euler solution supported on finitely manyFourier modes.

I Question : What is the structure of the solution of 2–d Euleron T2 that is supported on finitely many Fourier modes ?

I Answer : Independent of time and is supported either on a linepassing through the origin or on a circle centered at the origin.

I It must be steady state !

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The structure of Euler solution supported on finitely manyFourier modes.

I 2–d Euler can be written in Fourier components as

d

dtω(m, t) = − 1

∑k+l=m

(k1l2−k2l1)

(1

|k |2− 1

|l |2

)ω(k, t)ω(l , t) .

I Suppose the support is a finite set of Fourier modes S .

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The structure of Euler solution supported on finitely manyFourier modes.

I An (unordered) pair {k , l} of two distinct points k = (k1, k2),l = (l1, l2) ∈ Z2\{(0, 0)} is called degenerate if either k , l lieon the same circle centered at the origin (i.e.k2

1 + k22 = l2

1 + l22 ), or k, l lie on the same line passing through

the origin (i. e. k1l2 − k2l1 = 0). In the former case we call thepair to be c–degenerate, and in the latter case we call the pairto be l–degenerate. A pair which is not degenerate is callednon–degenerate.

I If the set S is not degenerate, then there exists a non–zeroelement m ∈ Z2\S such that m = k + l for exactly onenon–degenerate (unordered) pair {k , l} ∈ S .

I S “wakes up” those modes that are not in S .

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Figure: Proof that S is symmetric w.r.t. O (if not a line).

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Figure: Proof that S is degenerate.

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Reference.

Elgindi, T., Hu, W., Sverak, V., On 2d incompressible Eulerequations with partial damping. Communications inMathematical Physics, 355, Issue 1, October 2017, pp.145-159.

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Model problem : AB–model.

I We consider here a model problem{dxt = −xtytdt ,dyt = x2

t dt .

I A phase picture is shown in the next Figure.

I We see that the whole line OyA contains stable equilibriumsand the whole line OyB contains unstable equilibriums. This isdifferent from the cases considered in the classicalFreidlin-Wentzell theory.

I In this case we can understand the symmetry of our model ina more rough way : the stable and unstable equilibriums aresymmetric with respect to shifts in the directions of OyA andOyB , respectively.

I Our model preserves the energy E (x , y) = x2 + y 2. Thedriving vector field b(x , y) = (−xy , x2) is degenerate onx = 0.

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Figure: The AB model.

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Randomly perturbed AB–model.

I We add friction and random perturbation to this system{dX εt = −X εt Yεt dt − εX εt dt +

√εdW 1

t , X ε0 = x0 ,dYεt = (X εt )2dt − εYεt dt +

√εdW 2

t , Yε0 = y0 .

I Here W 1t and W 2

t are two independent standard1–dimensional Brownian motions ;

I The small parameter ε > 0 is the intensity of the friction, andthe small parameter

√ε > 0 is the intensity of the noise.

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Randomly perturbed AB–model.

I Question : What is the long–time behavior of the system(X εt ,Yεt ) as t →∞ and ε ↓ 0 ?

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Randomly perturbed AB–model : Background.

I Our model problem here differs from the set–up in the classicalFreidlin–Wentzell theory in that the point–like asymptoticallystable attractor is replaced by a manifold. We can view ourlimiting process as a “process–level attractor” of our system.

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Randomly perturbed AB–model : Background.

I Finite dimensional models for the inviscid stochastic 2–dNavier–Stokes equations of the form

∂ω

∂t+u·∇ω−ν∆ω =

√νξ(t, x) , u = ∇>∆−1ω , ω(x , 0) = ω0(x) ,

in which ξ(t, x) is a noise, and positive parameter ν → 0.

I Invariant measure for Euler equations.

I Our consideration of the unperturbed system mimics theattractor for the 2–d Euler system, that has continuous sets ofsteady states.

I In fact, systems that arise in hydrodynamics, such as in thecontext of Euler’s equation, typically possess equilibriumpoints that belong to an infinite dimensional “manifold” ofother equilibria.

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Limit as ε→ 0 of the perturbed AB–model.

I We come back to our system{dX εt = −X εt Yεt dt − εX εt dt +

√εdW 1

t , X ε0 = x0 ,dYεt = (X εt )2dt − εYεt dt +

√εdW 2

t , Yε0 = y0 .

I We do a time rescaling t → t

εand we let

(X εt ,Y

εt ) = (X εt/ε,Y

εt/ε) .

I ThendX ε

t = −1

εX εt Y ε

t dt − X εt dt + dW 1

t , X ε0 = x0 ,

dY εt =

1

ε(X ε

t )2dt − Y εt dt + dW 2

t , Y ε0 = y0 .

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Limit as ε→ 0 of the perturbed AB–model.

I System (X εt ,Y

εt ) :

dX εt = −1

εX εt Y ε

t dt − X εt dt + dW 1

t , X ε0 = x0 ,

dY εt =

1

ε(X ε

t )2dt − Y εt dt + dW 2

t , Y ε0 = y0 .

I Separation of slow and fast motions.

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Limit as ε→ 0 of the perturbed AB–model.

I In the limit as ε ↓ 0, the process (X εt ,Y

εt ) is pushed by the

flow onto OyA, and will be close to π(x0, y0) in short time.

I There, the Y –component Y εt behaves as a 2–dimensional

linearly damped radial Bessel process (damped–BES(2)) onOyA :

dYt =

(1

2Yt− Yt

)dt + dW 2

t ,Y0 = yπ(x0, y0) .

I What is the heuristic ?

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Figure: The AB model.

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Limit as ε→ 0 of the perturbed AB–model.

I The radial process r εt =√

(X εt )2 + (Y ε

t )2.

I By applying Ito’s formula we see that

dr εt =

(1

2r εt− r εt

)dt + dW r

t , r ε0 =√

(X ε0 )2 + (Y ε

0 )2

I As ε ↓ 0 the process X εt is pushed by the fast flow to be close

to 0 when Y εt ≥ δ > 0.

I δ = ε1/10.

I Near the OyA axis we have r εt ≈ Y εt as ε ↓ 0.

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Limit as ε→ 0 of the perturbed AB–model.

I What happened when Y εt < δ ?

I The above comparison with the radial process will not work.

I If (X εt ,Y

εt ) is close to the origin O = (0, 0), we look at our

systemdX ε

t = −1

εX εt Y ε

t dt − X εt dt + dW 1

t , X ε0 = x0 ,

dY εt =

1

ε(X ε

t )2dt − Y εt dt + dW 2

t , Y ε0 = y0 .

I In the limit as ε ↓ 0 the positive drift1

ε(X ε

t )2 → 1

2Yt.

I Recall that the damped BES(2) process takes the form

dYt =

(1

2Yt− Yt

)dt + dW 2

t ,Y0 = yπ(x0, y0) .

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Limit as ε→ 0 of the perturbed AB–model.

I This suggests that the origin O is inaccessible and as ε→ 0the limit process Y ε

t → Yt lives only on OyA axis.

I Also supported by simulation results.

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Figure: Sample pathes of the X εt and Y ε

t processes, as well as thelimiting Y –process (driven by W 2

t ) starting from (X ,Y ) = (0, 2) in15000 steps for stepsize= 0.0001, that is rescaled to [0, 1]. (a) ε = 0.1 ;(b) ε = 0.01 ; the red curves are the sample pathes for Yt , the bluecurves are the sample pathes for Y ε

t . (c) ε = 0.1 ; (d) ε = 0.01 ; the blackcurves are the sample pathes for X ε

t .

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Limit as ε→ 0 of the perturbed AB–model.

I To prove this we have to carefully analyze the behavior of theprocess (X ε

t ,Yεt ) near the origin O = (0, 0).

I We introduce the angular variable θεt = arctan

(Y εt

X εt

).

I dθεt =

1

εX εt dt +

1

(r εt )2dW θ

t , θε0 = arctan

(Y ε

0

X ε0

),

dr εt =

(1

2r εt− r εt

)dt + dW r

t , rε0 =

√(X ε

0 )2 + (Y ε0 )2 .

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Limit as ε→ 0 of the perturbed AB–model.

I δ = ε1/10.

I Set the slow time clock t = (δ/ε)t and let us consider atime–rescaled pair of processes Θε

t = θε(ε/δ)t and Rεt = r ε(ε/δ)t.

I Then the stochastic differential equations satisfied by (Θεt ,R

εt )

are given bydΘε

t =X ε

(ε/δ)t

δdt +

√ε

δ· 1

(Rεt )2

dW θt , Θε

0 = θε0

dRεt =

ε

δ

(1

2Rεt

− Rεt

)dt +

√ε

δdW r

t ,Rε0 = r ε0 .

I Θεt is fast motion and Rε

t is slow motion.

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Limit as ε→ 0 of the perturbed AB–model.

I This analysis enables us to conclude that for any initialcondition |X ε

0 | ≥ 2δ, the flow will quickly bring the particleback to the region Y ≥ δ, and during this process the|X |–value is less or equal than 3δ.

I In particular, this implies that

P (|X εt | ≤ 3δ for 0 ≤ t ≤ T )→ 1

as ε ↓ 0.

I As ε→ 0 the process X εt will be localized near 0, and the

process Y εt lives on {Y ≥ δ}.

I We also need to do some exit time analysis and the proof oftightness for {Y ε

t : 0 ≤ t ≤ T}ε>0.

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Limit as ε→ 0 of the perturbed AB–model.

TheoremLet T > 0 and initial condition (x0, y0) ∈ R2. Then(a) For any bounded continuous function F : R2 → R that isuniformly Lipschitz continuous with a Lipschitz constantLip(F ) <∞ we have

limε↓0

E [F (X εT ,Y

εT )− F (0,Y ε

T )] = 0 .

(b) The measures on C[0,T ](R) induced by the process Y εt converge

weakly as ε ↓ 0 to the measure induced by Yt with Y0 = yπ(x0, y0).

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Limit as ε→ 0 of the perturbed AB–model.

I The proof makes use of the martingale problem framework forMarkov processes.

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Metastable behavior.

I For fixed ε > 0, at a subexponential time scale, excursions ofthe process (X ε

t ,Yεt ) moving from OyA towards a level set

y = −a will be observed.

I This induces jumps from points in OyB to points in OyA.

I As ε becomes smaller, motions of the process (X εt ,Y

εt ) to

OyB and jumping back become more and more rare, and inthe limit no more such jumps appear, so that we come to thelimiting process Yt which cannot penetrate through O.

I Thus as ε > 0 is close to 0, the description of the“metastable” behavior of system involves both a diffusion partand a jump part.

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Figure: The AB model.

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Relation with the partial damping model.

I System (X εt ,Y

εt ) (with rescaled time) :

dX εt = −1

εX εt Y ε

t dt − X εt dt + dW 1

t , X ε0 = x0 ,

dY εt =

1

ε(X ε

t )2dt − Y εt dt + dW 2

t , Y ε0 = y0 .

I Limit of the Y –process is given by

dYt =

(1

2Yt− Yt

)dt + dW 2

t ,Y0 = yπ(x0, y0) .

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Relation with the partial damping model.

I Remove the damping in Y –directiondX ε

t = −1

εX εt Y ε

t dt − X εt dt + dW 1

t , X ε0 = x0 ,

dY εt =

1

ε(X ε

t )2dt + dW 2t , Y ε

0 = y0 .

I Limit of the Y –process is given by

dYt =1

2Ytdt + dW 2

t ,Y0 = yπ(x0, y0) .

I A positive drift in the Y –direction !

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A few more remarks.

I Motion on the cone of invariant measures of the unperturbedsystem.

I One can use this result to analyze behavior of elliptic operator

L =1

εL0 + L1, where

L0 = −xy∂

∂x+ x2 ∂

∂y

and

L1 = −x∂

∂x− y

∂y+

1

2

∂2

∂x2+

1

2

∂2

∂y 2.

I The underlying 2d system is related to Euler equation - it is,in fact, the Euler-Arnold equation on a Lie algebra of the 2dLie group of affine transformations of the line.

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Reference.

Hu, W., On the long time behavior of a perturbedconservative system with degeneracy. Journal of TheoreticalProbability, in revision.

Page 46: On 2{d Euler equations with partial damping and some ...web.mst.edu/~huwen/slides_2dEulerModel.pdf · The structure of Euler solution supported on nitely many Fourier modes. I 2{d

Open Problem : Invariant measure for 2–d Euler equations.

I Kuksin, S., Shirikyan, A., Rigorous results in space–periodictwo–dimensional turbulence, Physics of Fluids, 29 :125106,2017.

I 2–d Euler equation with damping and noise :

ωt+(u·∇)ω−ν∆ω =√νη(x , t) , u = ∇⊥∆−1ω , ω(x , 0) = ω0(x) .

I “random force” η(x , t) =∑

k∈Λ∗cke ikxdwk(t) where Λ∗ is the

lattice Λ∗ = 2πa Z⊕ 2π

b Z.

I invisid scaling ν → 0.

I Change of variable t → νt :

ωt+1

ν(u·∇)ω−∆ω = η(x , t) , u = ∇⊥∆−1ω , ω(x , 0) = ω0(x) .

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Open Problem : Invariant measure for 2–d Euler equations.

I Under some natural (and very mild) assumptions on the noiseη, a well known result of Hairer–Mattingly 5 asserts that thereis a unique stationary measure µν for the stochastic processωt .

I As ν → 0 one can show that one has sufficient control overthe measures µν to establish that one might choose asubsequence µνn of these measures which converges to a(nontrivial) probability measure µ.

I µ is an invariant measure for the system with µ = 0 : theinvariant measure for the 2d Euler equations.

I The more sets the measure charges, the more possibilities wehave for the long–time behavior of 2d Euler equations.

I The simplest situation would be for µ to be supported on asubset of the steady–state solutions of the Euler equation.

5. Hairer, M., Mattingly, J.C., Ergodicity of the 2D Navier–Stokes equationswith degenerate stochastic forcing, Annals of Mathematics, (2) 164 (2006), no.3., pp. 993–1032.

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Open Problem : Invariant measure for 2–d Euler equations.

I Finite–dimensional Hamiltonian system

x + εΓx = J∇H +√εBw

where w is the m–dimensional Brownian motion and B is asuitable matrix, and Γ is the matrix describing damping.

I Change of variable t → εt :

x + Γx =1

εJ∇H +

√εBw .

I Invariant measure µε.

I Assuming that the limit µ = limε→0

µε exists (at least for a

subsequence), it is easy to see that µ is an invariant measurefor the original Hamiltonian system x = J∇H : Liouvillemeasure or Gibbs measure.

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Open Problem : Invariant measure for 2–d Euler equations.

I At the infinite–dimensional level one does not have a referencemeasure such as the Liouville or Gibbs measure.

I Situation can be quite different from finite–dimensional case.

I “inviscid damping” effects cause solution to approach (in theweak–∗) sense the solution which no longer exhibit invisciddamping (such as steady–state or a stable periodic solution).

I The long–time behavior can be in some sense a mixturebetween the behavior of a dissipative system (with thedissipation replaced by the inviscid damping) and thefinite–dimensional Hamiltonian behavior (for which one doesnot generically expect to converge to an equilibrium, butrather motion along an ergodic component).

I Better finite–dimensional models...

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Thank you for your attention !