Controllability implies ergodicity Armen Shirikyan Department of - - PowerPoint PPT Presentation

Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Controllability implies ergodicity

Armen Shirikyan

Department of Mathematics University of Cergy–Pontoise Evolution Equations: Long-time behaviour and Control Université Savoie Mont-Blanc 15 June 2015

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Outline

Introduction Preliminaries Controllability Stationary measures and mixing Randomly forced differential equations Mixing in the total variation metric Main result Differential equations on a compact manifold Mixing in the Kantorovich–Wasserstein metric Navier–Stokes system with a random boundary force Main result Open problems

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Control problem and its stochastic counterpart

Let X be a compact metric space, let E be separable Hilbert space, and let S : X × E → X be a continuous mapping. Let us consider the discrete-time control system uk = S(uk−1, ζk), k ≥ 1, (1) supplemented with the initial condition u0 = u. (2) Here u ∈ X is an initial state, {uk} ⊂ X is the trajectory of the system, and {ζk} ⊂ E are control functions at our disposal. We say that (1) is globally approximately controllable if for any u, ˆ u ∈ X and any ε > 0 one can find an integer m ≥ 1 and a finite sequence ζ1, . . . , ζk ∈ E such that d(um, ˆ u) < ε. (3)

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Control problem and its stochastic counterpart

Let X be a compact metric space, let E be separable Hilbert space, and let S : X × E → X be a continuous mapping. Let us consider the discrete-time control system uk = S(uk−1, ζk), k ≥ 1, (1) supplemented with the initial condition u0 = u. (2) Here u ∈ X is an initial state, {uk} ⊂ X is the trajectory of the system, and {ζk} ⊂ E are control functions at our disposal. We say that (1) is globally approximately controllable if for any u, ˆ u ∈ X and any ε > 0 one can find an integer m ≥ 1 and a finite sequence ζ1, . . . , ζk ∈ E such that d(um, ˆ u) < ε. (3)

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Control problem and its stochastic counterpart

Let us take a probability measure ℓ on E and consider the stochastic system uk = S(uk−1, ηk), k ≥ 1, (4) where {ηk} is a sequence of random variables in E distributed according to the law ℓ. In this case, for any initial state u, the trajectory of (4), (2) is a stochastic process in X. Very often, one is interested in the asymptotic behaviour of the law of uk as k → +∞. For instance, an interesting problem is to find a stationary measure µ, so that D(u) = µ = ⇒ D(uk) = µ for any k ≥ 0, (5) and to study the asymptotic stability of µ as k → ∞. GOAL: To investigate the relationship between controllability properties of (1) and the long-time behaviour of trajectories for (4).

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Control problem and its stochastic counterpart

Let us take a probability measure ℓ on E and consider the stochastic system uk = S(uk−1, ηk), k ≥ 1, (4) where {ηk} is a sequence of random variables in E distributed according to the law ℓ. In this case, for any initial state u, the trajectory of (4), (2) is a stochastic process in X. Very often, one is interested in the asymptotic behaviour of the law of uk as k → +∞. For instance, an interesting problem is to find a stationary measure µ, so that D(u) = µ = ⇒ D(uk) = µ for any k ≥ 0, (5) and to study the asymptotic stability of µ as k → ∞. GOAL: To investigate the relationship between controllability properties of (1) and the long-time behaviour of trajectories for (4).

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Control problem and its stochastic counterpart

Let us take a probability measure ℓ on E and consider the stochastic system uk = S(uk−1, ηk), k ≥ 1, (4) where {ηk} is a sequence of random variables in E distributed according to the law ℓ. In this case, for any initial state u, the trajectory of (4), (2) is a stochastic process in X. Very often, one is interested in the asymptotic behaviour of the law of uk as k → +∞. For instance, an interesting problem is to find a stationary measure µ, so that D(u) = µ = ⇒ D(uk) = µ for any k ≥ 0, (5) and to study the asymptotic stability of µ as k → ∞. GOAL: To investigate the relationship between controllability properties of (1) and the long-time behaviour of trajectories for (4).

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Approximate and solid controllability

Let K ⊂ E be a closed subset and let ˆ u ∈ X. We say that (1) is globally approximately controllable to ˆ u with a K-valued control if for any ε > 0 there is m ≥ 1 such that, given an initial point u ∈ X, one can find ζ1, . . . , ζm ∈ K satisfying the inequality d

• Sm(u; ζ1, . . . , ζm), ˆ

u

• ≤ ε,

(6) where Sk(u; ζ1, . . . , ζk) stands for the trajectory of (1), (2). We say that (1) is solidly controllable from u0 if there is a compact set Q ⊂ K, a non-degenerate ball B ⊂ X, and ε > 0 such that, for any continuous mapping Φ : Q → X satisfying sup

ζ∈Q

d

• Φ(ζ), S(u0, ζ)
• ≤ ε,

(7) we have Φ(Q) ⊃ B.

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Approximate and solid controllability

Let K ⊂ E be a closed subset and let ˆ u ∈ X. We say that (1) is globally approximately controllable to ˆ u with a K-valued control if for any ε > 0 there is m ≥ 1 such that, given an initial point u ∈ X, one can find ζ1, . . . , ζm ∈ K satisfying the inequality d

• Sm(u; ζ1, . . . , ζm), ˆ

u

• ≤ ε,

(6) where Sk(u; ζ1, . . . , ζk) stands for the trajectory of (1), (2). We say that (1) is solidly controllable from u0 if there is a compact set Q ⊂ K, a non-degenerate ball B ⊂ X, and ε > 0 such that, for any continuous mapping Φ : Q → X satisfying sup

ζ∈Q

d

• Φ(ζ), S(u0, ζ)
• ≤ ε,

(7) we have Φ(Q) ⊃ B.

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Local stabilisability

Let Dδ = {(u, u′) ∈ X × X : d(u, u′) ≤ δ}. We say that (1) is locally stabilisable if for any R > 0 there is a finite-dimensional subspace E ⊂ E, positive numbers C, δ, α ≤ 1, and q < 1, and a continuous mapping Φ : Dδ × BE(R) → E, (u, u′, η) → η′, which is continuously differentiable in η and satisfies the following inequalities for any (u, u′) ∈ Dδ: sup

ζ∈BE(R)

d

• S(u, ζ), S(u′, ζ + Φ(u, u′, ζ))
• ≤ q d(u, u′),

(8) sup

ζ∈BE(R)

• Φ(u, u′, ζ)E + DζΦ(u, u′, ζ)L(E)
• ≤ C d(u, u′)α. (9)

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Metrics on the space of probability measures

Let us denote by P(X) the space of probability measures on X, let C(X) be the space of continuous functions f : X → R, and let L(X) be the subspace of f ∈ C(X) satisfying fL := sup

u∈X

|f(u)| + sup

u=v

|f(u) − f(v)| d(u, v) < ∞ Total variation metric: µ1 − µ2var = sup

Γ∈B(X)

|µ1(Γ) − µ2(Γ)| = 1 2 sup

f∞≤1

|(f, µ1) − (f, µ2)|. Kantorovich–Wasserstein metric: µ1 − µ2∗

L = sup fL≤1

|(f, µ1) − (f, µ2)|.

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Discrete-time Markov processes

The stochastic system (4), (2) defines a discrete-time Markov process in X, provided that {ηk} is an i.i.d. sequence in E. We denote by Pk(u, Γ) its transition function, Pk(u, Γ) = Pu{uk ∈ Γ}, and by Pk : C(X) → C(X) and P∗

k : P(X) → P(X) the

corresponding Markov operators: (Pkf)(u) =

• X

Pk(u, dv)f(v), (P∗

kµ)(Γ) =

• X

Pk(u, Γ)µ(du). Let us recall that (Pkf)(u) = Euf(uk), (P∗

kµ)(Γ) = Pµ{uk ∈ Γ}.

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Exponential mixing

Definition

A measure µ ∈ P(X) is said to be stationary for (4) if P∗

1µ = µ

= ⇒ P∗

kµ = µ for all k ≥ 1.

A stationary measure µ for (4) is said to be exponentially mixing in the total variation metric if there are positive numbers C and γ such that P∗

kλ − µvar ≤ Ce−γk

for all k ≥ 0, λ ∈ P(X). (10) Similarly, a stationary measure µ is said to be exponentially mixing in the Kantorovich–Wasserstein metric if there are positive numbers C and γ such that (10) holds with · var replaced by · ∗

L.

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Exponential mixing

Definition

A measure µ ∈ P(X) is said to be stationary for (4) if P∗

1µ = µ

= ⇒ P∗

kµ = µ for all k ≥ 1.

A stationary measure µ for (4) is said to be exponentially mixing in the total variation metric if there are positive numbers C and γ such that P∗

kλ − µvar ≤ Ce−γk

for all k ≥ 0, λ ∈ P(X). (10) Similarly, a stationary measure µ is said to be exponentially mixing in the Kantorovich–Wasserstein metric if there are positive numbers C and γ such that (10) holds with · var replaced by · ∗

L.

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Exponential mixing

Definition

A measure µ ∈ P(X) is said to be stationary for (4) if P∗

1µ = µ

= ⇒ P∗

kµ = µ for all k ≥ 1.

A stationary measure µ for (4) is said to be exponentially mixing in the total variation metric if there are positive numbers C and γ such that P∗

kλ − µvar ≤ Ce−γk

for all k ≥ 0, λ ∈ P(X). (10) Similarly, a stationary measure µ is said to be exponentially mixing in the Kantorovich–Wasserstein metric if there are positive numbers C and γ such that (10) holds with · var replaced by · ∗

L.

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Differential equations on with random forcing

Let us consider a differential equation of the form ∂tu = F(u, η(t)), u(0) = u0, (11) where u(t) ∈ X, η(t) is a random process with range in a Hilbert space E0, and F is a nonlinear (differential) operator acting from X × E0 to X. We assume that the Cauchy problem (11) is well posed. Setting uk = u(k) and ηk = η|[k−1,k), we can write (formally) uk = S(uk−1, ηk), k ≥ 1, (12) where S is resolving operator for (11) on the time interval [0, 1]: S : (u0, {η, t ∈ [0, 1)}) → u(1), u(t) is the solution of (11).

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Differential equations on with random forcing

Let us consider a differential equation of the form ∂tu = F(u, η(t)), u(0) = u0, (11) where u(t) ∈ X, η(t) is a random process with range in a Hilbert space E0, and F is a nonlinear (differential) operator acting from X × E0 to X. We assume that the Cauchy problem (11) is well posed. Setting uk = u(k) and ηk = η|[k−1,k), we can write (formally) uk = S(uk−1, ηk), k ≥ 1, (12) where S is resolving operator for (11) on the time interval [0, 1]: S : (u0, {η, t ∈ [0, 1)}) → u(1), u(t) is the solution of (11).

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Reduction to a discrete-time Markov process

Let us assume that the random force η has the form η(t) =

∞

• k=1

I[k−1,k)(t)ηk(t − k + 1), (13) where {ηk} is a sequence of i.i.d. random variables in the space E = L2(J, E0) with J = (0, 1). In this case, (12) defines a discrete-time Markov process in X. We also assume that the law ℓ of {ηk} is decomposable: ηk(t) =

∞

• j=1

bjξjkej(t), (14) where {ej} is an ONB in E, ξjk = ξω

jk are independent scalar

random variables with smooth positive densities , and {bj} is a sequence of positive numbers going to zero sufficiently fast.

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Reduction to a discrete-time Markov process

Let us assume that the random force η has the form η(t) =

∞

• k=1

I[k−1,k)(t)ηk(t − k + 1), (13) where {ηk} is a sequence of i.i.d. random variables in the space E = L2(J, E0) with J = (0, 1). In this case, (12) defines a discrete-time Markov process in X. We also assume that the law ℓ of {ηk} is decomposable: ηk(t) =

∞

• j=1

bjξjkej(t), (14) where {ej} is an ONB in E, ξjk = ξω

jk are independent scalar

random variables with smooth positive densities , and {bj} is a sequence of positive numbers going to zero sufficiently fast.

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Recurrence and coupling

Let us consider a discrete-time Markov process in X with transition function Pk(u, Γ) and the corresponding Markov

• perators Pk and P∗
• k. Assume that the following two conditions

are satisfied for a point ˆ u ∈ X and a number δ > 0. Recurrence: There is p > 0 and an integer m ≥ 1 such that Pm

• u, B(ˆ

u, δ)

• ≥ p

for any u ∈ X. (15) Coupling: There is ε > 0 such that P1(u, ·) − P1(u′, ·)var ≤ 1 − ε for u, u′ ∈ B(ˆ u, δ). (16) The recurrence condition is a consequence of approximate controllability to ˆ u, while the validity of coupling follows from solid controllability and non-degeneracy of the random force.

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Recurrence and coupling

Let us consider a discrete-time Markov process in X with transition function Pk(u, Γ) and the corresponding Markov

• perators Pk and P∗
• k. Assume that the following two conditions

are satisfied for a point ˆ u ∈ X and a number δ > 0. Recurrence: There is p > 0 and an integer m ≥ 1 such that Pm

• u, B(ˆ

u, δ)

• ≥ p

for any u ∈ X. (15) Coupling: There is ε > 0 such that P1(u, ·) − P1(u′, ·)var ≤ 1 − ε for u, u′ ∈ B(ˆ u, δ). (16) The recurrence condition is a consequence of approximate controllability to ˆ u, while the validity of coupling follows from solid controllability and non-degeneracy of the random force.

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Recurrence and coupling

Let us consider a discrete-time Markov process in X with transition function Pk(u, Γ) and the corresponding Markov

• perators Pk and P∗
• k. Assume that the following two conditions

are satisfied for a point ˆ u ∈ X and a number δ > 0. Recurrence: There is p > 0 and an integer m ≥ 1 such that Pm

• u, B(ˆ

u, δ)

• ≥ p

for any u ∈ X. (15) Coupling: There is ε > 0 such that P1(u, ·) − P1(u′, ·)var ≤ 1 − ε for u, u′ ∈ B(ˆ u, δ). (16) The recurrence condition is a consequence of approximate controllability to ˆ u, while the validity of coupling follows from solid controllability and non-degeneracy of the random force.

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Main result

Theorem

Suppose that a Markov process in X satisfies the recurrence and coupling conditions. Then it has a unique stationary measure µ ∈ P(X), which is exponentially mixing in the total variation metric.

Proposition

In addition to the above hypotheses assume that bj > 0 and ρj(r) > 0 for all j ≥ 1 and r ∈ R. (a) If there is ˆ u ∈ X such that (1) is globally approximately controllable to ˆ u, then the recurrence property holds. (b) If (1) is solidly controllable from a point ˆ u ∈ X, then the coupling property holds. In particular, if (1) globally approximately controllable to ˆ u and solidly controllable from ˆ u, then (4) has a unique stationary measure, which is exponentially mixing in the total variation.

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Weak and strong Hörmander conditions

Let X be a compact Riemannian manifold without boundary and let V0, . . . , Vn be smooth vector fields on X. Consider the control system ˙ u = V0(u) +

n

• j=1

ζj(t)Vj(u), (17) where ζ1, . . . , ζn are real-valued control functions. Denote by Lie(V1, . . . , Vn) the minimal Lie algebra containing V1, . . . , Vn and by Lie0(V0; V1, . . . , Vn) the minimal Lie ideal in Lie(V0, V1, . . . , Vn) containing the vector fields V1, . . . , Vn. In other words, Lie(V1, . . . , Vn) and Lie0(V0; V1, . . . , Vn) are the vector spaces spanned, respectively, by the families

• V1, . . . , Vn; [Vi, Vj], 1 ≤ i, j ≤ n; [[Vi, Vj], Vk], 1 ≤ i, j, k ≤ n; . . .
• ,
• V1, . . . , Vn; [Vi, Vj], 0 ≤ i, j ≤ n; [[Vi, Vj], Vk], 0 ≤ i, j, k ≤ n; . . .
• .

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Weak and strong Hörmander conditions

The following result provides sufficient conditions on the vector fields V0, . . . , Vn for approximate and solid controllability.

Proposition

(a) Suppose that the restriction of Lie(V1, . . . , Vn) to any point u ∈ X coincides with the tangent space TuX. Then, for any u ∈ X, system (17) is globally approximately controllable to u. (b) Suppose that the restriction of Lie0(V0; V1, . . . , Vn) to some point ˆ u ∈ X coincides with Tˆ

• uX. Then system (17) is solidly

controllable from ˆ u.

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Main results for ODE on manifolds

Let us consider the following ODE with random coefficients on a compact Riemannian manifold X: ˙ u = V0(u) +

n

• j=1

ηj(t)Vj(u). (18) We assume that η = (η1, . . . , ηn) has the form η(t) =

∞

• k=1

I[k−1,k)(t)ηk(t − k + 1), where {ηk} is a sequence of i.i.d. random variables in L2(J, Rn) with a decomposable law: ηk(t) =

∞

• j=1

bjξjkej(t), {ej} is an ONB in L2(J, Rn).

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Main results for ODE on manifolds

Theorem

Suppose that the strong Hörmander condition holds for the vector fields V1, . . . , Vn, the coefficients bj > 0 decay to zero sufficiently fast, and the laws of ξjk have positive C1 densities. Then the Markov process generated by (18) has a unique stationary measure, which is exponentially mixing in the total variation metric. In the case when ηj(t) are independent white noises, this type

• f results were obtained by Arnold–Kliemann (1984),

Veretennikov (1988), and many others.

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Sufficient conditions for mixing

Theorem

Suppose that a Markov process satisfies the following two conditions for some ˆ u ∈ X, C > 0, and q ∈ (0, 1) : Recurrence: For any δ > 0 there are p > 0, m ≥ 1 such that Pm(u, B(ˆ u, δ)) ≥ p for any u ∈ X. (19) Squeezing: For any u, u′ ∈ X there are X-valued random variables v, v′ such that P

• d(v, v′) ≤ q d(u, u′)
• ≥ 1 − C d(u, u′),

(20) D(v) = P1(u, ·), D(v′) = P1(u′, ·). (21) Then the Markov process has a unique stationary measure, which is exponentially mixing in the Kantorovich–Wasserstein metric.

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Squeezing in terms of controllability

Proposition

Suppose that the control system (1) is locally stabilisable and the law of ηk is decomposable: ηk =

∞

• j=1

bjξjkej, where {ej} is an ONB in E, bj are positive numbers going to zero sufficiently fast, and the laws of ξjk possess C1-smooth densities supported by [−1, 1]. Then the squeezing property holds. The above results apply to various randomly forced PDE’s, including the Ginzburg–Landau and Navier–Stokes equations with a bounded noise. Moreover, there exist analogues of these results for unbounded random perturbations.

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Navier–Stokes system with a boundary noise

Let D ⊂ R2 be a bounded domain with smooth boundary and let D1 = (0, 1) × D. Consider the problem ∂tu + u, ∇u − ν∆u + ∇p = h(t, x), div u = 0, (22) u

• ∂D = η(t, x),

(23) u(0, x) = u0(x). (24) Here h is a 1-periodic function belonging to H1

loc(R+ × D) and

η(t, x) is a space-time localised noise of the form η(t, x) =

∞

• k=1

I[k−1,k)(t)η(t − k + 1, x), (25) where {ηk} is a sequence i.i.d. random variables in the space E =

• g ∈ H5/2(∂xD1) :
• ∂D

g(t), ndσx ≡ 0, supp g ⊂ Q ⋐ ∂xD1

• .

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Main result

The restrictions of trajectories of (22), (23) to the integer lattice form a Markov process in the space H = {u ∈ L2(D, R2) : div u = 0 in D, u, n = 0 on ∂D} Suppose that the law of ηk is decomposable and denote by K the support of the law of ηk, which is a compact subset in E.

Theorem

In addition to the above hypotheses, let us assume that (22), (23) is globally approximately controllable to some point ˆ u ∈ H with a K-valued control. Then for any ν > 0 there is N ≥ 1 such that if bj = 0 for 1 ≤ j ≤ N, then the Markov process defined by the Navier–Stokes system has a unique stationary measure µ ∈ P(H), which possesses the property of exponentially mixing in the Kantorovich–Wasserstein metric.

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Open problems

Mixing for Fourier-localised noise Let us consider the Navier–Stokes system with the RHS f(t, x) = h(t, x) +

N

• j=1

bjηj(t)ej(x), (26) where ej is an orthonormal basis in H. It was proved by Hairer–Mattingly (2006–2011) that if h ≡ 0, ηj are independent white noises, D = T2, and finitely many bj are nonzero, then the problem is exponentially mixing for any ν > 0. A similar result is a challenging open problem in the following two cases: Nontrivial limiting dynamics: h is not identically equal to zero. Non-Gaussian noise: ηj are smooth in time random processes. Note that, due to Agrachev–Sarychev (2005), the problem is globally approximately controllable for any ν > 0.

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Introduction Preliminaries Strong mixing Differential equations Weak mixing Navier–Stokes system

Open problems

Approximate controllability by a bounded force Our result on mixing requires global approximate controllability by a control taking values in the support of η. This condition is trivially satisfied if h ≡ 0, and the support of η contains zero. Due to Coron–Fursikov–Imanuvilov (1996–1999), the problem is globally approximately controllable by an unbounded C∞

• force. The following question remains completely open:

Nontrivial uncontrolled dynamics: Given a smooth function h, find a compact (or even bounded) subset K ⊂ E and a point ˆ u ∈ H such that the Navier–Stokes system is globally approximately controllable to ˆ u with a K-valued control.

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