## Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations (2006)

Citations: | 15 - 7 self |

### BibTeX

@TECHREPORT{Hairer06spectralgaps,

author = {Martin Hairer and Jonathan C. Mattingly},

title = {Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations},

institution = {},

year = {2006}

}

### OpenURL

### Abstract

We develop a general method that allows to show the existence of spectral gaps for Markov semigroups on Banach spaces. Unlike most previous work, the type of norm we consider for this analysis is neither a weighted supremum norm nor an Ł p-type norm, but involves the derivative of the observable as well and hence can be seen as a type of 1–Wasserstein distance. This turns out to be a suitable approach for infinite-dimensional spaces where the usual Harris or Doeblin conditions, which are geared to total variation convergence, regularly fail to hold. In the first part of this paper, we consider semigroups that have uniform behaviour which one can view as an extension of Doeblin’s condition. We then proceed to study situations where the behaviour is not so uniform, but the system has a suitable Lyapunov structure, leading to a type of Harris condition. We finally show that the latter condition is satisfied by the two-dimensional stochastic Navier-Stokers equations, even in situations where the forcing is extremely degenerate. Using the convergence result, we show shat the stochastic Navier-Stokes equations ’ invariant measures depend continuously on the viscosity and the structure of the forcing. 1

### Citations

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Citation Context ...ce between probability measures by ∫ ∫ d(µ1,µ2) = sup ∣ φ(x)µ1(dx) − φ(x)µ2(dx) ∣ , (4) Lip d (φ)≤1 where Lipd(φ) denotes the Lipschitz constant of φ in the metric d. By the Monge–Kantorovich duality =-=[44, 49]-=-, the right-hand side of (4) is equivalent to ∫ ∫ (5) d(µ1,µ2) = inf d(x,y)µ(dx,dy). µ∈C(µ1,µ2) (Note that the infimum is actually achieved; see [50], Theorem 4.1.) With these notation, one has the fo... |

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Citation Context ...of degenerately forced dissipative SPDEs, control of the gradient term on the right-hand side of Assumption 5 combines an argument strongly inspired by the probabilistic proofs of Hörmander’s theorem =-=[24]-=- based on Malliavin’s calculus [33, 41, 48], together with the infinitesimal equivalent of the Foias– Prodi-type estimate, namely the fact that the linearized flow contracts all but finitely many dire... |

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Citation Context ...ce between probability measures by ∫ ∫ d(µ1,µ2) = sup ∣ φ(x)µ1(dx) − φ(x)µ2(dx) ∣ , (4) Lip d (φ)≤1 where Lipd(φ) denotes the Lipschitz constant of φ in the metric d. By the Monge–Kantorovich duality =-=[44, 49]-=-, the right-hand side of (4) is equivalent to ∫ ∫ (5) d(µ1,µ2) = inf d(x,y)µ(dx,dy). µ∈C(µ1,µ2) (Note that the infimum is actually achieved; see [50], Theorem 4.1.) With these notation, one has the fo... |

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Citation Context ...pectrum of Pt viewed as a bounded operator on B. Remark 5.6. It follows from standard semigroup theory that the above statements imply that Pt possesses a generator L densely defined on B (see, e.g., =-=[10]-=-, Theorem 1.7) and that there exists g > 0 such that Re(λ) ≤ −g for every λ ∈ σ(L) \ {0} (see, e.g., [10], Theorem 2.16). Before we give the precise statement of our results, let us turn to the constr... |

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Citation Context ...m ‖φ‖∞ + ‖Dφ‖∞) is strictly smaller than 1. This is a well-known and often exploited fact 4 in the theory of dynamical systems. A bound like (2) is usually referred to as the Lasota– Yorke inequality =-=[30, 31]-=- or the Ionescu–Tulcea–Marinescu inequality [26] and is used to show the existence of absolutely continuous invariant measures when Pt is the transfer operator acting on densities. Usually, it is used... |

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Citation Context ...and C(α) with (20) ‖DPtφ(x)‖ ≤ C1V p √ (x)(C(α) (Pt|φ| 2 √ )(x) + α (Pt‖Dφ‖2 )(x)), for every x ∈ H and t ≥ T(α). Remark 3.3. While (20) is reminiscent of gradient estimates of the type considered in =-=[4]-=-, there does not seem to be an obvious link between the two approaches. The main reason is that (20) is really a statement about the long-time behavior of Pt whereas the bounds in [4] are statements a... |

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Citation Context ...he relationship between the results of the previous section and those of this section is analogous to the relationship between Doeblin’s condition mentioned in the last section and Harris’ conditions =-=[16, 22, 40]-=-. While the assumptions given in this section are heavily influenced by the known a priori bounds on the dynamics of the two-dimensional Navier–Stokes equations, we suspect the result will be useful m... |

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Citation Context ...SPDEs, control of the gradient term on the right-hand side of Assumption 5 combines an argument strongly inspired by the probabilistic proofs of Hörmander’s theorem [24] based on Malliavin’s calculus =-=[33, 41, 48]-=-, together with the infinitesimal equivalent of the Foias– Prodi-type estimate, namely the fact that the linearized flow contracts all but finitely many directions. This work has its intellectual root... |

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Citation Context .... Unless indicated otherwise, we will assume that the24 M. HAIRER AND J. MATTINGLY constant component ¯ f of the body force and the coefficients qk satisfy Assumption 1. It is well known (see, e.g., =-=[8, 17]-=-) that (27) has a unique solution under much weaker assumptions on the covariance operator Q. It is also well known that under similar conditions, (27) has an invariant measure µ⋆. The uniqueness of t... |

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Citation Context ...vergence results are direct descendants of those developed by many authors in, among others, [6, 14, 20, 28, 34, 37, 38, 43]. All of these works make use of a version of the Foias–Prodi-type estimate =-=[18]-=-, introduced in the stochastic context in [35]. The later papers also use a coupling construction to prove convergence. In particular, [20, 37, 38] developed a coupling construction to prove exponenti... |

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Citation Context ...‖ if α < −2 and ∇ · u = 0, ‖Kv‖α = ‖v‖α−1, ‖v‖ 2 β ≤ ε‖v‖ 2 α + ε −2((γ−β)/(β−α)) ‖v‖ 2 γ if 0 ≤ α < β < γ and ε > 0. We start with the following set of a priori bounds, most of which were taken from =-=[21]-=- and [39]. Lemma A.1. The solution wt of the 2D stochastic Navier–Stokes equations in the vorticity formulation satisfies the following bounds: 1. There exist constants C,η⋆,γ > 0, such that ( ∫ t Eex... |

40 | Hairer: Uniqueness of the invariant measure for a stochastic PDE driven by degenerate noise
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Citation Context ...nt measure can be obtained, provided that all the qk with |k| 2 ≤ N are nonzero, for some value N ≈ ∑ q 2 k /ν3 . To the best of the author’s knowledge, the only exception to this were the results of =-=[15]-=-, that indicated that the invariant measure µ⋆ should be unique provided that there exist R > 0 and α large enough such that all the qk with |k| ≥ R are bounded from above and from below by multiples ... |

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Citation Context ...n operator T can be characterized as the supremum over all λ > 0 such that there exists a singular sequence {fn}n≥0 with ‖fn‖ = 1 and ‖Tfn‖ ≥ λ for every n. A slightly different proof can be found in =-=[23]-=- and is directly based on the study of the essential spectral radius by Nussbaum [42]. It is, however, very close in spirit to the much earlier paper [26].8 M. HAIRER AND J. MATTINGLY To measure the ... |

38 | Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation
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Citation Context ...this invariant measure is a much harder problem and has been a field of intense research over the past decade. Early results can be found in [9, 17, 36]. Until recently, the consensus that emerged in =-=[5, 6, 14, 20, 28, 34, 37, 38]-=- was that the uniqueness of the invariant measure can be obtained, provided that all the qk with |k| 2 ≤ N are nonzero, for some value N ≈ ∑ q 2 k /ν3 . To the best of the author’s knowledge, the only... |

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Citation Context ...loser to the finite-dimensional setting. It is not expected that such estimates hold in the total variation norm in the setting of this article. We should also remark that previous estimates, such as =-=[20, 29, 37, 38]-=-, giving simply exponential convergence to equilibrium are weaker and the results in this article represent a significant and new extension of those results.SPECTRAL GAPS IN WASSERSTEIN DISTANCES 5 I... |

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Citation Context ...ts a singular sequence {fn}n≥0 with ‖fn‖ = 1 and ‖Tfn‖ ≥ λ for every n. A slightly different proof can be found in [23] and is directly based on the study of the essential spectral radius by Nussbaum =-=[42]-=-. It is, however, very close in spirit to the much earlier paper [26].8 M. HAIRER AND J. MATTINGLY To measure the convergence to equilibrium, we will use the following distance function on H: (3) d(x... |

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Citation Context ...as w and u = (u1,u2), then B(u,v) = (u · ∇)v. Setting S = {s = (s1,s2,s3) ∈ R 3 + :∑ si ≥ 1,s ̸= (1,0,0),(0,1,0),(0,0,1)} and keeping u, v and w as above, then the following relations are useful (cf. =-=[7]-=-): (35) (36) (37) (38) (39) 〈B(u,v),w〉 = −〈B(u,w),v〉 if ∇ · u = 0, |〈B(u,v),w〉| ≤ C‖u‖s1 ‖v‖1+s2 ‖w‖s3 , (s1,s2,s3) ∈ S, ‖B(u,v)‖α ≤ Cα‖u‖‖v‖ if α < −2 and ∇ · u = 0, ‖Kv‖α = ‖v‖α−1, ‖v‖ 2 β ≤ ε‖v‖ 2 ... |

29 | Lefevere: Ergodicity of the 2D NavierStokes equations with random forcing
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Citation Context ...this invariant measure is a much harder problem and has been a field of intense research over the past decade. Early results can be found in [9, 17, 36]. Until recently, the consensus that emerged in =-=[5, 6, 14, 20, 28, 34, 37, 38]-=- was that the uniqueness of the invariant measure can be obtained, provided that all the qk with |k| 2 ≤ N are nonzero, for some value N ≈ ∑ q 2 k /ν3 . To the best of the author’s knowledge, the only... |

28 |
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Citation Context ...this invariant measure is a much harder problem and has been a field of intense research over the past decade. Early results can be found in [9, 17, 36]. Until recently, the consensus that emerged in =-=[5, 6, 14, 20, 28, 34, 37, 38]-=- was that the uniqueness of the invariant measure can be obtained, provided that all the qk with |k| 2 ≤ N are nonzero, for some value N ≈ ∑ q 2 k /ν3 . To the best of the author’s knowledge, the only... |

26 | Exponential Mixing Properties of Stochastic PDEs Through Asymptotic
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Citation Context ...loser to the finite-dimensional setting. It is not expected that such estimates hold in the total variation norm in the setting of this article. We should also remark that previous estimates, such as =-=[20, 29, 37, 38]-=-, giving simply exponential convergence to equilibrium are weaker and the results in this article represent a significant and new extension of those results.SPECTRAL GAPS IN WASSERSTEIN DISTANCES 5 I... |

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Citation Context |

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Citation Context .... Unless indicated otherwise, we will assume that the24 M. HAIRER AND J. MATTINGLY constant component ¯ f of the body force and the coefficients qk satisfy Assumption 1. It is well known (see, e.g., =-=[8, 17]-=-) that (27) has a unique solution under much weaker assumptions on the covariance operator Q. It is also well known that under similar conditions, (27) has an invariant measure µ⋆. The uniqueness of t... |

23 |
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Citation Context ...N WASSERSTEIN DISTANCES 27 Proposition 5.3 immediately implies that Assumption 5 is satisfied for every choice of η, so that it remains to verify Assumption 6. This, however, follows immediately from =-=[13]-=-, Lemma 3.1, and Remark 5.4 above. As a consequence, we have just shown that: Theorem 5.5. If Assumption 1 holds, there exists η0 > 0 such that, for every η ≤ η0, the stochastic flow solving (27) sati... |

20 |
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Citation Context ...tions, (27) has an invariant measure µ⋆. The uniqueness of this invariant measure is a much harder problem and has been a field of intense research over the past decade. Early results can be found in =-=[9, 17, 36]-=-. Until recently, the consensus that emerged in [5, 6, 14, 20, 28, 34, 37, 38] was that the uniqueness of the invariant measure can be obtained, provided that all the qk with |k| 2 ≤ N are nonzero, fo... |

20 | Coupling approach to white-forced nonlinear PDEs
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Citation Context ...loser to the finite-dimensional setting. It is not expected that such estimates hold in the total variation norm in the setting of this article. We should also remark that previous estimates, such as =-=[20, 29, 37, 38]-=-, giving simply exponential convergence to equilibrium are weaker and the results in this article represent a significant and new extension of those results.SPECTRAL GAPS IN WASSERSTEIN DISTANCES 5 I... |

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Citation Context ...SPDEs, control of the gradient term on the right-hand side of Assumption 5 combines an argument strongly inspired by the probabilistic proofs of Hörmander’s theorem [24] based on Malliavin’s calculus =-=[33, 41, 48]-=-, together with the infinitesimal equivalent of the Foias– Prodi-type estimate, namely the fact that the linearized flow contracts all but finitely many directions. This work has its intellectual root... |

17 |
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Citation Context ...ghted by Lyapunov functions are used to prove the existence of solutions to infinite-dimensional Kolmogorov equations. The convergence of observables dominated by Lyapunov functions was also given in =-=[27, 38]-=- in the “essentially elliptic” case. The results obtained there were, however, far from what is needed to prove a spectral gap. The convergence results are direct descendants of those developed by man... |

17 |
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Citation Context ...is section we give a theorem guaranteeing convergence in a Wasserstein distance with assumptions analogous to Doeblin’s condition. A classical generalization of Doeblin’s condition was made by Harris =-=[22]-=- who showed how to combine the existence of a Lyapunov function and a Doeblin-like estimate localized to a sufficiently large compact set to prove convergence to equilibrium. We will give a “Harris-li... |

16 | Navier-Stokes equations: controllability by means of low modes forcing - Agrachev, Sarychev - 2005 |

14 |
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Citation Context ...a well-known and often exploited fact 4 in the theory of dynamical systems. A bound like (2) is usually referred to as the Lasota– Yorke inequality [30, 31] or the Ionescu–Tulcea–Marinescu inequality =-=[26]-=- and is used to show the existence of absolutely continuous invariant measures when Pt is the transfer operator acting on densities. Usually, it is used with two different Hölder norms on the right-ha... |

11 |
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Citation Context ...se developed by many authors in, among others, [6, 14, 20, 28, 34, 37, 38, 43]. All of these works make use of a version of the Foias–Prodi-type estimate [18], introduced in the stochastic context in =-=[35]-=-. The later papers also use a coupling construction to prove convergence. In particular, [20, 37, 38] developed a coupling construction to prove exponential convergence. Though in a less explicit way ... |

11 | Ergodicity of the finite dimensional approximation of the 3D Navier-Stokes equations forced by a degenerate noise - Romito |

11 | The ODE method and spectral theory of Markov operators
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Citation Context ...ws one to incorporate information about the pathwise contraction properties of the system. When the system is completely pathwise contracting, the story is relatively straightforward; see [36, 38] or =-=[25]-=- for the finite-dimensional setting. However, when the system is not pathwise contracting one must introduce a change of measure to make it contracting.36 M. HAIRER AND J. MATTINGLY This was one of t... |

11 | On recent progress for the stochastic Navier Stokes equations. Journées “Équations aux Dérivées Partielles”, Exp - Mattingly - 2003 |

10 | Ergodicity for the stochastic Complex Ginzburg–Landau equations, to appear in Annales de l’institut Henri-Poincar, Probabilits et Statistiques
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Citation Context ...exponential convergence to 0 of Ed1(xnt,ynt). There are a few subtle issues arising from the fact that τ is not adapted to the natural filtration of the process, and we refer the interested reader to =-=[20, 38, 43]-=- for examples on how to circumvent these technicalities by a specific construction of (xnt,ynt). Since our goal is only to sketch the argument, we do not concern ourselves with these issues here. Obse... |

10 |
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Citation Context ...ce. Note that (SNS) is not expected to have the strong Feller property, so that it is a fortiori not expected that P ∗ t µ → µ⋆ in the total variation topology if µ and µ⋆ are mutually singular. (See =-=[9]-=- for a general discussion of the strong Feller property in infinite dimensions and [21] for a discussion of its limitations in the present setting.) In order to state the main result of the present ar... |

10 |
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Citation Context ...ique invariant measure, namely Doeblin’s condition 3 : 3 Doeblin’s original condition was the existence of a probability measure ν and a constant ε > 0 such that P(x,A) ≥ ε whenever ν(A) > 1 − ε; see =-=[11, 12]-=-. It turns out that, provided that the Markov chain is aperiodic and ψ-irreducible, this is equivalent to the (in general stronger) condition given here; see [40], Theorem 16.0.2.6 M. HAIRER AND J. M... |

8 | Invariant measure and their properties. a functional analytic point of view
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(Show Context)
Citation Context ...m ‖φ‖∞ + ‖Dφ‖∞) is strictly smaller than 1. This is a well-known and often exploited fact 4 in the theory of dynamical systems. A bound like (2) is usually referred to as the Lasota– Yorke inequality =-=[30, 31]-=- or the Ionescu–Tulcea–Marinescu inequality [26] and is used to show the existence of absolutely continuous invariant measures when Pt is the transfer operator acting on densities. Usually, it is used... |

6 |
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Citation Context ...ique invariant measure, namely Doeblin’s condition 3 : 3 Doeblin’s original condition was the existence of a probability measure ν and a constant ε > 0 such that P(x,A) ≥ ε whenever ν(A) > 1 − ε; see =-=[11, 12]-=-. It turns out that, provided that the Markov chain is aperiodic and ψ-irreducible, this is equivalent to the (in general stronger) condition given here; see [40], Theorem 16.0.2.6 M. HAIRER AND J. M... |

5 |
The emergence of large-scale coherent structure under small-scale random bombardments
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Citation Context ...ndence of µ⋆ on these parameters. The results obtained in this article enable us to give a relatively simple argument that shows that µ⋆ depends in a continuous way on all the parameters involved. In =-=[32]-=-, Majda and Wang proved that in the setting where the dissipation dominates the dynamics, the invariant measure depends continuously on the viscosity. This is a reflection of the fact in this context ... |

5 | Malliavin calculus for infinite-dimensional systems with additive noise
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(Show Context)
Citation Context ...e ikx ||k| < N∗}, ¯ f is as in Assumption 1, and ∑ q 2 k < ∞, then all of the results of this paper hold. In particular, this allows infinitely many qk to be nonzero. Remark 1.2. Using the results in =-=[3]-=- one can remove the restriction that the forcing need consist of Fourier modes and replace it with the requirement that the forced functions span the Fourier modes required above. Since this is not th... |

5 |
A new approach to Kolmogorov equations in infinite dimensions and applications to the stochastic 2d Navier Sokes equation
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Citation Context ...ve been obtained for some time [40], but these estimates are of course not uniform when (SNS) is approximated by a sequence of finite-dimensional systems (say by spectral Galerkin approximations). In =-=[46]-=-, spaces of observables weighted by Lyapunov functions are used to prove the existence of solutions to infinite-dimensional Kolmogorov equations. The convergence of observables dominated by Lyapunov f... |

4 | Malliavin calculus for the Stochastic 2D Navier Stokes Equation
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(Show Context)
Citation Context ...hough we are confident that our results are valid for Q sufficiently smooth, we restrict ourselves to the case where Q is a trigonometric polynomial, so that we can make use of the bounds obtained in =-=[21, 39]-=-. Instead of considering the velocity (SNS) directly, we will consider the equation for the vorticity w = ∇ ∧ v = ∂1v2 − ∂2v1. Note that v is uniquely determined from w (we will write v = Kw) through ... |

4 | Exponential ergodicity for stochastic Burgers and 2D Navier-Stokes equations
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(Show Context)
Citation Context ... parameters allowed by Assumption 1. For every η small enough there exist constants C and γ such that ‖Ptφ − µ⋆φ‖η ≤ Ce −γt ‖φ‖η, for every Fréchet differentiable function φ:H → R and every t ≥ 0. In =-=[19]-=- a similar operator-norm estimate on Pt was obtained in a weighted total variation norm (‖ · ‖η without the ‖Dφ‖ term) when the forcing was spatially rough and nondegenerate. Our setting is quite diff... |

2 | Stability of random attractors under perturbation and approximation
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Citation Context ...e point (see [32, 36, 38]). Hence the continuous dependence of the32 M. HAIRER AND J. MATTINGLY invariant measure follows from the continuous dependence of the random attractor. This can be found in =-=[45]-=- in an abstract setting and [32] in this setting. In the setting of this article, the random attractor is not necessarily a single point, hence results for the attractor do not translate to results fo... |

2 |
Optimal transport, old and new (Saint-Flour 2005), Version of
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(Show Context)
Citation Context ...the metric d. By the Monge–Kantorovich duality [44, 49], the right-hand side of (4) is equivalent to ∫ ∫ (5) d(µ1,µ2) = inf d(x,y)µ(dx,dy). µ∈C(µ1,µ2) (Note that the infimum is actually achieved; see =-=[50]-=-, Theorem 4.1.) With these notation, one has the following convergence result: Theorem 2.5. Let (Pt)t≥0 be a Markov semigroup over a Banach space H satisfying Assumptions 2 and 3. Then, there exist co... |

1 | Controllability for the Navier-Stokes equation with small control - AGRACHËV, SARYCHEV |

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