Equilibrium Statistical Mechanics of Barotropic Quasi-Geostrophic Equations
aa r X i v : . [ m a t h . P R ] F e b EQUILIBRIUM STATISTICAL MECHANICSOF BAROTROPIC QUASI-GEOSTROPHIC EQUATIONS
FRANCESCO GROTTO AND UMBERTO PAPPALETTERA
Abstract.
We consider equations describing a barotropic inviscid flow ina channel with topography effects and beta-plane approximation of Coriolisforce, in which a large-scale mean flow interacts with smaller scales. Gibbsianmeasures associated to the first integrals energy and enstrophy are Gaussianmeasures supported by distributional spaces. We define a suitable weak for-mulation for barotropic equations, and prove existence of a stationary solutionpreserving Gibbsian measures, thus providing a rigorous infinite-dimensionalframework for the equilibrium statistical mechanics of the model. Introduction
Barotropic quasi-geostrophic equations in channel domains constitute a physi-cally relevant partial differential equation in oceanography and atmospheric mod-eling, with applications including for instance the Antarctic circumpolar current.Significance of the model is discussed for instance in [15, 7, 20, 11] and referencestherein, to which we refer.The presence of conserved quantities and their associated equilibrium statisticalmechanics constitute an important feature of the model, and our work will focus oninvariant measures and stationary solutions. Although numerical reasons naturallylead to consider Fourier truncated or other approximations of the stationary flow,as for instance in [19, Section 6], the full infinite-dimensional setting is of greatinterest because of its geophysical relevance and mathematical difficulty, as dis-cussed in [20]. The latter monography thoroughly discusses in Chapter 8 equationsfor fluctuations around the mean state for the truncated model, and then consid-ers a continuum limit by scaling parameters of invariant measures so to neglectfluctuations, obtaining a mean state description for the PDE model.Our contribution in a sense furthers their study: we will show how fluctuationscan be included in the continuum limit by defining a suitably weak notion of solu-tion, so to include the distributional regimes dictated by the full infinite-dimensionalinvariant measure, under which fluctuations of comparable order are observed atall scales.The model under consideration, for the derivation of whom we refer to [20,Chapter 1], is the following. We consider the rectangle R = [ − π, π ] × [0 , π ] as aspace domain, and denote z = ( x, y ) ∈ R its points; we also fix a finite interval fortime t ∈ [0 , T ]. The governing dynamics is the inviscid quasi-geostrophic equationfor the scalar potential vorticity q ( t, z ),(1.1) ∂ t q + ∇ ⊥ ψ · ∇ q = 0 , where ∇ ⊥ = ( − ∂ y , ∂ x ), and ψ ( t, z ) is the stream function determining the divergence-less velocity field ∇ ⊥ ψ . The channel geometry prescribes that velocity ∇ ⊥ ψ betangent to the top and bottom boundaries of R , and we further assume the flow to Date : February 10, 2020.
Key words and phrases. quasi-geostrophic equations, channel flow, equilibrium statistical mechan-ics, weak vorticity formulation. be periodic in the x coordinate. Such boundary conditions are encoded in terms of ψ as follows: ∂ x ψ ( t, x, π ) = ∂ x ψ ( t, x,
0) = 0 , (1.2) ∇ ⊥ ψ ( t, x + 2 π, y ) = ∇ ⊥ ψ ( t, x, y ) . (1.3)As a consequence, at fixed t the stream function ψ is constant on the impermeableboundaries y = 0 , π . Using the fact that ψ is defined up to an additive constant,possibly depending on time, we will set ψ ( t, x, ≡
0, from which it is easily seenthat ψ takes the form ψ = − V y + ψ ′ , with V ( t ) a function of time only describing a large-scale mean flow, and ψ ′ ( t, z )the scalar small-scale stream function , periodic in x and null at y = 0 , π at all times.Potential vorticity is then linked to ψ ′ by(1.4) q = ∆ ψ ′ + h + βy, where h ( z ) is a smooth scalar function modelling the effect of the underlying to-pography on the fluid, and βy , β ∈ R , is the beta-plane approximation of Coriolis’force.Dynamics of V ( t ) is derived by imposing conservation of total energy ,(1.5) E = 12 − Z R |∇ ⊥ ψ | dxdy = 12 V + 12 − Z R |∇ ⊥ ψ ′ | dxdy, from which one obtains an equation for time evolution of the mean flow,(1.6) dVdt = −− Z R ∂ x h ( z ) ψ ′ ( z ) dz, the right-hand side being usually referred to as topographic stress . This last relationcompletes our set of equations,(BQG) ∂ t q + ∇ ⊥ ψ · ∇ q = 0 ,q = ∆ ψ ′ + h + βy,ψ = − V y + ψ ′ , dVdt = − − R R ∂ x h ( z ) ψ ′ ( z ) dz. Since both ψ and ψ ′ can be recovered from V and q , taking into account theboundary conditions (1.2), (1.3) in solving Poisson’s equation (1.4), we will consider( V, q ) as the state variables of the system. This particular choice has the advantageof retaining the active scalar form for the dynamics (1.1) of q .Our study focuses on equilibrium statistical mechanics of (BQG) in the fullinfinite dimensional setting, generalising the well-established theory developed for2-dimensional Euler equations. Besides the total energy E , (BQG) preserve the large-scale enstrophy (1.7) Q ( V, q ) = βV + 12 − Z R ( q − βy ) . Due to the Hamiltonian nature of the fluid dynamics, it is thus expected that theGibbsian ensembles(1.8) dν α,µ ( V, q ) = 1 Z α,µ e − α ( µE ( V,q )+ Q ( V,q )) dV dq, α, µ > , are invariant measures for (BQG). Since Boltzmann’s exponents are quadratic func-tionals of the state variables ( V, q ), these are Gaussian measures. Unfortunately,they are only supported on spaces of distributions –they give null mass to any spaceof functions– so some effort is required to give meaning to the dynamics (BQG) inthe low-regularity regime dictated by ν α,µ . QUILIBRIUM STATISTICAL MECHANICS BAROTROPIC QUASI-GEOSTROPHIC 3
Inspired by works on Euler’s equations, [2, 12], we describe a weak formulation of(BQG) robust enough to admit samples of ν α,µ as fixed-time distributions, and thenproduce by means of a Galerkin approximation scheme such a solution, arriving atour main result: Theorem 1.1.
Let β = 0 and h as above. For any α, µ > there exists a stationarystochastic process ( V t , q t ) t ∈ [0 ,T ] with fixed-time marginals ν α,µ , whose trajectoriessolve (BQG) in the weak vorticity formulation of Definition 3.8. The reader should be aware that, although we often make use of terminology fromProbability theory, there is no noise or external randomness acting on the systemunder consideration, and we are only referring to the fact that an invariant measureof a deterministic evolution can be regarded as a random initial data producing astationary (deterministic) process.As in the case of 2-dimensional Euler’s equations in the Energy-Enstrophy sta-tionary regime, or more generally when fixed time marginals are absolutely contin-uous with respect to space white noise, see [13], uniqueness remains an importantopen problem. We will not discuss uniqueness of solutions of (BQG) in the abovestationary regime; thus, in particular, we are not able to state that the solutions weproduce form a flow, i.e. a one-parameter group of transformations of phase spaceindexed by time.The present paper is structured as follows. In section 2 we collect some prelim-inary material, including a short discussion on regularity regimes in which (BQG)are well-posed. In section 3 we thoroughly discuss the formulation of weak solu-tion required by our low-regularity setting, and finally in section 4 we will proveTheorem 1.1 by approximating the infinite-dimensional stationary solution withfinite-dimensional, stationary Galerkin truncations of (BQG).2.
Definitions and Preliminary Results
We consider mixed boundary conditions on R for the small scale stream function ψ ′ , that is periodicity in the x variable and Dirichlet boundary at y = 0 , π . In orderto simplify Fourier analysis, let us extend the space domain to the 2-dimensionaltorus D = [ − π, π ] with periodic boundary conditions on both x, y variables, ex-tending ψ ′ to D so that it becomes an odd function of y . We still denote points z = ( x, y ) ∈ D .To study (BQG) on D we also extend q, h in the same way; the extension of h might be discontinuous at y = 0, but this will not be relevant in the following.Indeed, it is not difficult to see that equations (BQG) preserve such condition.We also remark that the beta-plane term βy of (1.4) is coherent with the domainextension.Due to the (skew-)symmetry in y variable, it will be convenient to introduce thefollowing set of orthonormal functions of L ( D, C ),( e j ) j ∈ Z , ( e j s k , e j c k ) ( j,k ) ∈ Λ Λ = { ( j, k ) : j ∈ Z , k ∈ N \ { } , } e j ( x ) = 12 π e i jx , s k ( y ) = sin( ky ) , c k ( y ) = cos( ky ) . Since we work with real valued objects, Fourier coefficients relative to modes ( j, k )and ( − j, k ) will always be complex conjugated. With this relation between Fouriercoefficients, { e j , e j s k , e j c k } ( j,k ) ∈ Λ is a Hilbert basis of L = L ( D, R ).Odd functions of y only have non null Fourier coefficients relative to ( e j s k ) ( j,k ) ∈ Λ :we will denote those coefficients, say of ψ ′ , by F j,k ( ψ ′ ) = ˆ ψ ′ j,k = Z D ψ ′ ( x, y ) e − j ( x ) s k ( y ) dxdy. F. GROTTO AND U. PAPPALETTERA so that ψ ′ ( x, y ) = X ( j,k ) ∈ Λ ˆ ψ ′ j,k e j ( x ) s k ( y ) , ˆ ψ ′ j,k = ˆ ψ ′− j,k . For α ∈ R , we denote by H α = W α, ( D, R ) the L ( D, R )-based Sobolev spaces,which enjoy the compact embeddings H α ֒ → H β whenever β < α , the injectionsbeing furthermore Hilbert-Schmidt if α > β + 1. The scale of Sobolev spaces of odddistributions in y , H α = u = X ( j,k ) ∈ Λ ˆ u j,k e j s k : k u k H α = X ( j,k ) ∈ Λ | ˆ u j,k | ( j + k ) α < ∞ , clearly share the same properties. We denote with H the subspace of odd functionsof y in L ( D ), and more generally each H α is a closed subspace of H α . Brackets h· , ·i will denote H -based duality couplings.As a convention, C will denote a positive constant, possibly changing in everyoccurrence even in the same formula and depending only on its eventual subscripts.2.1. Well-posedness regimes.
Our main aim is to give meaning to (BQG) indistributional regimes dictated by the formally invariant Gibbs measures. Beforewe undertake that task, we briefly discuss, for the sake of completeness, moreregular regimes in which our equations are actually well-posed. Let us begin byintroducing the notion of weak solution.
Definition 2.1.
Given ( V , q ) ∈ R × L ∞ ( D ) , we say that ( V ( t ) , q ( t )) t ∈ [0 ,T ] ∈ L ∞ ([0 , T ] , R × D ) is a weak solution to (BQG) with initial datum ( V , q ) if for any ϕ ∈ C ([0 , T ] × D ) it holds Z D ϕ ( T, z ) q ( T, z ) dz − Z D ϕ (0 , z ) q ( z ) dz (2.1) = Z T Z D ( ∂ t ϕ ( s, z ) + ∇ ⊥ ψ ( s, z ) · ∇ ϕ ( s, z )) q ( s, z ) dzds,V ( t ) = V + Z t − Z D h ( z ) ∂ x ψ ′ ( z, s ) dzds, (2.2) ψ = − V y + ψ ′ , q = ∆ ψ ′ + h + βy. (2.3)Thanks to the fact that the equation for q is in the active scalar form, the methodof characteristics produces an existence result: a minor modification of the proofof [21, Ch.2,Theorem 3.1] leads to the following: Proposition 2.2.
Let ( V , q ) ∈ R × L ∞ ( D ) , and consider the Lagrangian formu-lation of (BQG) given by ( ddt φ t ( z ) = ∇ ⊥ ψ ( t, φ t ( z )) φ ( z ) = z , q ( t, z ) = q ( φ − t ( z )) , (2.4) together with equations (2.2) , (2.3) . There exists a unique solution ( φ, V, q ) of suchsystem, and moreover ( V, q ) is a weak solution of (BQG) in the sense of Defini-tion 2.1. The argument ultimately relies on the fact that ∇∇ ⊥ ∆ − is a singular kernel ofCalder´on-Zygmund type, so that its associated convolution operator is a bounded QUILIBRIUM STATISTICAL MECHANICS BAROTROPIC QUASI-GEOSTROPHIC 5 linear map from L ∞ ( D ) to the Bounded Mean Oscillation (BMO) space. Thisimplies that the vector field ∇ ⊥ ψ = V (cid:18) (cid:19) + ∇ ⊥ ∆ − ( q − h − βy )has gradient in BMO, and thus it is log-Lipschitz ( cfr. [21, Ch.2,Lemma 3.1]).The vector field ∇ ⊥ ψ then satisfies the Osgood condition ([22]) for the associatedCauchy problem (2.4), which is thus well-posed; it is not difficult to check that q ( t, z ) = q ( φ − t ( z )) satisfies the weak formulation (2.1). All these ideas date backto the celebrated work of Judoviˇc, [17], concerning well-posedness of Euler equationsfor initial vorticity in L ∞ . Proposition 2.3.
For any ( V , q ) ∈ R × L ∞ ( D ) , the weak solution of (BQG) inthe sense of Definition 2.1 is unique. Uniqueness can be obtained by energy estimates at the level of the velocityvector field v = ∇ ⊥ ψ . Such estimates are performed for instance in [18, Theorem8.2] for the 2D Euler equations ( h = 0, β = 0), and again they rely on the factthat ∇∇ ⊥ ψ is in BMO to arrive at Gronwall-type inequalities, something which isnot influenced by the addition of regular terms such as h + βy to q . We refer to [4]for a thorough discussion of uniqueness for a large class of active scalar equationssharing similar features. We also mention the recent work [5], where the argumentswe just sketched are applied to a barotropic quasi-geostrophic model closely relatedto ours: the difference consists in impermeable boundary conditions on the wholeboundary and the presence of a free surface effect instead of the fixed topography h we consider. The paper [6], moreover, is devoted to multi-layered barotropicquasi-geostrophic equations.2.2. Conserved Quantities and Gibbsian Measures.
Smooth solutions of (BQG)preserve the first integrals energy and enstrophy, E = 12 V + 12 − Z (cid:12)(cid:12) ∇ ⊥ ψ ′ (cid:12)(cid:12) , Q = βV + 12 − Z ( q − βy ) . We refer again to [20, Section 1.4] for a detailed discussion of conserved quantities.As already remarked, energy E can be seen as a functional of variables ( V, q ) bysolving the Poisson equation (1.4).In (1.8) above, we have formally introduced the Gibbsian measures dν α,µ ( V, q ) = 1 Z α,µ e − α ( µE + Q ) dV dq, α, µ > , the expression meaning that we consider the Gaussian measure whose inverse co-variance operator is given by the quadratic functional α ( µE + Q ) of ( V, q ).Let us now provide a rigorous framework: we define ν α,µ as the joint law of theGaussian variable V ∼ N (cid:16) − βµ , αµ (cid:17) and the Gaussian random field q indexed by H with mean and covariance given by, for f, g ∈ H , E [ h q, f i ] = h ¯ q, f i = (cid:28) µµ − ∆ h + βy, f (cid:29) E [ h q, f i h q, g i ] − h ¯ q, f i h ¯ q, g i = (cid:28) f, α (1 − µ ) g (cid:29) ,V and q being independent. Notice that α only plays a role in the variance. Thelink between the latter and the formal definition (1.8) is perhaps clearer thinkingof the formal reference measure dV dq as the infinite product of uniform measureson the infinite product space R × C Λ of Fourier modes (modulo the relation ˆ q j,k = F. GROTTO AND U. PAPPALETTERA ˆ q − j,k ), and considering the Boltzmann exponent e − α ( µE + Q ) as the infinite productof densities given by the Parseval expansion of the quadratic form α ( µE + Q ).In order to deal with centred variables we set(2.5) U = V + βµ , ω = q − ¯ q, the new variables satisfying equations of motion(2.6) ( ∂ t ω + ∇ ⊥ ∆ − ω · ∇ ω + Lω = 0 dUdt = − R D h∂ x ∆ − ω , where Lω collects all affine terms in ω , Lω = (cid:18) U − βµ (cid:19) ∂ x ω + U µ∂ x µ − ∆ h + ∇ ⊥ µ − ∆ h · ∇ ω + ∇ ⊥ ∆ − ω · µ ∇ µ − ∆ h + β∂ x ∆ − ω. The equivalence of (2.6) and (BQG) is intended for smooth solutions.We now define the purely quadratic pseudoenergy : for µ > S µ ( U, ω ) = µ U + 12 Z D ( ω − µ ∆ − ω ) ωdxdy so that the law of ( V, ω ) under ν α,µ is given by(2.8) dη α,µ ( U, ω ) = 1˜ Z α,µ e − αS µ ( U,ω ) dU dω, the latter to be interpreted analogously to the definition of ν α,µ above, (1.8): itis the joint law of the real Gaussian variable U ∼ N (cid:16) , αµ (cid:17) and the independentcentred Gaussian field ω with covariance operator α − (1 − µ ∆ − ) − . In order tolighten the exposition, we will abuse notation denoting by η α,µ ( dω ) the law of ω under η α,µ , and analogously for U . We will also denote σ := Z U dν α,µ ( U, ω ) = 1 αµ , σ j,k := Z | ˆ ω j,k | dη α,µ ( U, ω ) = j + k α ( µ + j + k ) . Indeed, under η α,µ the Fourier modes ˆ ω j,k are independent centred Gaussian vari-ables with the above covariances; notice that they are complex valued, but subjectto the condition ¯ˆ ω j,k = ˆ ω − j,k .We have considered ω under η α,µ as a Gaussian random field indexed by H (its reproducing kernel Hilbert space ): it is well known that it can also be identifiedwith a random distribution in a larger Hilbert space into which H has an Hilbert-Schmidt embedding, such as H − − δ for any δ >
0. In other terms, since all σ j,k ,( j, k ) varying in Λ, are of order 1, the random Fourier series ω = P ( j,k ) ∈ Λ ˆ ω j,k e j s k converges in L ( η α,µ ) in H − − δ for any δ >
0, but not for δ ≥ Lemma 2.4.
For any δ > , ( R × H − − δ , R × H , η α,µ ) is a (complex) abstractWiener space; equivalently, under η α,µ , ω can be identified with a H − − δ -valuedGaussian random variable. We refer to [10] for a complete treatment of the Gaussian analysis underlyingthe above discussion, and to [3] for its application to Enstrophy measures of 2-dimensional Euler equations.
QUILIBRIUM STATISTICAL MECHANICS BAROTROPIC QUASI-GEOSTROPHIC 7 Weak Solutions for Low-Regularity Marginals
We now discuss how to interpret (2.6) in the case when, at a fixed time, (
U, ω ) isa sample of η α,µ . Indeed, as we remarked above, in that case ω can be identified atbest as a distribution in H − − δ , δ >
0, and thus the main concern is the nonlinearterm of the evolution equation, the affine term Lω being easily defined pathwise asa distribution of class H − − δ .3.1. Fourier Expansion of the Nonlinear Term.
Let us fix δ >
0, and considerthe coupling between the nonlinear term ∇ ⊥ ∆ − ω · ∇ ω and a smooth test function φ ∈ C ∞ ( D ) . If ω ∈ H − − δ , we can define the tensor product ω ⊗ ω as a distributionon D × D via(3.1) h ω ⊗ ω, ϕ ⊗ ψ i := h ω, ϕ i h ω, ψ i , ϕ, ψ ∈ C ∞ ( D ) , where ϕ ⊗ ψ ( z, z ′ ) := ϕ ( z ) ψ ( z ′ ); it is easily observed that the resulting distribution ω ⊗ ω is of class H − − δ ( D × D ) (with H α ( D × D ) we denote the closed subspaceof H α ( D × D ) generated by vectors e j s k ⊗ e j ′ s k ′ ), so the expression(3.2) Z D × D H ( z, z ′ ) ω ( dz ) ω ( dz ′ ) = h ω ⊗ ω, H i is well defined via duality for every H ∈ H δ ( D × D ). Now, given any smoothtest function φ ∈ C ∞ ( D ), we look for a suitable function H φ such that(3.3) Z D ∇ ⊥ ∆ − ω ( z ) · ∇ ω ( z ) φ ( z ) dz = Z D × D H φ ( z, z ′ ) ω ( z ) ω ( z ′ ) dzdz ′ . We perform the computation in Fourier series. Let us also recall that we denotepoints of D by z = ( x, y ) , z ′ = ( x ′ , y ′ ). We thus have ω ( z ) = X ( j,k ) ∈ Λ ˆ ω j,k e j ( x ) s k ( y ) , ∆ − ω ( z ) = − X ( j,k ) ∈ Λ ˆ ω j,k j + k e j ( x ) s k ( y ) , ∇ ω ( z ) = X ( j,k ) ∈ Λ (cid:18) i js k ( y ) kc k ( y ) (cid:19) ˆ ω j,k e j ( x ) , ∇ ⊥ ∆ − ω ( z ) = X ( j,k ) ∈ Λ (cid:18) kc k ( y ) − i js k ( y ) (cid:19) ˆ ω j,k j + k e j ( x ) , ∇ ⊥ ∆ − ω ( z ) · ∇ ω ( z ) = X ( j,k ) ∈ Λ X ( j ′ ,k ′ ) ∈ Λ (( jk ′ − j ′ k ) s k + k ′ ( y ) + ( j ′ k + jk ′ ) s k − k ′ ( y )) × ˆ ω j,k ˆ ω j ′ ,k ′ j + k ) e j + j ′ ( x ) , so that equation (3.3) becomes Z D ∇ ⊥ ∆ − ω ( z ) · ∇ ω ( z ) φ ( z ) dz = X ( j,k ) ∈ Λ X ( j ′ ,k ′ ) ∈ Λ (cid:16) ( jk ′ − j ′ k ) ˆ φ − j − j ′ ,k + k ′ + ( j ′ k + jk ′ ) ˆ φ − j − j ′ ,k − k ′ (cid:17) ˆ ω j,k ˆ ω j ′ ,k ′ j + k )= X ( j,k ) ∈ Λ X ( j ′ ,k ′ ) ∈ Λ (cid:16) ( jk ′ − j ′ k ) ˆ φ − j − j ′ ,k + k ′ + ( j ′ k + jk ′ ) ˆ φ − j − j ′ ,k − k ′ (cid:17) × (cid:18) j + k − j ′ + k ′ (cid:19) ˆ ω j,k ˆ ω j ′ ,k ′ X ( j,k ) ∈ Λ( j ′ ,k ′ ) ∈ Λ F − j,k F − j ′ ,k ′ ( H φ )ˆ ω j,k ˆ ω j ′ ,k ′ , F. GROTTO AND U. PAPPALETTERA the second step consisting in a symmetrisation with respect to indices ( j, k ) and( j ′ , k ′ ). The last equality is the Fourier expansion of the right-hand side of (3.3),and becomes our definition of H φ : F j,k F j ′ ,k ′ H φ := (cid:16) ( j ′ k − jk ′ ) ˆ φ j + j ′ ,k + k ′ − ( j ′ k + jk ′ ) ˆ φ j + j ′ ,k − k ′ (cid:17) × (cid:18) j + k − j ′ + k ′ (cid:19)
14 i , where F j,k F j ′ ,k ′ is an abbreviation for the more rigorous notation F j,k ⊗ F j ′ ,k ′ , theFourier projector on e j s k ⊗ e j ′ s k ′ . We also adopt the conventionˆ φ j + j ′ ,k − k ′ := − ˆ φ j + j ′ ,k ′ − k whenever k − k ′ < . So far, H φ is defined only as a formal Fourier series: the forthcoming Lemmadiscusses the convergence of the latter, i.e. the regularity of H φ . Lemma 3.1.
For every φ ∈ H , H φ ∈ H ( D × D ) .Proof. To ease notation, we denote l = ( j, k ) and l ′ = ( j ′ , k ′ ). We have that H φ ∈ H ( D × D ) if and only if(3.4) X l,l ′ ∈ Λ |F l F l ′ ( H φ ) | < ∞ . The Fourier coefficients of H φ are given by two summands which we estimate sep-arately. The first one is F j + j ′ ,k + k ′ ( φ )( j ′ k − jk ′ ) (cid:18) j + k − j ′ + k ′ (cid:19) = F l + l ′ ( φ ) (cid:0) − l ⊥ · l ′ (cid:1) (cid:18) | l | − | l ′ | (cid:19) , where | l | = j + k , and similarly for l ′ ; taking squares and summing over l + l ′ = m ∈ Λ gives us(3.5) X m ∈ Λ |F m ( φ ) | X l ∈ Λ l = m (cid:18) l ⊥ · ( m − l ) (cid:18) | l | − | m − l | (cid:19)(cid:19) . We now resort to the following inequalities: l ⊥ · ( m − l ) = l ⊥ · m ≤ | l || m | , (3.6) | m − l | − | l | = m · ( m − l ) ≤ | m || m − l | ≤ | m | (cid:0) | m − l | + | l | (cid:1) / , (3.7)so that the inner summation in (3.5) can be estimated with X l ∈ Λ l = m (cid:18) | l | | m | | m − l | | l | | m − l | + | l | | m | | l | | m − l | (cid:19) = | m | X l ∈ Λ l = m (cid:18) | l | | m − l | + 1 | m − l | (cid:19) ≤ | m | X l ∈ Λ | l | . Modulo a multiplicative constant, (3.5) is therefore smaller or equal to X m ∈ Λ |F m ( φ ) | | m | , which is finite as soon as φ ∈ H . The other contribution is given by the terms ofthe form F j + j ′ ,k − k ′ ( φ )( j ′ k + jk ′ ) (cid:18) j + k − j ′ + k ′ (cid:19) , QUILIBRIUM STATISTICAL MECHANICS BAROTROPIC QUASI-GEOSTROPHIC 9 which after the change of variables ( j, k, j ′ , k ′ ) ( j, k, − j ′ , k ′ ) becomes F l − l ′ ( φ ) (cid:0) l ⊥ · l ′ (cid:1) (cid:18) | l | − | l ′ | (cid:19) , which can be estimated in a similar fashion taking the modulo square and summingover l − l ′ = m ∈ Λ. Thus (3.4) is proved. (cid:3)
Remark . Even though we will not need it in the following, for every δ < H φ ∈ H δ ( D × D ) if φ ∈ H δ . This in fact isthe optimal Sobolev regularity, since in general H φ / ∈ H δ ( D × D ) for δ ≥
1, evenfor more regular φ . Indeed, for φ ( x, y ) = sin( y ) the Fourier coefficients of H φ aregiven by F j,k F j ′ ,k ′ ( H φ ) = { j + j ′ =0 } { k − k ′ =1 } j (1 − k )( j + k )( j + ( k − ) , therefore H φ ∈ H δ ( D × D ) if and only if X ( j,k ) ∈ Λ (cid:0) j + k + ( k − (cid:1) δ j (1 − k ) ( j + k ) ( j + ( k − ) < ∞ , but the sum above can be estimated from below (modulo a positive multiplicativeconstant) by X ( j,k ) ∈ Λ j k ( j + k ) − δ , the latter converging if and only if δ < H φ does not belong to H δ ( D × D ), it is not possible todefine the nonlinear term of (2.6) pathwise , that is fixing a realisation of ω under η α,µ and taking products of distributions. It is at this point that we make essentialuse of the probabilistic approach to invariant measures.3.2. The Nonlinear Term as a Stochastic Integral.
Thanks to the peculiarform of the fluid-dynamic nonlinearity, which in our setting is reflected by thecoefficients of H φ , when ω is sampled from the Gaussian measure η α,µ it is possibleto define the nonlinear term as a (double) stochastic integral, that is, as an L ( η α,µ )-limit of suitable approximations.The following result finds analogues in [2, Lemma 1.3.2], see also [1], and in [12,Theorem 8] or the related [9, 14, 16, 13], all dealing with stationary solutions of2-dimensional Euler equations. Proposition 3.3.
Let H ∈ H ( D × D ) be a symmetric function. Consider functions ( H n ) n ∈ N ⊂ H δ ( D × D ) such that, for every ( j, k ) , ( j ′ , k ′ ) ∈ Λ , (3.8) lim n →∞ X ( j,k ) ∈ Λ F j,k F j,k ( H n ) σ j,k = 0 , F j,k F j ′ ,k ′ ( H n ) = F j ′ ,k ′ F j,k ( H n ) , and suppose that the sequence H n approximates H in the following sense: (3.9) lim n →∞ X ( j,k ) ∈ Λ( j ′ k ′ ) ∈ Λ ( F j,k F j ′ ,k ′ ( H n − H )) σ j,k σ j ′ ,k ′ = 0 . Under η α,µ , the sequence of random variables h ω ⊗ ω, H n i defined by (3.1) , con-verges in mean square. Moreover, the limit does not depend on the approximatingsequence H n . Proof.
To ease notation we denote l = ( j, k ) and l ′ = ( j ′ , k ′ ). For any function H ∈ H ( D × D ), we compute E h h ω ⊗ ω, H i i = E X l,l ′ ∈ Λ m,m ′ ∈ Λ F l F l ′ ( H ) F m F m ′ ( H )ˆ ω l ˆ ω l ′ ˆ ω m ˆ ω m ′ = X l,l ′ ∈ Λ m,m ′ ∈ Λ F l F l ′ ( H ) F m F m ′ ( H ) E (cid:2) ˆ ω l ˆ ω l ′ ˆ ω m ˆ ω m ′ (cid:3) . By Wick-Isserlis formula the expected value in the last summand is given by E (cid:2) ˆ ω l ˆ ω l ′ ˆ ω m ˆ ω m ′ (cid:3) = σ l σ m δ l,l ′ δ m,m ′ + σ l σ l ′ δ l,m δ l ′ ,m ′ + σ l σ l ′ δ l,m ′ δ l ′ ,m . Substituting and using the relations (3.8) one gets E h h ω ⊗ ω, H i i = X l ∈ Λ F l ( H ) σ l ! + 2 X l,l ′ ∈ Λ F l F l ′ ( H ) σ l σ l ′ . If conditions (3.8) and (3.9) hold, applying the latter equation to differences H m − H n we obtain that the sequence of random variables h ω ⊗ ω, H n i is a Cauchysequence in L (Ω). The independence of limit from the sequence ( H n ) follows fromtriangular inequality and (3.9). (cid:3) Remark . In [12], conditions (3.8) and (3.9) are replaced by H n symmetric, lim n →∞ Z H n ( z, z ) dz = 0 , lim n →∞ Z Z ( H n ( z, z ′ ) − H ( z, z ′ )) dzdz ′ = 0 , where integration is performed over the 2-dimensional torus. These conditions aresimpler than ours since we deal with coloured noise η α,µ rather then space whitenoise.Consider now a test function φ ∈ C ∞ ( D ): Proposition 3.3 allows us to define thenonlinearity (3.3) as the L ( η α,µ )-limit of D ω ⊗ ω, H nφ E for any sequence H nφ ap-proximating H φ in the above sense (for instance, progressive truncations of Fourierseries). To emphasize the peculiarity of its definition, we adopt a special notationfor this object. Definition 3.5.
For any H ∈ H ( D × D ) , and H n as in Proposition 3.3, (3.10) h ω ⋄ ω, H i := L ( η α,µ ) − lim n →∞ h ω ⊗ ω, H n i . We chose a distinct symbol because if we consider a smooth H and confront thenew object we define and coupling with tensor products (3.1), a straightforwardcomputation reveals that h ω ⋄ ω, H i = h ω ⊗ ω, H i − X ( j,k ) ∈ Λ F j,k F j,k ( H ) σ j,k . Indeed, let H n be the following approximation of H : F j,k F j ′ ,k ′ ( H n ) := F j,k F j ′ ,k ′ ( H ) − Snσ ,k { j = j ′ =0 ,k = k ′ =1 ,...,n } , QUILIBRIUM STATISTICAL MECHANICS BAROTROPIC QUASI-GEOSTROPHIC 11 where S := P ( j,k ) ∈ Λ F j,k F j,k ( H ) σ j,k < ∞ . Hence h ω ⋄ ω, H i = h ω ⊗ ω, H i − lim n →∞ n X k =1 ˆ ω ,k Snσ ,k = h ω ⊗ ω, H i − S. as an equality between random variables in L ( η α,µ ). Notice that the last summandin the latter expression diverges for a generic H ∈ H ( D × D ), according to thefact that the coupling with tensor product ω ⊗ ω can not be defined in that case. Remark . The present paragraph takes its name because the coupling h ω ⋄ ω, H i we defined in fact corresponds to the double It¯o-Wiener integral of H with respectto the Gaussian measure η α,µ .We now extend Proposition 3.3 to manage stochastic processes, rather than justrandom variables. Proposition 3.7.
On a probability space (Ω , F , P ) consider a stochastic process ( ω t ) t ∈ [0 ,T ] with trajectories in C ([0 , T ] , H − − δ ) such that the law of ω t is η α,µ ( dω ) for every t ∈ [0 , T ] . Let ( H nφ ) n ∈ N ⊆ H δ ( D × D ) be an approximation of H φ inthe sense of Proposition 3.3. Then the sequence of processes t D ω t ⊗ ω t , H nφ E converges in L ([0 , T ] , L ( P )) . Moreover, the limit does not depend on the approx-imating functions H nφ . The proof is a direct consequence of Proposition 3.3 and stationarity of theprocess ω . We are now ready to give the definition of solution we mentioned inTheorem 1.1. Definition 3.8.
A stochastic process ( U t , ω t ) t ∈ [0 ,T ] defined on a probability space (Ω , F , P ) , with trajectories in C ([0 , T ]; R × H − − δ ) , solves the reduced form (2.6) of (BQG) in the weak vorticity formulation if, for every test function φ ∈ C ∞ ( D ) , P -almost surely, for every t ∈ [0 , T ] , h ω t , φ i = h ω , φ i + Z t h ω s ⋄ ω s , H φ i ds + Z t h Lω s , φ i ds, (3.11) U t − U = Z t − Z D h (cid:18) ∂ x ∆ − ω s + ∂ x µ − ∆ h (cid:19) dzds, (3.12) where the process s
7→ h ω s ⋄ ω s , H φ i is defined by Proposition 3.7. In the remainder of the paper we will focus on equations for centred variables(
U, ω ), and thus prove the following corresponding version of Theorem 1.1, fromwhich the latter is straightforwardly recovered.
Theorem 3.9.
Let β = 0 and h as above. For any α, µ > there exists a stationarystochastic process ( U t , ω t ) t ∈ [0 ,T ] with trajectories in C ([0 , T ] , R × H − − δ ) and fixed-time marginals η α,µ , whose trajectories solve (2.6) in the weak vorticity formulationof Definition 3.8. A Galerkin Approximation Scheme
Let us define the finite-dimensional projection of L ( D ) onto the finite set ofmodes Λ N = (cid:8) ( j, k ) ∈ Λ : j + k ≤ N (cid:9) ,(4.1) Π N : L ( D ) ∋ f Π N f := X ( j,k ) ∈ Λ N F j,k ( f ) e j s k ∈ H N , where we can identify the finite dimensional codomain with H N = X ( j,k ) ∈ Λ N ξ j,k e j s k : ξ j,k = ξ − j,k ≃ (cid:8) ξ ∈ C Λ N : ξ j,k = ξ − j,k (cid:9) ≃ C ˜Λ N , ˜Λ N = (cid:8) ( j, k ) ∈ Λ : j ≥ , j + k ≤ N (cid:9) . Truncated Barotropic Quasi-Geostrophic Equations.
Let h N = Π N h :we consider the following truncated version of (2.6),(4.2) ( ∂ t ω N + Π N (cid:0) ∇ ⊥ ∆ − ω N · ∇ ω N (cid:1) + L N ω N = 0 dU N dt = − R D h N ∂ x ∆ − ω N , with L N ω N collecting affine terms in ω N : L N ω N = (cid:18) U N − βµ (cid:19) ∂ x ω N + U N µ∂ x µ − ∆ h N + Π N (cid:18) ∇ ⊥ µ − ∆ h N · ∇ ω N (cid:19) + Π N (cid:18) ∇ ⊥ ∆ − ω N · µ ∇ µ − ∆ h N (cid:19) + β∂ x ∆ − ω N . For the sake of simplicity, equations (4.2) can be rewritten in the compact form(4.3) ∂ t ( U N , ω N ) = B N ( U N , ω N ) , where ω N is the vector with components (ˆ ω Nj,k ) ( j,k ) ∈ ˜Λ N , and B N : R × H N → R × H N . Let us stress the fact that we can reduce ourselves to consider Fouriermodes in ˜Λ N thanks to ˆ ω Nj,k = ˆ ω N − j,k .Galerkin approximants (4.2) are globally well-posed, and truncation is such thatthey preserve the following projection of η α,µ , η Nα,µ = (Id R , Π N ) η α,µ . In other words, under η Nα,µ , U N has the same Gaussian distribution of U under η α,µ , while ω N is the projection of ω under η α,µ . More explicitly, we can define η Nα,µ by density with respect to the product Lebesgue measure on H N ≃ ˜Λ N , dη Nα,µ ( U N , ω N ) = 1 Z Nα,µ e − αµ ( U N ) dU N × Y ( j,k ) ∈ ˜Λ N exp (cid:18) − α | ˆ ω Nj,k | (cid:18) µj + k (cid:19)(cid:19) d ˆ ω Nj,k . Proposition 4.1.
For η Nα,µ -almost every initial datum ( U N , ω N ) , there exists aunique solution ( U Nt , ω Nt ) ∈ C ∞ ([0 , ∞ ) , R × H N ) to the ordinary differential equa-tion (4.2) . Moreover, the global flow preserves η Nα,µ .Proof.
The components of vector field B N are polynomials of U N , ˆ ω Nj,k , ( j, k ) ∈ ˜Λ N ,and thus B N and its derivatives have finite moments of all orders under η Nα,µ . Thethesis then follows from non-explosion results in [8, Section 3], as soon as we checkthat B N has null divergence with respect to η Nα,µ , i.e. η Nα,µ B N = ∂ U N B NU N + X ( j,k ) ∈ ˜Λ N ∂ j,k B Nj,k − αµU N B NU N − α X ( j,k ) ∈ ˜Λ N (cid:18) µj + k (cid:19) ˆ ω Nj,k B Nj,k , subscripts denoting components (and derivatives) relative to U N or ˆ ω j,k . In fact, [8]treats the case of a standard Gaussian measure on R n , but their results are easily QUILIBRIUM STATISTICAL MECHANICS BAROTROPIC QUASI-GEOSTROPHIC 13 extended to our case. Showing that B N is divergence-free with respect to η Nα,µ can be done by direct computation: the full computation in the case of completelyperiodic geometry can be found in [19, Section 6.2], to which we refer, the differenceswith our case being minimal. (cid:3)
The Truncated Nonlinear Term.
In the finite-dimensional Galerkin trun-cation (4.2) we can repeat the arguments of subsection 3.1 to cast couplings of thenonlinear term into a double integral formulation. For any φ ∈ C ∞ ( D ), expandingin Fourier series the equality (cid:10) Π N (cid:0) ∇ ⊥ ∆ − ω N · ∇ ω N (cid:1) , φ (cid:11) H = (cid:10) ∇ ⊥ ∆ − ω N · ∇ ω N , Π N φ (cid:11) H = (cid:10) ω N ⊗ ω N , H Nφ (cid:11) H ( D × D ) , we deduce a Fourier expansion of H Nφ , F j,k F j ′ ,k ′ H Nφ = ( j ′ k − jk ′ ) ˆ φ j + j ′ ,k + k ′ { ( j + j ′ ,k + k ′ ) ∈ Λ N } (cid:18) j + k − j ′ + k ′ (cid:19) − ( j ′ k + jk ′ ) ˆ φ j + j ′ ,k − k ′ { ( j + j ′ ,k − k ′ ) ∈ Λ N } (cid:18) j + k − j ′ + k ′ (cid:19) , the computation being completely analogous to the one in subsection 3.1.4.3. Compactness Results.
The first step towards taking the limit of Galerkinapproximants as N → ∞ is to provide estimates from which we can deduce relativecompactness of approximations.We begin by reviewing a deterministic compactness criterion due to Simon, whichallows us to control separately time and space regularity, in the spirit of Aubin-Lionscompactness Lemma. We refer to [23] for the result and the required generalitieson Banach-valued Sobolev spaces. Proposition 4.2 (Simon) . Assume that • X ֒ → B ֒ → Y are Banach spaces such that the embedding X ֒ → Y iscompact and there exists < θ < such that for all v ∈ X ∩ Y k v k B ≤ M k v k − θX k v k θY ; • s , s ∈ R are such that s θ = (1 − θ ) s + θs > .If F ⊂ W is a bounded family in W = W s ,r ([0 , T ] , X ) ∩ W s ,r ([0 , T ] , Y ) with r , r ∈ [0 , ∞ ] , and moreover s ∗ = s θ − − θr − θr > , then if F is relatively compact in C ([0 , T ] , B ) . Let us specialise this result to our framework. Take X = R × H − − δ/ , B = R × H − − δ , Y = R × H − − δ , with δ >
0: by Gagliardo-Niremberg estimates the interpolation inequality is sat-isfied with θ = δ/
2. Let us take moreover s = 0, s = 1, r = 2 and r = q ≥
1; ifwe can take q large such that s ∗ = δ − − δ q > , then the hypothesis are satisfied and obtain: Corollary 4.3.
Let δ > . If a family of functions { v n } ⊂ L q ([0 , T ] , R × H − − δ/ ) ∩ W , ([0 , T ] , R × H − − δ ) is bounded for any q ≥ , then it is relatively compact in C ([0 , T ] , R × H − − δ ) .As a consequence, if a sequence of stochastic processes u n : [0 , T ] → R × H − − δ , n ∈ N , defined on a probability space (Ω , F , P ) is such that, for any q ≥ , thereexists a constant C T,δ,q for which (4.4) sup n E h k u n ( t ) k pL q ([0 ,T ] , R ×H − − δ/ ) + k u n k W , ([0 ,T ] , H − − δ ) i ≤ C T,δ,q , then the laws of u n on C ([0 , T ] , R × H − − δ ) are tight. For the sake of completeness we remark that the second, probabilistic part ofthe latter statement follows from the deterministic one and a simple application ofChebyshev inequality.We want to apply Corollary 4.3 to the sequence of finite dimensional Galerkinapproximations we built in Proposition 4.1. To obtain the uniform bound (4.4), letus begin with the “space regularity” part: by stationarity of the process ( U N , ω N )we can swap expectations and time integrals, so that E h(cid:13)(cid:13) U N (cid:13)(cid:13) pL q ([0 ,T ]) + (cid:13)(cid:13) ω N (cid:13)(cid:13) pL q ([0 ,T ] , H − − δ/ ) i ≤ T E h(cid:12)(cid:12) U N (cid:12)(cid:12) p + (cid:13)(cid:13) ω N (cid:13)(cid:13) p H − − δ/ i ≤ T Z (cid:0) | U | p + k ω k p H − − δ/ (cid:1) dη α,µ ( dU, dω ) ≤ C T,p,α,µ . As for bounds on time regularity: starting with U N , by the evolution equation k U N k W , ([0 ,T ]) = k U N k L ([0 ,T ]) + (cid:13)(cid:13)(cid:13)(cid:13) dU N dt (cid:13)(cid:13)(cid:13)(cid:13) L ([0 ,T ]) ≤ Z T | U Nt | + (cid:12)(cid:12)(cid:12)(cid:12) − Z D h N ∂ x ∆ − ω Nt (cid:12)(cid:12)(cid:12)(cid:12) ! dt, from which we deduce, using that ω N has marginals η Nα,µ ( dω N ) for every fixed time t , and that h ∈ C ∞ ( D ), E h k U N k W , ([0 ,T ]) i ≤ C T,h (cid:0) E (cid:2) k ω N k H − − δ (cid:3)(cid:1) ≤ C T,h (cid:0) E (cid:2) k ω k H − − δ (cid:3)(cid:1) ≤ C T,α,µ,h . Let us now focus on time regularity of ω N : we have k ω N k W , ([0 ,T ] , H − − δ ) = k ω N k L ([0 ,T ] , H − − δ ) + k ∂ t ω N k L ([0 ,T ] , H − − δ ) ≤ Z T (cid:0) k ω Nt k H − − δ + k Π N (cid:0) ∇ ⊥ ∆ − ω Nt · ∇ ω Nt (cid:1) k H − − δ + k L N ω Nt k H − − δ (cid:1) dt. The affine term is controlled at any fixed time t by E (cid:2) k L N ω Nt k H − − δ (cid:3) ≤ C α,µ,h (cid:0) E (cid:2) k ω N k H − − δ (cid:3)(cid:1) ≤ C α,µ,h (cid:0) E (cid:2) k ω k H − − δ (cid:3)(cid:1) ≤ C α,µ,h . The quadratic term is the one forcing us to consider a large Hilbert space such as H − − δ . As above, we denote m = ( j, k ) ∈ Λ N . We set φ m = e j s k and consider E h(cid:10) ω N ⊗ ω N , H Nφ m (cid:11) i = E X l,l ′ ∈ Λ N F l F l ′ ( H φ m )ˆ ω Nl ˆ ω Nl ′ , QUILIBRIUM STATISTICAL MECHANICS BAROTROPIC QUASI-GEOSTROPHIC 15 where, by the expansion we derived in subsection 4.2, F l F l ′ ( H Nφ m ) = − l ⊥ · l ′ { l + l ′ = m } (cid:18) | l | − | l ′ | (cid:19) + l ⊥ · l ′ { l − l ′ = m } (cid:18) | l | − | l ′ | (cid:19) . We can consider only the first contribution of the latter sum, since, up to a constant,we can bound the contribution of the sum with the contributions of the sole firstterm, similarly to what we did in the proof of Lemma 3.1. We obtain: E h(cid:10) ω N ⊗ ω N , H Nφ m (cid:11) i ≤ C X l,h ∈ Λ N l,h = m l ⊥ · ( m − l ) (cid:18) | l | − | m − l | (cid:19) (4.5) × h ⊥ · ( m − h ) (cid:18) | h | − | m − h | (cid:19) E h ˆ ω Nl ˆ ω Nm − l ˆ ω Nh ˆ ω Nm − h i . By Wick-Isserlis Formula the expected value on the right-hand side is given by E h ˆ ω Nl ˆ ω Nm − l ˆ ω Nh ˆ ω Nm − h i = σ l σ h δ l,m − l δ h,m − h + σ l σ m − l δ l,h δ m − l,m − h + σ l σ h δ l,m − h δ m − l,h = σ l σ h δ l,m − l δ h,m − h + σ l σ m − l δ l,h + σ l σ h δ l,m − h . (4.6)Notice that if l = m − l we have l ⊥ ( m − l ) = 0, hence the first summand in (4.6)does not play any role in the computation of (4.5). Moreover, it is easy to checkthat the second and third terms give the same contribution, since l ⊥ · h = − h ⊥ · l .Therefore, applying inequalities (3.6),(3.7), E h(cid:10) ω N ⊗ ω N , H Nφ m (cid:11) i ≤ C X l ∈ Λ N l = m σ l σ m − l (cid:18) l ⊥ · ( m − l ) (cid:18) | l | − | m − l | (cid:19)(cid:19) ≤ C X l ∈ Λ N l = m σ l σ m − l (cid:18) | l | | m | | m − l | | l | | m − l | + | l | | m | | l | | m − l | (cid:19) = C | m | X l ∈ Λ N l = m σ l σ m − l (cid:18) | l | | m − l | + 1 | m − l | (cid:19) ≤ C | m | X l ∈ Λ N σ l σ m − l | l | . Recall now the expression for σ l : σ l = | l | α ( µ + | l | ) , which is smaller than α − for every l . Therefore P l ∈ Λ N σ l σ m − l | l | is bounded formabove uniformly in m ∈ Λ N , N ∈ N . Hence E (cid:2) k Π N (cid:0) ∇ ⊥ ∆ − ω N · ∇ ω N (cid:1) k H − − δ (cid:3) ≤ C X m ∈ Λ N | m | ) δ E h(cid:10) ω N ⊗ ω N , H Nφ m (cid:11) i ≤ c δ X m ∈ Λ N | m | (1 + | m | ) δ ≤ C δ , where C δ is a finite constant which does not depend on N . All in all, we arrive to E h k ω N k W , ([0 ,T ] , H − − δ ) i ≤ C T,α,µ,h (cid:0) E (cid:2) k ω k H − − δ (cid:3)(cid:1) . The estimates made so far, combined with Corollary 4.3, lead us finally to:
Lemma 4.4.
The laws Θ Nα,µ of the sequence of processes u N = ( U Nt , ω Nt ) t ∈ T definedby Proposition 4.1 are tight on C ([0 , T ] , R × H − − δ ) . The Continuous Limit.
By Prokhorov theorem there exists a subsequenceof Θ
Nα,µ –with a slight abuse of notation we will denote it with the same symbol–weakly converging to a probability measure Θ α,µ on C ([0 , T ] , R × H − − δ ). BySkorokhod theorem, there exists a new probability space ( ˜Ω , ˜ F , ˜ P ) and randomvariables ˜ u N , ˜ u with values in C ([0 , T ] , R × H − − δ ) such that: • the law of ˜ u N (resp. ˜ u ) is Θ Nα,µ (resp. Θ α,µ ); • ˜ u N converges to ˜ u ˜ P -almost surely.In order to lighten notation, we will drop tilde superscripts in the following.The aim of this final paragraph is to prove that the stochastic process u is aweak solution of (BQG) in the sense of Definition 3.8, thus concluding the proof ofTheorem 3.9. First of all, we make the following fundamental observation. Lemma 4.5.
The Galerkin approximations u N = ( U N , ω N ) solve (4.2) in the senseof Definition 3.8. More precisely, given any test function φ ∈ C ∞ ( D ) , (4.7) (cid:10) ω Nt , φ (cid:11) = (cid:10) ω N , φ (cid:11) + Z t (cid:10) ω Ns ⊗ ω Ns , H φ (cid:11) ds + Z t (cid:10) L N ω Ns , φ (cid:11) ds, Proof.
This follows from the discussion made in subsection 3.1. (cid:3)
Proof of Theorem 3.9.
All but the bilinear term in (4.7) converge almost surelybecause of the convergence of ω N → ω in C ([0 , T ] , H − − δ ) and continuity of dualitycoupling with φ . The almost sure convergence of U N to U solving (3.12) followssimilarly. Let us thus focus on convergence of the nonlinearity. For any given φ ∈ C ∞ ( D ) and M ∈ N it holds Z t (cid:10) ω Ns ⊗ ω Ns , H φ (cid:11) ds = Z t (cid:10) ω Ns ⊗ ω Ns , H φ − H Mφ (cid:11) ds + Z t (cid:10) ω Ns ⊗ ω Ns − ω s ⊗ ω s , H Mφ (cid:11) ds + Z t (cid:10) ω s ⊗ ω s , H Mφ (cid:11) ds. For the first term on the right-hand side we have the following L estimate: E (cid:2) | (cid:10) ω N ⊗ ω N , H φ − H Mφ (cid:11) | (cid:3) ≤ E " X m ∈ Λ | ˆ φ m | (cid:12)(cid:12)(cid:10) ω N ⊗ ω N , H φ m − H Mφ m (cid:11)(cid:12)(cid:12) ≤ X m ∈ Λ | ˆ φ m | (1 + | m | ) β ! / X m ∈ Λ E h(cid:12)(cid:12)(cid:12)D ω N ⊗ ω N , H φ m − H Mφ m E(cid:12)(cid:12)(cid:12)i (1 + | m | ) β / ≤ X m ∈ Λ | ˆ φ m | (1 + | m | ) β ! / X m ∈ Λ m/ ∈ Λ M E h(cid:10) ω N ⊗ ω N , H φ m (cid:11) i (1 + | m | ) β / ≤ C k φ k H β X m ∈ Λ m/ ∈ Λ M | m | (1 + | m | ) β / → M → ∞ for β > . QUILIBRIUM STATISTICAL MECHANICS BAROTROPIC QUASI-GEOSTROPHIC 17
For the last term, Proposition 3.7 implies the convergence in L ([0 , T ] , L (Ω)) Z t (cid:10) ω s ⊗ ω s , H Mφ (cid:11) ds → Z t h ω s ⋄ ω s , H φ i ds as long as we check that H Mφ is an approximation of H φ in the sense of Propo-sition 3.3. But this last property is easily implied by the definition of H Mφ andLemma 3.1. The second term in the right-hand side goes to zero as N → ∞ for everyfixed M , since ω N ⊗ ω N converges almost surely to ω ⊗ ω in C ([0 , T ] , H − − δ ( D × D )),and H Mφ belongs to C ∞ ( D × D ). Thus, up to subsequences, we have the almostsure convergence Z t (cid:10) ω Ns ⊗ ω Ns , H φ (cid:11) ds → Z t h ω s ⋄ ω s , H φ i ds. Therefore, taking the almost sure limit in (4.7) we get h ω t , φ i = h ω , φ i + Z t h ω s ⋄ ω s , H φ i ds. + Z t h Lω s , φ i ds. (cid:3) References [1] S. Albeverio, M. Ribeiro de Faria, and R. H¨oegh-Krohn. Stationary measures for the periodicEuler flow in two dimensions.
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