Nathan GLATT-HOLTZ Roger TEMAM Chuntian WANG

(Dedicated to Haim Brézis on the occasion of his 70th birthday)

1 Introduction

The primitive equations(PEs for short)of the oceans and the atmosphere are a fundamental model for the large scale fluid flows forming the analytical core of the most advanced general circulation models(GCMs for short)in use today.In recent years,these systems have been a subject of considerable interest in the mathematical community not only because of their wide significancein geophysical applications but also for their delicatenonlinear,nonlocal,anisotropic structure and as a cousin to the other basic equations of mathematical fluid dynamics,namely the incompressible Navier-Stokes and Euler equations.

In this paper,we study a stochastic version of the PEsand develop techniques which may be viewed as a first step toward their numerical analysis.From the point of view of applications,this work is motivated by a plea from the geophysical community to further develop the theory of nonlinear stochastic partial differential equations(SPDEs for short)in a large scale fluid dynamics context and in general(see[63]).Indeed,in view of the many sources of uncertainty both physical and numerical which are typically encountered by the modeler,stochastic techniques are playing an increasingly central role in the study of geophysical fluid dynamics(see,e.g.,[9,23,45,52,55,59,62,75]and also[32]for a small sampling of this vast literature).

The primitive equations trace their origins to the beginning of the 20th century with the seminal works of Bjerknes and Richardson[6,61]and have played a central role in the development of climate modeling and weather prediction since that time(see[56]).To the best of our knowledge,the development of the mathematical theory for the deterministic PEs began in the early 1990’s with a series of articles by Lions,Temam and Wang[46–48].This direction in mathematical geophysics is now a fairly well developed subject with results guaranteeing the global existence of weak solutions which are bounded in(see[47]),and the global existence and uniqueness of strong solutions,i.e.,solutions evolving continuously in(see[13,38–39,43]).Of course,these latter developments stand in striking contrast to the current state of the art for the Navier-Stokes equations as proving the global existence and uniqueness of strong solutions is tantamount to solving the famous Clay problem.For further background on the deterministic mathematical theory,see the recent surveys[60,64].

Recently,significant efforts have been made to establish suitable analogues of the above(deterministic)mathematical results in a stochastic setting.In a series of works[15–16,24,27,30–31,35],the mathematical theory of strong,pathwise1Here pathwise refers to the fact that solutions are found relative to a prescribed driving noise.In this paper,we use the terms “pathwise” and “martingale” as opposed to the alternate terminology of“weak” and “strong”solutions to avoid confusion with the typical PDE terminology for which weak solutions are,roughly speaking,those inand strong solutions are those insolutions has been developed.These recent works more or less bring this aspect of the subject to the state of the art,that is they establish,in increasingly physically realistic settings,the global existence and uniqueness of solutions evolving continuously in

Notwithstanding the above cited body of works,many aspects of the stochastic theory still need further consideration.In this paper,we develop existence results for weak solutions,which remain bounded in time only inThis is a direction which,to the best of our knowledge,remained unaddressed previously.Since such “weak solutions” are not expected to be unique,even in the deterministic setting,it is natural to work within the framework of martingale solutions.In other words,we consider below solutions which are weak in both the sense of PDE theory and stochastic analysis.

One particular advantage of this weak-martingale setting is that it allows us to consider physical situations unattainable so far in the above cited works on strong(or strong-pathwise)solutions.From the deterministic point of view,we obtain results for the case of in homogenous,physically realistic boundary conditions.On the other hand,from the stochastic viewpoint,our results cover a very general class of state-dependent(multiplicative)noise structures.In particular,these noise terms may be interpreted in either the It? or Stratonovich sense.The latter Stratonovich interpretation of noiseisimportant as it may bemorerealistic in geophysical settings(see,e.g.,[37,57]for further details).Note that we develop our analysis in a slightly abstract setting which at once allows us to treat the PEs of the oceans,the atmosphere and the coupled oceanic/atmospheric system.2We have previously taken such an abstract approach in other work on the stochastic primitive equations(see[15]).There however our focus was on the local existence of strong,pathwise solutions and that framework was,by necessity,more restrictive with respect to domains,noise structures,etc.

While the results established here take an important further step in the development of the analytical theory for the PEs,we believe that the main contribution of this article relates to numerical considerations.The approach below centers on an implicit Euler(i.e.,time discrete)scheme,and we choose this set-up mainly because it may be seen as a mathematical setting suitable for the development of tools needed for the numerical analysis of the stochastic PEs and other nonlinear SPDEs arising in fluid dynamics.Note that while discrete time approximation was previously employed in[14,17],these works treat hyperbolic type systems and only address the case of an additive noise.As such,a number of the techniques developed here,play a crucial role in a work related to the stability and consistency of a class of numerical schemes(both explicit and semi-implicit)for the 2D and 3D stochastic Navier-Stokes equations(see[33]).

Let us now finally turn to sketch some of the main technical challenges and contributions of the article.In fact,the first main difficulty is to justify the validity of the implicit scheme on which our analysis centers.While classical arguments involving the Brouwer fixed point theorem can be used to establish the existence of sequences satisfying the implicit scheme,we crucially need that these sequences are adapted to the driving noise.To address this concern,we rely on a specifically chosen filtration and a suitable measurable selection theorem from[10](see also[11,41]).

With suitable solutions to these mi-implicit scheme in hand,basic uniform estimates proceed analogously to the continuous time case with the use of martingale inequalities,etc.In contrast to previous works on Martingale solutions(see,e.g.,[4,15,25,34,51]),we circumvent the need for higher moments with suitable stopping time arguments.Another difficulty related to the concern that solutions are adapted appears when we associate continuous time processes with the discrete time schemes in pursuit of compactness and the passage to the limit.In contrast to the deterministic case(see[53,71]),we must introduce processes which are lagged by a time step.While these processes are indeed adapted,we obtain a time evolution equation with troublesome error terms.In turn,these error terms prevent us from addressing compactness directly from the equations and force us to carry out the compactness arguments for a series of interrelated processes.

1.1 Organization of this pap er

The exposition is organized as follows.In Section 2,we outline an abstract,functional analytic framework for the stochastic primitive equations(and related evolution systems)which may be seen as an “axiomatic”basis for the rest of the work.The section concludes by recalling the basic notion of Martingale solutions within the context of this framework.In Section 3,we introduce an implicit Euler scheme which discretizes the equations in time.The details of the existence of suitable solutions(adapted to the specific filtration)of this implicit scheme along with associated uniform estimates are given in Propositions 3.1 and 3.2,respectively.In Section 4,we study some continuous time processes associated with the implicit Euler scheme introduced in Section 3.Section 5 then outlines the compactness(tightness)arguments that allow us to pass to the limit and derive the existence of solutions from these approximating continuous time processes.Finally,Section 7 provides extended details connecting the abstract results that we just derived with the concrete example of the primitive equations of the oceans.In this section,we also provide a number of examples of possible types of nonlinear state dependent noises covered under the main abstract results.In the interest of making the manuscript as self-contained as possible,an Appendix(Section A)collects various technical tools used in the course of our analysis.

After this work was completed,we heard of[3]which we regrettably overlooked.In this paper,the authors study the space and time discretization of the incompressible Navier-Stokes equations with multiplicative random forcing in space dimension 2 or 3.The space discretization of the equations is made by finite elements and the time discretization by an implicit Euler scheme.In this paper,we only perform discretization in time,also by an implicit Euler scheme.However,the issue of time and space discretization will bead dressed in a forthcoming paper[33].Note that[33]is still distinct from[3]because we also discuss in this paper the discretization of the Navier-Stokes equations by an explicit or semi-implicit scheme which raises issues of a.s.stability,a question not addressed in[3].

We continue with some additional remarks and comparisons between[3]and the present paper,and leave to[33]some further comparisons of[3]with our work.

(i)Regarding the equations considered,we study here a class of“abstract” fluid mechanics equations as in[16],and this class of equations covers the Navier-Stokes equations as well as the primitive equations of the atmosphere and the oceans(see,e.g.,[46–48]).[3]dealt only with the Navier-Stokes equations.Because of the difficulty of constructing divergence free finite elements,the authors of[3]chose to deal with weakly incompressible finite elements,using the antisymmetrized form of the nonlinear term introduced in[67–68]to overcome the difficulties arising from handling approximate functions which are not exactly divergence free(see[3,33]for further aspects of the spatial discretization).

(ii)In[3],the authors construct martingale solutions to the 3D Navier-Stokes equations and pathwise solutions to the 2D Navier-Stokes equations,also called weak and strong solutions in the probabilistic sense.All solutions are weak solutions in the PDE sense that is correspond to L∞(L2)and L2(H1)solutions.In our case,the framework is general enough to include the 3D Navier-Stokes equations and therefore we only obtain martingale solutions;we do not specialize our results to the 2D case.

(iii)The tools are generally the same in both articles:Existence of approximate solutions Un≈ U(n?t)by a fixed point method,energy a priori estimates,and compactness argument to pass to the limit.However,the construction of the approximate solutions Un≈ U(n?t),raises a delicate question of measurability which we fully address in this paper.We did not see how this issue of measurability is addressed or bypassed in[3].This issue of measurability was also overlooked in[14]to which[3]refers.The authors of this paper thank Debussche for helping them resolve this measurability difficulty.

(iv)In[3],the authors derived estimates on higher moments after assuming that U0is deterministic,which implies that U0is uniformly bounded in the probability space and in turn makes the derivation of the higher moments estimate possible.However,we assumed that the initial data belongs to only L2in the probability space and thereby were forced to develop some techniques to overcome the lack of higher moment estimates when e.g.,establishing the compactness argument.

(v)In both papers,the passage to the limit is based on the construction of auxiliary approximate processes.We use very different arguments than that in[3].However,it is not clear whether the methods are interchangeable in both circumstances,as again the lack of higher moment estimates in our case may matter,so much so that the more probabilistic approach of[3]may fail.Another difference is that,in 2D space,[3]provided the convergence to the unique solution using a monotonicity argument.This argument is inspired from the theory of montone operators of Minty and Browder[8,49](see also[7,44]for pseudo monotone operators).The argument was extended to the stochastic context in[50](see also[28,51]).However,this argument implies uniqueness and therefore it cannot be applied to the general framework that we study which includes the 3D Navier Stokes equations.

This paper is dedicated to Haim Brézis on the occasion of his 70th birthday with admiration and friendship and(for RT)warm recollection of many years of interaction.

2 The Abstract Problem Set-up

We begin by describing the setting for the abstract evolution equation that we will study below(see(2.13)at the end of this section).As we note in the introduction,we take this point of view in order to systematically treat the existence of weak solutions to a class of geophysical fluids equations including but not limited to the example(7.1)–(7.4)developed below in Section 7.For further details about how to cast other related equations of geophysical fluid dynamics in the following abstract formulation,we refer the reader to[60]and the references therein.

Throughout what follows,we fix a Gelfand-Lions inclusion of Hilbert spaces

Each space is densely,continuously and compactly embedded in the next one.We denote the norms for H and V by|·|and ∥·∥,respectively,and the remaining spaces simply by e.g.When the context is clear,we denote the dual pairing betweenby

2.1 Basic op erators

We now outline the main elements,a collection of abstract operators,which we use to build the stochastic evolution(2.13)below.We suppose that the following are given.

(1)A linear continuous operatorwhich defines a bilinear continuous formon V.We assume that a is coercive,i.e.,

This term will typically capture the diffusive terms in the concrete equations:Molecular and eddy viscosity,diffusion of heat,salt,humidity,etc.3In previous works on the Stochastic PEs(see[15,30–31]),we required that this a is symmetric.In particular,such a symmetry was strongly used in these previous works so that we could apply the spectral theorem to the inverse of an associated operator A?1.This is not needed for the arguments presented here,and we therefore revert to the more general weak formulation of the PEs given in[60].

(2)A second linear operator E continuouson both H and V;E defines a bilinear continuous form e(U,U?):=(EU,U?)on H(which is also continuous on V).We suppose furthermore that e is antisymmetric,that is,

This term E appears in applications to account for the Coriolis(rotational)forces coming from the rotation of the earth.

(3)A bilinear form B which continuously maps V×V intoB givesriseto an associated trilinear formwhich satisfies the estimates

Moreover,we assume the antisymmetry property

Note that,in particular,we may infer from(2.4)that

Furthermore,from(2.4)–(2.5),we may assume that B is continuous from V ×V(2)into V′and satisfies

Finally,we impose some additional technical convergence conditions on b.Firstly,we suppose that when Ukconverges weakly to U in V then,up to a subsequence k′,

Similarly,we assume that if,for some T>0,

then,again up to a subsequence k′,

B accounts for the main nonlinear(convective)terms in the equations.

(4)An externally given element ?.We consider ? to be random in general;it is specified only as a probability distribution on(0,∞;V′)subject to the second moment condition(2.17)given below.This term ? captures various inhomogeneous elements,i.e.,externally determined body forcings,boundary forcings,etc.

In order to define the operators involving the“stochastic terms” in the equations,we consider an auxiliary space U,on which the underlying driving noise,a cylindrical Brownian motion W evolves(see Subsection 2.2).We suppose that U is a separable Hilbert space and useto denote the space of Hilbert-Schmidt operators frominto X,where,for example X=H,V or R.Sometimes,we abbreviate and write

Returning to the list of operators,we suppose that we have defined the following.

(1)A(possibly nonlinear)continuous mapWe suppose thatσ is uniformly sublinear,i.e.,

where the constant c3>0 is independent of t∈ [0,∞).For economy of notation,we will frequently drop the dependence on t in the exposition below.We defineaccording tofor U,U?∈ H.The elementσ determines the structure of the(volumic)stochastic forcing applied to the equations.These stochastic terms typically appear to account for various sources of physical,empirical and numerical uncertainty as we described in the introduction.

(2)A continuous map ξ:[0,∞)×H 7→ H which is subject to the uniform sublinear condition

where c4>0 does not depend on t≥ 0.We defineby

for U,U?∈ H.Weincludeξin the abstract formulation to allow,in particular,for the treatment of a class of Stratonovich noises; ξarises when we convert from a Stratonovich into an It? type noise.This term S therefore allows us to carry out the forthcoming analysis entirely within the It? framework(see Remarks 2.1,7.3 below).

With the above abstract framework now in place,we may reduce the problem(7.1)–(7.4)below(and related equations)to studying the following abstract stochastic evolution equation in,namely,

This system is to be interpreted in the It? sense which we recall immediately below in Subsection 2.2.

Note that U0and ? in(7.1)are considered to be random in general.Indeed,since we are studying Martingale solutions to(2.13)where the underlying stochastic elements in the problem are considered as unknowns,we will specify U0and ? only as probability distributions on H and L2(0,T;V′)(see Definition 2.1 and Remark 2.1).Note also that,for brevity of notation,we sometimes write

in the course of the exposition below.When the context is clear,we sometimes drop the dependence on t and simply write N(U).

2.2 Some elements of stochastic analysis and abstract probability theory

Of course,(2.13)is understood relative to a stochastic basisP,that is a filtered probability space with{Wk}k≥1,a sequence of independent standard 1D Brownian motions relative to Ft.Here we may define W on U by considering an associated orthonormal basis{ek}k≥1of U and takingW is thus a“cylindrical Brownian”motion evolving over U.

Note that the embedding of U?U0is Hilbert-Schmidt.Moreover,using standard martingale arguments with the fact that each Wkis almost surely continuous,we have that,for almost every ω ∈ ?,W(ω)∈ C([0,T],U0).

Since(2.13)is actually short hand for a stochastic integral equation,we next briefly recall some elements of the theory of It? stochastic integration in infinite dimensional spaces.We choose an arbitrary Hilbert space X and,as above,we use L2(U,X)to denote the collection of Hilbert-Schmidt operators from U into X.Given an X-valued predictable4For a given stochastic basis S,letΦ=?×[0,∞)and take G to be the sigma algebra generated by the sets of the formRecall that an X valued process U is called predictable(with respect to the stochastic basis S)if it is measurable from(Φ,G)into(X,B(X))where B(X)denotes the family of Borelian subsets of X.process G∈the(It?)stochastic integral

is defined as an element inthe space of all X-valued square integrable martingales(see[58,Subsections 2.2–2.3]).For further details on the general theory of infinite-dimensional stochastic integration and stochastic evolution equations,we refer the reader to[19,58].

Since we will be working in the setting of Martingale solutions,where the data in the problem(2.13)are specified only as a probability distribution(over an appropriate function space),it is convenient to introduce some further notations around Borel probability measures.Let(H,ρ)be a complete metric space and denote the family of Borel probability measures on H by Pr(H).Given a Borel measurable functionand an element μ ∈ Pr(H),we sometimes write μ(f)forwhen the associated integral makes sense.In particular,we write

We review some basic properties related to convergence and compactness of subsets of Pr(H)in the Appendix below(see Section A.1).We refer the reader to[5]for an extended treatment of the general theory of probability measures on Polish spaces which include Hilbert spaces such as H and V.

2.3 Definition of martingale solutions and statement of the main result

We turn now to give a rigorous meaning for the so-called weak-martingale solutions to(2.13)which are defined as follows.

Definition 2.1(Weak-Martingale Solutions)FixμU0,μ?Borel measures respectively on H andwith

A weak-martingale solutionto(2.13)consists of a stochastic basisand processesand(defined relative toadapted toThis triplewill enjoy the following properties:

(i)For every T>0,

(ii)For every t>0 and each test function

almost surely.

(iii)Finally,andhave the same laws asμU0,μ?,i.e.,

With this definition in hand,we now state one of the main results of the work as follows.

Theorem 2.1LetμU0,μ?be a given pair of Borel measures on respectively H and∞;V′)which satisfy the moment conditions(2.17).Then,relative to this data,there exists a martingale solutionto(2.13)in the sense of Definition 2.1.

Remark 2.1Depending on the structure ofσthe application of noise leads to a variety of different effects on the behavior of the solutions.In particular,σcan be chosen so that the noise either provides a damping or an exciting effect.It is therefore unsurprising that the structure of the stochastic terms in e.g.(7.1)remains a subject of ongoing debate among physicists and applied modelers.In any case,viewed as a proxy for physical and numerical uncertainty,the structure of the noise would be expected to vary by application.With this debate in mind we have therefore sought to treat a very general class of state-dependent noise structures inσ requiring only the sublinear condition(2.10).We have illustrated some interesting examples covered under this condition in Subsection 7.3 below.

Actually,the Stratonovich interpretation of white noise driven forcing may often be more appropriate for applications in geophysics(see,e.g.,[37,57]for extended discussions on this connection).Note that although(2.13)is considered in ansense,an additional,state dependent drift termξis added to the equations which allows us to treat a class of Stratonovich noises with(2.13)via the standard “conversion formula” between It? and Stratonovich evolutions(see,e.g.,[1]and also Subsection 7.3 where we present one such example of Stratonovich forcing in detail).

3 A Discrete Time Approximation Scheme

We now describe in detail the semi-implicit Euler scheme,(3.3),which we use to approximate(2.13).This system is given rigorous meaning in Definition 3.1.We then recall a specific stochastic basis in Subsection 3.2 and establish the existence of solutions to(3.3)in Proposition 3.1 relative to this basis.We conclude this section by providing certain uniform bounds(energy estimates)independent of the time step of the discretization in Proposition 3.2.

3.1 The implicit scheme

Fix a stochastic basisand elements∞;V′)),U0∈ L2(?;H)whose distributions correspond to the externally given.For a given T>0 and any integer N,let

along with the associated stochastic increments

Using an implicit Euler time discretization scheme,we would then like to approximate(2.13)by considering sequencessatisfying

infor n=1,···,N.For how to choose,see Remark 3.1.The termsare given by

and the operatoris any approximation ofσwhich satisfies

for every t≥0 and every U∈H.Additionally,we suppose that,for any t≥0,

For the existence of such σN,see Remark 3.1.We write5The choice of a “time explicit” term in is needed to obtain the correctstochastic integral in the limit as?t→0.Actually,this adaptivity(measurability)concern also leads us to introduce the approximations ofσin(3.3)(see Remark 3.1,(4.6),(4.21)).Note that,as explained in this remark approximations ofσ satisfying(3.5)–(3.7)can always be found via an elementary functional-analytic construction.

We make the notion of suitable solutions to(3.3)precise in the following definition.

Definition 3.1We consider a stochastic basisGiven N ≥ 1 and an elementwhich ismeasurable and a process?=adapted to,we say that a sequenceis an admissible solution of the Euler Scheme(3.3),if

(i)for eachandis Fnadapted,where Fn:=Ftn,n=0,···,N;

(ii)every pairsatisfies

almost surely for all U?∈V(2);

(iii)for eachandsatisfy the “energy inequality”,almost surely on?:

for n=1,···,N and where c1is the constant from(2.2).

Remark 3.1At first glance the dependence on N in both the initial condition and the noise term involvingσmay seem strange.Indeed,in the deterministic setting,when we approximate(2.13)with(3.3),we would simply taketo be equal to the initially given U0for all N.Similarly,if we were to add deterministic sublinear terms analogous toσto the governing equations,no approximation as in(3.5)–(3.7)would be necessary.However,the situation is,in general,more complicated in the stochastic setting as we shall see in detail in Section 4,Proposition 4.1.This is essentially because we must construct continuous time processes from the’s which are adapted to a given filtration(see(4.6),(4.15)–(4.17)and(4.21)for specific details).

For now let us describe how we can achieve suitable approximations in theand σN’s.

(1)For a given initial probability distributions μU0,on H(withand having fixed a suitable stochastic basis and an element U0∈ L2(?;H),F0-measurable,with distributionwe then pick a sequencesuch thatin L2(?;H)but subject to the restriction given in(4.3)below.Such a sequence can be found with a simple density argument.Indeed,since V(2)is dense in H,we may initially approximate U0in L2(?,H)with a sequenceWe then define M(N)=max{M ≥ 1:and defineSinceapproximates U0in L2(?;H)while maintaining the constraint(4.3).

(2)We may construct elements σNfrom σ satisfying(3.5)–(3.7)according to the following general functional analytic construction.For any U∈H,via Lax-Milgram we defineΨ(U)to be the unique solution in V of(Ψ(U),U?)=(U,U?)for all U?∈ V.Classically,Ψ is a compact,self-adjoint and injective linear operator on H.Thus,by the spectral theorem,we may find a complete orthonormal basis for H,{Φj}j≥1,which is made up of eigenfunctions ofΨ with a corresponding sequence of eigenvalues{γj}j≥1decreasing to zero.For any integer m,we let Pmto be the projection onto.Now choose a sequence mNincreasing to infinity but so thatIt is not hard to see that defined in this waysatisfies the requirements given in(3.5)–(3.7).

3.2 Existence of the

While the existence for a.e.ω∈? of solutions to(3.3)satisfying(3.9)follows along arguments similar to those found in[60,Lemma 2.3],some care is required to demonstrate the existence of sequenceswhich are adapted to the underlying stochastic basis.For this complication,we will make use of a“measurable selection theorem”(see Theorem A.2)from[10](see also the related earlier works[11,41]).In order to apply this result,we use a specific stochastic basis defined around the canonical Wiener space whose definition we recall next.

3.2.1 The Wiener measure and its filtration

We recall the canonical Wiener space as follows(see[42]for further details).Let

equipped with the Borelσ-algebra denoted as G.We equip(?,G)with the Wiener measure P.6Using the orthonormal basis{e k}k≥1 of U,P is obtained as the product of the independent Wiener measures each one defined on C([0,T];R).Then the evaluation map W(ω,t):= ω(t),ω ∈ ?,t∈ [0,T],is a cylindrical Wiener process on U0.The filtration is given by Gtdefined as follows:

the completion of the sigma algebra generated by the W(s)for s∈[0,t]with respect to P.Combining these elements SG=(?,G,{Gt}t≥0,P,W)gives a stochastic basis suitable for applying Theorem A.2.

3.2.2 Existence of the’s adapted to G t n

Proposition 3.1Suppose that

where c4is the constant arising in(2.11).Consider the stochastic basis SGdefined as in Subsection 3.2,an N ≥ N0,and an elementwhich is G0-measurable and a process ?= ?(t)∈ L2(?;L2(0,T;V′))measurable with respect to the sigma algebra generated by the W(s)for s∈[0,t].Then there exists a sequencewhich is an admissible solution to the Euler scheme(3.3)in the sense of Definition 3.1.

The rest of this subsection is devoted to the proof of Proposition 3.1.Below we will construct the sequenceiteratively starting frombut we first need to take the preliminary step of establishing the existence of a certain Borel measurable mapwhich is used at the heart of this construction.

We define the continuous mapaccording to

and,for each t∈[0,T]and F ∈V′we set

Using this family of sets defined by(3.12),we now establish the following lemma.

Lemma 3.1There exists a mapwhich is universally Radon measurable(Radon measurable for every Radon measure on(0,T)×V′),such that for every t∈ (0,T)and every F ∈V′,U:= Γ(t,F)∈ Λ(t,F).

ProofWe establish the existence of the desired Γ by showing that Λ satisfies the conditions of Theorem A.2.More precisely,we need to verify that7To apply Theorem A.2,we actually would like to defineΛon the Banach space R×V′.For this purpose,we may simply takeΛ(t,F)=Λ(T,F)when t>T,and when t<0 we letΛ(t,F)=Λ(0,F).

(i)for each t∈ [0,T],F ∈ V′,the setΛ(t,F)is non-empty,

(ii)Λ(t,F)is closed.In other words,we need to show that,given any sequences

such that,for every n,

we have

Observe that,for any Umof this form,using(2.2)–(2.3),(2.5)and(2.11),we estimate

We next seek bounds on the resulting sequence of Um’s in V independent of m.Starting from(3.13),we find that

Using once again the standing assumption(3.10),we have that Umis bounded in V independently of m.Passing to a subsequence as needed and using that V is compactly embedded in H,we infer the existence of an element U such thatweakly in V and strongly in H.

Returning to(3.14)and using the lower semicontinuity of weakly convergent sequences,we obtain thatTo show that U satisfiesfor everywe simply invoke(2.8)for B and the other continuity assumptions on A,E andξ,and obtain this identity forfor each k≥ 1.By linearity and density,we therefore infer the identify for arbitrary.With this we now have established(i).The second item,(ii),to show thatΛis closed,follows immediately from the continuity of G from[0,T]×V intoand the continuity ofξfrom[0,T]×H into H.The proof of Lemma 3.1 is therefore complete.

Construction of an adapted solution

Step 1We will build the desired sequenceinductively as follows:

withmeasurable for V equipped with B(V)and C([0,tn];U0)equipped with Gn:=Gtn(defined as in Subsection 3.2).

Suppose that we have obtainedfor some n ≥ 2.Since Gn?1is the completion ofwith respect to the Wiener measure8We observe that the sigma algebra generated by the W(s)for s∈(0,t)is justwhereis the mapping(see[42]).is P-measurable.Now we defineby setting

Then we can define

SinceσNis a continuous map,clearlyis a continuous map.Moreover,Γ is universally Radon measurable thanks to Lemma 3.1,hence Corollary A.1 applies and we infer thatχis universally Radon measurable from the Borel sigma algebra onto the Borel sigma algebra on V.

Since?= ?(t)is a process assumed to be measurable with respect to the sigma algebra generated by the W(s)for s ∈ [0,t],is measurable with respect to the sigma algebra generated by the W(s)for s∈[0,tn]thanks to(3.4).Hence by Theorem A.1 in the appendix with X as?,(Y,M)asψ asas V,we see that there exists a functionwhich is Borel measurable,such that

From(3.17)–(3.18),we infer

Sinceandare P-measurable,and κ is universally Radon measurable,Theorem A.3 applies and we infer thatis P-measurable,that isis measurable with respect to Gn.

Step 2We infer thatis measurable with respect to Gnas desired.

Observe moreover that,according to Lemma 3.1(see(3.12)),,for everyandwhich is to say thatandsatisfy(3.3)and(3.9).

It remains to show thatWe start from(3.9),now established forandand use the elementary identityand obtain

almost surely.To address the terms involving?,we have that(see(3.4))

where we defineaccording to

For the terms involving s defined as in(2.12),we simply infer from(2.11)

With H?lder’s inequality,we find

Then using that gNis linear in its second argument,we have

Using these observations forand s,we rearrange and infer that,up to a set of measure zero,

Using(2.10),(3.6)and thatis Gn?1-measurable,in L2(?;H),we have

From this observation,(3.25)and(3.10),we infer

which implies that,as needed.

We have thus established the iterative step in the construction of.The base case,n=1,is established in an identical fashion to the iterative steps.The proof of Proposition 3.1 is now complete.

Remark 3.2Although necessary for the establishment of the existence of the’s in Proposition 3.1,it is not necessary to assume the underlying stochastic basis to be SG(defined in Subsection 3.2)in the results through out Subsection 3.3 to Subsection 5.1.The reason is that these results are true whenever such’s defined as in Definition 3.1 exist.In other words they are independent of the choice of the underlying stochastic basis.Similarly,it is not necessary at this point to assume that U0and ? have laws which coincide with those of the externally givenμU0 andμ?for these results.

However,it is necessary that we resume these assumptions of SG,μU0and μ?starting in Subsection 5.2.

3.3 Uniform “energy” estimates for the

Starting from(3.9)we next determine certain uniform bounds,independent of N,for(suitable)sequencessatisfying(3.3)as follows.

Proposition 3.2Let

where c3and c4are from(2.10)and(2.11),respectively.Letbe the given stochastic basis and assume thatis measurable with respect to Ft.For each N ≥N1,we assume thatmeasurable and such that

Then for each N ≥ N1,consider the sequenceswhich satisfy(3.3)starting fromand relative to?in the sense of Definition 3.1.Then

ProofThe starting point for the estimates leading to(3.28)is of course(3.9)and from this inequality,we can use the same proof as in Proposition 3.1 to obtain(3.25).In order to make suitable estimates for the final two terms in(3.25),we need to take advantage of some martingale structure in the terms involving σN.For any 1≤m≤n≤N,we define the stochastic processes

Summing(3.25)for 1≤m≤n=k≤l≤N,we find

Sinceis adapted to Fn:=Ftn,it is easy to see thatis a martingale relative towithWe would like to apply a discrete version of the Burkholder-Davis-Gundy inequality,recalled here as in Lemma 3.2 to obtain estimates forUnfortunately,it is not clear thatis square integrable,so we have to apply a localization argument to make proper use of this inequality.For any K>0,we define the stopping times

Since

we have thatalmost surely asClearlyis a square-integrable martingale.For the moment,let us recall a discrete analogue of the Burkholder-Davis-Gundy inequality.This result and other related martingale inequalities can be found in e.g.[22].

Lemma 3.2Assume that{Mn}n≥0is a(discrete)martingale on a Hilbert space H(with norm|·|),relative to a given filtrationWe assume,additionally that M0≡ 0 and thatfor all n≥0.Then,for any q≥1 and any n≥1,

where cqis a universal positive constant depending only on q9We may often determine c q in(3.31)explicitly,and in particular,we have that c1=3.(which is independent of n and{Mm}m≥0),and Anis the quadratic variation defined by

Hence with the observation thatis-measurable,we compute the quadratic variation ofin view of(3.32)as follows:

Thus,by Lemma 3.2,(2.10)and(3.6),we infer

Hence,letting,we have,by the monotone convergence theorem,

On the other hand,sinceis adapted to Fn,given the condition(2.10)onσand(3.6),we infer that

We now use(3.33)–(3.34)with(3.30)and infer that

Rearranging it,we find that

for the constantwhich in particular depends only on c3,c4.Thus,subject to the condition

we have

whereThus,by iterating this inequality and noting from(3.21)that

we finally conclude that

Note carefully that,in view of(3.36),we need not iterate(3.37)more than,say,times to obtain(3.38).10Indeed,for N≥N 1,let N(N)be the minimum number of iterations of(3.37),subject to the constraint(3.36),which are needed to establish(3.38).Take F(N)to be the“fraction of the time interval that can be covered at each step”,namely, where the last inequality follows from the standing assumption(3.26).Since N(N)F(N)≤2,we finally estimate Here smallest integer that is larger than or equal to p.As such,we may takewhich,crucially,is independent of N.

We now return to(3.30).With(3.34),we infer

where we can takeAs such,(3.38)–(3.39)with(3.27)imply(3.28),completing the proof of Proposition 3.2.

4 Continuous Time Approximations and Uniform Bounds

In this section,we detail how the sequencesdefined in the sense of Definition 3.1 may be used to define continuous time processes that approximate(2.13).The details of establishing the compactness of the associated sequences of probability laws and of the passage to the limit are given further on in Section 5.

We now fix sequencessatisfying(3.3)in the sense of Definition 3.1.For N≥N1,with N1as in(3.26),let

Of course,we do not have any time derivatives of the UN’s(even fractional in time)as are typically needed for compactness.Furthermore,we would like to be able to associate an approximate stochastic equation for(2.13)with theseFor these dual concerns,we introduce further stochastic processes and consider

Remark 4.1The processes UNandare slightly different than those typically used in the deterministic case(see,e.g.,[71]).Actually,these processes are essentially their deterministic analoguese valuated at time t by their value at time t??t.With this choice,we crucially obtain processes which are adapted toNot surprisingly however the present definitions ofleads to bothersome error terms in(4.6)below.In turn these error terms dictate the additional convergences inσand U0when we initially defined the discrete scheme(3.3)(see(3.5)–(3.7)and Remark 3.1).These error terms also complicate compactness arguments further in Section 5(see Remark 4.2).

The rest of this section is now devoted to proving the following desirable properties of UNand

Proposition 4.1Letbe a stochastic basis,and let N1be as in(3.26)in Proposition 3.2.Consider a sequencebounded in L2(?,H)independent of N,with-measurable for each N and such that

for a constant c>0,independent of N.11The constraint(4.3)is necessary for(4.4)–(4.7).This is not a serious restriction when we pass to the limit in Section 5.As we described above in Remark 3.1,for any given U 0∈ L 2(?;H)we may obtain a sequenceapproximating U 0 which maintains(4.3).Suppose that we also have defined a process?=?(t) ∈ L2(?;L2(0,T;V′))adapted to

For each N≥N1,we consider sequenceswhich satisfy(3.3)starting fromin the sense of Definition 3.1.Once these sequencesexist,then we define the continuous time processesaccording to(4.1)and(4.2),respectively.Then,

(i)for eachand-adapted and

Moreover,we have that

(ii)UNandsatisfy a.s.and for every t≥0,

subject to error termswhich are defined explicitly in(4.15)–(4.16)below.

(iii)These error termssatisfy

respectively,and moreover,

We proceed to prove Proposition 4.1 in a series of subsections below.The proof of(i)is essentially a direct application of Proposition 3.2,and we provide the details in the subsection immediately following.In Subsection 4.2,we provide the details of the derivation of(4.6)and in particular explain the origin of the error terms.The final Subsection 4.3 provides details of the estimates for these error terms which lead to(4.7)–(4.9).

Remark 4.2It is not straightforward to obtain fractional in time estimates forfrom(4.6)in view of the error terms which have a rather complicated structure(see(4.15)–(4.16)).As such,we cannot establish sufficient compactness for the sequencedirectly to facilitate the passage to the limit.For this reason,we choose to introduce additional continuous time processes in Section 5 below.An alternate approach will be presented later on in the related work[33].

4.1 Uniform bound s and clustering

It is clear from(4.1)that UNis{Ft}t≥0-adapted and that

Thus,since(3.27)holds,we have the uniform bound(3.28)from Proposition 3.2,and we immediately infer that

with the integer N1appearing in(3.26).

As the UNabove,it is easy to see from(4.2)thatis adapted to{Ft}t≥0,and thatis adapted to Fn(=Ftn).Furthermore,direct calculations show that

Using(4.11),similarly to[71],we compute that

We thus infer(4.5)directly from this observation and(3.28).Based on similar considerations,we also have

Thus,once again due to(4.3)and(3.28),we finally have

With(4.10)and(4.12),we have now established the first item in Proposition 4.1.

4.2 The ap proximate stochastic evolution systems

We next derive the equation(4.6)relating UNandgiving explicit expressions forWe observe that,almost surely and for almost every t≥ 0(in fact for every

where χ(t1,t2)denotes the indicator function of(t1,t2).Recall thatand letin other words,we takesuch that

Working from(4.13)and(3.3),we therefore compute

where the “error terms”,,are defined as

and

respectively.To understand the origin of these error terms,we observe that

Moreover,using the definition of thein(3.4),we have

On the other hand,for the error termsinvolving σNin(4.16),we compute

4.3 The estimates for the error terms

We next proceed to make estimates on the error termsandas desired in(4.7),(4.9).Perusing(4.15),we begin with estimates forInvoking the bounds provided by(2.7)along with the continuity properties of the other operators making up N in(2.14)defined in Subsection 2.1,we have

As such,in view of the standing condition(4.3)(see Remark 3.1),we conclude that

Forwe estimate in L2(0,T;V′)

In summary,we have

and so we conclude(4.7)from(4.17)–(4.18).

We next turn to make estimates forWe begin with estimates in L2(0,T;H).Forwe observe with(2.10)and(3.6)(see(3.34))that

and infer from(3.28)in Proposition 3.2 that

On the other hand,with the It? isometry and another application of(2.10)and(3.6),we have

so that

By combining(4.19)–(4.20),we obtain(4.8).

We turn now to establishing the uniform bounds announced in(4.9).Estimates similar to those leading to(4.19)–(4.20),but which instead make use of the condition(3.5),yield bounds in L2(0,T;V),namely,

and similarly

so that,taken together we infer that

Finally,we supply a bound forin L∞(0,T;H).Forwe observe with(2.10),(3.6)that

To estimatewe use Doob’s inequality and(2.10)to infer

With these bounds and(3.28),we conclude that

In turn,(4.21)–(4.22)directly imply(4.9),and so the proof of Proposition 4.1 is now complete.

5 Compactness and the Passage to the Limit

In this section,we detail the compactness arguments that we use to prove the existence of martingale solutions of(2.13)using the processes UNanddefined in the previous section.As it is not clear how to obtain compactness directly from(see Remark 4.2),we must introduce further processes to achieve this end.

Recalling(4.1)–(4.2),(4.15)–(4.16),we define

and then consider the associated probability measures

Notice that,due to Proposition 4.1,are defined on the space X:=L2(0,T;H).Regarding the elements,we observe that,as a consequence of(4.6),

As a result of this identity and Proposition 4.1,the elementsmay be regarded as measures on the space

We will show below that μN(yùn)andconverge weakly to a common measure μ and then make careful usage of the Skorohod embedding theorem to pass to the limit in(5.3)on a new stochastic basis.The former compactness arguments,which rely on the intermediate measureswill be carried out in the next subsection and the details of the Skorohod embedding will be discussed in Subsection 5.2 further on.

5.1 Tightness arguments

In this section,we will establish the following compactness properties of theand

Proposition 5.1The assumptions are precisely those in Proposition 4.1.Defineandaccording to(4.1)and(5.1)and where N1is as in(3.26).Letbe the associated Borel measures on

defined according to(5.2).Then,there exists a Borel measure μ onsuch that,up to a subsequence,12We recall the notion of weak compactness of probability measures along with the equivalent notion of tightness in Appendix(see Section A.1).

The rest of this subsection is devoted to the proof of Proposition 5.1.We proceed as follows:First we show thatis tight(see Appendix A.1)in L2(0,T;H)by employing a suitable variant of the Aubin-Lions compactness theorem which we establish in Proposition A.4 below.We next show thatis tight in C([0,T];via an Arzelá-Ascoli type compact embedding from[25,70].We finally employ the estimates(4.5),(4.7)along with the general convergence results recalled in Lemma A.1 to finally infer(5.4)–(5.5).

5.1.1 Tightness for in L 2(0,T;H)

With the aid of Proposition A.4,we identify some compact subsets of X=L2(0,T;H)that,in conjunction with suitable estimates(see(5.10)–(5.13)immediately below)are used to establish the tightness ofin X.For U ∈ X,n>0,define

and,for each R>0,consider

It is not hard to show that each set BRis a closed subset of X.Perusing(5.6),it is clear that the condition(A.4)holds uniformly for elements in BR.Thus,as a consequence of Proposition A.4(ii),these sets BRare compact in X=L2(0,T;H)for each R>0.

Now,for each R>0,we have

As a consequence of(4.4),(4.9)and(5.1),we have

for some constant c independent of N.

with

To addresswe observe,with(2.6)and the standing assumptions on the operators that make up N in(2.14),that for any U∈V,

Furthermore,it is clear from(3.4)and H?lder’s inequality that,a.s.

Combining these observations,we infer that,a.s.

For the term,we estimate,for 0≤θ≤δ,

where the second line follows from Doob’s inequality and the standing assumptions(2.10)on σand(3.6)onσN:

The estimates(5.12)–(5.13)allow the second term in(5.8)to be treated as follows.Observe that according to(5.6)and(5.10),we have

For the first term,we observe with(5.12)that

Regarding the second term,we simply bound

so that forρ>0,sufficiently large,

We finally conclude that

Combining(5.8)–(5.9)and(5.16),we now conclude that(see Appendix A.1)

5.1.2 Tightness for

We next show thatis tight inFor this purpose,we make appropriate usage of a compact embedding from[25](see also[70]).Let us fix anysuch thatαp>1.According to[25],

that is,the embeddings are continuous and compact.We now define

for any R>0.With(5.18),it is clear that BRis compact infor every R>0.Observe moreover that,in view of(5.3),

and thus that

Hence we will infer thatis tight inif we can show thatconverge uniformly in N to zero as R↑∞.

Forwe estimate,with(5.11),

Thus we find(see(5.15))

We turn toFor this purpose,let us define for any R>0 the stopping times

UsingτR,we now estimate with the Chebyshev inequality that

Now in order to treat this final stochastic integral term,we recall the following generalization of the Burkholder-Davis-Gundy inequality from[25]:For a given Hilbert space X,p≥2 andwe have for all X-valued predictable

which holds with a constant c depending only onαand p.Continuing now from(5.21),we have

Combining the estimates(5.20),(5.22)with(4.4),we finally conclude

and hence infer that

Remark 5.1Let us observe that the tightness bounds forcould be carried out differently if we had available,for example,the uniform bounds on “higher moments” like

or equivalently that

Indeed,in numerous other previous works related to stochastic fluids equations(see,e.g.,[4,15,25,34,51])estimates analogous to(5.25)are established essentially via It?’s lemma in order to achieve tightness in the probability laws associated to a regularization scheme.

In the current situation,instead due to the way we carry out the estimates in(5.15),(5.21)–(5.22),we have adopted a different approach,namely,we establish tightness(compactness)estimates without recourse to such higher moment estimates.

A different method using higher moments will be shown in the related work[33].

5.1.3 Cauchy arguments and conclusions

With(5.17)and(5.23)now in hand,it is then simply a matter of collecting the various convergences above to complete the proof of Proposition 5.1.

By making use of Prohorov’s theorem(see Section A.1)with(5.17),we infer the existence of a probability measure μ such that,up to a subsequence,

Due to(5.1)with(4.5)and(4.8),it is clear thatconverges to zero in X=L2(0,T;H)and hence in L2(0,T;V′)a.s.Hence,by now invoking(4.7)and referring back once more to(5.1),we have thatconverges to zero inThus,invoking Lemma A.1,again up to a subsequence,we conclude that

In particular,this is the first desired convergence for(5.4).On the other hand,invoking Prohorov’s theorem with(5.23)and the convergence just established forin L2(0,T;V′),we see thatis tight inBy Prohorov’s theorem in the other direction and passing to a further subsequence as needed,we have

Since,clearly,,this yields the second desired item(5.5).The proof of Proposition 5.1 is therefore complete.

5.2 Proof of Theorem 2.1 conclusion:Almost sure convergence and the passage to the limit on the Skorokhod basis

We now have all of the ingredients to finally prove the main results of this article,namely Theorem 2.1.Suppose that we are givenandaccording to the conditions specified in Definition 2.1.As mentioned in Remark 3.2,now it is necessary to introduce the stochastic basis SG(defined as in Subsection 3.2),an element U0which is G0measurable and a process?= ?(t)measurable with respect to the sigma algebra generated by the W(s)for s ∈ [0,t],13Note that since the sigma algebra generated by the W(s)for s∈[0,t])is the smallest respect to which W(t)is measurable,?(t)is adapted to{F t}t≥0,and hence all the previous results apply.whose laws coincide with those ofμU0,μ?.Thus Proposition 3.1 applies,and we obtain the existence of theadapted to Gtn.

We then approximate U0∈ L2(?;H)with a sequence of elementswhich maintains the bound(4.3)as described in Remark 3.1 above.Proposition 4.1 applies,and hence we can use this sequence,the process ?,and the sequenceto define processesaccording to(4.1)and(5.1),respectively(N1is given by(3.26)).In order to pass to the limit in the associated evolution equation(5.3),we consider the product measures

which are defined on the space

where,as above,and U0is defined as in Subsection 2.2,(2.15).By invoking Proposition 5.1,we have that(passing to a subsequences as needed)μN(yùn)? μ on X andon Y,whereμN(yùn)andare defined as in(5.2).It follows,again up to passing to a subsequence,thatνNconverges weakly to a measureνon Z(defined in(5.27)).Furthermore,recalling(5.1)and making use of(4.5),(4.7)–(4.8),it is not hard to see that

Thus,by making use of the Skorokhod embedding theorem(see Section A.1),we obtain,relative to a new probability space,a sequence of random variables

Moreover,the uniform bounds forfrom Proposition 4.1,(4.4)imply that in addition to(5.28),we also have

Following a procedure very similar to[4],we may now show thatis a cylindrical Brownian motion relative to the filtrationdefined as the sigma algebra generated byfor s≤t and thatsatisfies(5.3)on the“Skorokhod space”viz.

Using the convergences in(5.28)–(5.29)with(5.30)it is standard14Note that,in particular,the stochastic terms involvingσN(U N)converge due to(3.7).to show thatsatisfies(2.18)–(2.20)relative to the stochastic basis,whereis defined as the sigma algebra generated by thefor s≤t andTherefore,is a martingale solution to(2.13)relative toμU0,μ?in the sense of Definition 2.1,and the proof of Theorem 2.1 is complete.

6 Convergence of the Euler Scheme

We conclude by reinterpreting from the point of view of numerical analysis,the study above as a result of convergence for the Euler scheme(3.3).

Theorem 6.1We assume given μU0∈ Pr(H)andaccording to Definition 2.1.We also assume given the stochastic basis SG(defined as in Subsection 3.2),an element U0which is G0measurable and a process ?= ?(t)measurable with respect to the sigma algebra generated by the W(s)for s ∈ [0,t],whose laws coincide with those of μU0,μ?.Let a sequences of elementsapproximate U0∈ L2(?;H)as described in Remark 3.1.Then the processes{UN}N≥N1defined according to(4.1)(N1is given by(3.26))adapted to{Gt}t≥0exist.

Moreover,the family{μN(yùn)}of probability laws of{UN},is weakly compact over the phase spaceand hence converges weakly to a probability measure μ on the same phase space up to a subsequence.Furthermore,there exists a probability spaceand a subsequence of random vectorswith values insuch that

(i)have the same probability distribution as(UNk,?,W).

(ii)converges almost surely asin the topology of Z1,to an elementParticularly,wherehas the probability distributionμ.

ProofThe existence offollows directly from the existence of theproven in Proposition 3.1.(i)and(ii)follow from the Skorokhod embedding theorem(see Section A.1)as shown in Subsection 5.2.

7 Applications for Equations in Geophysical Fluid Dynamics

In this section,we apply the above framework culminating in Theorems 2.1 and 6.1 to a stochastic version of the primitive equations.Our presentation here will focus on the case of the equations of the oceans.Note however that the abstract setting introduced above is equally well suited to derive results for analogous systems for the atmosphere or for the coupled oceanicatmospheric system(COA for short).15Via a suitable change of variables,the dynamical equations for the compressible gases which constitute the earth’s atmosphere may be shown to take a mathematical form essentially similar to the incompressible equations for the oceans.We refer the interested reader to[60]for further details on these other interesting situations.

7.1 The oceans equations

The stochastic primitive equations of the oceans take the form

Here,U:=(v,T,S)=(u,v,T,S);v,T,S,p,ρrepresent the horizontal velocity,temperature,salinity,pressure and density of the fluid under consideration,respectively;μv,νv,μT,νT,μS,νSare positive coefficients which account for the eddy and molecular diffusivities(viscosity)in the equations for v,T and S.The terms Fv,FT,FSare volumic sources of momentum,heat and salt which are zero in idealized situations but which we consider to be random in general.

The state dependent stochastic terms are driven by independent Gaussian white noise processeswhich are formally delta correlated in time.The stochastic terms may be written in the expansion

where the elementsare independent 1D white(in time)noise processes.We may interpret the multiplication in(7.2)in either the It? or the Stratonovich sense;as we detail in one example below that the classical correspondence between theand Stratonovich systems allows us to treat both situations within the framework of the It? evolution(2.13).We will describe some physically interesting configurations of these “stochastic terms” in detail below in Subsection 7.3.

The operatorsare the horizontal Laplacian and the gradient operator,respectively.Herethe operator?vcapturespart of the convective(material)derivative and is defined according to

Remark 7.1As given,the model(7.1),expresses the equations for oceanic flows in the“beta-plane approximation”,that is to say we make use of the fact that the earth is locally flat.This setting is suitable for regional studies,and we will focus on this case for the simplicity of presentation.With suitable adjustments to the definition of the operators?,?,?vand to the domain introduced below we could consider the evolutions in the full spherical geometry of the earth.We refer to[47](and also to[60])for further details on how to cast a global circulation model in the form of(2.13).

7.1.1 Domain and boundary conditions

The evolution(7.1)takes place on a bounded domain M?R3which we define as follows.Fix a bounded,open domainΓi? R2with sufficiently smooth boundary(C3,say);Γirepresents the surface of the ocean in the region under consideration.We suppose that we have defined a“depth”function h=h(x,y):Γi→ R which is at least C2and is subject to the restrictionWith these ingredients,we then let

The boundary ?M of M,is divided into its top Γi,lateralΓland bottom Γbboundaries.We denote the outward unit normal to?M by n and the normal toΓlin R2by nH.

Wenext prescribe the following,physically realistic boundary conditions for(7.1)considered in M(see,e.g.,[60]for further details).OnΓiwe suppose

where αv,αTare fixed positive constants,and τv,va,Taare in general random and nonconstant in space and time.Physically speaking,the first two equations in(7.4)account for a boundary layer model,where va,Tarepresent the values for velocity and temperature of the atmosphere at the surface of the oceans,respectively;τvaccounts for the shear of the wind.

At the bottom of the oceanΓb,we take

Finally for the lateral boundaryΓl,

Note that,in view of the Neumann(no-flux)boundary conditions imposed on S in(7.4)–(7.6),there is no loss in generality in assuming

(see[60]for further details).Finally,(7.1)–(7.7)are supplemented with initial conditions for v,T and S,that is,

7.1.2 A reformulation of the equations

Starting from the incompressibility condition,(7.1c)and the hydrostatic equation(7.1b),we may derive an equivalent form for(7.1)as follows:

This reformulation is desirable as,in particular,it is more suitable for the typical functional setting of the equations which we describe next.The unknowns and parameters in the equations are precisely those given above immediately after(7.1).Of course,(7.9)is subject to the same initial and boundary conditions as in(7.1),namely(7.4)–(7.8).For further details concerning the equivalence of(7.9)and(7.1)(see[60]).

7.2 The functional setting and connections with the abstract framework

We now proceed to introduce the basic function spaces associated with the primitive equations(7.9)(equivalently(7.1)),and then introduce and explain the variational formulation of the various terms in equation connecting them with the abstract assumptions laid out above in Section 2.

7.2.1 Basic function spaces

To begin,we define the smooth test functions

We now take H to be the closure of V in L2(M)4or,equivalently,H:=H1×H2,which is

On H,it is convenient to define the inner product and norm according to

The constants KT,KS>0,which are introduced for coercivity in the principal linear terms in the equations,are chosen in order to fulfill(2.2)for(7.14)below.We defineΠto be the orthogonal(Leray-type)projection from L2(M)4onto H.

We shall next define the H1type space V=V1×V2,which is

We endow V with the inner product and norm

where

From(7.11)–(7.12),we may deduce the Poincaré type inequality|U|≤ c∥U∥ for every U ∈ V.This justifies taking∥·∥as the norm for V(which is equivalent to the H1norm).Finally,we define

and simply endow V(2)and V(3)with the H2(M)and H3(M)norms,respectively.Let V′(resp.be the dual of V(resp.V(2),V(3))relative to the H inner product.

It is clear with the Rellich-Kondrachov theorem and standard facts about Hilbert spaces that the spaces introduced in(7.10)–(7.13)provide a suitable Gelfand-Lions inclusion as desired for(2.1).On this functional basis we now turn to describe the variational form of(7.9).

7.2.2 The variational form of the equations

To capture most of the linear structure in(7.9),we define the operator A as a continuous linear map from V to V′via the bilinear form

We observe that if KT,KSin(7.12)are chosen sufficiently large,then a is coercive,namely,it satisfies the condition required by(2.2).

where

To capture the rotation(Coriolis)term in(7.9a),we define E:H→H via

Note carefully that a,e and b satisfy the conditions imposed in Subsection 2.1 which we used in the abstract result Theorem 2.1.The in homogenous terms in(7.9)are given by the element ? defined according to

Note that va,τv,Ta,which represent the velocity,shear force of the wind and the temperature at the surface of ocean,have significant uncertainties and should thus be considered to have a random component in practice.

7.3 Some stochastic forcing regimes

It remains to complete the connection between(7.1)and(2.13)by describing various physically interesting scenarios forWe connect these“concrete descriptions”with the terms σandξin the abstract equation(2.13)(or equivalently to g,s in(2.19)).We consider three situations in detail below.In each case,we describe how to defineσUappearing in(7.9),and we then take

7.3.1 Additive noise

The most classical case is to consider an additive noise,where we suppose thatσUis independent of U=(v,T,S).In other wordsFor(2.10)to be satisfied,we would require that

Note that since the It? and Stratonovich interpretations of(7.2)coincide in the additive case,we may takeξ≡0 so that(2.11)is automatically satisfied.

We also observe that in this case we may give an explicit(if formal)characterization of the space-time correlation structure of the noise

where the correlation kernel K is given by

Remark 7.2Given the condition(7.18),the case of space-time white noise is ruled out under our framework.Of course such a space-time white noise is very degenerate in space(not even defined inand so such a situation is far from reach due to the highly nonlinear character of the PEs.Similar remarks apply to the 3D stochastic Navier-Stokes equations,but see[18]for the 2-D case.

7.3.2 Nemytskii type op erators

We next consider stochastic forcings of transformations of the unknown U as follows.Letand suppose,for simplicity,thatΨ is smooth.We denote the partial derivatives ofΨ with respect to the v,T,S variables by ?vΨ,?TΨ,?SΨ respectively and the gradient by?UΨ.Take a sequence of smooth functionsαk= αk(x):M → R and define

We may formally interpretwhere

(1)is a white in time Gaussian process with the spatial-temporal correction structure

(2)The“multiplication”Ψ(U)andmay be taken in either the It? or the Stratonovich sense.

We now connect(7.20)to(2.13)in theor the Stratonovich situations in turn illustrating conditions on Ψ and the αk’s guarantee that(2.10)holds and in the Stratonovich case that(2.11)holds.

The caseSuppose that

and for the elementsαk,we suppose that

The Stratonovich caseIf we understand the multiplicationΨ(U)˙ηin the Strantonovich sense,then we may convert back to an It? type evolution according to

where

Onecan refer to,e.g.,[1,40]for further details on this conversion formula.Under the additional assumption

we defineξU(U):=Πξ(U)for any U ∈H.It is clear thatξsatisfies(2.11).

Remark 7.3We note here that the relationship(7.23)is,for now,only formal;we prove the existence of martingale solutions to the system that results from a formal application of this conversion formula(see,e.g.,[1,40]).We leave the rigorous justification of(7.23)and the related issues of an approximation of Wong-Zakai type(see[74])of(2.13)for future work.Note however that(7.23)has already been explored in[12,29,72]in an infinite dimensional fluids context for pathwise solutions and in[73]for martingale solutions to a class of abstract,nonlinear,stochastic PDEs.

7.3.3 Stochastic forcing of functionals

Finally,we examine the case when we stochastically force functionals of the unknown,i.e.,terms which have a non-local dependence on the solution U.For example consider,for k≥1 continuous(not necessarily linear)and sufficiently smooth αk=We define

Here,we interpretin the It? sense.Subject to,for example,

we obtain aσ from(7.25)which satisfies(2.10).For a“concrete example”of aσ of the form(7.25)which satisfies(7.26),letbe a sequence of elements in L2(M)2withand letαk∈ V satisfying the sumability condition in(7.26).We takeand obtain

A Appendix:Technical Complements

We collect here,for the convenience of the reader,various technical results which have been used in the course of the analysis above.While some of the material may be considered to be somewhat “classical”by specialists,we believe that the stochastic type results will be useful to the non-probabilists and that the deterministic results will be helpful for the probabilists.

A.1 Some convergence properties of measures

We next briefly review some basic notations of convergence for collections of Borel probability measures.In particular,we highlight a certain abstract convergence lemma that has been used in a crucial way in the passage to the limit several times above.For further details concerning the general theory of convergence in spaces of probability measures,one can refer to,e.g.,[5,65].

Let(H,ρ)be a completemetric space and denote by Pr(H)the collection of Borel probability measures on H.We recall that a sequenceis said to converge weakly to a measure μ on H(denoted byif and only iffor every bounded continuous functionWe recall that a collectionΛ?Pr(H)is said to be weakly relatively compact if every sequencepossesses a weakly convergent subsequence.On the other hand,we say that Λ ? Pr(H)is tight if,for every?>0 there exists a compact set K?? H such thatμ(K?)≥ 1?? for each μ ∈Λ.The Prokhorov theorem asserts that these two notions,namely tightness and weak compactness of probability measures,are equivalent.

We also make use of the Skorokhod embedding theorem which states that,wheneveron H,then there exists a probability spaceand a sequence of random variables Xn:such thatand which converges a.s.to a random variablewith

The following convergence result,found in e.g.[5],relates roughly speaking weak convergence and clustering in probability,and was used to facilitate the proof of(5.26)in Subsection 5.1.3.

Lemma A.1Let(H,ρ)be an arbitrary metric space.Suppose that Xnand Ynare H-valued random variables,and letbe the associated sequences of the probability laws.If the sequence{μn}n≥0converges weakly to a probability measure μ and if,for all?>0,

then νnalso converges weakly to μ.

A.2 An extension of the Doob-Dynkin lemma

We extend the Doob-Dynkin lemma(see,e.g.,[54])to the case where the image space of the measurable functions are complete separable metric spaces.In order to achieve this goal,let us recall the following notions and results from[21].

If(?,F)is a measure space and E ? ?,let FE:={B∩E:B ∈F}.Then FEis a sigma algebra of subsets of E,and FEwill be called the relative sigma algebra(of F on E).

Proposition A.1Let(?,F)be any measurable space and E be any subset of ? (not necessarily in F).Let f be a function on E with values in a Polish space H and measurable with respect to FE.Then f can be extended to a function on all of ?,measurable with respect to F.

ProofThe proof is direct combining Theorem 4.2.5 and Proposition 4.2.6 in[21].

Now let(Y,M)be a measure space,X be any set,andψbe a function from X into Y.LetThen ψ?1[M]is a sigma algebra of subsets of X.

Theorem A.1Given a set X,a measure space(Y,M),and a functionψfrom X into Y,if a function ? on X with values in a Polish space H isψ?1[M]measurable,then there exists an M-measurable function L on Y such that?=L?ψ.

ProofWhen everψ(u)= ψ(v),we have?(u)= ?(v).Otherwise,let B be a Borel set in H with?(u)∈B but?(v)/∈B.Then??1(B)=ψ?1(C)for some C∈M,withψ(u)∈C but ψ(v)/∈C,a contradiction.Thus,?=L?ψfor some function L from D:=rangeψinto H.For any Borel set E ?H,ψ?1(L?1(E))= ??1(E)=ψ?1(F)for some F ∈ M,so F ∩D=L?1(E)and L is MDmeasurable.By Proposition A.1,L has an M-measurable extension to all of Y.

A.3 A measurable selection theorem

We turn now to restate the measurable selection theorem which was proven in[10]and is based on the earlier works(see[11,41]).We employed this result above to establish the existence of adapted solutions to(3.8)in Proposition 3.1.

Firstly,we recall the definition of a Radon measure.Let X be a locally compact Hausdorff spaces and B(X)be the Borel sigma algebra on X.A Radon measure on X is a measure defined on B(X)that is finite on all compact sets,outer regular on all Borel sets,and inner regular on all open sets(see[26,p.212]).

Theorem A.2Let X and Y be separable Banach spaces and suppose thatΛ is a“multivalued map”from X into Y,i.e.,a map from X into the subsets of Y.We assume thatΛ takes values in closed,non-empty subsets of Y and that its graph is closed viz.,

Then,Λ admits a universal Radon measurable section,Γ,that is there exists a mapΓ:X →Y such thatΓx∈Λx for every x,and such thatΓ is Radon measurable for every Radon measure on X.

Remark A.1Note that since X is a separable Banach space,any probability measure on X is Radon;this is because any separable Banach space is a Polish space(separable and complete metric space)and that every Polish space is a Radon space(a Hausdorff space X is called a Radon space if every finite Borel measure on X is a Radon measure,i.e.,is inner regular(see[66]).

The following results are from[20,66].The final goal is to establish Corollary A.1 below,which we have employed in the article to prove that the mapχdefined in(3.17)(see Subsection 3.2)is universally Radon measurable.For that purpose,we need to introduce the following results(from Proposition A.2 to Theorem A.3).

Definition A.1(Lusinμ-Measurable)Let X be a topological space.Letμ be a Radon measure on X and let h map X into Y,where Y is a Hausdorff topological space.Then the mapping h is said to be Lusinμ-measurable if,for every compact set K ?X and everyδ>0,there exists a compact set Kδ?K withμ(K?Kδ)≤δsuch that h restricted to Kδis continuous.

Proposition A.2A function whose restriction to every compact set is continuous,is Lusin measurable for every Radon measure(see[66,p.25]).

Proposition A.3The assumptions are the same as in Definition A.1.If h:is Lusin μ-measurable,then h isμ-measurable,and conversely,if Y is metrizable and separable,then everyμ-measurable function is also Lusin μ-measurable(see[66,p.26]).

Theorem A.3Let X,Y and Z be separable Banach spaces andμbe a Radon measure on X.Letφ:be aμ-measurable mapping.LetΓ:be universally Radon measurable.Then G:=Γ?φ isμ-measurable on X.

ProofFrom Proposition A.3,φ is Lusinμ-measurable.Then Theorem A.3 follows from the proof of Theorem 3.2 in[10].

Corollary A.1Let X,Y and Z be separable Banach spaces andbe a continuous mapping.Letbe universally Radon measurable.Then G:=Γ?φis universally Radon measurable.

ProofThis can be directly deduced from Proposition A.2 and Theorem A.3.

A.4 Compact embedding results

In order to establish the compactness of a sequence of probability measures associated with the solutions to(3.3),we made use of the following compact embedding theorem which is close to that found in[69]and of course generalizes the classical Aubin-Lions compactness theorem(see[2]).

Proposition A.4Let Z??Y?X be a collection of three Banach spaces with Z compactly embedded in Y and Y continuously embedded in X.

(i)Suppose thatis a bounded subset of Lp(R,Z)∩L∞(R,Y),where 1

uniformly forand that there exists L>0 such that

Then,the setis relatively compact in Lp(R,Y).

(ii)For T>0,if G is a bounded subset of Lp(0,T,Z)∩L∞(0,T,Y)and

uniformly for elements in G,then G is relatively compact in Lp(0,T,Y).

ProofThe proof is a fairly straightforward generalization of[70,Theorem 13.2].Observe that if q>p,then(A.2)and(A.3)together imply that

uniformly for g∈G.Therefore there is no loss of generality in supposing that q≤p in what follows.

For a>0,define the averaging operator Jaaccording to

We takeArguing exactly as in[69],we have,for a>0,thatis relatively compact in Lp(R;Y).

To show thatis itself relatively compact in Lp(R;Y),we prove that it is a totally bounded subset of Lp(R;Y);in other words,we prove that,for every?>0,there exists finitely many elements g1,···,gNin Lp(R,Y)such that G is contained in the union of the ?balls centered at these points.

Again,arguing exactly as in[69],we have that,as a consequence of(A.2),for everyδ>0 there exists a=a(δ)>0 such that

On the other hand,from[71,Chapter 3,Lemma 2.1],we infer that,for everyη>0,there exists Cη>0 such that,for every g∈Lp(R,Z),

The last inequality follows from the fact thatfor all f∈Lp(R,Z).Now,on the other hand,we have

Fix?>0.Letand letand pick a>0,sufficiently small,so that(A.5)holds forwhere Cηis the constant corresponding toηin(A.6).Using thatis precompact inwe next choose a finite collectionsuch that the-balls centered atcoverNow,with these various choices,we have that for anythere existssuch thatAs such,we employ(A.6)withfollowed by(A.7)and estimate

Since,?>0 is arbitrary to begin with,this shows that G is a totally bounded subset of Lp(R;Y),and we thus infer(i).The second item(ii)follows directly from(i)as in[69].The proof of Proposition A.4 is therefore complete.

AcknowledgementsThe authors have benefited from the hospitality of the department of mathematics Virginia Tech and from the Newton institute for mathematical sciences,University of Cambridge where the final stage of the writing was completed.The authors wish to thank Arnaud Debussche for his help with the use of the universally Radon measurable selection theorem.The authors also thank the referee(s)for bringing their attention to the article[3]which they regrettably overlooked.

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