On ill- and well-posedness of dissipative martingale solutions to stochastic 3D Euler equations
aa r X i v : . [ m a t h . P R ] S e p ON ILL- AND WELL-POSEDNESS OF DISSIPATIVE MARTINGALESOLUTIONS TO STOCHASTIC 3D EULER EQUATIONS
MARTINA HOFMANOV ´A, RONGCHAN ZHU, AND XIANGCHAN ZHU
Abstract.
We are concerned with the question of well-posedness of stochastic three dimensionalincompressible Euler equations. In particular, we introduce a novel class of dissipative solutionsand show that (i) existence; (ii) weak–strong uniqueness; (iii) non-uniqueness in law; (iv) existenceof a strong Markov solution; (v) non-uniqueness of strong Markov solutions; all hold true withinthis class. Moreover, as a byproduct of (iii) we obtain existence and non-uniqueness of proba-bilistically strong and analytically weak solutions defined up to a stopping time and satisfying anenergy inequality.
Contents
1. Introduction 22. Notations 52.1. Function spaces 52.2. Noise 62.3. Path spaces 73. Class of dissipative solutions 83.1. Preliminary discussion 83.2. Dissipative martingale solutions 93.3. Dissipative probabilistically weak solutions 124. Weak–strong uniqueness, stability and existence 134.1. Weak–strong uniqueness 134.2. Stability 154.3. Existence 195. Non-uniqueness in law 205.1. Construction by convex integration 205.2. Extension of solutions 335.3. Application to the convex integration solutions 37
Date : September 23, 2020.2010
Mathematics Subject Classification.
Key words and phrases. stochastic Euler system, dissipative solutions, non-uniqueness in law, convex integration,strong Markov selection.Supported in part by NSFC (No. 11671035, No. 11771037, No. 11922103). Financial support by the DFG throughthe CRC 1283 “Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics andtheir applications” and support by key Lab of Random Complex Structures and Data Science, Chinese Academy ofScience are gratefully acknowledged.
6. Existence and non-uniqueness of strong Markov solutions 426.1. Selection of strong Markov processes 436.2. Application to the stochastic Euler equations 446.3. Non-uniqueness of strong Markov solutions 47Appendix A. Young integration 48References 481.
Introduction
The mathematics community is kept intrigued by the questions of ill- and/or well-posedness ofequations in fluid dynamics. A substantial progress has been experienced in the past ten years or so,starting from the groundbreaking results by De Lellis and Sz´ekelyhidi Jr. [DLS09, DLS10, DLS13].In these works, the method of convex integration was used in order to prove non-uniqueness of weaksolutions to Euler equations. Furthermore, non-uniqueness was established among weak solutionsdissipating energy which is one of the well-accepted criteria for the selection of physically relevantsolutions. The method was then further developed and successfully applied to a number of other fluiddynamics models. In particular also to the Navier–Stokes system in three dimensions by Buckmasterand Vicol [BV19b] and Buckmaster, Colombo and Vicol [BCV18], where it permitted to prove non-uniqueness for weak solutions not satisfying the energy inequality, i.e. not Leray solutions. We referto the excellent review articles by Buckmaster and Vicol [BV19a, BV20] for a gentle introductionand further references.In view of these issues, there has been a hope that a certain stochastic perturbation can providea regularizing effect of the underlying PDE dynamics. And indeed, some positive results havebeen achieved. Flandoli and Luo [FL19] showed that a noise of transport type prevents a vorticityblow-up in the Navier–Stokes equations. Flandoli, Hofmanov´a, Luo and Nilssen [FHLN20] thenshowed that the regularization is even provided by deterministic vector fields. A linear multiplicativenoise prevents the blow up of the velocity with high probability for the three dimensional Eulerand Navier–Stokes system as well, as shown by Glatt-Holtz and Vicol [GHV14] and R¨ockner, Zhuand Zhu [RZZ14], respectively. Noise also has a beneficial impact when it comes to long timebehavior and ergodicity. Da Prato and Debussche [DPD03] obtained a unique ergodicity for threedimensional stochastic Navier–Stokes equations with non-degenerate additive noise. The theoryof Markov selections by Flandoli and Romito [FR08] provides an alternative approach which alsoallowed to prove ergodicity for every Markov solution, see Romito [Ro08].Quite the contrary, in our previous work [HZZ19] we proved a negative result: non-uniquenessin law holds for the stochastic three dimensional Navier–Stokes equations with additive or linearmultiplicative noise in a class of analytically weak solutions. This in particular shows that the noiseof [DPD03, FR08, Ro08, RZZ14] mentioned above is of no help for the initial value problem inthis solution framework. The proof relies on a stochastic variant of the convex integration methodtogether with a general probabilistic construction developed in order to extend solutions defined upto a stopping time to the whole time interval [0 , ∞ ). This way, the convex integration permits toconstruct solutions which fail the corresponding energy inequality. The approach of [HZZ19] wasapplied by Yamazaki [Ya20a, Ya20b] to obtain non-uniqueness in law for stochastic Navier–Stokesequations in a three dimensional hyperviscous and a two dimensional fractional dissipative setting. N ILL- AND WELL-POSEDNESS TO STOCHASTIC 3D EULER EQUATIONS 3
For completeness, let us also mention that prior to [HZZ19] the convex integration has alreadybeen applied in a stochastic setting, namely, to the isentropic Euler system by Breit, Feireisl and Hof-manov´a [BFH20b] and to the full Euler system by Chiodaroli, Feireisl and Flandoli [CFF19]. Earlyworks on stochastic Euler equations treat only the two dimensional case (see e.g. [Be99, BF99, BP01,GHV14, BFM16]). The three-dimensional case has been treated in [MV00, Ki09, GHV14, CFH19].In particular, Glatt-Holtz and Vicol [GHV14] obtained local well-posedness of strong solutions tostochastic Euler equations in two and three dimensions, global well-posedness in two dimensions foradditive and linear multiplicative noise, and the above mentioned regularization by linear multiplica-tive noise in the three dimensional setting. Local well-posedness for three dimensional stochasticcompressible Euler equations was proved by Breit and Mensah [BM19].In the present paper, we are concerned with stochastic Euler equations governing the time evo-lution of the velocity u of an inviscid fluid on the three dimensional torus T . The system readsas d u + div( u ⊗ u )d t + ∇ P d t = G ( u )d B, div u = 0 , (1.1)where P stands for the corresponding pressure and the right hand side represents a random externalforce acting on the fluid. It is driven by a cylindrical Wiener process B defined on some probabilityspace and the diffusion coefficient G satisfies suitable assumptions, see Section 2.2 for more details.Our goal is to investigate the ill/well-posedness of this system from various perspectives. Moreprecisely, we aim at finding one stable solution concept which provides a suitable framework to studythe questions of existence, (non-)uniqueness as well as Markov selections. To this end, we introducethe notion of dissipative martingale solution . Roughly speaking, it corresponds to measure–valuedsolutions weak in the probabilistic sense and satisfying a version of an energy inequality. Althoughmeasure–valued solutions have been extensively studied in the deterministic literature (see e.g.[DPM87, Li96]), their formulation becomes rather challenging in the stochastic setting. A firstattempt in the context of the Euler equations was done by Breit and Moyo [BM20], who also provedexistence and weak–strong uniqueness. Nevertheless, it turns out that the question of existence of aMarkov selection poses severe restrictions on the definition of solution. Consequently, the frameworkfrom [BM20] cannot be used and new ideas are required.The key insight which we put forward is twofold. On the one hand, we present a novel formulationof the energy inequality and, on the other hand, we include an energy variable as a part of thesolution. This is necessary due to the presence of the so-called energy sinks which is an intrinsicproperty of the Euler equations. More precisely, the L -norm of the initial value itself does notcontain all the necessary information on the actual energy in order to restart the system. In thissense, including an additional variable is a legitimate step which reflects the nature of the equations.The principle ideas behind our definition of solution can be found in Section 3.1, where we refer thereader for more details and further notations.With this in hand, let us summarize our main results. The precise formulations can be foundin the respective sections and in particular the precise assumptions on the noise coefficient G arealso stated there. At this stage, let us only mention that all our results apply to the additive noisecase with a sufficient regularity of G , while some results also allow for certain possibly nonlinearcoefficient G ( u ).(i) Existence:
We prove that dissipative martingale solutions exist globally in time for diver-gence-free initial conditions in L . The proof relies on a compactness argument combinedwith Jakubowski–Skorokhod’s representation theorem. Due to the limited compactness of MARTINA HOFMANOV´A, RONGCHAN ZHU, AND XIANGCHAN ZHU the Euler system it is necessary to work with dissipative rather than analytically weaksolutions. Indeed, the latter one can only be shown to exists for certain initial conditionsup to a certain stopping time. See Section 4.2 and Section 4.3.(ii)
Weak–strong uniqueness:
We show that dissipative martingale solutions satisfy a weak–strong uniqueness principle. More precisely, if for some initial value there is an analyticallystrong solution defined on the canonical path space up to a stopping time, then it coincideswith all dissipative martingale solutions having the same initial value. See Section 4.1.(iii)
Non-uniqueness in law:
We apply the method of convex integration in order to constructinfinitely many solutions to the stochastic Euler system which live up to a certain stoppingtime, are analytically weak, probabilistically strong and satisfy an energy inequality. Usingthe general probabilistic extension of solutions which we developed in [HZZ19], we extendthese solutions beyond the stopping time to the whole time interval [0 , ∞ ). Even thoughanalytically weak before the stopping time, these solutions become only dissipative, i.e.,measure–valued, after the stopping time due to the limitations of the general existenceresult. See Section 5.(iv) Existence of a strong Markov solution:
Our notion of dissipative martingale solutionpermits to select a system of solutions to the stochastic Euler system satisfying the strongMarkov property. In addition, the solutions fulfil a version of the principle of maximal energydissipation proposed by Dafermos [Da79] in order to select the physically relevant solutions.Our proof makes use of the abstract Markov selection procedure by Krylov [Kr73]. Themain idea is to include an additional datum into the selection procedure. See Section 6.2.(v)
Non-uniqueness of strong Markov solutions:
Finally, we combine the result of non-uniqueness in law with the existence of a strong Markov solution and deduce non-uniquenessof strong Markov selections. See Section 6.3.The main contribution of our paper lies in the points (iii), (iv) and (v) and some novelties are alsopresent in the point (ii). In particular, compared to the weak–strong uniqueness in the deterministicsetting, the additional difficulties in the stochastic setting originate in the fact that the times of theenergy sinks are generally random. Consequently, evaluation of the energy inequality in its usualform at the stopping time, up to which the analytically strong solution lives, becomes delicate. Weovercome this issue by using a different form of the relative energy, and accordingly also a differentform of the energy inequality in the definition of a solution, with the help of a continuous stochasticprocess z rather than the kinetic energy itself, which is defined only a.e. in time.Regarding the point (iv), we recall that applications of Krylov’s Markov selection to SPDEscan be found in [FR08, GRZ09, BFH18a]. In particular, Flandoli and Romito [FR08] introduceda weaker notion of Markov property, the so-called almost sure Markov property. It means thatthe Markov property holds up to an exceptional set of deterministic times in (0 , ∞ ) having zeroLebesgue measure. They were able to show that the stochastic Navier–Stokes equations admit analmost sure Markov solution. The same issue appears for the stochastic compressible Navier–Stokessystem in [BFH18a] as well as for the models in [GRZ09]. However, in these works it was notpossible to obtain the usual Markov property, let alone the strong Markov property.Needless to say that the stochastic Euler equations (1.1) represent a number of additional diffi-culties when it comes to the construction of Markov solutions. More precisely, due to the presenceof random energy sinks together with the limited compactness the method of [FR08] does not apply.Our approach not only works for the Euler equations, it even permits to obtain the strong Markovproperty. This is precisely where the process z as part of the solution plays an essential role. Itgives the necessary control of the kinetic energy of the system after solution trajectories are shifted N ILL- AND WELL-POSEDNESS TO STOCHASTIC 3D EULER EQUATIONS 5 in time, required by the so-called disintegration property. We note that the same ideas can also beapplied in the (simpler) settings of [FR08, GRZ09, BFH18a] in order to construct strong Markovselections.The core of the non-uniqueness in law in (iii) is the method of convex integration. Unlike inour previous work [HZZ19] where the convex integration relied on an iteration procedure, we relyhere on a Baire category argument by De Lellis and Sz´ekelyhidi Jr. [DLS10]. The first applicationof this method in the stochastic setting was done in [BFH20b]. In the present paper, we extendthe stochastic approach further in several aspects. In particular, we present two new versionsof oscillatory lemmas, see Lemma 5.2 and Lemma 5.5. The former one permits to construct asubsolution with a prescribed energy at time t = 0. That is, we are able to eliminate the initialjump of the energy present in [BFH20b], which is essential in order to obtain solutions satisfyingthe energy inequality and weak–strong uniqueness principle. The latter oscillatory lemma is appliedin order to enforce the energy inequality at a given stopping time. In order to combine thesetwo, we need to construct suitable energy function by applying the theory of Young’s integrationand a control of the iterated stochastic integral of the Wiener process before a stopping time, seeLemma 5.4. This way we are able to construct infinitely many analytically weak solutions beforethe stopping time satisfying a.e. in time the usual energy inequality with a prescribed energy defect.Moreover, the prescribed energy depends on the solution itself.Furthermore, our convex integration solutions are probabilistically strong, i.e. adapted to the givenWiener process. This is the key property needed to extend these solutions as dissipative martin-gale solutions beyond the stopping time and to deduce the non-uniqueness in law. Compared tothe Navier–Stokes setting from [HZZ19] where the energy inequality could not be fulfilled by theconvex integration solutions, this point requires a careful treatment in the extension of solutions inSection 5.3. Since we have to control the iterated stochastic integral of the Wiener process beforethe stopping time as mentioned above, we also need to define the corresponding stopping time onthe canonical path space. But now the difficulty lies in how to define the iterated stochastic integralon the path space without the use of any probability measure. Indeed, due to the low time reg-ularity of the Wiener process, the stochastic integral cannot be defined by purely analytical toolsand probability theory is required in a nontrivial way. We overcome this issue by using the energyequality obtained in the convex integration step to identify the corresponding stochastic integral.Applying the theory of Young’s integration we are able to define the necessary stopping time onthe path space and to transfer the convex integration solutions to the path space where they canbe extended to dissipative martingale solutions on [0 , ∞ ).To conclude this introductory part, we note that the Markov property corresponds to the semiflowproperty in the deterministic setting. In other words, as a simple observation, our results translatedto the deterministic setting imply in particular non-uniqueness of the associated semiflow, which tothe best of our knowledge has not been known before, see Remark 6.13 for more details.2. Notations
Function spaces.
Throughout the paper, we use the notation a . b if there exists a constant c > a cb , and we write a ≃ b if a . b and b . a . Given a Banach space E with anorm k · k E and T >
0, we write C T E = C ([0 , T ]; E ) for the space of continuous functions from[0 , T ] to E , equipped with the supremum norm k f k C T E = sup t ∈ [0 ,T ] k f ( t ) k E . We also use CE todenote the space of C ([0 , ∞ ); E ). For α ∈ (0 ,
1) we define C αT E as the space of α -H¨older continuousfunctions from [0 , T ] to E , endowed with the seminorm k f k C αT E = sup s,t ∈ [0 ,T ] ,s = t k f ( s ) − f ( t ) k E | t − s | α . Wedenote by L pT E the set of L p -integrable functions from [0 , T ] to E . For α ∈ (0 , p ∈ [1 , ∞ ), we MARTINA HOFMANOV´A, RONGCHAN ZHU, AND XIANGCHAN ZHU define W α,pT E as the Sobolev space of all f ∈ L pT E such that R T R T k f ( t ) − f ( s ) k pE | t − s | αp d t d s < ∞ endowedwith the norm k f k pW α,pT E := R T k f ( t ) k pE d t + R T R T k f ( t ) − f ( s ) k pE | t − s | αp d t d s . We also use C α loc ([0 , ∞ ); E )and W α,p loc ([0 , ∞ ); E ), respectively, to denote the space of functions f satisfying for every T > f | [0 ,T ] ∈ C αT E and f | [0 ,T ] ∈ W α,pT E , respectively.We use L p to denote the set of the standard L p -integrable functions from T to R . Set L σ = { u ∈ L , div u = 0 } . For s > p > W s,p := { f ∈ L p ; k ( I − ∆) s f k L p < ∞} with the norm k f k W s,p = k ( I − ∆) s f k L p . For s > H s := W s, ∩ L σ . For s < W s,q to be the dual spaceof W − s,p with p + q = 1. Let { e i } i ∈ N be a complete orthonormal system in L σ . For a domain D we use D ′ ( D ) to denote the dual space of C ∞ c ( D ).2.2. Noise.
For a Hilbert space U let L ( U, L σ ) be the space all Hilbert–Schmidt operators from U to L σ with the norm k · k L ( U,L σ ) . Let G : L σ → L ( U, L σ ) be B ( L σ ) / B ( L ( U, L σ )) measurable.In the sequel, we assume the following.(G1) There exists C > x ∈ L σ k G ( x ) k L ( U,L σ ) C (1 + k x k L ) . (G2) If x n → x weakly in L σ thenlim n →∞ k G ( x n ) − G ( x ) k L ( U,L σ ) = 0 . Remark 2.1.
We note that if the noise is additive, i.e., G ∈ L ( U, L σ ) does not depend on thesolution, then the conditions (G1) and (G2) hold. Moreover, for the multiplicative case, we havefor instance the following example: for x ∈ L σ , u ∈ U and { l k } k ∈ N being the orthonormal basis in U , let G ( x ) u = ∞ X k =1 h u, l k i U Π k xf k , where Π k y = P kj =1 h y, e k i e k and f k ∈ L ∞ ( T ) satisfying P ∞ k =1 | k | s k f k k L ∞ < ∞ for some s > k G ( x n ) − G ( x ) k L ( U,L σ ) = ∞ X k =1 k Π k ( x n − x ) f k k L σ k x n − x k H − s/ ∞ X k =1 | k | s k f k k L ∞ , where k x n − x k H − s/ → k x by g (Π ℓ x ) for some fixed ℓ and a Nemytskii operator g given by a Lipchitzfunction. Remark 2.2.
At first sight, it may seem unsatisfactory that the simple linear multiplicative noiseof the form G ( u )d W = u d β with a real-valued Brownian motion β is not covered by our theory.Indeed, it does not satisfy the condition (G2). However, let us point out that for this kind of noiseone can perform a suitable transformation and a random rescaling of time in the spirit of [CFF19]in order to rewrite the stochastic Euler system as a deterministic one. In other words, the results ofthe deterministic theory can be translated into this stochastic setting and actually much more canbe proved in this case.For the weak–strong uniqueness principle we assume the following Lipschitz condition.(Glip) There exists a constant L such that k G ( x ) − G ( y ) k L ( U,L σ ) L k x − y k L . N ILL- AND WELL-POSEDNESS TO STOCHASTIC 3D EULER EQUATIONS 7
And additional assumptions are required for the convex integration method and the resultingnon-uniqueness in law. Suppose there is another Hilbert space U such that the embedding U ⊂ U is Hilbert–Schmidt. In particular, we assume that the noise is additive and the following holds.(G3) G ∈ L ( U, H (3+ σ ) / ) for some σ > G : U → H / σ is a bounded operator for some σ > Path spaces.
In Section 3.1, we introduce four notions of solutions: martingale and prob-abilistically weak solutions as well as simplified martingale and simplified probabilistically weaksolutions. Thus, as a first step, we define four corresponding path spaces, which we denote by Ω M ,Ω P W , Ω SM and Ω SP W , respectively.2.3.1.
Martingale solutions.
Let α ∈ (2 / , k ∈ (3 / , ∞ ), q ∈ (1 , ∞ ) be such that αq > α/ − /q >
1. DefineΩ SM := (cid:8) ( x, y ) ∈ C ([0 , ∞ ) , H − × ( M + ( T , R × ) , w )); y ∈ W α,q loc ([0 , ∞ ); W − k, ( T , R × )) (cid:9) . Here M + ( T , R × ) denotes the space of (symmetric) positive semidefinite matrix valued finitemeasures m so that for every ξ ∈ R m : ξ ⊗ ξ is a non-negative finite measure on T , where A : B = P i,j =1 a ij b ij denotes the matrix inner product of symmetric matrices A = ( a ij ) i,j =1 , B = ( b ij ) i,j =1 . By ( M + ( T , R × ) , w ) we denote this space equipped with the weak topology, whichis completely separably metrizable by the Prokhorov metric, see [Ka17, Lemma 4.3, Lemma 4.5].As a consequence, the path space Ω SM is a Polish space. The parameters α, q are chosen in away to permit the definition of a certain Young integral, see the discussion after (2.1) below formore details. The parameter k is chosen so that the embedding M + ⊂ W − k, holds in three spacedimensions.Let P (Ω SM ) denote the set of all probability measures on (Ω SM , B SM ) with B SM being the Borel σ -algebra coming from the topology of locally uniform convergence on Ω SM . Let ( x, y ) : Ω SM → H − × M + ( T , R × ) denote the canonical process on Ω SM given by( x t ( ω ) , y t ( ω )) = ω ( t ) . Similarly, for t > σ -algebra B tSM = σ { ( x ( s ) , y ( s )) , s > t } . Finally, we define thecanonical filtration B SM,t := σ { ( x ( s ) , y ( s )) , s t } , t >
0, as well as its right continuous version B SM,t := ∩ s>t B SM,s , t > X := { ( x , y , z ) ∈ L σ × M + ( T , R × ) × R ; k x k L z } .Ω M := (cid:8) ( x, y, z ) | ( x, y, z ) ∈ C ([0 , ∞ ) , H − × ( M + ( T , R × ) , w ) × R ); y ∈ W α,p loc ([0 , ∞ ); W − k, ( T , R × )) (cid:9) , and Ω tM := (cid:8) ( x, y, z ) | ( x, y, z ) ∈ C ([ t, ∞ ) , H − × ( M + ( T , R × ) , w ) × R ); y ∈ W α,p loc ([ t, ∞ ); W − k, ( T , R × )) (cid:9) . Let P (Ω M ) denote the set of all probability measures on (Ω M , B M ) with B M being the Borel σ -algebra coming from the topology of locally uniform convergence on Ω M . Let ( x, y, z ) : Ω M → H − × M + ( T , R × ) × R denote the canonical process on Ω M given by( x t ( ω ) , y t ( ω ) , z t ( ω )) = ω ( t ) . MARTINA HOFMANOV´A, RONGCHAN ZHU, AND XIANGCHAN ZHU
Similarly, for t > σ -algebra B tM = σ { ( x ( s ) , y ( s ) , z ( s )) , s > t } . Finally, we definethe canonical filtration B M,t := σ { ( x ( s ) , y ( s ) , z ( s )) , s t } , t >
0, and its right continuous version B M,t := ∩ s>t B M,s , t >
0. For given probability measure P we use E P to denote the expectationunder P .2.3.2. Probabilistically weak solutions.
For the same parameters α, q, k as before defineΩ
SP W := (cid:8) ( x, y, b ) ∈ C ([0 , ∞ ); H − × ( M + ( T , R × ) , w ) × U ); y ∈ W α,q loc ([0 , ∞ ); W − k, ( T , R × )) , b ∈ W α/ ,q loc ([0 , ∞ ); U ) (cid:9) . (2.1)Similarly to the above, the path space Ω SP W is a Polish space. The parameters α, q are chosenin a way to permit the definition of a certain Young integral, which will be needed below. Recallthat for g ∈ C β and f ∈ C γ the Young integral t R t g r d f r is well-defined provided β + γ > W α,q ⊂ C β valid in one dimension for β ∈ (0 , α − /q ), the above Young integral is well-defined provided g ∈ W α,q and f ∈ W α/ ,q suchthat 3 α/ − /q > P (Ω SP W ) denote the set of all probability measures on (Ω
SP W , B SP W ) with B SP W being theBorel σ -algebra coming from the topology of locally uniform convergence on Ω SP W . Let ( x, y, b ) :Ω
SP W → H − × M + ( T , R × ) × U denote the canonical process on Ω SP W given by( x t ( ω ) , y t ( ω ) , b t ( ω )) = ω ( t ) . For t > σ -algebra B tSP W = σ { ω ( s ) , s > t } . Finally, we define the canonical filtration B SP W,t := σ { ω ( s ) , s t } , t >
0, and its right continuous version B SP W,t := ∩ s>t B SP W,s , t > P W := (cid:8) ( x, y, z, b ) ∈ C ([0 , ∞ ); H − × ( M + ( T , R × ) , w ) × R × U ); y ∈ W α,q loc ([0 , ∞ ); W − k, ( T , R × )) , b ∈ W α/ ,q loc ([0 , ∞ ); U ) (cid:9) , and the canonical process on Ω P W and ( B P W,t ) t > , ( B P W,t ) t > . Class of dissipative solutions
Preliminary discussion.
It turns out that a good stable notion of solution to the Eulersystem (1.1) has to include more information than what is provided by the velocity field itself. Letus first discuss the main ideas on an informal level which also permits us to fix the notation.In the definitions below we denote by x the velocity field and rewrite the Euler system (1.1) asd x + div R d t + ∇ p d t = G ( x ) d b, div x = 0 , x (0) = x , (3.1)satisfied in a distributional sense. Here b is a cylindrical Wiener process in U . This way we introducea matrix-valued variable R , which is considered as part of the solution. Furthermore, we require acompatibility condition, namely, that the so-called Reynolds stress satisfies N := R − x ⊗ x > . (3.2)Observe that if R = x ⊗ x , the definition reduces to the usual notion of analytically weak solution.However, due to the lack of compactness for the Euler equations, weak solutions are not stable underapproximations and this is the reason for weakening the notion of solution further by introducing R . Since R is only L ∞ with respect to the time variable, we also work with its primitive functiongiven by ∂ t y = R , y (0) = y . (3.3) N ILL- AND WELL-POSEDNESS TO STOCHASTIC 3D EULER EQUATIONS 9
We aim for a class of solutions satisfying a weak–strong uniqueness principle, thus, we includean energy inequality into our definition. In the deterministic setting, this corresponds to the notionof dissipative measure–valued solution extensively studied in the literature. However, we choosea different formulation which overcomes various difficulties present in the stochastic setting. Inparticular, we introduce a new variable z satisfyingd z = 2 h x, G ( x )d b i + k G ( x ) k L ( U,L ) d t, z (0) = z . (3.4)Note that if x is an analytically weak solution possessing sufficient spatial regularity then the energyequality holds, that is, z ( t ) = k x ( t ) k L for all t >
0. This requirement has to be relaxed and so wepostulate instead the compatibility condition Z T dtr R ( t ) = k x ( t ) k L + Z T dtr N ( t ) z ( t ) for a.e. t > , (3.5)which only permits to show that z ( t ) > k x ( t ) k L for all t > z = k x k L . However, a generalinitial value z has to be allowed for the Markov selection in order to obtain a notion of solutionstable under shifts on trajectories.Roughly speaking, a dissipative martingale solution defined below is the probability law of ( x, y, z )satisfying (3.1), (3.2), (3.3), (3.4), (3.5) in a suitable sense. A dissipative probabilistically weaksolution is then the probability law of ( x, y, z, b ) such that (3.1), (3.2), (3.3), (3.4), (3.5) hold. Inthe case of z = k x k L , it follows from (3.4) that z is a function of the other variables. Therefore, wedefine a simplified dissipative martingale solution as the law of ( x, y ) under a dissipative martingalesolution and a simplified dissipative probabilistically weak solution as the law of ( x, y, b ) under adissipative probabilistically weak solution.3.2. Dissipative martingale solutions.
Throughout the paper, the meaning of the variables x, b, R , N , y, z remains the same as in Section 3.1. Definition 3.1.
Let ( x , y , z ) ∈ X . A probability measure P ∈ P (Ω M ) is a dissipative martingalesolution to the Euler system (1.1) with the initial value ( x , y , z ) at time s provided (M1) P ( x ( t ) = x , y ( t ) = y , z ( t ) = z , t s ) = 1 , P ( N ∈ L ∞ loc ([ s, ∞ ); M + ( T ; R × ))) = 1 . (M2) For every e i ∈ C ∞ ( T ) ∩ L σ and t > the process M it,s := h x ( t ) − x ( s ) , e i i − Z ts Z T ∇ e i : d R ( r )d r is a continuous square integrable ( B M,t ) t > s -martingale under P with the quadratic variation processgiven by R ts k G ( x ( r )) ∗ e i k U d r . (M3) P-a.s. for every t > sz ( t ) = z ( s ) + 2 M Et,s + Z ts k G ( x ( r )) k L ( U,L ) d r, where M Et,s = P ∞ i =1 R ts h x ( r ) , e i i d M ir,s is a continuous ( B M,t ) t > s -martingale and P (cid:18)Z T dtr R ( t ) z ( t ) for a.e. t > s (cid:19) = 1 . The following result shows how to derive a priori estimates for dissipative solutions.
Lemma 3.2.
Suppose that (G1) holds. Let P be a dissipative martingale solution to (1.1) . ThenP-a.s. for every t > s and every p ∈ [2 , ∞ ) z p ( t ) = z p ( s ) + 2 p Z ts z p − ( r )d M Er + p Z ts z p − ( r ) k G ( x ( r )) k L ( U,L ) d r + 2 p ( p − Z ts z p − ( r ) k G ( x ( r )) ∗ x ( r ) k U d r, (3.6) and P ( k x ( t ) k L z ( t ) for all t > s ) = 1 . (3.7) Consequently, for every N ∈ N and p ∈ [2 , ∞ ) there exists a universal constant C N,p > such that E P " sup t ∈ [0 ,N ] k x ( t ) k pL + E P " esssup t ∈ [0 ,N ] (cid:18)Z T dtr R ( t ) (cid:19) p C N,p ( z p ( s ) + 1) . Proof.
First, we observe that according to (M1), (M3) it holds P -a.s. for a.e. t > s that z ( t ) > t > s by continuity of t z ( t ). Then (3.6) is an application ofItˆo’s formula.Next, we realize that by (M1), (M3) it holds P -a.s. for a.e. t > s Z T dtr R ( t ) z ( t ) , k x ( t ) k L = Z T dtr R ( t ) − Z T dtr N ( t ) z ( t ) . Since the function t
7→ k x ( t ) k L is lower semicontinuous due to the continuity of x in H − and t z ( t ) is continuous, we deduce (3.7).To prove the last claim, we first choose stopping time τ R := inf { t > , z ( t ) > R } and by (M3)we know τ R → ∞ as R → ∞ . In fact by (M3) and using the Burkholder–Davis–Gundy inequality,Young’s inequality, the linear growth assumption (G1) on G and Gronwall’s lemma we obtain E P " sup t ∈ [0 ,N ] z ( t ) . z ( s ) + 1 . Then we estimate the right hand side of (3.6) using the Burkholder–Davis–Gundy inequality, Young’sinequality, the linear growth assumption (G1) on G and Gronwall’s lemma to deduce E P " sup t ∈ [0 ,N ∧ τ R ] z p ( t ) . z p ( s ) + 1 , where the implicit constant only depends on N, p and the constant in (G1). Letting R → ∞ , theresult follows. (cid:3) We note that a dissipative martingale solution in the sense of Definition 3.1 may contain aninitial jump of the energy in the sense that z > k x k L . However, we are even able to constructsolutions without the initial energy jump, and this will be seen in the construction by compactnessin Section 4.3 as well as in the construction by convex integration in Section 5.1. In addition, theweak–strong uniqueness principle requires the assumption z = k x k L . The reason for relaxing thisin our main definition of a solution is the Markov selection in Section 6. More precisely, a notionof solution without an initial energy jump is not stable under shifts on trajectories, which is one ofthe main ingredients required by the Markov selection.We observe that the notion of dissipative solution simplifies in case of no initial energy jump. N ILL- AND WELL-POSEDNESS TO STOCHASTIC 3D EULER EQUATIONS 11
Definition 3.3.
Let ( x , y ) ∈ L σ × M + ( T ; R × ) . A probability measure P ∈ P (Ω SM ) is asimplified dissipative martingale solution to the Euler system (1.1) with the initial value ( x , y ) attime s provided (M1) P ( x ( t ) = x , y ( t ) = y , t s ) = 1 , P ( N ∈ L ∞ loc ([ s, ∞ ); M + ( T ; R × ))) = 1 . (M2) For every e i ∈ C ∞ ( T ) ∩ L σ and t > the process M it,s := h x ( t ) − x ( s ) , e i i − Z ts Z T ∇ e i : d R ( r )d r is a continuous square integrable ( B SM,t ) t > s -martingale under P with the quadratic variation processgiven by R ts k G ( x ( r )) ∗ e i k U d r . (M3) P-a.s. define for every t > sz ( t ) := k x k L + 2 M Et,s + Z ts k G ( x ( r )) k L ( U,L ) d r, where M Et,s = P ∞ i =1 R ts h x ( r ) , e i i d M ir,s is a continuous ( B SM,t ) t > s -martingale and P (cid:18)Z T dtr R ( t ) z ( t ) for a.e. t > s (cid:19) = 1 . For this definition we do not need z in the path space and we can prove that if z = k x k L thenthese two definitions are equivalent. Corollary 3.4.
Let P be a dissipative martingale solution with the initial value ( x , y , z ) at time s such that z = k x k L . Then the canonical process z under P is a function of x, y . In other words, P is fully determined by the joint probability law of x, y and can be identified with a probabilitymeasure on the reduced path space Ω SM . Hence P is a simplified dissipative martingale solutionwith the initial value ( x , y ) at time s .Conversely, let ( x , y ) ∈ L σ × M + ( T ; R × ) be given and let P ∈ P (Ω SM ) be a simplifieddissipative martingale solution and define P -a.s. for t > sz ( t ) := k x k L + 2 M Et,s + Z ts k G ( x ( r )) k L ( U,L ) d r. Let Q be the law of ( x, y, z ) under P . Then Q ∈ P (Ω M ) gives raise to a dissipative martingalesolution starting from the initial value ( x , y , k x k L ) at the time s .Proof. It follows from (M2) that M is a function of x, y and consequently from (M3) we deducethat z is determined by x, y and the initial value z . This gives the first claim whereas the secondclaim is immediate. (cid:3) We also observe that if P is a dissipative martingale solution to (1.1) with initial value ( x , y , z )then the law of the process ( x, y + c , z + c ) under P is again a dissipative martingale solution to(1.1) with initial value ( x , y + c , z + c ) for every c ∈ R , c >
0. Indeed, the initial value y doesnot have any influence on the actual dynamics: it was introduced artificially by including y into thepath space rather than R = ∂ t y , which is not continuous in time. The role of the initial value z ismore delicate. It is related to the so-called energy sinks discussed more in detail in Section 6.In order to further verify that our definition of dissipative solution is reasonable, we first provethat a sufficiently regular dissipative solution is a solution in the classical sense. Proposition 3.5.
Suppose that (G1) holds. Let t be a ( B SM,t ) t > s -stopping time. If P is a simplifieddissipative martingale solution to (1.1) with the initial value ( x , y ) at time s such that x ( · ∧ t ) ∈ C ([ s, ∞ ); C ( T )) P -a.s. , then P ( R = x ⊗ x for a.e. t > t > s ) = 1 . In other words, under P the canonical process x satisfies (1.1) before t in the analytically strongsense.Proof. Due to the sufficient spatial regularity the canonical process x under P , we may apply Itˆo’sformula to obtain for an arbitrary ( B SM,t ) t > s -stopping time τ t k x ( t ∧ τ ) k L = k x ( s ) k L + 2 Z t ∧ τs Z T ∇ x : d N ( r )d r + 2 ∞ X i =1 Z t ∧ τs h x ( r ) , e i i d M ir,s + Z t ∧ τs k G ( x ( r )) k L ( U,L σ ) d r. (3.8)Now, we subtract (3.8) from z ( t ) and use the positive semidefinitness of N as well as (M3) todeduce E P (cid:2) ( z − k x k L )( t ∧ τ ) (cid:3) = − E P (cid:20)Z t ∧ τs Z T ∇ x : d N ( r )d r (cid:21) . E P (cid:20)Z t ∧ τs k∇ x k L ∞ Z T dtr N ( r )d r (cid:21) E P (cid:20)Z t ∧ τs k∇ x k L ∞ ( z − k x k L )( r )d r (cid:21) E P (cid:20)Z ts k∇ x k L ∞ ( z − k x k L )( r ∧ τ )d r (cid:21) . (3.9)Here in the last step we used (3.7). Let us now define the stopping times τ R = inf { t > s ; k∇ x ( t ) k L ∞ > R } ∧ t . Then τ R → t P -a.s. and it follows from (3.9) that E P (cid:2) ( z − k x k L )( t ∧ τ R ) (cid:3) R Z ts E P (cid:2) ( z − k x k L )( r ∧ τ R ) (cid:3) d r. By Gronwall’s inequality and sending R → ∞ we obtain for every t > t > sP ( z ( t ∧ t ) = k x ( t ∧ t ) k L ) = 1 , hence the claim follows by the continuity of t z ( t ) as well as t
7→ k x ( t ) k L under P and (M3). (cid:3) Dissipative probabilistically weak solutions.
We conclude this section with the definitionof dissipative probabilistically weak solution.
Definition 3.6.
Let ( x , y , z ) ∈ X . A probability measure P ∈ P (Ω P W ) is a dissipative prob-abilistically weak solution to the Euler system (1.1) with the initial value ( x , y , z , b ) at time s provided (M1) P ( x ( t ) = x , y ( t ) = y , z ( t ) = z , b ( t ) = b , t s ) = 1 , P ( N ∈ L ∞ loc ([ s, ∞ ); M + ( T ; R × ))) = 1 . N ILL- AND WELL-POSEDNESS TO STOCHASTIC 3D EULER EQUATIONS 13 (M2)
Under P , b is a cylindrical ( B P W,t ) t > s -Wiener process in U starting from b at time s andfor every e i ∈ C ∞ ( T ) ∩ L σ and t > s h x ( t ) − x ( s ) , e i i − Z ts Z T ∇ e i : d R ( r )d r = Z ts h e i , G ( x ( r ))d b ( r ) i . (M3) P-a.s. for every t > sz ( t ) = z ( s ) + 2 Z ts h x ( r ) , G ( x ( r ))d b ( r ) i + Z ts k G ( x ( r )) k L ( U,L ) d r and P (cid:18)Z T dtr R ( t ) z ( t ) for a.e. t > s (cid:19) = 1 . Similar as above we can also introduce the following simplified dissipative probabilistically weaksolution if z = k x k L . Definition 3.7.
Let ( x , y ) ∈ L σ × M + ( T ; R × ) . A probability measure P ∈ P (Ω SP W ) is asimplified dissipative probabilistically weak solution to the Euler system (1.1) with the initial value ( x , y , b ) at time s provided (M1) P ( x ( t ) = x , y ( t ) = y , b ( t ) = b , t s ) = 1 , P ( N ∈ L ∞ loc ([ s, ∞ ); M + ( T ; R × ))) = 1 . (M2) Under P , b is a cylindrical ( B SP W,t ) t > s -Wiener process in U starting from b at time s andfor every e i ∈ C ∞ ( T ) ∩ L σ and t > s h x ( t ) − x ( s ) , e i i − Z ts Z T ∇ e i : d R ( r )d r = Z ts h e i , G ( x ( r ))d b ( r ) i . (M3) P-a.s. define for every t > sz ( t ) = k x k L + 2 Z ts h x ( r ) , G ( x ( r ))d b ( r ) i + Z ts k G ( x ( r )) k L ( U,L ) d r. Then P (cid:18)Z T dtr R ( t ) z ( t ) for a.e. t > s (cid:19) = 1 . Remark 3.8. (i) From this definition it is easy to see that the law of ( x, y, z ) under a dissipativeprobabilistically weak solution P gives a dissipative martingale solution in the sense of Definition 3.1.(ii) Similarly to Corollary 3.4 these two definitions of dissipative probabilistically weak solutionsare equivalent under the condition that z = k x k L .4. Weak–strong uniqueness, stability and existence
Weak–strong uniqueness.
As the next step, we show that dissipative solutions satisfy aweak–strong uniqueness principle.
Theorem 4.1.
Suppose that (G1) and (Glip) hold. Let P be a simplified dissipative probabilisticallyweak solution to (1.1) starting from the initial value ( x , y , b ) at time s > . Assume that onthe stochastic basis (Ω SP W , B SP W , ( B SP W,t ) t > , P ) together with the ( B SP W,t ) t > s -Wiener process b ,there exists an ( B SP W,t ) t > s -adapted process u which is an analytically strong solution to (1.1) up to a ( B SP W,t ) t > s -stopping time t such that u ( · ∧ t ) ∈ C ([ s, ∞ ); C ( T )) P -a.s. and P ( u ( s ) = x ( s )) = 1 . Then P (cid:0) x ( t ∧ t ) = u ( t ∧ t ) for all t > s (cid:1) = 1 . Proof.
First of all, we note that since u is regular enough, the usual a priori estimate for the Eulerequations holds true. In particular, we may apply Itˆo’s formula to the function u
7→ k u k L andestimate using Burkholder–Davis–Gundy’s inequality, the linear growth assumption (G1) on G andGronwall’s lemma to obtain for any N ∈ N E P " sup t ∈ [ s,N ] k u ( t ∧ t ) k L < ∞ . (4.1)In order to establish the weak–strong uniqueness principle, we introduce a modification of theso-called relative energy between the two solutions x and u , which is adapted to our definition ofdissipative solution. Namely, for t ∈ [ s, t ] we let E rel ( t ) := 12 z ( t ) − h x ( t ) , u ( t ) i + 12 k u ( t ) k L = 12 k x ( t ) − u ( t ) k L + 12 (cid:0) z ( t ) − k x ( t ) k L (cid:1) . As a consequence of (3.7), we obtain P ( E rel ( t ∧ t ) > t > s ) = 1 . (4.2)Let τ t be a ( B SP W,t ) t > s -stopping time. Using the regularity of u , we may apply Itˆo’s formulato obtain h x ( t ∧ τ ) , u ( t ∧ τ ) i = h x ( s ) , u ( s ) i + ∞ X i =1 Z t ∧ τs h u ( r ) , e i ih e i , G ( x ( r ))d b r i + Z t ∧ τs Z T ∇ u ( r ) : d R ( r )d r − Z t ∧ τs Z T x ( r ) · div( u ( r ) ⊗ u ( r ))d ξ d r + ∞ X i =1 Z t ∧ τs h x ( r ) , e i ih e i , G ( u ( r ))d b r i + Z t ∧ τs h G ( x ( r )) , G ( u ( r )) i L ( U,L ) d r = h x ( s ) , u ( s ) i + ∞ X i =1 Z t ∧ τs h u ( r ) , e i ih e i , G ( x ( r ))d b r i + Z t ∧ τs Z T ∇ u ( r ) : d N ( r )d r + Z t ∧ τs Z T ∇ u ( r ) : ( x ⊗ x )d ξ d r − Z t ∧ τs Z T x ( r ) · div( u ( r ) ⊗ u ( r ))d ξ d r + ∞ X i =1 Z t ∧ τs h x ( r ) , e i ih e i , G ( u ( r ))d b r i + Z t ∧ τs h G ( x ( r )) , G ( u ( r )) i L ( U,L ) d r. Furthermore, we have Z t ∧ τs Z T ∇ u ( r ) : d N ( r )d r + Z t ∧ τs Z T ∇ u ( r ) : ( x ⊗ x )d ξdr − Z t ∧ τs Z T x ( r ) · div( u ( r ) ⊗ u ( r ))d ξ d r = Z t ∧ τs Z T Du ( r ) : d N ( r )d r + Z t ∧ τs Z T ( x − u ) · Du ( r )( x − u )d ξ d r. N ILL- AND WELL-POSEDNESS TO STOCHASTIC 3D EULER EQUATIONS 15
Here Du = ( ∇ u + ∇ u t ). Taking expectation and using Lemma 3.2 and (4.1) we obtain E P [ E rel ( t ∧ τ )] = 12 E P (cid:2) k u ( t ∧ τ ) k L (cid:3) + 12 E P [ z ( t ∧ τ )] − E P [ h x ( s ) , u ( s ) i ] − E P (cid:20)Z t ∧ τs h G ( x ( r )) , G ( u ( r )) i L ( U,L ) d r (cid:21) − E P (cid:20)Z t ∧ τs Z T Du ( r ) : d N ( r )d r (cid:21) − E P (cid:20)Z t ∧ τs Z T ( x − u ) · Du ( r )( x − u )d ξ d r (cid:21) . Hence combining this with (M3) and the energy equality for u , namely, E P [ k u ( t ∧ τ ) k L ] = E P [ k u ( s ) k L ] + E P (cid:20)Z t ∧ τs k G ( u ( r )) k L ( U,L ) d r (cid:21) , implies that E P [ E rel ( t ∧ τ )] E P [ k x ( s ) − u ( s ) k L ] + E P (cid:20)Z t ∧ τs k Du k L ∞ ( z − k x k L )( r )d r (cid:21) + 12 E P (cid:20)Z t ∧ τs k G ( x ( r )) − G ( u ( r )) k L ( U,L ) d r (cid:21) + E P (cid:20)Z t ∧ τs k Du k L ∞ k x ( r ) − u ( r ) k L d r (cid:21) . Choose τ as the stopping time τ R = inf { t > s ; k Du ( t ) k L ∞ > R } ∧ t and using similar argument as in the proof of Theorem 3.5 together with (4.2), we obtain P (cid:0) x ( t ∧ t ) = u ( t ∧ t ) (cid:1) = 1 for all t > s, which implies the result by the time continuity of x and u . (cid:3) Stability.
The following result provides a stability of the set of all probabilistically weaksolutions with respect to the initial time and the initial condition. We denote by C P W ( s, x , y , z , b )the set of all dissipative martingale solutions with the initial condition ( x , y , z , b ) and the initialtime s . Theorem 4.2.
Suppose that (G1) , (G2) hold. Let ( x n , y n , z n ) ∈ X , s n ∈ [0 , ∞ ) , n ∈ N , and assumethat ( s n , x n , y n , z n , b n ) → ( s , x , y , z , b ) in [0 , ∞ ) × L σ × M + ( T , R × ) × [0 , ∞ ) × U as n → ∞ and let P n ∈ C P W ( s n , x n , y n , z n , b n ) . Then there exists a subsequence n k such that thesequence ( P n k ) k ∈ N converges weakly to some P ∈ C P W ( s , x , y , z , b ) .Proof. Step 1: Tightness. In the first step, we show that ( P n ) n ∈ N is tight in Ω P W . Since for every n ∈ N the measure P n is a dissipative probabilistically weak solution to (1.1) starting from theinitial condition ( x n , y n , z n , b n ) at time s n in the sense of Definition 3.6, the process b ( · + s n ) − b n is under P n a cylindrical Wiener process on U starting at time 0 from the initial value 0. Usingthe fact that the law of a cylidrical Wiener process is unique and tight on C α/ ε loc ([0 , ∞ ); U ) for α/ ε < /
2, the same argument as in the proof of Theorem 5.1 in [HZZ19] implies that the lawof b under the family of measures ( P n ) n ∈ N is tight on C ([0 , ∞ ); U ) ∩ W α/ ,q loc ([0 , ∞ ); U ) for α, q asin the Section 2.3. Next, Lemma 3.2 yields all the necessary uniform estimates for the remaining variables. Moreprecisely, in view of Lemma 3.2 and (M2), (M3) we deduce for all N ∈ N and κ ∈ (0 , / n ∈ N E P n " sup t ∈ [0 ,N ] k x ( t ) k L + sup r = t ∈ [0 ,N ] k x ( t ) − x ( r ) k H − | t − r | κ < ∞ , sup n ∈ N E P n " sup t ∈ [0 ,N ] | z ( t ) | + sup r = t ∈ [0 ,N ] | z ( t ) − z ( r ) || t − r | κ < ∞ . By the compact embedding (see Theorem 1.8.5 in [BFH18b]) L ∞ (0 , N ; L ( T )) ∩ C κ ([0 , N ]; H − ) ⊂ C ([0 , N ]; L w ( T )) , this implies tightness of the law of x and z , respectively, under the family of measures ( P n ) n ∈ N on C ([0 , ∞ ); L w ) and C ([0 , ∞ )), respectively.In order to prove tightness of y under ( P n ) n ∈ N we recall that y ( t ) = y n + Z ts n R ( r )d r P n -a.s.and that R ( r ) is a positive semidefinite matrix-valued measure by (M1). Accordingly, it followsfrom Lemma 3.2sup n ∈ N E P n " sup t ∈ [0 ,N ] k y ( t ) k M ( T , R × ) + sup r = t ∈ [0 ,N ] k y ( t ) − y ( r ) k M ( T , R × ) | t − r | . sup n ∈ N E P n " sup t ∈ [0 ,N ] Z T dtr R ( t ) + sup n ∈ N k y n k M ( T , R × ) < ∞ . Therefore, we deduce the tightness of y under ( P n ) n ∈ N on C ([0 , ∞ ); ( M ( T , R × ) , w )) ∩ W α,q loc ([0 , ∞ ); W − k, ( T , R × )) . Without loss of generality, we may assume that P n converges weakly to some probability measure P on Ω P W .As the next step, we apply Jakubowski–Skorokhod’s representation theorem (cf. Theorem 2.7.1in [BFH18b]). After passing to a subsequence, we deduce that on some probability space (Ω , F , P )there are random variables (˜ x n , ˜ y n , ˜ z n , ˜ b n ) as well as (˜ x, ˜ y, ˜ z, ˜ b ) such that(i) the law of (˜ x n , ˜ y n , ˜ z n , ˜ b n ) under P is given by P n for each n ∈ N and the law of (˜ x, ˜ y, ˜ z, ˜ b )under P is given by P .(ii) (˜ x n , ˜ y n , ˜ z n , ˜ b n ) → (˜ x, ˜ y, ˜ z, ˜ b ) in Ω P W , and ˜ x n → ˜ x in C ([0 , ∞ ); L w ) . Step 2: Identification of the limit.
Our goal is to show that P := Law(˜ x, ˜ y, ˜ z, ˜ b ) is a dissipativeprobabilistically weak solution starting from the initial condition ( x , y , z , b ) at time s . First, weobserve that as the initial conditions are deterministic, it follows immediately that P (cid:16) ˜ x ( t ) = x , ˜ y ( t ) = y , ˜ z ( t ) = z , ˜ b ( t ) = b , t ∈ [0 , s ] (cid:17) = 1 . Since the weak formulation of (1.1) in (M2) as well as the energy equality in (M3) is satisfiedby ( x, y, z, b ) under each measure P n , and (˜ x n , ˜ y n , ˜ z n , ˜ b n ) has the same law under P , it followsfrom Theorem 2.9.1 in [BFH18b] that (M2), (M3) are also satisfied by (˜ x n , ˜ y n , ˜ z n , ˜ b n ) under P .More precisely, ˜ b n is a cylindrical Wiener process on U starting from b n at time s n with respect to σ ((˜ x n , ˜ y n , ˜ z n , ˜ b n )( s ) , s t ). Taking the limit it is easy to see that ˜ b is a cylindrical Wiener process on N ILL- AND WELL-POSEDNESS TO STOCHASTIC 3D EULER EQUATIONS 17 U starting from b at time s with respect to σ ((˜ x, ˜ y, ˜ z, ˜ b )( s ) , s t ). Since P is supported on Ω P W , b isa cylindrical Wiener process on U under P with respect to ( B P W,t ) t > . By Lemma 2.6.6 in [BFH18b]and (G2), we are able to pass to the limit in the stochastic integrals in (M2) and (M3) in L (0 , T )in probability. Indeed, since ˜ x n → ˜ x in C ([0 , ∞ ); L w ) , we obtain sup n sup t ∈ [0 ,T ] k ˜ x n ( t ) k L < ∞ forany T > P -a.s. The convergence of the quadratic variation term Z ts n k G (˜ x n ( r )) k L ( U,L ) d r → Z ts k G (˜ x ( r )) k L ( U,L ) d r P -a.s.also follows from (G2) and (G1). For the stochastic integral in (M3), we have k [ s n ,T ] G (˜ x n ( r )) ∗ ˜ x n − [ s,T ] G (˜ x ( r )) ∗ ˜ x k U [ s n ,s ) k G (˜ x n ( r )) ∗ ˜ x n || U + 1 [ s,T ] k ( G (˜ x n ( r )) − G (˜ x ( r ))) k L ( U,H ) k ˜ x n k H + k [ s,T ] ( G (˜ x ( r )) ∗ (˜ x n − ˜ x )) k U , and the last term goes to zero by the compactness of G ∗ and weak convergence of ˜ x n to ˜ x . Thenby (G1) and (G2) and dominated convergence theorem we deduce P -a.s. Z T k [ s n ,T ] G (˜ x n ( r )) ∗ ˜ x n − [ s,T ] G (˜ x ( r )) ∗ ˜ x k U d r → . Thus by Lemma 2.6.6 in [BFH18b] we can pass the limit of the stochastic integral in (M3). For thestochastic term in (M2) the argument is similar and actually easier.Regarding the compatibility conditions on the stresses N and R in (M1) and (M3) and theconvergence of the stress term in (M2), let us denote consistently˜ R n := ∂ t ˜ y n , ˜ N n := ˜ R n − ˜ x n ⊗ ˜ x n , ˜ R := ∂ t ˜ y, ˜ N := ˜ R − ˜ x ⊗ ˜ x. (4.3)In order to pass to the limit in the stress term in (M2), we note that by (4.3) and using theconvergence of ˜ y n → ˜ y in C ([0 , ∞ ); ( M + ( T , R × ) , w )) P -a.s. we obtain Z ts Z T ∇ e i : d ˜ R n ( r )d r = h∇ e i , ˜ y n ( t ) − ˜ y n ( s ) i → h∇ e i , ˜ y ( t ) − ˜ y ( s ) i = Z ts Z T ∇ e i : d ˜ R ( r )d r, where the convergence takes place in C ([0 , ∞ )) P -a.s. Thus (M2) follows.Since ˜ R n is a measurable function of ˜ y n and hence ˜ N n is a measurable function of (˜ x n , ˜ y n ), weobtain from the equality of joint laws P (cid:18)Z T dtr ˜ R n ( t ) ˜ z n ( t ) for a.e. t > s n (cid:19) = P n (cid:18)Z T dtr R ( t ) z ( t ) for a.e. t > s n (cid:19) = 1 , (4.4) P (cid:16) ˜ N n ∈ L ∞ loc ([ s n , ∞ ); M + ( T , R × )) (cid:17) = P n (cid:0) N ∈ L ∞ loc ([ s n , ∞ ); M + ( T , R × )) (cid:1) = 1 , (4.5)and as in Lemma 3.2 for every N ∈ N sup n ∈ N E P " esssup t ∈ [0 ,N ] (cid:18)Z T dtr ˜ R n ( t ) (cid:19) = sup n ∈ N E P n " esssup t ∈ [0 ,N ] (cid:18)Z T dtr R ( t ) (cid:19) < ∞ . (4.6)Hence, by Banach–Alaoglu’s theorem applied in the dual space (see [MNRR96, Theorem 2.11]and [Ed65, Theorem 8.20.3]) L w (Ω; L ∞ w (0 , N ; M ( T ; R × ))) ≃ (cid:0) L (Ω; L (0 , N ; C ( T ; R × )) (cid:1) ∗ where the subscript w stands for weak-star measurable mappings, we deduce that there exists F ∈ L w (Ω; L ∞ w (0 , N ; M ( T ; R × ))) such that˜ R n → F weak-star in L w (Ω; L ∞ w (0 , N ; M ( T ; R × ))) . (4.7)On the other hand, as a consequence of the convergence of ˜ y n to ˜ y in (ii) we deduce˜ R n → ˜ R in D ′ ((0 , ∞ ) × T ) P -a.s.Thus, we get F = ˜ R . Even though F = F N is defined for times t ∈ [0 , N ], taking N → ∞ we mayextend F to [0 , ∞ ) using uniqueness of the limit. Since all ˜ R n are positive semidefinite, the sameremains valid for F and the corresponding norm of F is bounded by the left hand side of (4.6) byweak-star lower semicontinuity. Therefore we have in particular P (cid:18)Z T dtr ˜ R ∈ L ∞ loc ([ s, ∞ )) (cid:19) = 1 . Therefore, as ˜ z n − Z T dtr ˜ R n is a non-negative distribution P -a.s. by (4.4), the same remains valid for the limit˜ z − Z T dtr ˜ R . Since this is an L ∞ loc ([ s, ∞ ))-function P -a.s., we deduce that P (cid:18)Z T dtr ˜ R ( t ) ˜ z ( t ) for a.e. t > s (cid:19) = 1 . Hence (M3) follows. Finally, it remains to verify the second condition in (M1). To this end, we take η ∈ R and write˜ N : ( η ⊗ η ) = ˜ R : ( η ⊗ η ) − | ˜ x · η | = lim n →∞ [ ˜ R n : ( η ⊗ η ) − | ˜ x n · η | ] + lim n →∞ [ | ˜ x n · η | − | ˜ x · η | ] , where the limit is taken in the sense of distributions D ′ ([0 , ∞ ) × T ) P -a.s. According to (4.5), thefirst limit on the right hand side is non-negative, whereas the second limit is a non-negative due tothe weak lower semicontinuity of the convex function x
7→ | x · η | . Thus, ˜ N is a positive semidefinitematrix-valued measure. In addition, by weak lower semicontinuity and equality of laws, we obtain E P " sup t ∈ [0 ,N ] k ˜ x ( t ) k L lim inf n →∞ E P " sup t ∈ [0 ,N ] k ˜ x n ( t ) k L = lim inf n →∞ E P n " sup t ∈ [0 ,N ] k x ( t ) k L < ∞ . Hence, in view of the definition of ˜ N we conclude P (cid:16) ˜ N ∈ L ∞ loc ([ s, ∞ ); M + ( T , R × )) (cid:17) = 1 , which completes the proof. (cid:3) In the same way, we can prove stability for dissipative martingale solutions. To this end, wedenote by C ( s, x , y , z ) the set of all dissipative martingale solutions with the initial condition( x , y , z ) and the initial time s . N ILL- AND WELL-POSEDNESS TO STOCHASTIC 3D EULER EQUATIONS 19
Theorem 4.3.
Suppose that (G1) , (G2) hold. Let ( x n , y n , z n ) ∈ X , s n ∈ [0 , ∞ ) , n ∈ N , and assumethat ( s n , x n , y n , z n ) → ( s , x , y , z ) in [0 , ∞ ) × L σ × M + ( T , R × ) × [0 , ∞ ) as n → ∞ and let P n ∈ C ( s n , x n , y n , z n ) . Then there exists a subsequence n k such that the sequence ( P n k ) k ∈ N converges weakly to some P ∈ C ( s , x , y , z ) .Proof. The proof is a consequence of Theorem 4.2. More precisely, by the martingale representationtheorem, see [DPZ92], the dissipative martingale solution P n gives raise to a dissipative proba-bilistically weak solution Q n ∈ C P W ( s n , x n , y n , z n , Q n on the firstthree components is P n . By Theorem 4.2 there is a subsequence ( Q n k ) k ∈ N converging weakly tosome Q ∈ C P W ( s , x , y , z , Q on the first threecomponents belongs to C ( s , x , y , z ). (cid:3) Existence.
Based on the proof of stability in Theorem 4.2, we also obtain existence of dissipa-tive probabilistically weak solutions and as a corollary existence of dissipative martingale solutions.
Theorem 4.4.
Suppose that (G1) , (G2) hold. For every s ∈ [0 , ∞ ) , ( x , y , z ) ∈ X and b ∈ U ,there exists P ∈ P (Ω M ) which is a dissipative probabilistically weak solution to the Euler system (1.1) starting at time s from the initial condition ( x , y , z , b ) .Proof. Consider the following Galerkin approximation through stochastic Navier–Stokes equationswith vanishing viscosityd u n − n ∆ u n d t + Π n P div( u n ⊗ u n )d t = Π n G ( u n )d B, div u n = 0 ,u n ( t ) = Π n x , t s, where Π n and P , respectively, are the Galerkin and Leray projection operator, respectively. It isclassical to show that a solution exists on some probability space (Ω , F , P ) with a cylindrical Wienerprocess B on U starting from b at time s . For notational simplicity and without loss of generality,we may assume that the probability space and the Wiener process do not depend on n .Define for t > sz n ( t ) := z + 2 Z ts h u n , G ( u n ( r ))d B ( r ) i + Z ts k Π n G ( u n ( r )) k L ( U,L ) d r, R n := u n ⊗ u n , y n ( t ) := y + Z ts R n ( r )d r, (4.8)and denote by P n the joint law of ( u n , y n , z n , B ). By an application of Itˆo’s formula to the function u
7→ k u k L we obtain in particular P (cid:0) k u n ( t ) k L z n ( t ) for all t > s (cid:1) = P (cid:18)Z T dtr R n ( t ) z n ( t ) for all t > s (cid:19) = 1 , and estimating the right hand side of the energy equality in (4.8) by Burkholder–Davis–Gundy’sinequality and (G1), we obtain for every N ∈ N sup n ∈ N E P " sup t ∈ [0 ,N ] k u n ( t ) k pL + n Z Ns k∇ u n ( r ) k L d r ! p < ∞ . Therefore, the processes ( u n , y n , z n , B ), n ∈ N , satisfy exactly the same uniform bounds as in theproof of Theorem 4.2 and their tightness follows exactly in the same way. The identification of thelimit is also the same, the only difference being the artificial viscosity term which vanishes in theasymptotic limit. (cid:3) Non-uniqueness in law
This section is devoted to the proof of non-uniqueness in law in the case of an additive noise. Inparticular, we consider the stochastic Euler systemd u + div( u ⊗ u )d t + ∇ P d t = G d B, div u = 0 , (5.1)where the coefficient G satisfies the hypotheses (G3), (G4). The proof follows in three main steps.First, in Section 5.1 we apply the convex integration method based on Baire’s category theorem inorder to construct infinitely many adapted weak solutions to (5.1) satisfying an energy inequality.Based on a general construction developed in Section 5.2, we show in Section 5.3 that these convexintegration solutions give raise to simplified probabilistically weak solutions defined on the full timehorizon [0 , ∞ ). With this in hand, we are able to complete the proof of the main result of thissection which reads as follows. Theorem 5.1.
Suppose that (G3) , (G4) hold. Then simplified dissipative martingale solutions to (5.1) are not unique. Moreover, for any given T > , non-uniqueness holds on [0 , T ] . We note that the restriction to the additive noise case satisfying (G3) is required for the con-struction by convex integration performed in Section 5.1. The results of Section 5.2 apply to a moregeneral multiplicative noise whereas the assumption (G4) is required for their application to theconvex integration solutions in Section 5.3.5.1.
Construction by convex integration.
In this subsection, we use the convex integrationmethod in order to find an initial condition which gives raise to infinitely many weak solutionssatisfying energy inequality, all adapted to the canonical filtration generated by a given Wienerprocess B . In particular, we fix a probability space (Ω , F , P ) with a cylindrical Wiener process B on U satisfying B (0) = 0 and let ( F t ) t > be its normal filtration. We recall that ( F t ) t > isthe canonical filtration augmented by P -null sets and that it is right continuous. Therefore, the σ -algebra F is generated by the P -null sets, i.e., F = σ { A ∈ F ; P ( A ) = 0 } and, as a consequence,every F -measurable random variable is P -a.s. constant. Moreover, we restrict ourselves to theadditive noise case. In particular, we fix parameters0 < δ < / , p ∈ (1 , ∞ ) , β ∈ (0 , , − δ + 1 p < β < − δ, and note that under (G3), it holds for any T > k GB k C T H (3+ σ ) / < ∞ , k GB k C / − δT H < ∞ P -a.s. , (cid:13)(cid:13)(cid:13)(cid:13)Z · h GB, G d B i (cid:13)(cid:13)(cid:13)(cid:13) C / − δT . (cid:13)(cid:13)(cid:13)(cid:13)Z · h GB, G d B i (cid:13)(cid:13)(cid:13)(cid:13) W β,pT . (cid:13)(cid:13)(cid:13)(cid:13)Z · h GB, G d B i (cid:13)(cid:13)(cid:13)(cid:13) C / − δT < ∞ P -a.s.For a given L > T L = inf { t > k GB ( t ) k H (3+ σ ) / > L } ∧ inf n t > k GB k C / − δt H > L o ∧ L, N ILL- AND WELL-POSEDNESS TO STOCHASTIC 3D EULER EQUATIONS 21 T L = inf ( t > (cid:13)(cid:13)(cid:13)(cid:13)Z · h GB, G d B i (cid:13)(cid:13)(cid:13)(cid:13) W β,pt > L ) ∧ L,T L = T L ∧ T L , (5.2)and we let B L be the stopped Wiener process B L ( · ) = B ( · ∧ T L ).This leads us to the truncated Euler systemd u + div( u ⊗ u )d t + ∇ P d t = G d B L , div u = 0 , (5.3)which we consider on the time interval [0 , T ] for a fixed T >
0. The truncated system coincideswith the original system (5.1) on the random time interval [0 , T L ], hence the solutions constructedin this section solve the stochastic Euler system (5.1) up to the stopping time T L . In addition,for a suitable (deterministic) initial condition u and for every choice of an additional parameter l ∈ [2 , ∞ ], we construct at least one solution u satisfying the following energy equality for P -a.s.and a.e. t ∈ (0 , T L ]12 k u ( t ) k L = 12 k u k L + M Et, + (cid:18) − l (cid:19) ( t ∧ T L ) k G k L ( U,L ) , (5.4)where M Et, = R t h u, G d B L i . In view of the Definition 3.1, we therefore quantify the error in theenergy equality through the defect l k G k L ( U,L ) ( t ∧ T L ) . This permits us to conclude the existenceof infinitely many solutions, i.e., at least one solution for every l ∈ [2 , ∞ ]. Even though for a fixed l , the Baire category argument used below in fact yields the existence of infinitely many solutionsto the truncated equation (5.3) on [0 , T ], we are not able to deduce that the non-uniqueness holdsfor the original equation, i.e. already on the random time interval [0 , T L ] with probability one.Therefore, we use different values of l ∈ [2 , ∞ ] to conclude the non-uniqueness.We rewrite the Euler system (5.3) with additive noise by setting v = u − GB L , which gives ∂ t v + div(( v + GB L ) ⊗ ( v + GB L )) + ∇ P = 0 , div v = 0 . (5.5)This is the system we apply the convex integration to. We define the energy functional e l ( v )( t ) := 12 k v (0) k L + Z t h v + GB L , G d B L i + (cid:18) − l (cid:19) ( t ∧ T L ) k G k L ( U,L ) . (5.6)We note that if a sequence v n satisfies v n (0) = v (0) andesssup ω ∈ Ω k v n − v k C αT H − → α > / δ , then due to Lemma A.1 we obtainesssup ω ∈ Ω sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z t h v n − v, G d B L i (cid:12)(cid:12)(cid:12)(cid:12) . esssup ω ∈ Ω k v n − v k C αT H − esssup ω ∈ Ω k GB L k C / − δT H → , which implies that esssup ω ∈ Ω sup t ∈ [0 ,T ] | e l ( v n ) → e l ( v ) | → . (5.7)Let us now introduce some preliminary notations and definitions needed in the sequel. For thefiltration ( F t ) t > we define its extension to negative times by F t = F whenever t <
0. We denote by R × , sym the set of symmetric trace-less 3 × w, H ) ∈ R × R × , sym we denote e ( w, H ) = 32 λ max ( w ⊗ w − H ) , where λ max ( Q ) is the largest eigenvalue of a symmetric 3 × Q . By [DLS10, Lemma 3], wehave 12 | w | e ( w, H ) , k H k e ( w, H ) , (5.8)where k · k means the operator norm of the matrix. Finally, we denote by d a metric which metrizesthe weak topology on bounded sets of L . We say that a Borel random variable F : Ω → X rangingin a topological X has a compact range provided there is a (deterministic) compact set K ⊂ X suchthat F ∈ K P -a.s.The following result is a modification of Lemma 5.11 from [BFH20b]. In particular, it permitsus to construct an initial condition with a prescribed energy, which gives raise to a subsolution. Lemma 5.2.
Let [ e, w, H ] be an ( F t ) t > -adapted stochastic process such that [ e, w, H ] ∈ C ([0 , T ] × T ; (0 , ∞ ) × R × R × , sym ) P -a.s.with compact range and e ( w, H ) < e − δ, for all ( t, x ) ∈ [0 , T ] × T P -a.s. (5.9) for some deterministic constant δ > . Then for any ε ∈ (0 , T ) there exists a sequence [ w n , V n ] ∈ C ∞ c (( − T, T ) × T ; R × R × , sym ) enjoying the following properties:i) the process [ w n , V n ] is ( F t ) t ∈ R -adapted such that [ w n , V n ] ∈ C ([ − T, T ] × T ; R × R × , sym ) P -a.s.with compact range and supp( w n , V n ) ⊂ [ − ε, ε ] × T P -a.s.;ii) we have P -a.s. ∂ t w n + div V n = 0 , div w n = 0; iii) we have esssup ω ∈ Ω sup t ∈ [0 ,T ] d ( w n ( t ) , → as n → ∞ ; iv) we have e ( w + w n , H + V n ) < e − δ n , for all ( t, x ) ∈ [0 , T ] × T P -a.s.for some deterministic constant δ n > ;v) if ¯ e > is such that e ¯ e , then the following holds P -a.s. lim inf n →∞ Z T | w + w n | (0 , x )d x > Z T | w | (0 , x )d x + c ¯ e Z T (cid:18) e − | w | (cid:19) (0 , x )d x for some universal constant c > .Proof. Comparing to Lemma 5.11 in [BFH20b] we see that the new result is v) . In order to prove it,we need to construct oscillations around the time t = 0, which requires an extension of the functions[ e, w, H ] to negative times. The chosen value for negative times is not important provided theextended process remains adapted to ( F t ) t ∈ R . Therefore, we set [ e, w, H ]( t ) = [ e, w, H ](0) whenever t <
0. Note that since we extended the filtration to negative times in the same way, [ e, w, H ] is( F t ) t ∈ R -adapted and in particular [ e, w, H ]( t ) is P -a.s. deterministic for t N ILL- AND WELL-POSEDNESS TO STOCHASTIC 3D EULER EQUATIONS 23
As the proof of Lemma 5.11 in [BFH20b], we intend to approximate [ e, w, H ] by piecewise constantfunctions [ e ap , w ap , H ap ] such that for a certain (deterministic) γ > | [ e ap , w ap , H ap ]( t, x ) − [ e, w, H ]( t, x ) | < γ (5.10) P -a.s. for all ( t, x ) ∈ [ − T, T ] × T . Moreover, we choose γ > (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) e ap − | w ap | (cid:19) ( t, x ) − (cid:18) e − | w | (cid:19) ( t, x ) (cid:12)(cid:12)(cid:12)(cid:12) < δ . (5.11) P -a.s. for all ( t, x ) ∈ [ − T, T ] × T . Choosing such a γ > e, w, H ]. Indeed,it implies that [ e, w, H ] is uniformly continuous with respect to ( t, x ) (essentially) uniformly in ω .Therefore, for a given ε > − ε, ε ] × T into finitely many disjointboxes of the form ( t j , t j +1 ] × ( a i , b i ] =: ( t j , t j +1 ] × K i , where { t j ; j = 1 , . . . , j max } is a partition of[ − ε, ε ], a i = ( a i, , a i, , a i, ) , b i = ( b i, , b i, , b i, ) ∈ T for i = 1 , . . . , i max , and ( a i , b i ] = Π ℓ =1 ( a i,ℓ , b i,ℓ ].On each of these boxes, the piecewise constant approximation is defined as[ e ap , w ap , H ap ]( t, x ) := [ e, w, H ]( t j , y i ) for some y i ∈ K i whenever ( t, x ) ∈ ( t j , t j +1 ] × K i . Note that this choice in particular preserves adaptedness. Inaddition, the boxes are chosen in a way that the time t = 0 appears in the middle of one of the timeintervals, say ( t j ∗ , t j ∗ +1 ], and that P -a.s. (5.10) holds true.Using this approximation in the proof of [BFH20b, Lemma 5.11], i)-iv) from the statement ofthe present lemma follow and it only remains to prove v) . To this end, we observe that on each ofthe above boxes we can construct oscillations by following the approach of [BFH20b, Lemma 5.6].Moreover, if w n , n ∈ N , is the sequence of oscillations on the box ( t j ∗ , t j ∗ +1 ] × K i then the argumentsof Step 5 in the proof of [BFH20b, Lemma 5.11] also implylim inf n →∞ Z K i | w n | (0 , x )d x > ce ( t j ∗ , y i ) (cid:18) e ( t j ∗ , y i ) − | w ( t j ∗ , y i ) | (cid:19) | K i | , (5.12)where c > w n , n ∈ N , of w ap on the whole domain ( − ε, ε ) × T . As a consequence, it holdslim inf n →∞ Z T | w n | (0 , x )d x = lim inf n →∞ Z ∪ i max i =1 K i | w n | (0 , x )d x = lim inf n →∞ i max X i =1 Z K i | w n | (0 , x )d x > i max X i =1 lim inf n →∞ Z K i | w n | (0 , x )d x > i max X i =1 ce (cid:18) e − | w | (cid:19) ( t j ∗ , y i ) | K i | > c ¯ e Z T (cid:18) e ap − | w ap | (cid:19) (0 , x )d x. (5.13) Thus, we obtainlim inf n →∞ Z T | w + w n | (0 , x )d x = Z T | w | (0 , x )d x + lim inf n →∞ Z T | w n | (0 , x )d x > Z T | w | (0 , x )d x + c e Z T (cid:18) e ap − | w ap | (cid:19) (0 , x )d x > Z T | w | (0 , x )d x + c e Z T (cid:18) e − | w | − δ (cid:19) (0 , x )d x, where in the first equality we used iii) and the last inequality follows from (5.11) together with thefact that by (5.9) and (5.8) we have (cid:18) e − | w | (cid:19) (0 , x ) > δ P -a.s. for all x ∈ T . The same argument permits to finally concludelim inf n →∞ Z T | w + w n | (0 , x )d x > Z T | w | (0 , x )d x + c e Z T (cid:18) e − | w | (cid:19) (0 , x )d x, which completes the proof. (cid:3) Let e be a given ( F t ) t > -adapted energy such that e ∈ C ([0 , T ] × T ; (0 , ∞ )) P -a.s. with acompact range. We define the collection of subsolutions corresponding to e by X ,e = n v : Ω → C loc ((0 , T ] × T ; R ) ∩ C ([0 , T ] , L w ); v is ( F t ) t > -adapted,there exists H : Ω → C loc ((0 , T ] × T ; R × , sym ) ( F t ) t > -adapted such that ∂ t v + div H = 0 , div v = 0 , for every ε ∈ (0 , T ) there is deterministic δ ε > e ( v + GB L , H ) < e − δ ε for all t ∈ [ ε, T ] , ( v, H ) as a function in C ([ ε, T ] × T ; R × R × , sym ) is of compact range o . The next result shows how to find a subsolution with a prescribed energy at time t = 0. Lemma 5.3.
Let e be ( F t ) t > -adapted such that e ∈ C ([0 , T ]; (0 , ∞ )) P -a.s. with compact rangeand for some deterministic δ > e ( GB L , < e − δ for all t ∈ [0 , T ] . (5.14) Then there exists ¯ v ∈ X ,e such that Z T | ¯ v (0) | d x = e (0) P -a.s . Proof.
The idea of the proof follows from [DLS10, Section 5]. We proceed iteratively and applyLemma 5.2. In particular, letting v = 0, H = 0 we have v ∈ X ,e . By the definition of thestopping time T L we know that GB L ∈ C ([0 , T ] × T ) P -a.s. is of compact range. Then applyingrepeatedly Lemma 5.2 to [ e, v k + GB L , H k ] we construct a sequence v k = v k − + w k,n for n largeenough, k ∈ N , so that v k ∈ X ,e , esssup ω ∈ Ω sup t ∈ [0 ,T ] d ( v k , v k − ) < k , e ( v k + GB L , H k ) < e − δ k , t ∈ [0 , T ] , N ILL- AND WELL-POSEDNESS TO STOCHASTIC 3D EULER EQUATIONS 25 ( v k , H k ) as a function in C ([0 , T ] × T ; R × R × , sym ) is of compact range,esssup ω ∈ Ω sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z T ( v k − v k − ) · v m d x (cid:12)(cid:12)(cid:12)(cid:12) < k for all m = 0 , . . . , k − , supp( v k − v k − ) ⊂ [ − / k , / k ] × T , (5.15)and P -a.s. Z T | v k | (0 , x )d x > Z T | v k − | (0 , x )d x + cα k , (5.16)where α k = Z T (cid:18) e − | v k − | (cid:19) (0 , x ) d x > . We recall that since the σ -algebra F is generated by the P -null sets, the random variables e , v k ,and consequently also α k , k ∈ N , are P -a.s. deterministic at time t = 0.Therefore, the sequence Z T | v k | (0 , x )d x is P -a.s. non-decreasing and there exists ¯ v adapted such that esssup ω ∈ Ω sup t ∈ [0 ,T ] d ( v k , ¯ v ) → k → ∞ . By (5.15), we know that for any ε > k ∈ N such that¯ v ( t ) = v k ( t ) = v k ( t ) t ∈ [ ε, T ] , k > k , which implies that ¯ v ∈ X ,e . By the construction of v k and (5.8) and the compact range of e weobtain in particular sup k ∈ N esssup ω ∈ Ω Z T | v k (0 , x ) | d x < ∞ . Hence R T | v k (0 , x ) | d x has a P -a.s. limit Y and by (5.16) we deduce α k → P -a.s. Consequently,it follows from the definition of α k that P -a.s. Z T | v k | (0)d x ↑ e (0) . Next, we have for all n > m Z T | v n − v m | (0)d x = Z T | v n | (0)d x − Z T | v m | (0)d x − Z T
12 (( v n − v m ) · v m )(0)d x, where Z T
12 (( v n − v m ) · v m )(0)d x = n − m − X k =0 Z
12 (( v m + k +1 − v m + k ) · v m )(0)d x < n − m − X k =0 m + k → P -a.s. as m → ∞ . Accordingly, we obtain the P -a.s. strong convergence v k (0) → ¯ v (0) in L , whichimplies Z T | ¯ v | (0)d x = e (0) P -a.s.and completes the proof. (cid:3) Now, we define the collection of subsolutions with variable energy by taking e l given in (5.6) andsetting X = n v : Ω → C loc ((0 , T ] × T ; R ) ∩ C ([0 , T ] , L w ); v is ( F t ) t > -adapted with v (0) = ¯ v (0),there exists H : Ω → C loc ((0 , T ] × T ; R × , sym ) ( F t ) t > -adapted such that ∂ t v + div H = 0 , div v = 0 , for every ε ∈ (0 , T ) there is deterministic δ ε > e ( v + GB L , H ) < e l ( v ) − δ ε for all t ∈ [ ε, T ] , ( v, H ) as a function in C ([ ε, T ] × T ) is of compact range o . As the next step, we find a suitable energy e for the application of Lemma 5.3. Lemma 5.4.
Suppose that (G3) holds. There exists a deterministic function e ∈ C ([0 , T ]; (0 , ∞ )) satisfying (5.14) such that the associated ¯ v constructed in Lemma 5.3 belongs to X .Proof. The constructed energy e has to satisfy (5.14) as well as e ( t ) e l (¯ v )( t ) = 12 k ¯ v (0) k L + Z t h ¯ v + GB L , G d B L i + (cid:18) − l (cid:19) ( t ∧ T L ) k G k L ( U,L ) , (5.17)for all t ∈ (0 , T ]. Lemma 5.3 then in particular implies that ¯ v has the desired energy at time 0, i.e.,12 k ¯ v (0) k L = e (0) = e l (¯ v )(0) . In order to find a suitable energy e so that this can be achieved, we shall therefore find a lowerbound for e l (¯ v ) based on the a priori information we get for ¯ v .To this end, we recall that if ¯ v was constructed from Lemma 5.3 with a given energy e then inparticular there is ¯ H such that ∂ t ¯ v + div ¯ H = 0 in D ′ ((0 , T ) × T ) , e (¯ v + GB L , ¯ H ) < e − δ ε for t ∈ [ ε, T ] . Thus, by (5.8) it follows12 | ¯ v + GB L | e on (0 , T ] , | ¯ H | . e on (0 , T ] , and we deduce | ¯ v | . √ e + C ( L ) , sup s,t ∈ [0 ,T ] ,s = t k ¯ v ( t ) − ¯ v ( s ) k H − | t − s | . sup t ∈ [0 ,T ] k ¯ H k L . sup t ∈ [0 ,T ] e, which by interpolation implies that for any β ∈ (0 , k ¯ v k C βT H − . sup t ∈ [0 ,T ] e β + C ( L ) . All the implict constants are independent of ¯ v . Next, for β > / δ we estimate one stochasticintegral by Lemma A.1 using the definition of the stopping time T L in (5.2) as (cid:12)(cid:12)(cid:12)(cid:12)Z t h ¯ v, G d B L i (cid:12)(cid:12)(cid:12)(cid:12) C ( L ) sup t ∈ [0 ,T ] e β + 1 ! t / − δ while for the other stochastic integral we get by the definition of the stopping time T L (cid:12)(cid:12)(cid:12)(cid:12)Z t h GB L , G d B L i (cid:12)(cid:12)(cid:12)(cid:12) C ( L ) t / − δ . Finally, we observe that 0 (cid:18) − l (cid:19) ( t ∧ T L ) k G k L ( U,L ) . t. N ILL- AND WELL-POSEDNESS TO STOCHASTIC 3D EULER EQUATIONS 27
Even though positive, this term behaves linearly in t hence it cannot compensate for the (possiblynegative) martingale part for small times. As a consequence, (5.17) leads us to the requirement forsome β ∈ (1 / δ, e ( t ) e (0) − C ( L ) sup t ∈ [0 ,T ] e β + 1 ! t / − δ . This in particular implies that e ( t ) e (0) and therefore we deduce that for a given L >
T > e (0) sufficiently large so that for some β ∈ (1 / δ, e ( t ) := e (0) − C ( L )( e (0) β + 1) t / − δ satisfies all the above requirements as well as (5.14) since the L ∞ -norm of GB L is bounded beforethe stopping time by the Sobolev embedding theorem. The proof is complete. (cid:3) The following version of the oscillatory lemma permits us to construct solutions which satisfy theenergy equality at a given stopping time τ . Lemma 5.5.
Let ε ∈ (0 , T ) , α > and let τ T be an ( F t ) t > -stopping time. Let [ e, w, H ] be an ( F t ) t > -adapted stochastic process such that [ e, w, H ] ∈ C ([ ε, T ] × T ; (0 , ∞ ) × R × R × , sym ) P -a.s.with compact range and e ( w, H ) < e − δ, for all ( t, x ) ∈ [ ε, T ] × T P -a.s.for some deterministic constant δ > . Let I ε [ w ] = E P (cid:20) { τ > ε } Z T (cid:18) | w | − e (cid:19) ( τ ) dx (cid:21) < − α . (5.18) Then there exists a sequence [ w n , V n ] ∈ C ∞ ([ ε, T ] × T ; R × R × , sym ) , satisfying ( w n , V n )( ε, x ) = 0 , x ∈ T , and enjoying the following properties:i) the process [ w n , V n ] is ( F t ) t > -adapted and [ w n , V n ] ∈ C ([ ε, T ] × T ; R × R × , sym ) P -a.s. withcompact range;ii) we have ∂ t w n + div V n = 0 , div w n = 0; iii) we have for any α ∈ (0 , ω ∈ Ω k w n k C αT H − → as n → ∞ , iv) we have e ( w + w n , H + V n ) < e − δ n , for all ( t, x ) ∈ [ ε, T ] × T P -a.s.for some deterministic constant δ n > .v) the following holds lim inf n →∞ I ε [ w + w n ] > I ε [ w ] + cα , with c depending only on k e k L ∞ . Proof.
The proof follows the lines of [DLS10, Section 4.5] and therefore we only discuss the stepsthat need to be done differently. Let us define the shifted grid of size h as in [DLS10, Section 4.5]and use the same notation for C ζ,i , Ω hν , τ hν , ν = 1 ,
2. In order to be consistent with our firstoscillatory lemma, Lemma 5.2, we use a different notation for the velocity and the correspondingoscillations: the lemma is applied to w (instead of v in [DLS10]) and the oscillations are denoted by w n (instead of ˜ v N in [DLS10]). We extend [ e, w, H ] to [ T, T + ε ] such that [ e, w, H ]( t ) = [ e, w, H ]( T )for t ∈ [ T, T + ε ].Similarly to Lemma 5.2, the compact range of the involved stochastic processes permits to choosethe piecewise constant approximations uniformly in ω . On the other hand, in the stochastic settingwe are not free to assign an arbitrary value on each of the small cylinders C ζ,i , nor to choose theadmissible segment arbitrarily: in order to preserve adaptedness, we shall always assign the valueat the minimal time of C ζ,i and choose the admissible segment by a measurable selection as inLemma 5.6 in [BFH20b].Let us now define E h ( t, x ) = E h ( t i , x ζ ) = 12 | w ( t i , x ζ ) | − e ( t i , x ζ ) for ( t, x ) ∈ C ζ,i , where t i is the minimal time and x ζ is an arbitrary spatial point in the cylinder C ζ,i . Since [ e, w ] isof compact range, lim h → Z Ω hν E h ( t )d x = 12 (cid:18) (cid:19) Z T (cid:18) | w | − e (cid:19) ( t )d x, uniformly in t ∈ [ ε, T ] P -a.s. and this implies for ν ∈ { , } lim h → E P " { τ > ε }∩{ τ ∈ τ hν } Z Ω hν E h ( τ )d x = 12 (cid:18) (cid:19) E P (cid:20) { τ > ε }∩{ τ ∈ τ hν } Z T (cid:18) | w | − e (cid:19) ( τ )d x (cid:21) . Moreover, we have X ν =1 E P (cid:20) { τ > ε }∩{ τ ∈ τ hν } Z T (cid:18) | w | − e (cid:19) ( τ )d x (cid:21) I ε [ w ] . According to (5.18), there exists ν ∈ { , } such that E P (cid:20) { τ > ε }∩{ τ ∈ τ hν } Z T (cid:18) | w | − e (cid:19) ( τ )d x (cid:21) − α / , hence given [ e, w ] we may choose h sufficiently small so that E P " { τ > ε }∩{ τ ∈ τ hν } Z Ω hν | E h | ( τ )d x > cα (5.19)for a universal constant c >
0. Denoting by w n the oscillatory sequence, we observe in particularthat (60) in [DLS10] rewrites aslim n →∞ Z Ω hν | w n | ( t )d x > cM Z Ω hν | E h | ( t )d x (5.20) N ILL- AND WELL-POSEDNESS TO STOCHASTIC 3D EULER EQUATIONS 29 uniformly in t ∈ τ hν ∩ [2 ε, T ] P -a.s. Here M = sup Ω × [ ε,T ] × T e . Now, we havelim inf n →∞ I ε [ w + w n ] = lim inf n →∞ E P (cid:20) { τ > ε } Z T (cid:18) | w + w n | − e (cid:19) ( τ )d x (cid:21) = lim inf n →∞ E P (cid:20) { τ > ε } (cid:18)Z T (cid:18) | w | − e (cid:19) ( τ )d x + Z T | w n | ( τ )d x (cid:19)(cid:21) > I ε [ w ] + E P " { τ > ε }∩{ τ ∈ τ h } lim inf n →∞ Z Ω h | w n | ( τ )d x + E P " { τ > ε }∩{ τ ∈ τ h } lim inf n →∞ Z Ω h | w n | ( τ )d x > I ε [ w ] + cM E P " { τ > ε }∩{ τ ∈ τ h } Z Ω h | E h | ( τ )d x + cM E P " { τ > ε }∩{ τ ∈ τ h } Z Ω h | E h | ( τ )d x . (5.21)Here in the last inequality we used (5.20) and H¨older’s inequality.Thus the result in v) follows from (5.21) and (5.19).Comparing to Lemma 5.2, it only remains to prove iii) . By a modification of (5.29) and (5.31)in [BFH20b] we obtain esssup ω ∈ Ω k w n k C T H − C, esssup ω ∈ Ω k w n k C T H − → , which implies iii) by interpolation. (cid:3) Remark 5.6. i) Although our proof of Lemma 5.5 follows the approach of [DLS10], we cannotobtain the corresponding result for the linear functional E P (cid:20) inf t ∈ [ ε,T ] Z T (cid:18) | w | − e (cid:19) ( t )d x (cid:21) since we cannot change the order of E P and inf. ii) By a modification of [BFH20b, Lemma 5.11], we can also deduce that [BFH20b, Lemma 5.11]holds with iii) in [BFH20b, Lemma 5.11] strengthened to the following statement: for any α ∈ (0 , ω ∈ Ω k w n k C αT H − → n → ∞ . Finally, we have all in hand to prove the main result of this section.
Theorem 5.7.
Suppose that (G3) holds. Let τ be a strictly positive ( F t ) t > -stopping time suchthat τ T . Let e ∈ C ([0 , T ]; (0 , ∞ )) be the energy constructed in Lemma 5.4 and let ¯ v ∈ X bethe corresponding velocity constructed in Lemma 5.3. There exist infinitely many ( F t ) t > -adaptedsolutions v l to the system (5.5) on [0 , T ] with the initial condition ¯ v (0) satisfying P -a.s. e l ( v l ) = 12 k v l + GB L k L for a.e. t ∈ (0 , T ) ,e l ( v l )(0) = 12 k v l (0) k L = 12 k ¯ v (0) k L , and e l ( v l )( τ ) = 12 k ( v l + GB L )( τ ) k L . (5.22) In particular, the results hold for T = L, τ = T L .Proof. Step 1: L ∞ -bound for subsolutions. First, we show that the subsolutions in X are boundedessentially uniformly in all the variables ω, t, x and the bound is only determined by k ¯ v (0) k L andthe constants L, T . In other words, it is independent of the particular subsolution. To this end, werecall that if v ∈ X then there exists H such that ∂ t v + div H = 0 , and by (5.8) for t ∈ (0 , T ]12 k v ( t ) k L ∞ C ( L ) + k ¯ v (0) k L + Z t h v + GB L , G d B L i + (cid:18) − l (cid:19) ( t ∧ T L ) k G k L ( U,L ) ,c k H ( t ) k L ∞ k ¯ v (0) k L + Z t h v + GB L , G d B L i + (cid:18) − l (cid:19) ( t ∧ T L ) k G k L ( U,L ) . Therefore, for t ∈ [0 , T ] and β ∈ (1 / δ,
1) and using Lemma A.112 k v k L ∞ T,x . C ( L, T ) + k ¯ v (0) k L + k v k C βT H − C ( L, T ) , and k v k C T H − . k H k L ∞ T L + k ¯ v (0) k L . C ( L, T ) + k ¯ v (0) k L + k v k C βT H − C ( L, T ) . C ( L, T ) + k ¯ v (0) k L + k v k βC T H − k v k − βL ∞ T,x C ( L, T ) . Hence substitute the first estimate into the second one and using Young’s inequality we obtain k v k C T H − . C ( L, T ) + k ¯ v (0) k L , and accordingly 12 k v k L ∞ T,x . C ( L, T ) + k ¯ v (0) k L . This is the desired uniform L ∞ -bound on the subsolutions in X . This also implies that e l ( v ) ∈ C ([0 , T ]) is of compact range for v ∈ X . Step 2: Definition of functionals.
Let X be the completion of X with respect to the metric D ( v, w ) := E P " sup t ∈ [0 ,T ] d ( v ( t ) , w ( t )) . Thus X is a complete metric space. For ε ∈ (0 , T / I ε [ v ] := E P (cid:20) Z Tε Z T (cid:18) | v + GB L | − e l ( v ) (cid:19) d x d t (cid:21) ,I τ,ε [ v ] := E P (cid:20) { τ > ε } Z T (cid:18) | v + GB L | − e l ( v ) (cid:19) ( τ, x )d x (cid:21) . Since E P " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z t h v n − v, G d B L i (cid:12)(cid:12)(cid:12)(cid:12) . E P "Z T k G ∗ ( v n − v ) k U d s → N ILL- AND WELL-POSEDNESS TO STOCHASTIC 3D EULER EQUATIONS 31 by compactness of G ∗ and the L ∞ -bound obtained in Step 1 , we obtain e l ( v n ) → e l ( v ) in L (Ω; C ([0 , T ]))whenever v n → v in X . Due to the continuity of the energy e l with respect to the topology on X ,the functionals I ε , I τ,ε are lower semi-continuous on the space X . Step 3: I ε vanishes at points of continuity. As the next step, we prove that at each point ofcontinuity v of I ε on X it holds I ε [ v ] = 0. The idea of proof comes from [CFK15, Section 3]. Weproceed by contradiction: fix ε ∈ (0 , T ) and let v ∈ X be a point of continuity of I ε and supposethat I ε ( v ) < . By definition of the space X , there is a sequence v m ∈ X such that D ( v m , v ) → , I ε [ v m ] → I ε [ v ] , I ε [ v m ] < − α for some α >
0. Now, we can find an ( F t ) t > -adapted function e m ∈ C ([0 , T ]) P -a.s. and δ m > T − ε ) δ m < α / e m < e l ( v m ) , t ∈ (0 , T ] , e m (0) = e l ( v m )(0) , e m = e l ( v m ) − δ m , t ∈ [ ε, T ] , and v m ∈ X ,e m . Then due to
Step 1 , e m ∈ C ([ ε, T ]; (0 , ∞ )) is of compact range. Moreover, by the definition of thestopping time we know that GB L ∈ C ([0 , T ] × T ; R ) is of compact range. Define I ε,e m [ v ] = E P (cid:20) Z Tε Z T (cid:18) | v + GB L | − e m (cid:19) d x d t (cid:21) . For v m we can choose δ m small enough such that I ε,e m [ v m ] < − α /
2. Then by using [BFH20b,Lemma 5.11] and Remark 5.6 for ( e m , v m + GB L , H m ) on [ ε, T ] we obtain an oscillatory sequence w m,n ∈ C ∞ c (( ε, T ) × T ; R ). In particular, we can extend these functions to the whole [0 , T ] by w m,n ( t ) = 0 for t ∈ [0 , ε ]. By Remark 5.6 we have for α ∈ (0 , ω ∈ Ω k w m,n k C αT H − → n → ∞ , (5.23)and lim inf n →∞ I ε,e m [ v m + w m,n ] > I ε,e m [ v m ] + cα = I ε [ v m ] + cα + ( T − ε ) δ m . As the next step, we observe that v m + w m,n ∈ X ,e m , which follows from the fact that v m + w m,n is of compact range as a function in C ([ ε , T ] × T ; R ) for any ε ∈ (0 , T ) since w m,n is of compactsupport in ( ε, T ). Moreover, we have I ε,e m [ v m + w m,n ] − I ε [ v m + w m,n ] = E "Z Tε Z T ( e l ( v m + w m,n ) − e l ( v m ) + δ m ) d x d t → ( T − ε ) δ m as n → ∞ , which implies that lim inf n →∞ I ε [ v m + w m,n ] > I ε [ v m ] + cα . Now, we prove that v m + w m,n ∈ X . The statements for t ∈ [0 , ε ) are obvious since v m + w m,n = v m . For t ∈ [ ε, T ] we use the convergence in (5.23) to obtain (5.7). In particular, we deduce e ( v m + w m,n + GB L , H m + V m,n ) < e m = e l ( v m ) − δ m e l ( v m + w m,n ) − δ m / , for n large enough. Here, V m,n satisfies ∂ t w m,n + div V m,n = 0, cf. Remark 5.6. This way, we founda sequence ˜ v m = v m + w m,n ( m ) ∈ X such that D (˜ v m , v ) → , lim inf m →∞ I ε [˜ v m ] > I ε [ v ] , which contradicts the assumption that v is a point of continuity of I ε . Step 4: I τ,ε vanishes at points of continuity. Now, we prove that at each point v of continuity of I τ,ε on X we have I τ,ε [ v ] = 0. The proof is similar to Step 3 . For a contradiction, let us supposethat v is a point of continuity and I τ,ε ( v ) < . Then there is a sequence v m ∈ X such that D ( v m , v ) → , I τ,ε [ v m ] → I τ,ε [ v ] , I τ,ε [ v m ] < − α for some α >
0. We can find an adapted function e m ∈ C ([0 , T ]; (0 , ∞ )) P -a.s. such that for δ m < α / e m < e l ( v m ) , t ∈ (0 , T ] , e m (0) = e l ( v m )(0) , e m = e l ( v m ) − δ m , t ∈ [ ε, T ] , and v m ∈ X ,e m . Define I τ,ε,e m [ v ] = E P (cid:20) { τ > ε } Z T (cid:18) | v + GB L | − e m (cid:19) ( τ )d x (cid:21) . We apply Lemma 5.5 to w = v m + GB L and obtain a sequence w m,n ∈ X ,e m such that for α ∈ (0 , v m + w m,n ∈ X ,e m , w m,n ( t ) = 0 , t ∈ [0 , ε ) , esssup ω ∈ Ω k w m,n k C αT H − → n → ∞ , and lim inf n →∞ I τ,ε,e m [ v m + w m,n ] > I τ,ε,e m [ v m ] + cα = I τ,ε [ v m ] + cα + δ m P ( τ > ε ) . Moreover, I τ,ε,e m [ v m + w m,n ] − I τ,ε [ v m + w m,n ] = E P (cid:20) { τ > ε } Z T ( e l ( v m + w m,n )( τ ) − e l ( v m )( τ ) + δ m )d x (cid:21) → δ m P ( τ > ε ) as n → ∞ , which implies that lim inf n →∞ I τ,ε [ v m + w m,n ] > I τ,ε [ v m ] + cα . As in
Step 3 , we obtain v m + w m,n ∈ X and therefore, we have a sequence ˜ v m = v m + w m,n ( m ) suchthat D (˜ v m , v ) → , lim inf n →∞ I τ,ε [˜ v m ] > I τ,ε ( v ) , which contradicts the assumption that v is a point of continuity of I τ,ε . Step 5: Conclusion.
According to the oscillatory lemma, Remark 5.6, the set of subsolutions X has an infinite cardinality and consequently the same is valid for X . Moreover, due to lowersemicontinuity of the functionals I ε and I τ,ε , it follows that their points of continuity form residualsets in X . Therefore, the set C = ∩ m ∈ N (cid:8) v ∈ X ; I /m [ v ] is continuous (cid:9) ∩ (cid:8) v ∈ X ; I τ, /m [ v ] is continuous (cid:9) , is residual as it is an intersection of a countable family of residual sets. Therefore, C has infinitecardinality. In addition, as in [BFH20b, Lemma 6.2] we obtain that if I /m [ v ] = 0 for all m ∈ N then v solves the truncated system (5.5) and P -a.s. e l ( v )( t ) = k ( v + B L )( t ) k L for a.e. t ∈ (0 , T ).If I τ, /m [ v ] = 0 for all m ∈ N , we have e l ( v )( τ ) = k v + B L k L ( τ ) P -a.s. Therefore, it follows from Step 3 and
Step 4 that there are infinitely many solutions of (5.5) satisfying the conditions in thestatement of the theorem. (cid:3)
N ILL- AND WELL-POSEDNESS TO STOCHASTIC 3D EULER EQUATIONS 33
Remark 5.8.
The fact that the energy equality holds at a given stopping time τ , namely, that(5.22) holds in Theorem 5.7 is actually not necessary for the extension of the convex integrationsolutions in Section 5.3 below. Besides, by a simple modification the result of Theorem 5.7 can bestrengthened further so that (5.22) holds true for an arbitrary sequence of stopping times.5.2. Extension of solutions.
In this section we present a general approach which permits toextend solutions defined up to a stopping time τ to the whole time interval [0 , ∞ ). The constructionadapts the ideas of [HZZ19, Section 5.2] to the Euler system. The arguments of this section applyto the setting of a general multiplicative noise coefficient G satisfying (G1) and (G2). For simplicity,we restrict ourselves to the simplified probabilistically weak solutions as this is what is needed forthe application to the convex integration solutions presented in Section 5.3 below.First, we introduce the notion of dissipative probabilistically weak solution up to a stoppingtime. We note that unlike in our previous definitions of solution, the filtration here is the rightcontinuous version ( B SP W,t ) t > of the canonical filtration ( B SP W,t ) t > . This is required by theconvex integration solutions from Section 5.1, where the corresponding stopping times can only beshown to satisfy the stopping time property with respect to ( B SP W,t ) t > , cf. Section 5.3 below. Tothis end, for a ( B SP W,t ) t > -stopping time τ we defineΩ SP W,τ := (cid:8) ω ( · ∧ τ ( ω )); ω ∈ Ω SP W (cid:9) . Then Ω
SP W,τ is a Borel subset of Ω
SP W . Definition 5.9.
Let ( x , y , b ) ∈ L σ × M + ( T ; R × ) × U and let τ > s be a ( B SP W,t ) t > -stoppingtime. A probability measure P ∈ P (Ω SP W,τ ) is a simplified dissipative probabilistically weak solutionto the Euler system (1.1) on [ s, τ ] with the initial value ( x , y , b ) at time s provided (M1) P ( x ( t ) = x , y ( t ) = y , b ( t ) = b , t s ) = 1 , P ( N ∈ L ∞ loc ([ s, τ ]; M + ( T ; R × ))) = 1 . (M2) Under P , for every l ∈ U , h b ( · ∧ τ ) , l i U is a continuous square integrable ( B SP W,t ) t > s -martingale starting from b at time s with quadratic variation process given by ( · ∧ τ − s ) k l k U andfor every e i ∈ C ∞ ( T ) ∩ L σ and t > s h x ( t ∧ τ ) − x ( s ) , e i i − Z t ∧ τs Z T ∇ e i : d R ( r )d r = Z t ∧ τs h e i , G ( x ( r ))d b ( r ) i . (M3) P-a.s. define for every s t τz ( t ) := k x k L + 2 Z ts h x ( r ) , G ( x ( r ))d b ( r ) i + Z ts k G ( x ( r )) k L ( U,L ) d r, then P (cid:18)Z T dtr R ( t ) z ( t ) for a.e. s t τ (cid:19) = 1 . The following result is a generalization of [HZZ19, Proposition 5.2]. Here and in the sequel, fora ( B SP W,t ) t > -stopping time τ we denote by B SP W,τ the associated σ -algebra. Proposition 5.10.
Let τ be a bounded ( B SP W,t ) t > -stopping time. Then for every ω ∈ Ω :=Ω
SP W ∩ Ω SP W, with Ω SP W, = { ( x, y, b ); x ∈ L ∞ loc ([0 , ∞ ); L ) } there exists Q ω ∈ P (Ω SP W ) suchthat Q ω (cid:0) ω ′ ∈ Ω SP W ; ω ′ ( t ) = ω ( t ) for 0 t τ ( ω ) (cid:1) = 1 , (5.24) and Q ω ( A ) = R τ ( ω ) , ( x,y,b )( τ ( ω ) ,ω ) ( A ) for all A ∈ B τ ( ω ) SP W , (5.25) where R τ ( ω ) , ( x,y,b )( τ ( ω ) ,ω ) ∈ P (Ω SP W ) is a simplified dissipative probabilistically weak solution to theEuler system (1.1) starting at time τ ( ω ) from the initial condition ( x, y, b )( τ ( ω ) , ω ) . Furthermore,for every B ∈ B SP W the mapping ω Q ω ( B ) is B SP W,τ -measurable.Proof.
According to the stability with respect to the initial time and the initial condition in The-orem 4.2, for every ( s, x , y , b ) ∈ [0 , ∞ ) × L σ × M + ( T , R × ) × U the set C ( s, x , y , b ) ofall associated simplified dissipative probabilistically weak solutions is compact with respect to theweak convergence of probability measures. Let Comp( P (Ω SP W )) denote the space of all compactsubsets of P (Ω SP W ) equipped with the Hausdorff metric. Using the stability from Theorem 4.2again together with [SV79, Lemma 12.1.8] we obtain that the map[0 , ∞ ) × L σ × M + ( T , R × ) × U → Comp( P (Ω SP W )) , ( s, x , y , b ) C ( s, x , y , b ) , is Borel measurable. Accordingly, [SV79, Theorem 12.1.10] gives the existence of a measurableselection. More precisely, there exists a Borel measurable map[0 , ∞ ) × L σ × M + ( T , R × ) × U → P (Ω SP W ) , ( s, x , y , b ) R s,x ,y ,b , such that R s,x ,y ,b ∈ C ( s, x , y , b ).As the next step, we recall that the canonical process ω on Ω SP W is continuous in H − × M + ( T , R × ) × U , hence ( x, y, b ) : [0 , ∞ ) × Ω SP W → H − × M + ( T , R × ) × U is progressively measurable withrespect to the canonical filtration ( B SP W,t ) t > and consequently it is also progressively measurablewith respect to the right continuous filtration ( B SP W,t ) t > . Define ˜ x : [0 , ∞ ) × Ω SP W → L σ by˜ x ( t, ω ) = x ( t, ω ) if x ( t, ω ) ∈ L σ and ˜ x ( t, ω ) = 0 if x ( t, ω ) ∈ H − \ L σ . Furthermore, L σ ⊂ H − continuously and densely, by Kuratowski’s measurable theorem we know L σ ∈ B ( H − ) and B ( L σ ) = B ( H − ) ∩ L σ . For A ∈ B ( L σ ) with 0 ∈ A then ˜ x | − ,t ] ( A ) = x | − ,t ] ( A ) ∪ x | − ,t ] ( H − \ L σ ). For A ∈ B ( L σ )with 0 / ∈ A then ˜ x | − ,t ] ( A ) = x | − ,t ] ( A ). This implies that(˜ x, y, b ) : [0 , ∞ ) × Ω SP W → L σ × M + ( T , R × ) × U is progressively measurable with respect to the right continuous filtration ( B SP W,t ) t > .In addition, τ is a stopping time with respect to the same filtration ( B SP W,t ) t > . Therefore, itfollows from [SV79, Lemma 1.2.4] that both τ and (˜ x, y, b )( τ ( · ) , · ) are B SP W,τ -measurable. Combin-ing this fact with the measurability of the selection ( s, x , y , b ) R s,x ,y ,b constructed above,we deduce that Ω SP W → P (Ω SP W ) , ω R τ ( ω ) , (˜ x,y,b )( τ ( ω ) ,ω ) (5.26)is B SP W,τ -measurable as a composition of B SP W,τ -measurable mappings. Recall that for every ω ∈ Ω SP W this mapping gives a simplified probabilistically weak solution starting at the deterministictime τ ( ω ) from the deterministic initial condition (˜ x, y, b )( τ ( ω ) , ω ). In other words, R τ ( ω ) , (˜ x,y,b )( τ ( ω ) ,ω ) (cid:0) ω ′ ∈ Ω SP W ; (˜ x, y, b )( τ ( ω ) , ω ′ ) = (˜ x, y, b )( τ ( ω ) , ω ) (cid:1) = 1 . Now, we apply [SV79, Lemma 6.1.1] and deduce that for every ω ∈ Ω SP W there is a uniqueprobability measure Q ω = δ ω ⊗ τ ( ω ) R τ ( ω ) , (˜ x,y,b )( τ ( ω ) ,ω ) ∈ P (Ω SP W ) , such that for ω ∈ Ω (5.24) and (5.25) hold.This permits to concatenate, at the deterministic time τ ( ω ), the Dirac mass δ ω with the proba-bilistically weak solution R τ ( ω ) , (˜ x,y,b )( τ ( ω ) ,ω ) . N ILL- AND WELL-POSEDNESS TO STOCHASTIC 3D EULER EQUATIONS 35
In order to show that the mapping ω Q ω ( B ) is B SP W,τ -measurable for every B ∈ B SP W , itis enough to consider sets of the form A = { ( x, y, b )( t ) ∈ Γ , . . . , ( x, y, b )( t n ) ∈ Γ n } where n ∈ N ,0 t < · · · < t n , and Γ , . . . , Γ n ∈ B ( H − × M + ( T , R × ) × U ). Then by the definition of Q ω ,we have Q ω ( A ) = [0 ,t ) ( τ ( ω )) R τ ( ω ) , (˜ x,y,b )( τ ( ω ) ,ω ) ( A )+ n − X k =1 [ t k ,t k +1 ) ( τ ( ω )) Γ (( x, y, b )( t , ω )) · · · Γ k (( x, y, b )( t k , ω )) × R τ ( ω ) , (˜ x,y,b )( τ ( ω ) ,ω ) (cid:0) ( x, y, b )( t k +1 ) ∈ Γ k +1 , . . . , ( x, y, b )( t n ) ∈ Γ n (cid:1) + [ t n , ∞ ) ( τ ( ω )) Γ (( x, y, b )( t , ω )) · · · Γ n (( x, y, b )( t n , ω )) . Here the right hand side is B SP W,τ -measurable as a consequence of the B SP W,τ -measurability of(5.26) and τ . The proof is complete. (cid:3) We proceed with a counterpart of [HZZ19, Proposition 5.3].
Proposition 5.11.
Let τ be a bounded ( B SP W,t ) t > -stopping time and let P be a simplified dissipa-tive probabilistically weak solution to the Euler system (1.1) on [0 , τ ] starting at the time from theinitial condition ( x , y , b ) . Suppose that there exists a Borel set N ⊂ Ω SP W,τ such that P ( N ) = 0 and for every ω ∈ N c it holds Q ω (cid:0) ω ′ ∈ Ω SP W ; τ ( ω ′ ) = τ ( ω ) (cid:1) = 1 . (5.27) Then the probability measure P ⊗ τ R ∈ P (Ω SP W ) defined by P ⊗ τ R ( · ) := Z Ω SPW Q ω ( · ) P (d ω ) satisfies P ⊗ τ R = P on Ω SP W,τ and is a simplified dissipative probabilistically weak solution to theEuler system (1.1) on [0 , ∞ ) with initial condition ( x , y , b ) .Proof. First, by using similar argument as in the proof of Lemma 3.2 and the fact that Ω is aBorel subset of Ω
SP W , we know that P (Ω) = 1. We observe that due to (5.27) and (5.24) and P (Ω SP W, ) = 1, it holds P ⊗ τ R ( A ) = P ( A ) for every Borel set A ⊂ Ω SP W,τ . It remains to verifythat P ⊗ τ R satisfies (M1), (M2) and (M3) in Definition 3.7 with s = 0. The first condition in (M1)follows easily since by construction P ⊗ τ R ( x (0) = x , y (0) = y , b (0) = b ) = P ( x (0) = x , y (0) = y , b (0) = b ) = 1 . Similarly, by definition we have P ⊗ τ R (cid:0) N ∈ L ∞ loc ([0 , ∞ ); M + ( T , R × )) (cid:1) = Z Ω SPW { N ∈ L ∞ loc ([0 ,τ ( ω )]; M + ( T , R × )) } Q ω (cid:0) N ∈ L ∞ loc ([ τ ( ω ) , ∞ ); M + ( T , R × )) (cid:1) d P ( ω ) = 1 . Now we shall verify (M2). First, we recall that since P is a probabilistically weak solution on[0 , τ ], for l ∈ U the process h b ( · ∧ τ ) , l i U is a continuous square integrable ( B SP W,t ) t > -martingaleunder P with the quadratic variation process given by k l k U ( · ∧ τ ) . On the other hand, since forevery ω ∈ Ω, the probability measure R τ ( ω ) , ( x,y,b )( τ ( ω ) ,ω ) is a weak solution starting at the time τ ( ω ) from the initial condition ( x, y, b )( τ ( ω ) , ω ), the process h b ( · ) − b ( · ∧ τ ( ω )) , l i U is a continuoussquare integrable ( B SP W,t ) t > τ ( ω ) -martingale under R τ ( ω ) , ( x,y,b )( τ ( ω ) ,ω ) with the quadratic variationprocess given by ( t − τ ( ω )) k l k U , t > τ ( ω ). Also P (Ω) = 1. Then by the same arguments as inthe proof of [HZZ19, Proposition 5.3] we deduce that under P ⊗ τ R , h b, l i U is a continuous square integrable ( B SP W,t ) t > -martingale with the quadratic variation process given by t k l k U , t >
0, whichimplies that b is cylindrical ( B SP W,t ) t > -Wiener process on U . Since b is adapted to ( B SP W,t ) t > , b is cylindrical ( B SP W,t ) t > -Wiener process on U .Furthermore, we have under P for every e i ∈ C ∞ ( T ) ∩ L σ , and for t > h x ( t ∧ τ ) − x (0) , e i i − Z t ∧ τ h R , ∇ e i i d r = Z t ∧ τ h e i , G ( x ( r ))d b ( r ) i . For ω ∈ Ω, under R τ ( ω ) , ( x,y,b )( τ ( ω ) ,ω ) it holds for t > τ ( ω ) h x ( t ) − x ( τ ( ω )) , e i i − Z tτ ( ω ) h R , ∇ e i i d r = Z tτ ( ω ) h e i , G ( x ( r ))d b ( r ) i . Hence, we have P ⊗ τ R (cid:26) h x ( t ) − x (0) , e i i − Z t h R , ∇ e i i d r = Z t h e i , G ( x ( r ))d b ( r ) i , e i ∈ C ∞ ( T ) ∩ L σ , t > (cid:27) = Z Ω d P ( ω ) Q ω (cid:26) h x ( t ) − x ( t ∧ τ ( ω )) , e i i − Z tt ∧ τ ( ω ) h R , ∇ e i i d r = Z tt ∧ τ ( ω ) h e i , G ( x ( r ))d b ( r ) i , h x ( t ∧ τ ( ω )) − x (0) , e i i − Z t ∧ τ ( ω )0 h R , ∇ e i i d r = Z t ∧ τ ( ω )0 h e i , G ( x ( r )) db ( r ) i , e i ∈ C ∞ ( T ) ∩ L σ , t > (cid:27) = Z d P ( ω ) Q ω (cid:26) h x ( t ∧ τ ( ω )) − x (0) , e i i − Z t ∧ τ ( ω )0 h R , ∇ e i i d r = Z t ∧ τ ( ω )0 h e i , G ( x ( r ))d b ( r ) i , e i ∈ C ∞ ( T ) ∩ L σ , t > (cid:27) . By using (5.24) we have Z d P ( ω ) Q ω (cid:26) h x ( t ∧ τ ( ω )) − x (0) , e i i − Z t ∧ τ ( ω )0 h R , ∇ e i i d r = Z t ∧ τ ( ω )0 h e i , G ( x ( r ))d b ( r ) i , e i ∈ C ∞ ( T ) ∩ L σ , t > (cid:27) = P (cid:26) h x ( t ∧ τ ) − x (0) , e i i − Z t ∧ τ h R , ∇ e i i d r = Z t ∧ τ h e i , G ( x ( r ))d b ( r ) i , e i ∈ C ∞ ( T ) ∩ L σ , t > (cid:27) = 1 . Then (M2) follows.Finally, we prove (M3). Define P -a.s. z ( t ) := k x (0) k L + 2 M Et, + Z t k G ( x ( r )) k L ( U,L σ ) d r with M Et, = R t h x, G ( x ( r ))d b ( r ) i . It holds that k x ( t ) k L z ( t ) for a.e. t ∈ [0 , τ ] P -a.s. which bylower semi-continuity implies k x ( τ ) k L z ( τ ) = k x (0) k L + 2 M Eτ, + Z τ k G ( x ( r )) k L ( U,L σ ) d r. (5.28) N ILL- AND WELL-POSEDNESS TO STOCHASTIC 3D EULER EQUATIONS 37
Thus we have Z d P ( ω ) Q ω (cid:26) Z T dtr R ( t ) z ( t ) a.e. t > (cid:27) = Z d P ( ω ) Q ω (cid:26) Z T dtr R ( t ) z ( t ) a.e. t ∈ [0 , τ ( ω )] , Z T dtr R ( t ) z ( t ) a.e. t > τ ( ω ) (cid:27) = Z d P ( ω )1 { R T dtr R ( t ) z ( t ) a.e. t ∈ [0 ,τ ] } Q ω (cid:26) Z T dtr R ( t ) z ( t ) a.e. t > τ ( ω ) (cid:27) = Z d P ( ω )1 { R T dtr R ( t ) z ( t ) a.e. t ∈ [0 ,τ ] } Q ω (cid:26) Z T dtr R ( t ) z ( t ) − z ( τ ( ω )) + z ( τ ( ω )) a.e. t > τ ( ω ) (cid:27) > Z d P ( ω )1 { R T dtr R ( t ) z ( t ) a.e. t ∈ [0 ,τ ] } Q ω (cid:26) Z T dtr R ( t ) z ( t ) − z ( τ ( ω )) + k x ( τ ( ω ) , ω ) k L a.e. t > τ ( ω ) (cid:27) = 1 , where in the last second step we used (5.24) and (5.28) to deduce for P -a.s. ωQ ω ( ω ′ : k x ( τ ( ω ) , ω ) k L z ( τ ( ω ) , ω ) = z ( τ ( ω ) , ω ′ )) = 1and in the last step we used (5.27) and (5.28) together with the fact that the corresponding process z for a solution starting from the initial time τ ( ω ) from the initial condition ( x, y, b )( τ ( ω ) , ω ) as in(5.25) is z ( t ) − z ( τ ( ω )) + k x ( τ ( ω ) , ω ) k L . (cid:3) Application to the convex integration solutions.
As in Section 5.1, we restrict ourselvesto the additive noise case satisfying the assumption (G3). Our goal is then to apply the constructionfrom Section 5.2 to the convex integration solutions obtained in Theorem 5.7, in order to concludethe non-uniqueness in law. Moreover, we assume (G4) on G .In this subsection we fix the parameters l ∈ [2 , ∞ ], L >
1. As the first step, it is necessary todefine a process on the path space Ω
SP W , that is, a function of the variables ( x, y, b ) defined withoutthe usage of any probability measure, such that under the law of the convex integration solution u = u l = v l + GB L from Section 5.1 it becomes the stochastic integral M E from the energy equalityin (5.4). In particular, since it holds M Et, = Z t h u, G d B L i = Z t h v, G d B L i + Z t h GB L , G d B L i , where v solves the transformed system (5.5), we see that the only part requiring probability is thelast term, namely the iterated integral of GB L . Indeed, the first term on the right hand side canbe defined without any probability as a Young integral due to the fact that v necessarily has bettertime regularity than u . This motivates the following definition: for every ω = ( x, y, b ) ∈ Ω SP W welet ¯ M ( x,y,b ) t, := 12 k x ( t ) k L − k x (0) k L − (cid:18) − l (cid:19) t k G k L ( U,L ) − Z t h x (0) − P div( y ( s ) − y (0)) , G d b ( s ) i , (5.29) where in view of (5.5) and (G4) and discussions in Section 2.3.2 the last integral is well-definedYoung integral, cf. Lemma A.1. In other words, we will show below that under the law of thecorresponding convex integration solution the process ¯ M is a.s. the iterated integral of GB L .For n ∈ N , L > δ ∈ (0 , /
12) to be determined below we define τ n, L = inf (cid:26) t > , k Gb ( t ) k H σ > L − n (cid:27) ∧ inf (cid:26) t > , k Gb k C − δt H > L − n (cid:27) ∧ L,τ n, L = inf (cid:26) t > , k ¯ M k W β,pt > L − n (cid:27) ∧ L, τ nL = τ n, L ∧ τ n, L . We observe that the sequence ( τ nL ) n ∈ N is nondecreasing and define τ L := lim n →∞ τ nL . (5.30)Note that without an additional regularity of the trajectory ω , it holds true that τ nL ( ω ) = 0.By [HZZ19, Lemma 3.5] we obtain that τ n, L is ( B SP W,t ) t > -stopping time. Here ω
7→ k ¯ M k W β,pt is ( B SP W,t ) t > -adapted since the fact that ( x, y, b ) is progressively measurable with respect to( B SP W,t ) t > implies that ¯ M is progressively measurable with respect to ( B SP W,t ) t > . Since t ¯ M k W β,pt is increasing, it holds for τ = inf n t > , k ¯ M k W β,pt > L o that { τ > t } = ∩ ∞ m =1 (cid:26) k ¯ M k W β,pt − m L (cid:27) ∈ B SP W,t . Indeed, it is obvious that the right hand side is contained in the left hand side. In addition, the set { τ > t } is also contained in the right hand side. For { τ = t } we also have for every m > k ¯ M k W β,pt − m L. Therefore, τ n, L is ( B SP W,t ) t > -stopping time. Thus also τ L is a ( B SP W,t ) t > -stopping time as anincreasing limit of stopping times with respect to a right continuous filtration.Now, we fix a GG ∗ -Wiener process B defined on a probability space (Ω , F , P ) and we denoteby ( F t ) t > its normal filtration, i.e. the canonical filtration of B augmented by all the P -negligiblesets. We recall that this filtration is right continuous. On this stochastic basis, for fixed parameters l ∈ [2 , ∞ ], L >
1, apply Theorem 5.7 and denote by u = u l = GB L + v l the corresponding solutionto the Euler system (1.1) on [0 , T L ], where the stopping time T L was defined in (5.2). We recall that u is adapted with respect to ( F t ) t > which is essential to show the martingale property in (M2) inProposition 5.12 below. We denote by P the law of ( u, R · u ⊗ u d s, B ) and prove the following result. Proposition 5.12.
The probability measure P is a simplified dissipative probabilistically weak so-lution to the Euler system (1.1) on [0 , τ L ] in the sense of Definition 5.9, where τ L was defined in (5.30) .Proof. Recall that the stopping time T L was defined in (5.2) in terms of the process B . Theorem 5.7yields the existence of a solution u = v l + GB L to the Euler system (1.1) on [0 , T L ] such that (cid:18) u, Z · u ⊗ u d s, B (cid:19) ( · ∧ T L ) ∈ Ω SP W P -a.s.In the following, we write ( u, B ) for notational simplicity to denote ( u, R · u ⊗ u d s, B ). N ILL- AND WELL-POSEDNESS TO STOCHASTIC 3D EULER EQUATIONS 39
We will now prove that τ L ( u, B ) = T L P -a.s . (5.31)To this end, we observe that due to the definition of ¯ M in (5.29) together with the fact that u solvesthe Euler system (1.1) on [0 , T L ] and k u k L = e l ( u − GB L ) for a.e. t ∈ [0 , T L ], we have P -a.s. b ( u,B ) ( t ) = B ( t ) for t ∈ [0 , T L ] , (5.32)and ¯ M ( u,B ) ( t ) = Z t h GB ( s ) , G d B ( s ) i for a.e. t ∈ [0 , T L ] . (5.33)Since GB ∈ CH σ ∩ C − δ loc H P -a.s. and R · h GB ( s ) , G d B ( s ) i ∈ C − δ loc P -a.s., the trajectories ofthe processes t
7→ k GB ( t ) k H σ and t
7→ k GB k C − δt H and t (cid:13)(cid:13)(cid:13)(cid:13)Z · h GB ( s ) , G d B ( s ) i (cid:13)(cid:13)(cid:13)(cid:13) W β,pt are P -a.s. continuous. It follows from the definition of T L that one of the following four statementsholds P -a.s.: either T L = L or k GB ( T L ) k H σ > L or k GB k C − δTL H > L, or (cid:13)(cid:13)(cid:13)(cid:13)Z · h GB ( s ) , G d B ( s ) i (cid:13)(cid:13)(cid:13)(cid:13) W β,pTL > L. Therefore, as a consequence of (5.32) and (5.33), we deduce that τ L ( u, B ) T L P -a.s. Supposenow that τ L ( u, B ) < T L holds true on a set of positive probability P . Then it holds on this set that k GB ( τ L ) k H σ = k Gb ( τ L ) k H σ > L or k Gb k C − δτL H = k GB k C − δτL H > L, or k ¯ M k W β,pτL = (cid:13)(cid:13)(cid:13)(cid:13)Z · h GB ( s ) , G d B ( s ) (cid:13)(cid:13)(cid:13)(cid:13) W β,pτL > L. which however contradicts the definition of T L . Hence we have proved (5.31).Recall that τ L is a ( B SP W,t ) t > -stopping time. We intend to show that P is a simplified dissipativeprobabilistically weak solution to the Euler system (1.1) on [0 , τ L ] in the sense of Definition 5.9.First, we observe that it can be seen from the construction in Theorem 5.7 that the initial value u (0) = ¯ v (0) + GB (0) = ¯ v (0) is indeed deterministic. Moreover, the convex integration solutions areanalytically weak, that is, N ≡ P -a.s. Hence the condition (M1) follows. Since ( u, B ) satisfiesthe Euler equations before T L , the equation in (M2) follows from (5.32) and (5.31). By usingadaptedness of u and a similar argument as in [HZZ19, Proposition 3.7] we obtain h b ( · ∧ τ L ) , l i U is a continuous square integrable ( B SP W,t ) t > s -martingale starting from b at time s with quadraticvariation process given by ( · ∧ τ L − s ) k l k U . In order to verify (M3), we recall R T dtr R ( t ) = k x ( t ) k L for t ∈ [0 , τ L ] P -a.s. and that by Theorem 5.7 we have for Z ( t ) := k u (0) k L + 2 Z t h u, G d B i + t k G k L ( U,L ) that P (cid:0) k u ( t ) k L Z ( t ) for a.e. t ∈ [0 , T L ] (cid:1) = 1 . Thus (M3) follows from (5.32) and (5.31). (cid:3)
At this point, we are already able to deduce that simplified dissipative probabilistically weaksolutions on [0 , τ L ] in the sense of Definition 5.9 are not unique. However, we aim at a strongerresult, namely that globally defined simplified dissipative probabilistically weak solutions on [0 , ∞ )in the sense of Definition 3.7 are not unique. Moreover, we are able to prove non-uniqueness on anarbitrary time interval [0 , T ], T > Proposition 5.13.
The probability measure P ⊗ τ L R is a simplified dissipative probabilistically weaksolution to the Euler system (1.1) on [0 , ∞ ) in the sense of Definition 3.7.Proof. In light of Proposition 5.10 and Proposition 5.11, it only remains to establish (5.27). Due to(5.31) (5.32) and (5.33), we know that P (cid:16) ω : Gb ω ( · ∧ τ L ( ω )) ∈ CH σ ∩ C − δ loc H (cid:17) = P (cid:16) Gb ( u,B ) ( · ∧ τ L ( u, B )) ∈ CH σ ∩ C − δ loc H (cid:17) = P (cid:16) GB ( · ∧ T L ) ∈ CH σ ∩ C − δ loc H (cid:17) = 1 ,P (cid:16) ω : ¯ M ω ( · ∧ τ L ( ω )) ∈ W β,pτ L ( ω ) (cid:17) = P (cid:16) ¯ M ( u,B ) ( · ) ∈ W β,pτ L ( u,B ) (cid:17) = P (cid:18)Z · h GB, G d B i ∈ W β,pT L (cid:19) = 1 . In other words, there exists a P -measurable set N ⊂ Ω SP W such that P ( N ) = 0 and for ω ∈ N c Gb ω ·∧ τ L ( ω ) ∈ CH σ ∩ C − δ loc H , ¯ M ω ·∧ τ L ( ω ) ∈ W β,pτ L ( ω ) . (5.34)Using (5.24) and (5.25) it holds that for all ω ∈ Ω , with Ω defined in the statement of Proposi-tion 5.10, Q ω (cid:16) ω ′ ∈ Ω SP W ; Gb ω ′ · ∈ CH σ ∩ C / − δ loc H , (cid:17) = Q ω (cid:16) ω ′ ∈ Ω SP W ; Gb ω ′ ·∧ τ L ( ω ) ∈ CH σ ∩ C / − δ loc H , Gb ω ′ · − Gb ω ′ ·∧ τ L ( ω ) ∈ CH σ ∩ C / − δ loc H (cid:17) = δ ω (cid:16) ω ′ ∈ Ω SP W ; Gb ω ′ ·∧ τ L ( ω ) ∈ CH σ ∩ C / − δ loc H (cid:17) × R τ L ( ω ) , ( x,y,b )( τ L ( ω ) ,ω ) (cid:16) ω ′ ∈ Ω SP W ; Gb ω ′ · − Gb ω ′ ·∧ τ L ( ω ) ∈ CH σ ∩ C / − δ loc H (cid:17) . Here the first factor on the right hand side equals to 1 for all ω ∈ N c due to (5.34). Since R τ L ( ω ) , ( x,y,z )( τ L ( ω ) ,ω ) is a simplified dissipative probabilistically weak solution starting at the de-terministic time τ L ( ω ) from the deterministic initial condition ( x, y, b )( τ L ( ω ) , ω ), the process ω ′ b ω ′ · − b ω ′ ·∧ τ L ( ω ) is a cylindrical ( B SP W,t ) t > -Wiener process on U starting from τ L ( ω ) under the measure R τ L ( ω ) , ( x,y,b )( τ L ( ω ) ,ω ) . Thus we deduce that also the second factor equals to 1.To summarize, we have proved that for all ω ∈ N c ∩ Ω, with Ω defined in the statement ofProposition 5.10, Q ω (cid:16) ω ′ ∈ Ω SP W ; Gb ω ′ · ∈ CH σ ∩ C / − δ loc H (cid:17) = 1 , and similarly we obtain Q ω (cid:16) ω ′ ∈ Ω SP W ; ¯ M ω ′ ·∧ τ L ( ω ) ∈ W β,pτ L ( ω ) (cid:17) = 1 . N ILL- AND WELL-POSEDNESS TO STOCHASTIC 3D EULER EQUATIONS 41
As a consequence, for all ω ∈ N c ∩ Ω there exists a measurable set N ω such that Q ω ( N ω ) = 0 andfor all ω ′ ∈ N cω the trajectory t Gb ω ′ ( t ) belongs to CH σ ∩ C / − δ loc H and t ¯ M ω ′ ( t ) belongsto W β,pτ L ( ω ) . Therefore, by (5.30) and continuity we obtain that { ω ′ ∈ N cω : τ L ( ω ′ ) = τ L ( ω ) } = { ω ′ ∈ N cω : ¯ τ L ( ω ′ ) = τ L ( ω ) } , where ¯ τ L ( ω ′ ) = inf n t > , k Gb ω ′ ( t ) k H σ > L o ∧ inf n t > , k Gb ω ′ k C / − δt H > L o ∧ L, ¯ τ L ( ω ′ ) = inf (cid:26) t > , k ¯ M ω ′ k W β,pt ∧ τL ( ω ) > L (cid:27) ∧ L, ¯ τ L = ¯ τ L ∧ ¯ τ L . Also for t < L { ω ′ ∈ N cω , ¯ τ L ( ω ′ ) t } = (cid:26) ω ′ ∈ N cω , sup s ∈ Q ,s t k Gb ω ′ ( s ) k H σ > L (cid:27) ∪ ( ω ′ ∈ N cω , sup s = s ∈ Q ∩ [0 ,t ] k Gb ω ′ ( s ) − Gb ω ′ ( s ) k H | s − s | − δ > L ) ∪ (cid:26) ω ′ ∈ N cω , k ¯ M ω ′ k W β,pt ∧ τL ( ω ) > L (cid:27) =: N cω ∩ A t . (5.35)Finally, we deduce that for all ω ∈ N c ∩ Ω Q ω (cid:0) ω ′ ∈ Ω SP W ; τ L ( ω ′ ) = τ L ( ω ) (cid:1) = Q ω (cid:0) ω ′ ∈ N cω ; τ L ( ω ′ ) = τ L ( ω ) (cid:1) = Q ω (cid:0) ω ′ ∈ N cω ; ω ′ ( s ) = ω ( s ) , s τ L ( ω ) , ¯ τ L ( ω ′ ) = τ L ( ω ) (cid:1) = 1 , where we used (5.24) and the fact that (5.35) implies { ω ′ ∈ N cω ; ¯ τ L ( ω ′ ) = τ L ( ω ) } = N cω ∩ ( A τ L ( ω ) \ ( ∪ ∞ n =1 A τ L ( ω ) − n )) ∈ N cω ∩ B SP W,τ L ( ω ) , and Q ω ( A τ L ( ω ) \ ( ∪ ∞ n =1 A τ L ( ω ) − n )) = 1. This verifies the condition (5.27) in Proposition 5.11 and asa consequence P ⊗ τ L R is a simplified dissipative probabilistically weak solution to the Euler system(1.1) on [0 , ∞ ) in the sense of Definition 3.7. (cid:3) Finally, we have all in hand to prove the main result of this section.
Proof of Theorem 5.1.
By Proposition 5.13 we obtain the existence of infinitely many simplifieddissipative probabilistically weak solutions P = P l on [0 , ∞ ) starting from the initial value (¯ v (0) , , l ∈ [2 , ∞ ]. In addition, it holds P l -a.s. k x ( t ∧ τ L ) k L = k ¯ v (0) k L + 2 Z t ∧ τ L h x, G d B i + (cid:18) − l (cid:19) ( t ∧ τ L ) k G k L ( U,L ) for a.e. t ∈ [0 , T ] . Thus, we obtain that the law of ( x, y, b ) on [0 , T ] is not unique. In addition, by (M2) we knowthat under each P l the law of b is determined by ( x, y ). Therefore, we deduce that the law ( x, y ) isnot unique which completes the proof of non-uniqueness in law for simplified dissipative martingalesolutions. (cid:3) Existence and non-uniqueness of strong Markov solutions
We aim at showing existence of a strong Markov dissipative solution to (1.1). The approachrelies on the abstract Markov selection introduced by Krylov [Kr73] and presented by Stroock,Varadhan [SV79]. Applications to SPDEs can be found in [FR08, GRZ09, BFH18a] where existenceof almost sure Markov solutions was proved for several SPDEs including stochastic Navier–Stokesequation in the incompressible as well as compressible setting. In these works, the existence ofsolutions satisfying the usual Markov property, was left open. Motivated by the recent constructionof solution semiflows to deterministic isentropic and complete Euler system in [BFH20a, BFH20c],we put forward a construction which not only applies to the stochastic Euler equations (1.1) but iteven permits to obtain strong Markov solutions.The principal idea is to include an additional datum into the selection procedure. In [BFH20a,BFH20c], this was done through the energy which in our notation corresponds to E ( t ) := R T dtr R ( t ) . In the deterministic setting, this function is non-increasing hence of finite variation. Thus, it admitsleft- and right-limits at all times and is continuous except for an at most countable set of times.Nevertheless, as in the stochastic setting E is not decreasing due to the martingale part and even ifit admits left- and right-limits these exceptional times become random, it does not seem possible toget existence of Markov selections by including solely the variable E . As suggested in Section 3.1,we rather include the variable z which plays the same role but in addition has time continuoustrajectories.As uniqueness of simplified dissipative martingale solutions was disproved in Section 5, it isnecessary to find additional selection criteria in order to select physically relevant solutions. A well-accepted criterion in the case of the deterministic Euler equations is the maximization of energydissipation or, equivalently, minimization of the total energy of the system proposed by Dafermos[Da79]. In the same spirit, we aim at selecting only solutions which minimize the average totalenergy. To be more precise, let P and Q be two dissipative martingale solutions starting from thesame initial data ( x , y , z ). We introduce the relation P ≺ Q ⇔ E P (cid:20)Z T dtr R ( t ) (cid:21) E Q (cid:20)Z T dtr R ( t ) (cid:21) for a.e. t ∈ (0 , ∞ ) , which leads to the following admissibility condition. Definition 6.1 (Admissible dissipative martingale solution) . We say that a dissipative martingalesolution P starting from ( x , y , z ) is admissible if it is minimal with respect to the relation ≺ , i.e.,if there is another dissipative martingale solution Q starting from ( x , y , z ) such that Q ≺ P , then E Q (cid:20)Z T dtr R ( t ) (cid:21) = E P (cid:20)Z T dtr R ( t ) (cid:21) for a.e. t ∈ (0 , ∞ ) . We remark that this definition is consistent with the admissibility condition in the deterministiccompressible setting introduced in [BFH20a, BFH20c]. In these works, it allowed to establishstability of deterministically stationary states, i.e., time independent solutions: if the system reachessuch an equilibrium then it remains there for all times.
Remark 6.2.
It is interesting to see what admissibility means for the convex integration solutionsfrom Section 5.1. Let T L be the stopping time defined in (5.2). Then the solutions u l = v l + GB L , l ∈ [2 , ∞ ], constructed through Theorem 5.7 satisfy E P (cid:2) k u l ( t ) k L (cid:3) = k u k L + (cid:18) − l (cid:19) E P [( t ∧ T L )] k G k L ( U,L ) . N ILL- AND WELL-POSEDNESS TO STOCHASTIC 3D EULER EQUATIONS 43
We observe that the right hand side is minimal when l = 2. However, we note that apart from theconvex integration solutions parametrized by l ∈ [2 , ∞ ], the same convex integration constructiongives raise to other convex integration solution where the defect in the energy is prescribed differently.Such solutions does not have to be comparable by the admissibility criterion. Furthermore, itactually seems that solutions obtained by convex integration as in Section 5.1 cannot be admissible.6.1. Selection of strong Markov processes.
Once a suitable notion of solution has been found,the abstract framework of Markov selections can be applied. For readers’ convenience, we recallsome of the key results in this respect presented by Stroock and Varadhan [SV79]. First, basedon [SV79, Lemma 1.3.3, Theorem 1.3.4] we obtain a disintegration result, that is, the existence ofa regular conditional probability distribution. We use a ( t, ω ) to denote ( x ( t, ω ) , y ( t, ω ) , z ( t, ω )) forsimplicity. Theorem 6.3.
Given P ∈ P (Ω M ) and τ a finite ( B M,t ) t > -stopping time, there exists a regularconditional probability distribution (abbreviated as r.c.p.d.) P ( ·|B M,τ )( ω ) , ω ∈ Ω M , of P with respectto B M,τ such that(1) For every ω ∈ Ω M , P ( ·|B M,τ )( ω ) is a probability measure on (Ω M , B M ) .(2) For every A ∈ B M , the mapping ω P ( A |B M,τ )( ω ) is B M,τ -measurable.(3) There exists a P -null set N ∈ B M,τ such that for any ω / ∈ NP (cid:0) { ˜ ω ; τ (˜ ω ) = τ ( ω ) , a ( s, ˜ ω ) = a ( s, ω ) , s τ (˜ ω ) }|B M,τ (cid:1) ( ω ) = 1 . (4) For any Borel set A ∈ B M,τ and any Borel set B ⊂ Ω M P (cid:0) a | [0 ,τ ] ∈ A, a | [ τ, ∞ ) ∈ B (cid:1) = Z ˜ ω ∈ A P (cid:0) B |B M,τ (cid:1) (˜ ω )d P (˜ ω ) . According to [SV79, Theorem 6.1.2] we obtain the following reconstruction result.
Theorem 6.4.
Let τ be a finite ( B M,t ) t > -stopping time. Let ω Q ω be a mapping from Ω M to P (Ω M ) such that for any A ∈ B M , ω Q ω ( A ) is B M,τ -measurable and for any ω ∈ Ω M Q ω (cid:0) ˜ ω ∈ Ω M : a ( τ ( ω ) , ˜ ω ) = a ( τ ( ω ) , ω ) (cid:1) = 1 . Then for any P ∈ P (Ω M ) , there exists a unique P ⊗ τ Q ∈ P (Ω M ) such that ( P ⊗ τ Q )( A ) = P ( A ) for all A ∈ B M,τ , and for P ⊗ τ Q -almost all ω ∈ Ω M δ ω ⊗ τ ( ω ) Q ω = ( P ⊗ τ Q ) (cid:0) ·|B M,τ (cid:1) ( ω ) . Recall that X = { ( x , y , z ) ∈ L σ × M + ( T , R × ) × R ; k x k L z } . We say P ∈ P X (Ω M ) ⊂ P (Ω M ) is concentrated on the paths with values in X if there exists A ∈ B with P ( A ) = 1 suchthat A ⊂ { ω ∈ Ω M ; ω ( t ) ∈ X for all t > } . It is clear that B ( P X (Ω M )) = B ( P (Ω M )) ∩ P X (Ω M ).We also use Comp( P X (Ω M )) to denote the space of all compact subsets of P X (Ω M ) and by C ( a )we denote the set of dissipative martingale solutions starting from a ∈ X at time s = 0. The shiftoperator Φ t : Ω M → Ω tM is defined byΦ t ( ω )( s ) := ω ( s − t ) , s > t. Now, we have all in hand to recall the definition of a strong Markov process.
Definition 6.5.
A family ( P a ) a ∈ X of probability measures in P X (Ω M ) , is called a strong Markovfamily provided (1) for every A ∈ B , the mapping a P a ( A ) is B ( X ) / B ([0 , -measurable,(2) for every finite ( B M,t ) t > -stopping time τ , every a ∈ X and for P a -a.s. ω ∈ Ω M P a (cid:0) ·|B M,τ (cid:1) ( ω ) = P a ( τ ( ω ) ,ω ) ◦ Φ − τ ( ω ) . A strong Markov family can be obtained from a so-called pre-Markov family through a selectionprocedure.
Definition 6.6.
Let the mapping X → Comp( P X (Ω M )) , a C ( a ) , be Borel measurable. We saythat ( C ( a )) a ∈ X forms a pre-Markov family if for each a ∈ X , P ∈ C ( a ) and every finite ( B M,t ) > -stopping time τ there holds(1) (Disintegration) there is a P -null set N ∈ B M,τ such that for ω / ∈ N , a ( τ ( ω ) , ω ) ∈ X , P (cid:0) Φ τ ( ω ) ( · ) |B M,τ (cid:1) ( ω ) ∈ C ( a ( τ ( ω ) , ω )) , (2) (Reconstruction) if a mapping Ω M → P X (Ω M ) , ω Q ω , satisfies the assumptions ofTheorem 6.4 and there is a P -null set N ∈ B M,τ such that for all ω / ∈ Na ( τ ( ω ) , ω ) ∈ X , Q ω ◦ Φ τ ( ω ) ∈ C ( a ( τ ( ω ) , ω )) , then P ⊗ τ Q ∈ C ( a ) . Finally, we recall the following abstract Markov selection theorem, cf. [SV79, Theorem 12.2.3].The generalization to our setting of Polish spaces can be found in [GRZ09]. One difference is that inthe definition of the path space Ω M we need to include y ∈ W α,q loc ([0 , ∞ ); W − k,p ( T , R × )). It canbe checked that the proof [SV79, Theorem 12.2.3] applies also to this setting. The only missing pointis then the maximization of a given functional, however, this can be achieved easily by choosing thisfunctional in the selection procedure as the first functional to be maximized. Theorem 6.7.
Let ( C ( a )) a ∈ X be a pre-Markov family. Suppose that for each a ∈ X , C ( a ) is non-empty and convex. Then there exists a measurable selection X → P X (Ω M ) , a P a , such that P a ∈ C ( a ) for every a ∈ X and ( P a ) a ∈ X is a strong Markov family. In addition, if F : X → R is a bounded continuous function and λ > then the selection can be chosen for every a ∈ X tomaximize E P (cid:20)Z ∞ e − λs F ( x ( s ) , y ( s ) , z ( s ))d s (cid:21) (6.1) among all dissipative martingale solutions P with the initial condition a . Application to the stochastic Euler equations.
Our goal is to verify the assumptionsof Theorem 6.7 and to choose a suitable function F so that the Markov selection only containsadmissible dissipative martingale solutions. To be more precise, we aim at proving the following. Theorem 6.8.
Suppose that (G1) , (G2) hold. The family ( C ( a )) a ∈ X admits a measurable strongMarkov selection. In other words, there exists a strong Markov family ( P a ) a ∈ X solving the stochasticEuler equations (1.1) . Moreover, for every a ∈ X , the dissipative martingale solution P a is admissiblein the sense of Definition 6.5. Remark 6.9.
We note that by choosing a different function F from the one needed in the proofof Theorem 6.8, we could find a strong Markov selection which is not necessarily admissible. Thisfact will be explored in Section 6.3 where combined with the non-uniqueness in law from Section 5it permits to prove non-uniqueness of strong Markov selections. N ILL- AND WELL-POSEDNESS TO STOCHASTIC 3D EULER EQUATIONS 45
In the remainder of this subsection, we prove Theorem 6.8. First of all, we recall that Theorem 4.3yields stability of dissipative martingale solutions and as a consequence of Theorem 4.4 we obtaintheir existence. Thus, we deduce that for every a ∈ X the set C ( a ) is a non-empty subset ofComp( P X (Ω M )) and that the mapping X → Comp( P X (Ω M )), a C ( a ), is Borel measurable. Werefer to [FR08] for more details on this step. The convexity of C ( a ) is immediate as the conditions(M1), (M2), (M3) only contain integration with respect to probability measures.As the next step, we verify that ( C ( a )) a ∈ X has the disintegration as well as the reconstructionproperty from Definition 6.6. Lemma 6.10.
The family ( C ( a )) a ∈ X satisfies the disintegration property in Definition 6.6.Proof. Fix a ∈ X , P ∈ C ( a ) and a finite ( B M,t ) t > -stopping time τ . Let P ( ·|B M,τ )( ω ) be a r.c.p.d.of P with respect to B M,τ . We want to show that there is a P -null set N ∈ B M,τ such that for all ω / ∈ N a ( τ ( ω ) , ω ) ∈ X , P (cid:0) Φ τ ( ω ) ( · ) |B M,τ (cid:1) ( ω ) ∈ C ( a ( τ ( ω ) , ω )) . Due to (M3) and continuity of x we know P (cid:0) k x ( τ ) k L z ( τ ) (cid:1) = 1 . Let N = {k x ( τ ) k L z ( τ ) } c . Then P ( N ) = 1 and N ∈ B M,τ .Next, we shall verify that P (Φ τ ( ω ) ( · ) |B M,τ )( ω ) satisfies the conditions (M1), (M2), (M3) in Defini-tion 3.1 with the initial condition a ( τ ( ω ) , ω ) and the initial time 0 or alternatively that P ( ·|B M,τ )( ω )satisfies (M1), (M2), (M3) in Definition 3.1 with the initial condition a ( τ ( ω ) , ω ) and the initial time τ ( ω ).(M1): Due to (3) from Theorem 6.3, it follows that outside of a P -null set in B M,τ , it holds P (cid:0) { ˜ ω ; a ( τ ( ω ) , ˜ ω ) = a ( τ ( ω ) , ω ) }|B M,τ (cid:1) ( ω ) = 1 . In other words, P ( ·|B M,τ )( ω ) has the correct initial value at the initial time τ ( ω ). By using (4) inTheorem 6.3 we deduce1 = P (cid:0) N ∈ L ∞ loc ([0 , ∞ ); M + ( T , R × )) (cid:1) = P (cid:0) N ∈ L ∞ loc ([0 , τ ]; M + ( T , R × )) , N ∈ L ∞ loc ([ τ, ∞ ); M + ( T , R × )) (cid:1) = Z ω ∈{ N ∈ L ∞ loc ([0 ,τ ]; M + ( T , R × )) } P (cid:0) N ∈ L ∞ loc ([ τ, ∞ ); M + ( T , R × ) |B M,τ (cid:1) ( ω )d P ( ω ) , which implies that P (cid:0) N ∈ L ∞ loc ([ τ, ∞ ); M + ( T , R × )) |B M,τ (cid:1) = 1 P -a.s. ω ∈ Ω M . Moreover, the corresponding P -null set belongs to B M,τ . We denote the union of the above two P -null sets by N .(M2): Using [SV79, Theorem 1.2.10] (cf. [GRZ09, Lemma B.3]), there exists a P -null set N ∈B M,τ such that for all ω / ∈ N , P (Φ τ ( ω ) ( · ) |B M,τ )( ω ) satisfies (M2).(M3): Similarly, there exists a P -null set N ∈ B M,τ such that for all ω / ∈ N , the process M E isa continuous ( B M,t ) t > -martingale under P (Φ τ ( ω ) ( · ) |B M,τ )( ω ), and P (cid:18)Z T dtr R ( t ) z ( t ) for a.e. t ∈ [ τ ( ω ) , ∞ ) |B M,τ (cid:19) ( ω ) = 1 . Recall that under P we have for every t > z ( t ) = z (0) + 2 Z t d M Er + Z t k G ( x ( r )) k L ( U,L ) d r. Now, we split this equation into two parts, i.e., z ( t ) = z (0) + 2 Z t d M Er + Z t k G ( x ( r )) k L ( U,L ) d r, t τ ( ω ) , (6.2) z ( t ) = z ( τ ( ω )) + 2 Z tτ ( ω ) d M Er + Z tτ ( ω ) k G ( x ( r )) k L ( U,L ) d r, τ ( ω ) t < ∞ , (6.3)and consider the sets R τ ( ω ) = (cid:8) ˜ ω ∈ Ω M ; ˜ ω | [0 ,τ ( ω )] satisfies (6.2) (cid:9) ,R τ ( ω ) = (cid:8) ˜ ω ∈ Ω M ; ˜ ω | [ τ ( ω ) , ∞ ) satisfies (6.3) (cid:9) . We obtain for P P ( R τ ∩ R τ ) = Z R τ P (cid:16) R τ ( ω ) |B M,τ (cid:17) ( ω )d P ( ω ) , where we used (3) from Theorem 6.3. Consequently, there is a P -null set N ∈ B M,τ such that forevery ω / ∈ N it holds P ( R τ ( ω ) |B M,τ )( ω ) = 1. Hence (M3) follows.We complete the proof by choosing the null set N = ∪ i =0 N i . (cid:3) Lemma 6.11.
The family ( C ( a )) a ∈ X satisfies the reconstruction property in Definition 6.6.Proof. Fix a ∈ X , P ∈ C ( a ), a finite ( B M,t ) t > -stopping time τ and Q ω as in Definition 6.6 (2).We shall show that P ⊗ τ Q ∈ C ( a ) hence we need to verify the conditions (M1), (M2), (M3) fromDefinition 3.1. The proof follows the lines of Proposition 5.11 using the fact that δ ω ⊗ Q ω (cid:0) ω ′ : τ ( ω ′ ) = τ ( ω ) (cid:1) = 1 , since for a ( B M,t ) t > -stopping time τ , { ω ′ : τ ( ω ′ ) = τ ( ω ) } ∈ B M,τ ( ω ) . (cid:3) Since we intend to select those solutions that are admissible, we shall find a suitable functionalof the form (6.1) which achieves this. By integration by parts formula it holds E P (cid:20)Z ∞ e − λs (cid:18) − Z T dtr y ( s ) (cid:19) d s (cid:21) = − λ Z T dtr y (0) + 1 λ E P (cid:20)Z ∞ e − λs (cid:18) − Z T dtr R ( s ) (cid:19) d s (cid:21) . (6.4)Since the initial datum y is fixed, the minimization of E P (cid:2)R T dtr R ( t ) (cid:3) required for the admissibilitycondition is equivalent to maximization of the functional on the left hand side of (6.4). Indeed,if Q ≺ P and E P (cid:2)R ∞ e − λs (cid:0)R T dtr R ( s ) (cid:1) d s (cid:3) is the minimal one, we obtain E P (cid:2)R T dtr R ( s ) (cid:3) = E Q (cid:2)R T dtr R ( s ) (cid:3) for a.e. s ∈ (0 , ∞ ).Even though the function F : ( x, y, z )( s ) (cid:18) − Z T dtr y ( s ) (cid:19) N ILL- AND WELL-POSEDNESS TO STOCHASTIC 3D EULER EQUATIONS 47 is not bounded on X , it follows from Lemma 3.2 using the definition of y together with (M3) that E P " sup s ∈ [0 ,t ] Z T dtr y ( s ) y + ( z + Ct ) e Ct , where C > λ > P as well as its initial value, such thatthe functional in (6.4) is well-defined and can be employed in the proof of Theorem 6.7.Finally, we have all in hand to apply Theorem 6.7 which completes the proof of Theorem 6.8.6.3. Non-uniqueness of strong Markov solutions.
Finally, combining the existence of a strongMarkov solution from Theorem 6.8 with the non-uniqueness in law from Section 5 we obtain non-uniqueness of strong Markov solutions. This holds under the assumptions of Section 5, in particular,for an additive noise.
Theorem 6.12.
Suppose that (G3) , (G4) hold. Then strong Markov families associated to thestochastic Euler system (5.1) are not unique. More precisely, apart from the admissible strongMarkov family ( P a ) a ∈ X constructed in Theorem 6.8, there exists a possibly non-admissible strongMarkov family.Proof. The proof follows the lines of [SV79, Theorem 12.2.4]. In particular, in view of the non-uniqueness in law from Theorem 5.1 there exists an initial value x ∈ L σ such that for z := k x k L and for every y there is infinitely many dissipative martingale solutions with the initial condition( x , y , z ) at the time 0. In particular, there are dissipative martingale solutions P , Q and afunction F : X → R such that E P (cid:20)Z ∞ e − λs F ( x ( s ) , y ( s ) , z ( s ))d s (cid:21) > E Q (cid:20)Z ∞ e − λs F ( x ( s ) , y ( s ) , z ( s ))d s (cid:21) . Now, applying Theorem 6.7 once with F and once with − F we obtain selections ( P + a ) a ∈ X and( P − a ) a ∈ X , respectively. In particular, for a = ( x , y , z ) it holds E P + a (cid:20)Z ∞ e − λs F ( x ( s ) , y ( s ) , z ( s ))d s (cid:21) > E P (cid:20)Z ∞ e − λs F ( x ( s ) , y ( s ) , z ( s ))d s (cid:21) , and E Q (cid:20)Z ∞ e − λs F ( x ( s ) , y ( s ) , z ( s ))d s (cid:21) > E P − a (cid:20)Z ∞ e − λs F ( x ( s ) , y ( s ) , z ( s ))d s (cid:21) . In other words, the two strong Markov selections are different. The admissibility cannot be guar-anteed since for this it is necessary to choose the first functional as in (6.4). (cid:3)
Remark 6.13.
The approach of Theorem 6.12 translated to the deterministic setting implies non-uniqueness of the semiflow associated to the incompressible Euler equations. To be more precise,existence of a semiflow selection can be proved by the same procedure as in [BFH20a, BFH20c],whereas non-uniqueness of solutions satisfying the energy equality was proved in [DLS10]. Thus,by a straightforward modification of the proof of Theorem 6.12 the claim follows. A challengingquestion which remains open in both deterministic and stochastic setting is whether the selectioncan be chosen admissible, i.e., whether it satisfies the principle of maximal energy dissipation.
Appendix A. Young integration
We present an auxiliary lemma used in Section 5.1.
Lemma A.1.
Let X be a Banach space and X ∗ be its topological dual. Let α, β ∈ (0 , be suchthat α + β > . Assume that f ∈ C α ([0 , T ]; X ) and g ∈ C β ([0 , T ]; X ∗ ) . Then the Young integral t Z t h g r , d f r i is well-defined and satisfies (cid:13)(cid:13)(cid:13)(cid:13)Z · h g r , d f r i (cid:13)(cid:13)(cid:13)(cid:13) C αT . k g k C βT X ∗ k f k C α X . Proof.
For 0 s θ t T we use the notation f st = f t − f s and for a two-index map δh sθt = h st − h sθ − h θt . We consider the following local approximation of the integral Z ts h g r , d f r i = h g s , f st i + g ♮st , (A.1)where g ♮st is expected to be a sufficiently regular remainder. Then we obtain for Ξ st = h g s , f st i| δ Ξ sθt | = |h g sθ , f θt i| k g k C βT X ∗ k f k C α X | t − s | α + β . Hence the sewing lemma [DGHT19, Lemma 2.2] yields | g ♮st | . k g k C βT X ∗ k f k C α X | t − s | α + β . Thus, (A.1) impliessup s,t ∈ [0 ,T ] ,s = t (cid:12)(cid:12)(cid:12)R ts h g r , d f r i (cid:12)(cid:12)(cid:12) | t − s | α sup s,t ∈ [0 ,T ] ,s = t |h g s , f st i|| t − s | α + sup s,t ∈ [0 ,T ] ,s = t | g ♮st || t − s | α . k g k C βT X ∗ k f k C α X and the claim follows. (cid:3) References [Be99] H. Bessaih. Martingale solutions for stochastic Euler equations.
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Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld, D-33501 Bielefeld, Germany
E-mail address : [email protected] (R. Zhu) Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China; Fakult¨atf¨ur Mathematik, Universit¨at Bielefeld, D-33501 Bielefeld, Germany
E-mail address : [email protected] (X. Zhu) Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190,China; Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld, D-33501 Bielefeld, Germany
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