Existence of martingale solutions and the incompressible limit for stochastic compressible flows on the whole space
aa r X i v : . [ m a t h . A P ] M a y EXISTENCE OF MARTINGALE SOLUTIONS AND THEINCOMPRESSIBLE LIMIT FOR STOCHASTIC COMPRESSIBLEFLOWS ON THE WHOLE SPACE
PRINCE ROMEO MENSAH
Abstract.
We give an existence and asymptotic result for the so-called finiteenergy weak martingale solution of the compressible isentropic Navier–Stokessystem driven by some random force in the whole spatial region. In particular,given a general nonlinear multiplicative noise, we establish the convergence tothe incompressible system as the Mach number, representing the ratio betweenthe average flow velocity and the speed of sound, approaches zero. Introduction
In continuum mechanics, the motion of an isentropic compressible fluid is de-scribed by the density ̺ = ̺ ( t, x ) and velocity u = u ( t, x ) in a physical domain in R satisfying the mass and momentum balance equations given respectively by(1.1) ∂ t ̺ + div( ̺ u ) = 0 ,∂ t ( ̺ u ) + div( ̺ u ⊗ u ) = div T + ̺ f . Here f is some external force and T the stress tensor . By Stokes’ law , T satisfies T = S − p I where p = p ( ̺ ) is the pressure and S = S ( ∇ u ) the viscous stress tensor .In following Newton’s law of viscosity , we assume that S satisfies S = ν (cid:0) ∇ u + ∇ T u (cid:1) + λ div u I with viscosity coefficients satisfying ν > λ + ν ≥ . For the pressure, we supposethe γ -law p = 1Ma ̺ γ where Ma > γ > , the adiabatic exponent. In orderto study the existence of solutions to system (1.1), it has to be complemented byinitial and boundary conditions (very common are periodic boundary conditions,no-slip boundary conditions and the whole space). The existence of weak solutionsto (1.1) has been shown in the fundamental book by Lions [23] and extended tophysical reasonable situations by Feireisl [11, 15], giving a compressible analogue ofthe pioneering work by Leray [22] on the incompressible case. These results involvethe concept of weak solutions where derivatives have to be understood in the sense Mathematics Subject Classification.
Key words and phrases.
Isentropic flows, Stochastic compressible fluid, Navier–Stokes, Machnumber, Martingale solution.The author would like to acknowledge the financial support of the Department of Mathematics,Heriot–Watt University, through the James–Watt scholarship. He will also like to thank D. Breitfor recommending this work and for his many useful discussions. of distributions . This concept has since become an integral technique in the studyof nonlinear PDE’s.In recent years, there has been an increasing interest in random influences on fluidmotions. It can take into account, for example, physical, empirical or numericaluncertainties and is commonly used to model turbulence in the fluid motion.As far as we know, the first result on the existence of solution to the stochasticcompressible system is due to [34]. This was done in 1-D and later for a specialperiodic 2-D case in [33]. The latter mostly relied on existence arguments developedin [35]. In [13], a semi-deterministic approach based on results on multi-valuedfunctions is used and follows in line with the incompressible analogue shown in [1].A fully stochastic theory has been developed in [5]. The existence of martingalesolutions has been shown in the case of periodic boundary conditions. This hasbeen extended to Dirichlet boundary conditions in [32].Compared to the stochastic compressible model, the incompressible system hasbeen studied much more intensively. It first appeared in the seminal paper by Ben-soussan and Temam [1] which is based on a semi-deterministic approach. Later, theconcept of a martingale solution of this system was then introduced by Flandoli andGatarek [16]. For a recent survey on the stochastic incompressible Navier–Stokesequations, we refer the reader to [30] or to [29] for the general survey includingdeterministic results.The aim of this paper is to look at the situation on the whole space R . This isparticularly important for various applications and especially for those in which thecomparative size of the fluids domain far exceeds the speed of sound accompanyingthe fluid. See [14] for more details. Difficulties arise due to the lack of certaincompactness tools which are available in the case of bounded domains. We shallstudy the system(1.2) d ̺ + div( ̺ u )d t = 0 , d( ̺ u ) + [div( ̺ u ⊗ u − S ( ∇ u )) + ∇ p ( ̺ )]d t = Φ( ̺, ̺ u )d W, in Q T = (0 , T ) × R . A prototype for the stochastic forcing term will be given by(1.3) Φ( ̺, ̺ u )d W ≈ ̺ d W + ̺ u d W where W and W is a pair of independent cylindrical Wiener processes. We referto Sect. 2 for the precise assumptions on the noise and its coefficients.The first main result of the present paper is the existence of finite energy weakmartingale solutions to (1.2). The precise statement is given in Theorem 2.4. Weapproximate the system on the whole space by a sequence of periodic problems(where the period tends to infinity). After showing uniform a priori estimates,we use the stochastic compactness method based on the Jakubowski-Skorokhodrepresentation theorem. In contrast to previous works, we adapt it to the situationon the whole space taking carefully into account, the lack of compact embeddings.In order to pass to the limit in the nonlinear pressure term, we use properties ofthe effective viscous flux originally introduced by Lion [23] similar to [5].A fundamental question in compressible fluid mechanics is the relation to theincompressible model. If the Mach number is small, the fluid should behave asymp-totically like an incompressible one, provided velocity and viscosity are small, andwe are looking at large time scales, see [21]. The problem has been studied rig-orously in the deterministic case in [24, 25, 26], as a singular limit problem. Amajor problem to overcome is the rapid oscillation of acoustic waves due to the XISTENCE AND LOW-MACH LIMIT FOR STOCHASTIC COMPRESSIBLE FLOWS 3 lack of compactness. A stochastic counterpart of this theory has very recently beenestablished in [3]. The limit ε of the system(1.4) d ̺ ε + div( ̺ ε u ε )d t = 0 , d( ̺ ε u ε ) + [div( ̺ ε u ε ⊗ u ε − S ( ∇ u ε )) + ∇ ̺ γε ε ]d t = Φ( ̺ ε , ̺ ε u ε )d W, has been analyzed under periodic boundary conditions. Given a sequence of theso-called finite energy weak martingale solution for (1.4) (see next section for defi-nition) where ε ∈ (0 ,
1) , its limit (as ε →
0) is indeed a weak martingale solution to the following incompressible system:(1.5) div( u ) = 0 , d( u ) + [div( u ⊗ u ) − ν ∆ u + ∇ ˜ p ]d t = P Φ(1 , u )d W. Here ˜ p is the associated pressure and P is the Helmholtz projection onto the spaceof solenoidal vector fields.A major drawback in the approach in [3] is that the noise coefficient Φ( ̺, ̺ u )has to be linear in the momentum ̺ u . This is due to the aforementioned lack ofcompactness of momentum when ε passes to zero. This cannot even be improvedin the deterministic case. The situation on the whole space, however, is muchbetter as a consequence of dispersive estimates for the acoustic wave equations, seeProposition 4.8. We apply them to the stochastic wave equation and hence are ableto prove strong convergence of the momentum, see Lemma 4.11. Based on this,we are able to prove the convergence of (1.4) to (1.5) under much more generalassumptions on the noise coefficients. See Theorem 2.6 for details.In Sect. 2, we state the required assumptions satisfied by the various quantitiesused in this paper, as well as some useful function space estimates. We definethe concept of a solution, state the required boundary condition applicable in oursetting and finally state the main results.In Sect. 3, we are concerned with the proof of Theorem 2.4, giving existence ofmartingale solutions on the whole space. Based on this result, we devote Sect. 4to the proof of Theorem 2.6; the low-Mach number limit on the whole space.2. Preliminaries
Throughout this paper, the spatial dimension is N = 3 and we assume that(Ω , F , ( F t ) t ≥ , P ) is a stochastic basis with a complete right-continuous filtration, W is a ( F t )-cylindrical Wiener process, that is, there exists a family of mutuallyindependent real-valued Brownian motions ( β k ) k ∈ N and orthonormal basis ( e k ) k ∈ N of a separable Hilbert space U such that W ( t ) = X k ∈ N β k ( t ) e k , t ∈ [0 , T ] . We also assume that ̺ ∈ L γ loc ( R ), ̺ ≥
0, and u ∈ L ( R ) so that √ ̺ u ∈ L ( R ).Now let set q = ̺ u and assume that there exists a compact set K ⊂ R andsome functions g k : R × R × R → R such that g k ∈ C ( K ) , for any k ∈ N , (2.1)and in addition, satisfies the following growth conditions:(2.2) X k ∈ N | g k ( x, ̺, q ) | ≤ c (cid:0) ̺ + | q | (cid:1) , X k ∈ N |∇ ̺, q g k ( x, ̺, q ) | ≤ c. PRINCE ROMEO MENSAH
Then if we define the map Φ( ̺, ̺ u ) : U → L ( K ) by Φ( ̺, ̺ u ) e k = g k ( · , ̺ ( · ) , ̺ u ( · )),we can use the embedding L ( K ) ֒ → W − l, ( K ) where l > , to show that k Φ( ̺, ̺ u ) k L ( U ; W − l, ( K )) is uniformly bounded provided ̺ ∈ L γ loc ( R ) and √ ̺ u ∈ L ( R ). See [5, Eq. 2.3].As such, the stochastic integral ´ · Φ( ̺, ̺ u )d W is a well-defined ( F t )-martingaletaking value in W − l, ( R ).Lastly, we define the auxiliary space U ⊃ U via U = u = X k ≥ c k e k ; X k ≥ c k k < ∞ and endow it with the norm k u k U = X k ∈ N c k k , u = X k ∈ N c k e k . Then it can be shown that W has P -a.s. C ([0 , T ]; U ) sample paths with the Hilbert–Schmidt embedding U ֒ → U . See [7].2.1. Sobolev inequalities for the homogeneous Sobolev space.
As we shallsee shortly, the compactness techniques used in this paper involves certain estimateswhose constants must necessarily be independent of the size of the domain. Wetherefore require the homogeneous Sobolev space D ,q ( O ) = ( u ∈ D ′ ( O ) : u ∈ L q − q ( O ) , ∇ u ∈ L q ( O ) if 1 ≤ q < u = { u + c } c ∈ R : u ∈ L q loc ( O ) , ∇ u ∈ L q ( O ) if q ≥ O is an exterior or an unboundeddomain, for example O = R . In particular, given a function u ∈ D ,q ( O ), we havethat for any 1 ≤ q < k u k L q − q ( O ) ≤ c q k∇ u k L q ( O ) (2.3)See [17, Chapter II] for more details. Note that the constant above is independentof the size of O , unlike in the case of the usual Sobolev–Poinc´are’s inequality.To continue, let us define the concept of a solution used in this paper. Definition 2.1.
If Λ is a Borel probability measure on L γ ( R ) × L γγ +1 ( R ), thenwe say that [(Ω , F , ( F t ) , P ); ̺, u , W ](2.4)is a finite energy weak martingale solution of Eq. (1.4) with initial law Λ provided:(1) (Ω , F , ( F t ) , P ) is a stochastic basis with a complete right-continuous filtra-tion,(2) W is a ( F t )-cylindrical Wiener process,(3) the density ̺ satisfies ̺ ≥ , t → h ̺ ( t, · ) , φ i ∈ C [0 , T ] for any φ ∈ C ∞ c ( R ) P − a.s., the function t
7→ h ̺ ( t, · ) , φ i is progressively measurable, and E " sup t ∈ [0 ,T ] k ̺ ( t, · ) k pL γ ( K ) < ∞ for all 1 ≤ p < ∞ , and for all K ⊂ R with K compact, XISTENCE AND LOW-MACH LIMIT FOR STOCHASTIC COMPRESSIBLE FLOWS 5 (4) the momentum ̺ u satisfies t → h ̺ u , φ i ∈ C [0 , T ] for any φ ∈ C ∞ c ( R ) P − a.s., the function t
7→ h ̺ u , φ i is progressively measurable, and for all 1 ≤ p < ∞ E " sup t ∈ [0 ,T ] k√ ̺ u k pL ( K ) < ∞ , E " sup t ∈ [0 ,T ] k ̺ u k pL γγ +1 ( K ) < ∞ , for all K ⊂ R , K compact,(5) the velocity field u is ( F t )-adapted, u ∈ L p (cid:16) Ω; L (cid:16) , T ; W , (cid:0) R (cid:1)(cid:17)(cid:17) and, E " ˆ T k u k W , ( K ) d t ! p < ∞ for all 1 ≤ p < ∞ , for all K ⊂ R , K compact,(6) Λ = P ◦ ( ̺ (0) , ̺ u (0)) − ,(7) for all ψ ∈ C ∞ c ( R ) and φ ∈ C ∞ c ( R ) and all t ∈ [0 , T ], it holds P − a.s. h ̺ ( t ) , ψ i = h ̺ (0) , ψ i + ˆ t h ̺ u , ∇ ψ i d s, h ̺ u ( t ) , φ i = h ̺ u (0) , φ i + ˆ t h ̺ u ⊗ u , ∇ φ i d s − ν ˆ t h∇ u , ∇ φ i d s − ( λ + ν ) ˆ t h div u , div φ i d s + 1Ma ˆ t h ̺ γ , div φ i d s + ˆ t h Φ( ̺, ̺ u )d W, φ i , (8) for any 1 ≤ p < ∞ , the energy estimate(2.5) E " sup t ∈ [0 ,T ] ˆ R ̺ | u | H ( ̺ ) ! ( t ) d x p + E (cid:20) ˆ Q T S ( ∇ u ) : ∇ u d x d s (cid:21) p ≤ c p (cid:18) E (cid:20) ˆ R (cid:18) | q | ̺ + H ( ̺ (0 , · )) (cid:19) d x (cid:21) p (cid:19) , holds where Q T := (0 , T ) × R and where H ( ̺ ) = aγ − (cid:0) ̺ γ − γ̺ γ − ( ̺ − ̺ ) − ̺ γ (cid:1) . (2.6) is the pressure potential for constants a, ̺ > holds in the renormalized sense. That is, for any φ ∈D ′ ( R ) and b ∈ C [0 , ∞ ) ∩ C (0 , ∞ ) such that | b ′ ( t ) | ≤ ct − λ , t ∈ (0 , λ < | b ′ ( t ) | ≤ ct λ , t ≥ c > − < λ < ∞ , we havethat d h b ( ̺ ) , φ i = h b ( ̺ ) u , ∇ φ i d t − h ( b ( ̺ ) − b ′ ( ̺ ) ̺ ) div u , φ i d t. (2.7) Remark . The definition above also holds for functions defined on the periodicspace T L = ([ − L, L ] | {− L,L } ) = ( R | L Z ) for any L ≥
1, rather than on the wholespace R . In that case, it even suffices to consider just smooth test functions whichare not necessarily compactly supported. See for example [4, 3, 5]. PRINCE ROMEO MENSAH
Definition 2.3.
If Λ is a Borel probability measure on L div ( R ), then we say that[(Ω , F , ( F t ) , P ) , u , W ] is a weak martingale solution of Eq. (1.5) with initial law Λprovided:(1) (Ω , F , ( F t ) , P ) is a stochastic basis with a complete right-continuous filtra-tion,(2) W is a ( F t )-cylindrical Wiener process,(3) u is ( F t )-adapted, u ∈ C w (cid:0) [0 , T ]; L ( R ) (cid:1) ∩ L (0 , T ; W , ( R )) P − a.s. and, E " sup (0 ,T ) k u k L ( R ) p + E " ˆ T k u k pW , ( R ) d t ! p < ∞ for all 1 ≤ p < ∞ , (4) Λ = P ◦ ( u (0)) − ,(5) for all φ ∈ C ∞ c, div ( R ) and all t ∈ [0 , T ], it holds P − a.s. h u ( t ) , φ i = h u (0) , φ i + ˆ t [ h u ⊗ u , ∇ φ i − ν h∇ u , ∇ φ i ] d s + ˆ t hP Φ(1 , u )d W, φ i , Existence of weak martingale solutions as defined in Definition 2.3 has beenshown to exist under suitable growth conditions on the noise term. We refer thereader to [27], albeit stated in the Stratonovich sense. A global-in-space existenceresult stated in the Itˆo form appears to be absent from the literatures although itis certainly expected. However, this is a by product of the singular limit problemthat we study in this paper. See Theorem 2.6 below. For bounded domains, see forexample, [6, 16].2.2.
Prescribed boundary conditions.
Let assume that the right-hand side ofthe energy inequality (2.5) is finite. Then we can deduce from (2.6) thatlim | x |→∞ ̺ ( x ) = ̺ (2.8)for some ̺ >
0. This is because if we apply Taylor’s expansion around the constant ̺ for the function f ( ̺ ) = ̺ γ , we can rewrite (2.6) as H ( ̺ ) = aγz γ − ̺ − ̺ ) , z ∈ [ ̺, ̺ ] or z ∈ [ ̺, ̺ ](2.9)and so the boundedness of the left-hand side of (2.5) means that the difference ̺ − ̺ ∈ L p (Ω; L ∞ (0 , T ; L min { ,γ } ( R ))) when (2.9) is substituted into (2.5).Furthermore, we also have that ̺ | u | ∈ L (Ω; L ∞ (0 , T ; L ( R ))) and as such,lim | x |→∞ ̺ ( x ) | u ( x ) | = 0 . (2.10)By combining (2.8) and (2.10) (keeping in mind that ̺ = 0), it is reasonable toimpose the boundary condition lim | x |→∞ u ( x ) = 0 . (2.11)2.3. Main results.
We now state the main results of this paper.
XISTENCE AND LOW-MACH LIMIT FOR STOCHASTIC COMPRESSIBLE FLOWS 7
Theorem 2.4.
Let γ > and let Λ be a probability law on L γ ( R ) × L γγ +1 ( R ) satisfying Λ n ( ̺, q ) ∈ L γ ( R ) × L γγ +1 ( R ) : ̺ ≥ ,M K ≤ ˆ K ̺ d x ≤ M K , q | { ̺ =0 } = 0 , (cid:12)(cid:12)(cid:12)(cid:12) ̺ − ε (cid:12)(cid:12)(cid:12)(cid:12) ≤ M K o = 1 , ˆ L γx × L γγ +1 x (cid:13)(cid:13)(cid:13)(cid:13) | q | ̺ + H ( ̺ ) (cid:13)(cid:13)(cid:13)(cid:13) pL x dΛ( ̺, q ) ≤ c p < ∞ , for all ≤ p < ∞ and any compact set K ⊂ R with constants < M K < M K whichare independent of ε ∈ (0 , . Also assume that (2.1) and (2.2) holds. Then thereexists a finite energy weak martingale solution of (1.4) in the sense of Definition2.1, with initial law Λ .Remark . The assumption (cid:12)(cid:12) ̺ − ε (cid:12)(cid:12) ≤ M K given in the law above is not restrictiveand can actually be dropped. However, it is needed in the proof of Theorem 2.6below. Theorem 2.6.
Let Λ be a given Borel probability measure on L ( R ) and for ε ∈ (0 , , we let Λ ε be a Borel probability measure on L γ ( R ) × L γγ +1 ( R ) where γ > / is such that the initial law in Theorem 2.4 holds and where the marginallaw of Λ ε corresponding to the second component converges to Λ weakly in the senseof measures on L γγ +1 ( R ) . If [(Ω ε , F ε , ( F εt ) , P ε ); ̺ ε , u ε , W ε ] is a finite energy weakmartingale solution of (1.4) with initial law Λ ε , then ( ̺ ε − → in law in L ∞ (0 , T ; L min { ,γ } ( R )) u ε → u in law in (cid:16) L (0 , T ; W , ( R )) , w (cid:17) ̺ ε u ε → u in law in L (0 , T ; L r loc ( R )) where u is a weak martingale solution of (1.5) in the sense of Definition 2.3 withthe initial law Λ and r ∈ ( , . Proof of Theorem 2.4
Let ̺ L and u L be some density and velocity fields defined d P × d t a.e. ( ω, t ) ∈ Ω × [0 , T ] on the space T L such that ̺ L and u L satisfies the so-called dissipative estimate; existence of which is shown in [4, Eq. 3.2] for the particular choice of L = 1.We observe that [4, Eq. 3.2] is translation invariant and as such, holds true forany fixed L ≥
1. Also, the inequality is preserved if we replace H δ ( ̺ ) by H ( ̺ ). Assuch if we consider ψ = χ [0 ,t ] , then we obtain the inequality:(3.1) ˆ t ˆ T L S ( ∇ u L ) : ∇ u L d x d s + ˆ T L (cid:20) ̺ L ( t ) | u L ( t ) | H ( ̺ L ( t )) (cid:21) d x ≤ ˆ T L (cid:20) | ( ̺ L u L )(0) | ̺ L (0) + H ( ̺ L (0)) (cid:21) d x + ˆ t ˆ T L u L · Φ( ̺ L , ̺ L u L )d x d W + ˆ t ˆ T L X k ∈ N | g k ( ̺ L , ̺ L u L ) | ̺ L d x d s PRINCE ROMEO MENSAH
However, due to (2.1), there is a compact set
K ⊂ R such that for any 1 ≤ p < ∞ ,we have that E sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ t ˆ T L X k ∈ N | g k ( ̺ L , ̺ L u L ) | ̺ L d x d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ E ˆ T ˆ T L X k ∈ N | g k ( ̺ L , ̺ L u L ) | ̺ L d x d s ! p ≤ c E ˆ T ˆ K ̺ − L (cid:0) ̺ L + | ̺ L u L | (cid:1) d x d s ! p ≤ c p E ˆ T ˆ K (1 + ̺ γL + ̺ L | u L | )d x ! p d s where c p is independent of both k and L and where we have used ̺ L ≤ ̺ γL .Also, by the use of the Burkholder–Davis–Gundy inequality, H¨older inequalityand Young’s inequality, we have that E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12) ˆ t ˆ T L u L · Φ( ̺ L , q L )d x d W (cid:12)(cid:12)(cid:12) p = E " sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12) ˆ t X k ∈ N ˆ T L u L · g k ( ̺ L , q L )d x d β k (cid:12)(cid:12)(cid:12) p ≤ c p E " ˆ T X k ∈ N ˆ T L u L · g k ( ̺ L , q L )d x ! d s p ≤ c p E " ˆ T X k ∈ N ˆ T L |√ ̺ L u L | d x ! ˆ T L (cid:12)(cid:12)(cid:12) g k ( ̺ L , q L ) √ ̺ L (cid:12)(cid:12)(cid:12) d x ! d s p ≤ ǫ E sup t ∈ [0 ,T ] ˆ T L |√ ̺ L u L | d x ! p + c p,ǫ E ˆ T ˆ K (1 + ̺ γL + ̺ L | u L | )d x ! p d s for an arbitrarily small ǫ > p th-moment of the supremum in (3.1) and applying Gronwall’slemma, we obtain the inequality(3.2) E " sup t ∈ [0 ,T ] ˆ T L ̺ L | u L | H ( ̺ L ) ! d x p + E " ˆ T ˆ T L S ( ∇ u L ) : ∇ u L d x d s p ≤ c p,ǫ, vol( K ) E " ˆ T L (cid:20) | q L, | ̺ L, + H ( ̺ L (0 , · )) (cid:21) d x p ! where c p,ǫ, vol( K ) is in particular, independent of L . Now by the assumptions onΛ, the right hand side of (3.2) is finite. As such, we obtain the following uniform XISTENCE AND LOW-MACH LIMIT FOR STOCHASTIC COMPRESSIBLE FLOWS 9 bounds in L (3.3) √ ̺ L u L ∈ L p (cid:0) Ω; L ∞ (0 , T ; L ( T L )) (cid:1) , ∇ u L ∈ L p (cid:0) Ω; L (0 , T ; L ( T L )) (cid:1) ,H ( ̺ L ) ∈ L p (cid:0) Ω; L ∞ (0 , T ; L ( T L )) (cid:1) , ( ̺ L − ̺ ) ∈ L p (cid:16) Ω; L ∞ (0 , T ; L min { ,γ } ( T L )) (cid:17) . Note that the estimates in (3.3) are global but unfortunately, do not include allnecessary quantities. In the following, we derive local estimates with respect toballs B r which will depend on the radius r >
0. A consequence of (3.3) is ̺ L ∈ L p (Ω; L ∞ (0 , T ; L γ ( B r )))(3.4)uniformly in L (but depending on r ). If B r ⊂ T L , this follows in an obviousway from the definition of H . Otherwise we cover B r ⊂ R by tori to which ̺ L is extended by means of periodicity. The number of necessary tori depends on r but is independent of L . To see this, we notice that since vol( B r ) ≈ c ( π ) r andvol( T L ) ≈ c ( π ) L , we will require O (cid:16) r L (cid:17) number of tori to cover B r . But since L ≥
1, we infact require O ( r ) (which is independent of L ) number of such tori tocover B r . Remark . We get (3.4) be making it the subject in (2.6) and using (3.3) , .However, we only obtain the estimate locally in space because of the constant term ̺ in the pressure potential (2.6). This will blow up with the size of the torus if wetry obtaining a global estimate.We observe that non of the bounds in (3.3) directly controls the amplitude of u L .However using the Sobolev-Poincar´e’s inequality and γ > , the following holds k ̺ k L ( B r ) | ( u L ) B r | = (cid:12)(cid:12)(cid:12)(cid:12) ˆ B r ̺ ( u L ) B r d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ c ˆ B r ̺ | ( u L ) B r − u L | d x + ˆ B r ̺ L | u L | d x ≤ c k ̺ L k L γ ( B r ) k ( u L ) B r − u L k L γ ′ ( B r ) + c k√ ̺ L k L ( B r ) k√ ̺ L u L k L ( B r ) ≤ c ( r ) k ̺ L k L γ ( B r ) k ( u L ) B r − u L k L ( B r ) + c k√ ̺ L k L γ ( B r ) k√ ̺ L u L k L ( B r ) ≤ c ( r ) k ̺ L k L γ ( B r ) k∇ u L k L ( B r ) + c k ̺ L k L γ ( B r ) + c (cid:13)(cid:13) ̺ L | u L | (cid:13)(cid:13) L ( B r ) , and, consequently, k ̺ k L ( B r ) ˆ τ | ( u L ) B r | d t ≤ c ( r ) sup t ∈ [0 ,τ ] k ̺ L k L γ ( B r ) ˆ τ k∇ u L k L ( B r ) d t + cτ sup t ∈ (0 ,τ ) (cid:16) k ̺ L k L γ ( B r ) + (cid:13)(cid:13) ̺ L | u L | (cid:13)(cid:13) L ( B r ) (cid:17) . (3.5)In view of the bounds established in (3.3), (3.4) and the assumptions on the initiallaw, we can conclude that u L ∈ L p (Ω; L (0 , T ; W , ( B r ))) . (3.6)uniformly in L . Furthermore, for r >
0, we can use the (uniform in L but not in r ) continuousembedding W , ( B r ) ֒ → L ( B r ) and H¨older’s inequality, to get for d P × d t a.e.( ω, t ) ∈ Ω × [0 , T ], k ̺ L u L k L γγ +1 ( B r ) ≤ k√ ̺ L k L γ ( B r ) k√ ̺ L u L k L ( B r ) = k ̺ L k L γ ( B r ) k√ ̺ L u L k L ( B r ) , k ̺ L u L ⊗ u L k L γ γ +3 ( B r ) ≤ k ̺ L u L k L γγ +1 ( B r ) k u L k L ( B r ) . Since the radius of the ball above is chosen arbitrarily, we may conclude that(3.7) ̺ L u L ∈ L p (Ω; L ∞ (0 , T ; L γγ +1 ( B r ))) ,̺ L u L ⊗ u L ∈ L p (Ω; L (0 , T ; L γ γ +3 ( B r ))) , uniformly in L for r > Higher integrability of density.
For reasons that will be clear in the sub-sequent sections, it is essential to improve the regularity of density. We give this inthe following lemma:
Lemma 3.2.
Let B r ⊂ R be a ball of radius r > . Then for all Θ ≤ γ − , wehave that E ˆ T ˆ B r a̺ γ +Θ L d x d t ≤ c (3.8) where the constant c , is independent of L (but depends on r ).Proof. If we set B r,L := B r ∩ T L , then it is enough to prove that E ˆ T ˆ B r,L a̺ γ +Θ L d x d t ≤ c (3.9)independently of L . The general case then follows by covering B r by sets of theform B ∩ T L for a ball B . First notice that by combining (2.3) with the continuityproperty of the Bogovski˘ı operator B ( ̺ Θ L ) = B (cid:2) ̺ Θ L − ffl ̺ Θ L d x (cid:3) , where B = B B r,L isas defined in [9, Theorem 5.2] for the set B r,L , we ensures that k B ( ̺ Θ L ) k L q − q ( B r,L ) ≤ c k ̺ Θ L k L q ( B r,L ) , r > L for 1 ≤ q < Remark . Note that infact the set B r,L is a bounded John domain and hencesatisfies the emanating chain condition with some constants σ and σ which areindependent of the size of the torus. The fact that the constant c in (3.10) isindependent of L therefore follows from the fact that the constant c in [9, Theorem5.2] only depends on σ , σ and q as well as the fact that c q is independent of L .The idea now is to test the momentum equation with B ( ̺ Θ ). To do this how-ever, we first replace the map ̺ ̺ Θ with the function b ( ̺ ) ∈ C c ( R ) andapply It´o formula to the function f ( b, q ) = ´ B r,L q · B ( b ( ̺ )) d x where B ( b ( ̺ )) = B (cid:2) b ( ̺ ) − ffl b ( ̺ ) d x (cid:3) . Since f is linear in q , no second-order derivative in this com-ponent exits. Also, the quadratic variance of b ( ̺ ) is zero since the renormalizedcontinuity equation is deterministic. XISTENCE AND LOW-MACH LIMIT FOR STOCHASTIC COMPRESSIBLE FLOWS 11
Now, notice that the Bogovski˘ı operator commutes with the time derivative (butnot with the spatial derivative) and since the continuity equation is satisfied in therenormalized sense, we have thatd [ B ( b ( ̺ L ))] = B [d ( b ( ̺ L ))] = − B [div( b ( ̺ L ) u L ) − ( b ′ ( ̺ L ) ̺ L − b ( ̺ L )) div u L ] d t. As such for b L := b ( ̺ L ), the following holds in expectation: ˆ t f b L ( b L , q L ) d b L = ¨ q L · ∂ b L ( B ( b L )) d b L d x = ¨ q L · d [ B ( b L )] d x. = − ¨ q L · B [div( b L u L )] d x d s − ¨ q L · B [( ̺ L b ′ L − b L ) div u L ] d x d s ˆ t f q L ( b L , q L ) d q L = ¨ B ( b L ) d q L d x = ¨ B ( b L ) [ − div( ̺ L u L ⊗ u L ) + ν ∆ u L + ( λ + ν ) ∇ div u L − a ∇ ̺ γL ] d x d s + ¨ B ( b L )Φ( ̺ L , ̺ L u L ) d W d x = ¨ h ( ̺ L u L ⊗ u L ) ∇ B ( b L )d x d s − ν ∇ u L : ∇ B ( b L ) − ( λ + ν ) b L div u L i d x d s + ¨ a̺ γL b L d x d s + ¨ B ( b L )Φ( ̺ L , ̺ L u L ) d W d x ˆ t f b L b L ( b L , q L ) d h b L i = ˆ t f q L q L ( b L , q L ) d h q L i = 0 since d h b L i = f q L q L = 0where we have integrated by parts and used the fact that B ( f ) solves the equationdiv v = f . It therefore follows that(3.11) E ˆ B r,L q L · B ( b L ) d x = E ˆ B r,L q L (0) · B [ b L (0)] d x − E ˆ t ˆ B r,L q L · B [div( b L u L )] d x d s − E ˆ t ˆ B r,L q L · B [ ̺ L b ′ L div u L ] d x d s + E ˆ t ˆ B r,L q L · B [ b L div u L ] d x d s + E ˆ t ˆ B r,L ( ̺ L u L ⊗ u L ) ∇ B ( b L ) d x d s − E ˆ t ˆ B r,L ν ∇ u L : ∇ B ( b L ) d x d s − E ˆ t ˆ B r,L ( λ + ν ) b L div u L d x d s + E ˆ t ˆ B r,L a̺ γL b L d x d s + E ˆ t ˆ B r,L B ( b L )Φ( ̺ L , ̺ L u L ) d W d x =: E X i =1 J i . To improve the regularity of ̺ , we aim at estimating J in terms of the rest. Todo this, we first set the left-hand side of (3.11) to E J . Then using (2.3), (3.3),(3.6), (3.7) and heavy reliance on H¨older inequalities, we can show just as in [5,Propositions 5.1, 6.1] for δ = 0 and noting that ∆ − ∇ and B enjoys the same continuity properties; E J i ≤ c, for all i ∈ { , , . . . , } \ { } for some constants c = c Θ ,γ which are in particular, independent of L . Remark . In estimating J , we use instead, the Bogovski˘ı operator in negativespaces which can be found in [18, Proposition 2.1], [2] or [10]. Also, note thecomment just after [18, Remark 2.2] about carrying over the properties of theBogovski˘ı operator from a star shaped domain onto more common domains treatedin the analysis of PDE’s.The result follows by making E J the subject and estimating it from above bythe estimates given by the rest. (cid:3) Compactness.
We now show that not only are our earlier estimates boundeduniformly on the torus T L but due to the fact that each constants obtained areuniform in L , they are indeed bounded locally on the whole space R . We thenproceed to show the usual compactness arguments. Lemma 3.5.
For any L ≥ , we have that u L ∈ L p (Ω; L (0 , T ; W , ( R ))) , √ ̺ L u L ∈ L p (cid:0) Ω; L ∞ (0 , T ; L ( R )) (cid:1) ,̺ L ∈ L p (cid:0) Ω; L ∞ (0 , T ; L γ loc ( R )) (cid:1) , ̺ L u L ∈ L p (Ω; L ∞ (0 , T ; L γγ +1 loc ( R ))) ,̺ L u L ⊗ u L ∈ L p (Ω; L (0 , T ; L γ γ +3 loc ( R ))) , ̺ L ∈ L p (Ω; L γ +Θ (0 , T ; L γ +Θloc ( R ))) . uniformly in L .Proof. We will only show the first uniform estimate as the rest can be done in asimilar manner in conjunction with (3.3), (3.7) and Lemma 3.2.Let
L, r ∈ N and let B r ⊂ R be the ball of radius r centered at the origin. If B r ⊂ T L , then we notice that we can directly deduce from (3.3) that u L ∈ L p (cid:0) Ω; L (0 , T ; W , ( B r )) (cid:1) (3.12)uniformly in L . Otherwise, we can use the same argument as in the justification of(3.4) above to get from (3.3) , k u L k L p (Ω; L (0 ,T ; W , ( B r ))) ≤ c ( p, r ) , ∀ r ∈ N (3.13)uniformly in L . That is, for any r ∈ N and any B r ⊂ R , (3.13) holds. By combining(3.12) and (3.13), we can deduce that u L ∈ L p (cid:16) Ω; L (0 , T ; W , ( R )) (cid:17) (3.14)uniformly in L . (cid:3) For the compactness result, let define the following path space χ = χ u × χ ̺ × χ ̺ u × χ W where χ u = (cid:16) L (0 , T ; W , ( R )) , ω (cid:17) ,χ ̺ = C ω (cid:0) [0 , T ]; L γ loc ( R ) (cid:1) ∩ ( L γ + θ (0 , T ; L γ + θ loc ( R )) , ω ) ,χ ̺ u = C ω (cid:18) [0 , T ]; L γγ +1 loc ( R ) (cid:19) ,χ W = C ([0 , T ]; U ) , XISTENCE AND LOW-MACH LIMIT FOR STOCHASTIC COMPRESSIBLE FLOWS 13 and let(1) µ u L be the law of u L on χ u ,(2) µ ̺ L be the law of ̺ L on the space χ ̺ ,(3) µ ̺ L u L be the law of ̺ L u L on the space χ ̺ u ,(4) µ W be the law of W on the space χ W ,(5) µ L be the joint law of u L , ̺ L , ̺ L u L and W on the space χ . Proposition 3.6.
For an arbitrary constant c , which is uniform in r ∈ N , L ≥ and R > , let us define the set A R := { u L ∈ L (0 , T ; W , ( R )) : k u L k L (0 ,T ; W , ( B r )) ≤ c ( r ) R, ∀ r ∈ N } . Then A R is compact in χ u Proof.
To see this, fix
R > { u n } n ∈ N ⊂ A R so that k u n k L (0 ,T ; W , ( B r )) ≤ c ( r ) R, ∀ n ∈ N and ∀ r ∈ N Then by the use of a diagonal argument, we can construct the sequence { u nn } n ∈ N ⊂{ u n } n ∈ N that is a common subsequence of all the sequences { u mn } n ∈ N for all m ∈{ } ∪ N where u n := u n . And by uniqueness of limits, we can therefore concludethat u nn ⇀ u in L (0 , T ; W , ( B r )) for every r ∈ N . This finishes the proof. (cid:3)
Proposition 3.7.
The family of measures { µ L ; L ≥ } is tight on χ .Proof. We first show that { µ u L ; L ≥ } is tight on χ u . To do this, we let R > A R ⊂ χ u . Now since( A R ) C := { u L ∈ L (0 , T ; W , ( R )) : k u L k L (0 ,T ; W , ( B r )) > c ( r ) R, for some r ∈ N } , for any measure µ u L ∈ { µ u L ; L ≥ } , there exists a r ∈ N such that: µ u L (cid:0) ( A R ) C (cid:1) = P (cid:0) k u L k L (0 ,T ; W , ( B r )) > c ( r ) R (cid:1) < c ( r ) R E (cid:0) k u L k L (0 ,T ; W , ( B r )) (cid:1) ≤ R → . as R → ∞ , where we have used (3.13) in the last inequality. This implies that { µ u L ; L ≥ } is tight on χ u .By using a similar argument adapted to suit the compactness arguments in [5,Sect. 6] we can show that { µ ̺ L ; L ≥ } and { µ ̺ L u L ; L ≥ } are also tight on χ ̺ and χ ̺u respectively. Furthermore, µ W is tight since its a Radon measure on thePolish space χ W . This finishes the proof. (cid:3) From Proposition 3.7, we cannot immediately use Skorokhod representation the-orem to deduce that { µ L ; L ≥ } is relatively compact (i.e. Prokhorov theo-rem), since the path space χ is not metrizable. However, we may use instead theJakubowski–Skorokhod representation theorem [20] that gives a similar result butfor more general spaces including quasi-Polish spaces, the space in which theselocally in space Sobolev functions live. Applying this yields the following result: Proposition 3.8.
There exists a subsequence µ n := µ L n for n ∈ N , a probabilityspace ( ˜Ω , ˜ F , ˜ P ) with χ -valued random variables (˜ u n , ˜ ̺ n , ˜ q n , ˜ W n ) , and their corre-sponding ‘limit’ variables (˜ u , ˜ ̺, ˜ q , ˜ W ) such that • the law of (˜ u n , ˜ ̺ n , ˜ q n , ˜ W n ) is given by µ n = Law( u L n , ̺ L n , ̺ L n u L n , W ) , n ∈ N , • the law of (˜ u , ˜ ̺, ˜ q , ˜ W ) , denoted by µ = Law( u , ̺, ̺ u , W ) is a Randon mea-sure, • (˜ u n , ˜ ̺ n , ˜ q n , ˜ W n ) converges ˜ P − a.s to (˜ u , ˜ ̺, ˜ q , ˜ W ) in the topology of χ . To extend this new probability space ( ˜Ω , ˜ F , ˜ P ) into a stochastic basis, we endowit with a filtration. To do this, let us first define a restriction operator r t define by r t : X → X | [0 ,t ] , f f | [0 ,t ] , (3.15)for t ∈ [0 , T ] and X ∈ { χ ̺ , χ u , χ W } . We observe that r t is a continuous map.We can therefore construct ˜ P − augmented canonical filtrations for (˜ ̺ n , ˜ u n , ˜ W n ) and(˜ ̺, ˜ u , ˜ W ) respectively, by setting˜ F nt = σ (cid:16) σ ( r t ˜ ̺ n , r t ˜ u n , r t ˜ W n ) ∪ { N ∈ ˜ F ; ˜ P ( N ) = 0 } (cid:17) , t ∈ [0 , T ] , ˜ F t = σ (cid:16) σ ( r t ˜ ̺, r t ˜ u , r t ˜ W ) ∪ { N ∈ ˜ F ; ˜ P ( N ) = 0 } (cid:17) , t ∈ [0 , T ] . The following result thus follows:
Lemma 3.9.
For any n > , [( ˜Ω , ˜ F , ( ˜ F nt ) t ≥ , ˜ P ) , ˜ ̺ n , ˜ u n , ˜ W n ] is a weak mar-tingale solution of (1.4) with initial law Λ . Furthermore, there exists b > and a W − b, ( R ) − valued continuous square integrable ( ˜ F t ) − martingale ˜ M and ˜ p ∈ L γ +Θ γ ( ˜Ω × Q ) , where Q = (0 , T ) × R , such that [( ˜Ω , ˜ F , ( ˜ F t ) t ≥ , ˜ P ) , ˜ ̺, ˜ u , ˜ p, ˜ M ] is a weak martingale solution of (3.16) d˜ ̺ + div(˜ ̺ ˜ u )d t = 0d(˜ ̺ ˜ u ) + [div(˜ ̺ ˜ u ⊗ ˜ u ) − ν ∆˜ u − ( λ + ν ) ∇ div˜ u + ∇ ˜ p ]d t = d ˜ M , in ˜Ω × Q with initial law Λ . Furthermore, (3.16) is satisfied in the renormalized sense.Proof. This follows in exactly the same manner as in [5, Proposition 5.6]. (cid:3)
Corollary 3.10.
The following ˜ P − a.s. convergence holds: (3.17) ˜ u n ⇀ ˜ u in L (0 , T ; W , ( R )) , ˜ ̺ n → ˜ ̺ in C ω ([0 , T ]; L γ loc ( R )) , ˜ ̺ n ⇀ ˜ ̺ in L γ +Θ (0 , T ; L γ +Θloc ( R )) , ˜ ̺ n ˜ u n → ˜ ̺ ˜ u in C ω ([0 , T ]; L γγ +1 loc ( R )) ∩ L (0 , T ; W − , ( R )) , ˜ ̺ n ˜ u n ⊗ ˜ u n ⇀ ˜ ̺ ˜ u ⊗ ˜ u in L (0 , T ; L ( R )) , ˜ W n → ˜ W in C ([0 , T ]; U ) , Proof.
The first three and the last is exactly contained in Proposition 3.8. For(3.17) , , see [5, Lemma 5.5, Proposition 6.3]. (cid:3) Proposition 3.11.
The limit process ˜ u in (3.17) is globally defined in space, i.e., ˜ u ∈ L (0 , T ; W , ( R )) .Proof. Let B r ⊂ R be an arbitrary ball of radius r >
0. Then from (3.17) , wehave that for ˜ P − a.s.,˜ u n ⇀ ˜ u in L (0 , T ; W , ( B r )) , for r > . XISTENCE AND LOW-MACH LIMIT FOR STOCHASTIC COMPRESSIBLE FLOWS 15
However, lower semicontinuity of norms means that for any such r > k χ B r ∇ ˜ u k L (0 ,T ; L ( R )) = k∇ ˜ u k L (0 ,T ; L ( B r )) ≤ lim inf n →∞ k∇ ˜ u n k L (0 ,T ; L ( B r )) ˜ P − a.s. Passing to the limit r → ∞ on either side of this inequality finishes theproof since by the Gagliardo–Nirenberg–Sobolev inequality, (2.3) then follows for q = 2. (cid:3) The effective viscous flux.
This section combines ideas from [15, 5] and[28, Chapter 7].Let ∆ − be the inverse Laplacian on R and let the global-in-space operators A i = ∆ − [ ∂ x i u ] , i = 1 , , ∂ i := ∂ x i and for some cutoff functions φ ( x ) , φ ( x ) ∈ C ∞ c ( R ), we may do a similar computation as in (3.11). That is, we apply Itˆo’sformula to the function f ( g, ˜ q ) = ´ R ˜ q · φ ( x ) A i [ φ ( x ) g ] d x where ˜ q = ˜ ̺ ˜ u and where g = T k (˜ ̺ ) and T k : [0 , ∞ ) → [0 , ∞ ) is given by T k ( t ) = (cid:26) t if 0 ≤ t < k,k if k ≤ t < ∞ . Or equivalently, by testing the momentum equation satisfied by the sequence ofweak martingale solution in Lemma 3.9 by ϕ i ( x ) = φ ( x ) A i [ φ ( x ) T k (˜ ̺ )]. We obtainthe following (by assuming that L is large enough such that sptφ ⊂ T L )(3.18) ˜ E ˆ R φ ˜ ̺ n ˜ u in A i (cid:2) φT k (˜ ̺ n ) (cid:3) d x = ˜ E ˆ R φ ˜ ̺ n ˜ u in (0) A i (cid:2) φT k (˜ ̺ n (0)) (cid:3) d x − ˜ E ˆ t ˆ R φ ˜ ̺ n ˜ u in A i [ φ ∂ j ( T k (˜ ̺ n )˜ u jn )]d x d s − ˜ E ˆ t ˆ R φ ˜ ̺ n ˜ u in A i (cid:2) φ ( T ′ k (˜ ̺ n ) ˜ ̺ n − T k (˜ ̺ n )) div ˜ u n (cid:3) d x d s + ˜ E ˆ t ˆ R ˜ ̺ n ˜ u in ˜ u jn ∂ j ( φ A i [ φT k (˜ ̺ n )]) d x d s + ν ˜ E ˆ t ˆ R φ A i [ φT k (˜ ̺ n )] ∆˜ u in d x d s + ˜ E ˆ t ˆ R [ a ˜ ̺ γn − ( λ + ν )div˜ u n ] ∂ i ( φ A i [ φ T k (˜ ̺ n )]) d x d s =: ˜ E X k =1 J k , i = 1 , , . where T k , as defined above, replaces b in the definition of the renormalized equationgiven by (2.7). Remark . Notice that since the approximate quantities in (3.17) are only definedlocally in space, to apply this globally defined operators A , it is essentially to pre-multiply our functions by some φ ∈ C ∞ c ( R ).Also, we observe that since our noise term is a martingale, it vanishes when wetake its expectation, as martingales are constant on average. Now notice that by integration by parts and the use of the properties of theoperators A i and R ij = ∂ i A j , we may rewrite J , J , J and J so that (3.18)becomes:(3.19) ˜ E ˆ t ˆ R [ a ˜ ̺ γn − ( λ + 2 ν )div˜ u n ] φ φ T k (˜ ̺ n ) d x d s = ˜ E ˆ R φ ˜ ̺ n ˜ u in A i (cid:2) φ T k (˜ ̺ n ) (cid:3) d x − ˜ E ˆ R φ ˜ ̺ n ˜ u in (0) A i (cid:2) φ T k (˜ ̺ n (0)) (cid:3) d x + ν ˜ E ˆ t ˆ R φ ˜ u in T k (˜ ̺ n ) ∂ i φ d x d s − ˜ E ˆ t ˆ R [ a ˜ ̺ γn − ( λ + ν )div˜ u n ] A i [ φ T k (˜ ̺ n )] ∂ i φ d x d s + ˜ E ˆ t ˆ R φ ˜ ̺ n ˜ u in A i (cid:2) φ ( T ′ k (˜ ̺ n ) ˜ ̺ n − T k (˜ ̺ n )) div ˜ u n (cid:3) d x d s + ˜ E ˆ t ˆ R ˜ u in (cid:0) R ij [ φ ˜ ̺ n ˜ u jn ] φ T k (˜ ̺ n ) − φ ˜ ̺ n ˜ u jn R ij [ φ T k (˜ ̺ n )] (cid:1) d x d s + ˜ E ˆ t ˆ R ˜ u jn (cid:0) A i [ φ ˜ ̺ n ˜ u in ] T k (˜ ̺ n ) ∂ j φ − ˜ ̺ n ˜ u in A i [ φT k (˜ ̺ n )] ∂ j φ (cid:1) d x d s =: ˜ E X k =1 I k , i = 1 , , . Remark . If we set the left-hand side of (3.19) to ˜ E I , then we point the readerto the difference in the viscosity constant in I and I .Similarly for the limit processes, we obtain(3.20) ˜ E ˆ t ˆ R [ a ˜ p − ( λ + 2 ν )div˜ u ] φ φ T k (˜ ̺ ) d x d s = ˜ E ˆ R φ ˜ ̺ ˜ u i A i h φ T k (˜ ̺ ) i d x − ˜ E ˆ R φ ˜ ̺ ˜ u i (0) A i h φ T k (˜ ̺ (0)) i d x + ν ˜ E ˆ t ˆ R φ ˜ u i T k (˜ ̺ ) ∂ i φ d x d s − ˜ E ˆ t ˆ R [ a ˜ p − ( λ + ν )div˜ u ] A i [ φ T k (˜ ̺ )] ∂ i φ d x d s + ˜ E ˆ t ˆ R φ ˜ ̺ ˜ u i A i h φ ( T ′ k (˜ ̺ ) ˜ ̺ − T k (˜ ̺ )) div ˜ u i d x d s + ˜ E ˆ t ˆ R ˜ u i (cid:16) R ij [ φ ˜ ̺ ˜ u j ] φ T k (˜ ̺ ) − φ ˜ ̺ ˜ u j R ij [ φ T k (˜ ̺ )] (cid:17) d x d s + ˜ E ˆ t ˆ R ˜ u j (cid:16) A i [ φ ˜ ̺ ˜ u i ] T k (˜ ̺ ) ∂ j φ − ˜ ̺ ˜ u i A i [ φT k (˜ ̺ )] ∂ j φ (cid:17) d x d s =: ˜ E X k =1 K k , i = 1 , , . where a ‘bar’ above a function represents the limit of the corresponding approximatesequence of functions. Lemma 3.14.
Let φ ( x ) , φ ( x ) ∈ C ∞ c ( R ) . Then the strong convergence R [ φ ˜ ̺ n ˜ u jn ] φT k (˜ ̺ n ) − φ ˜ ̺ n ˜ u jn R [ φT k (˜ ̺ n )] → R [ φ ˜ ̺ ˜ u j ] φT k (˜ ̺ ) − φ ˜ ̺ ˜ u j R [ φT k (˜ ̺ )] XISTENCE AND LOW-MACH LIMIT FOR STOCHASTIC COMPRESSIBLE FLOWS 17 holds in L (cid:16) ˜Ω × (0 , T ); W − , ( R ) (cid:17) where R := R ij .Proof. See [5, Sect. 6.1] or the deterministic counterpart in [28, Eq. 7.5.23]. (cid:3)
Now by using the weak-strong pair: (3.17) and Lemma 3.14, we can pass to thelimit in the crucial term I to get ˜ E I → ˜ E K .All other terms can be treated in a similar manner as in [5, Sect. 6.1] keepingin mind that the terms involving derivatives and cutoff functions are of lower orderand hence easier to handle. In particular, we obtain the convergence ˜ E I → ˜ E K by observing that R = ∂ j A i .We have therefore shown that(3.21) lim n → ˜ E ˆ Q [ a ˜ ̺ γn − ( λ + 2 ν )div ˜ u n ] φφT k (˜ ̺ n ) d x d t = ˜ E ˆ Q [ a ˜ p − ( λ + 2 ν )div ˜ u ] φφT k (˜ ̺ ) d x d t Identification of the pressure limit.
Showing that indeed ˜ p = ˜ ̺ γ or equiv-alently that ˜ ̺ n → ˜ ̺ strongly in L p ( ˜Ω × Q ) for all p ∈ [1 , γ + Θ) follows Feireisl’sapproach via the use of the so-called oscillation defect measure . This is a purelydeterministic argument even in our stochastic settings since it relies on the renor-malized continuity equation. To avoid repetition, we refer the reader to [28, Sect.7.3.7.3] or [11]. To confirm that it indeed applies in the stochastic setting, thereader may also refer to [5, Sect. 6.2 and 6.3].We now conclude with the following lemma which completes the proof of Theo-rem 2.4. Lemma 3.15. [( ˜Ω , ˜ F , ( ˜ F t ) t ≥ , ˜ P ) , ˜ ̺, ˜ u , ˜ W ] is a finite energy weak martingale solu-tion of (1.4) with initial law Λ . Furthermore, (1.4) is satisfied in the renormalizedsense. Proof of Theorem 2.6
For every ε >
0, let assume there exits a finite energy weak martingale solutionof Eq. (1.4) given by [(Ω ε , F ε , ( F εt ) , P ε ) , ̺ ε , u ε , W ε ] . Then by setting ̺ = 1 and a = ε in (2.6), and applying Taylor expansion to thefunction f ( ̺ ) = ̺ γ around ̺ = 1, we get E (cid:20) ˆ R (cid:18) | q ε (0) | ̺ ε (0) + H ( ̺ ε (0)) (cid:19) d x (cid:21) p = ˆ L γ × L γγ +1 (cid:13)(cid:13)(cid:13)(cid:13) | q ε | ̺ ε + γz γ − ε ( ̺ ε − (cid:13)(cid:13)(cid:13)(cid:13) pL ( R ) dΛ( ̺ ε , q ε ) ≤ c p,T for z ∈ [ ̺,
1] or z ∈ [1 , ̺ ] and where we have used the initial law in Theorem 2.4.Similar to Section 3, we can now collect the following uniform (in ε ) bounds(4.1) ϕ ε ∈ L p (Ω; L ∞ (0 , T ; L min { ,γ } ( R ))) , ∇ u ε ∈ L p (cid:0) Ω; L (0 , T ; L ( R )) (cid:1) , √ ̺ ε u ε ∈ L p (cid:0) Ω; L ∞ (0 , T ; L ( R )) (cid:1) ,̺ ε u ε ∈ L p (Ω; L ∞ (0 , T ; L γγ +1 loc ( R ))) ,̺ ε u ε ⊗ u ε ∈ L p (Ω; L (0 , T ; L γ γ +3 loc ( R ))) , where ϕ ε := ̺ ε − ε and where ̺ ε → L p (cid:0) Ω; L ∞ (0 , T ; L γ loc ( R )) (cid:1) . (4.2)cf. (3.3) (with ̺ = 1), (3.7) and [3, eqn. 3.6].4.1. Acoustic wave equation.
Let ∆ − represent the inverse of the Laplace oper-ator on R and let Q = ∇ ∆ − div and P be, respectively, the gradient and solenoidalparts according to Helmoltz decomposition. Then referring again to [3], we observethat by setting ϕ ε = ̺ ε − ε and Id = Q + P , we derive from equation (1.4):(4.3) ε d ϕ ε + div Q ( ̺ ε u ε )d t = 0 ,ε Q Φ( ̺ ε , ̺ ε u ε )d W − γ ∇ ϕ ε d t = ε d Q ( ̺ ε u ε ) − ε F ε d t, where F ε = div Q ( ̺ ε u ε ⊗ u ε ) − ν ∆ Q u ε − ( λ + ν ) ∇ div u ε + 1 ε ∇ [ ̺ γε − − γ ( ̺ ε − . Now let us observe that from (4.1) and the continuity of Q , we have thatdiv Q ( ̺ ε u ε ⊗ u ε ) ∈ L p (Ω; L (0 , T ; W − , γ γ +3 loc ( R )))(4.4)independently of ε . And that ν ∆ Q u ε + ( λ + ν ) ∇ div u ε ∈ L p (Ω; L (0 , T ; W − , ( R )))(4.5)uniformly in ε by virtue of (4.1) . Lastly, the choice of a = ε and ̺ ε = 1 in thepressure potential (2.6) of the energy estimate (2.5) and Taylor’s theorem meansthat for s := min { , γ } > ε ∇ [ ̺ γε − − γ ( ̺ ε − ∈ L p (Ω; L ∞ (0 , T ; W − ,s loc ( R ))) . (4.6)uniformly in ε . cf. (2.9) for ̺ = 1 and (4.1) . By combining (4.4), (4.5) and (4.6)with the embeddings W − , γ γ +3 ( B r ) ֒ → W − l, ( B r ) and W − ,s ( B r ) ֒ → W − l, ( B r ),where B r is a ball of radius r >
0, it holds that for l > / F ε ∈ L p (cid:16) Ω; L (0 , T ; W − l, ( R )) (cid:17) uniformly in ε . XISTENCE AND LOW-MACH LIMIT FOR STOCHASTIC COMPRESSIBLE FLOWS 19
Compactness.
To explore compactness for the acoustic equation, let firstdefine the path space χ = χ ̺ × χ u × χ ̺ u × χ W where χ ̺ = C ω (cid:0) [0 , T ]; L γ loc ( R ) (cid:1) , χ u = (cid:16) L (0 , T ; W , ( R )) , ω (cid:17) ,χ ̺ u = C ω (cid:18) [0 , T ]; L γγ +1 loc ( R ) (cid:19) , χ W = C ([0 , T ]; U ) , and let(1) µ ̺ ε be the law of ̺ ε on the space χ ̺ ,(2) µ u ε be the law of u ε on χ u ,(3) µ P ( ̺ ε u ε ) be the law of P ( ̺ ε u ε ) on the space χ ̺ u ,(4) µ W be the law of W on the space χ W ,(5) µ ε be the joint law of ̺ ε , u ε , P ( ̺ ε u ε ), and W on the space χ .Then the following lemma, the proof of which is similar to [3, Corollary 3.7], holdstrue. Lemma 4.1.
The sets { µ ε ; ε ∈ (0 , } is tight on χ . Now similar to Proposition 3.8, we apply the Jakubowski–Skorokhod represen-tation theorem [20] to get the following proposition.
Proposition 4.2.
There exists a subsequence µ ε (not relabelled), a probability space ( ˜Ω , ˜ F , ˜ P ) with χ -valued Borel measurable random variables (˜ ̺ ε , ˜ u ε , ˜ q ε , ˜ W ε ) , n ∈ N ,and (˜ ̺, ˜ u , ˜ q , ˜ W ) such that • the law of (˜ ̺ ε , ˜ u ε , ˜ q ε , ˜ W ε ) is given by µ ε , ε ∈ (0 , , • the law of (˜ ̺, ˜ u , ˜ q , ˜ W ) , denoted by µ is a Randon measure, • (˜ ̺ ε , ˜ u ε , ˜ q ε , ˜ W ε ) converges ˜ P − a.s to (˜ ̺, ˜ u , ˜ q , ˜ W ) in the topology of χ . To extend this new probability space ( ˜Ω , ˜ F , ˜ P ) into a stochastic basis, we en-dow it with the ˜ P − augmented canonical filtrations for (˜ ̺ ε , ˜ u ε , ˜ W ε ) and (˜ ̺, ˜ u , ˜ W ),respectively, by setting˜ F εt = σ (cid:16) σ ( r t ˜ ̺ ε , r t ˜ u ε , r t ˜ W ε ) ∪ { N ∈ ˜ F ; ˜ P ( N ) = 0 } (cid:17) , t ∈ [0 , T ] , ˜ F t = σ (cid:16) σ ( r t ˜ u , r t ˜ W ) ∪ { N ∈ ˜ F ; ˜ P ( N ) = 0 } (cid:17) , t ∈ [0 , T ] . where r t is the continuous function defined in (3.15) above adapted to the spacesdefined in this section.4.3. Identification of the limit.
We now verify that on this new probabilityspace, our new processes[( ˜Ω , ˜ F , ( ˜ F εt ) , ˜ P ) , ˜ ̺ ε , ˜ u ε , ˜ W ε ] and [( ˜Ω , ˜ F , ( ˜ F t ) , ˜ P ) , ˜ u , ˜ W ]are indeed finite energy weak martingale solutions and a weak martingale solutionrespectively for Eqs. (1.4) and (1.5). Proposition 4.3. [( ˜Ω , ˜ F , ( ˜ F εt ) t ≥ , ˜ P ) , ˜ ̺ ε , ˜ u ε , ˜ W ε ] is a finite energy weak martin-gale solution of Eq. (1.4) with initial law Λ ε for ε ∈ (0 , . The proof of this proposition is similar to [3, Proposition 3.10].
Consequently, the uniform bounds shown in (4.1), (4.2) and (4.7) earlier hold forthese corresponding random processes on this new space. In particular, we havethat(4.8) ˜ ϕ ε ∈ L p (cid:16) Ω; L ∞ (0 , T ; L min { ,γ } ( R )) (cid:17) , ˜ F ε ∈ L p (cid:16) Ω; L (0 , T ; W − l, ( R )) (cid:17) , ˜ u ε ∈ L p (cid:16) Ω; L (0 , T ; W , ( R )) (cid:17) , ˜ ̺ ε ˜ u ε ∈ L p (Ω; L ∞ (0 , T ; L γγ +1 loc ( R )))holds uniformly in ε for p ∈ [1 , ∞ ) and where l > /
2, ˜ ϕ ε = ˜ ̺ ε − ε and˜ F ε = div Q (˜ ̺ ε ˜ u ε ⊗ ˜ u ε ) − ν ∆ Q ˜ u ε − ( λ + ν ) ∇ div˜ u ε + 1 ε ∇ [˜ ̺ γε − − γ (˜ ̺ ε − Proposition 4.4. [( ˜Ω , ˜ F , ( ˜ F t ) t ≥ , ˜ P ) , ˜ u , ˜ W ] is a weak martingale solution of Eq. (1.5) with initial law Λ .Proof. The proof of this proposition will follow from the following lemmata andpropositions.
Lemma 4.5.
For all t ∈ [0 , T ] and φ ∈ C ∞ c ( R ) , we let M ( ̺, u , q ) t = h q ( t ) , φ i − h q (0) , φ i − ˆ t h q ⊗ u , ∇ φ i d s + ν ˆ t h∇ u , ∇ φ i d s + ( λ + ν ) ˆ t h div u , div φ i d s − ε ˆ t h ̺ γ , div φ i d s. Then M (˜ ̺ ε , ˜ u ε , ˜ ̺ ε ˜ u ε ) t → M (1 , ˜ u , ˜ u ) t ˜ P − a.s. as ε → .Proof. The proof of this lemma follows further from combining Proposition 4.2 withthe Lemmata 4.6, 4.9 and Proposition 4.8 below.
Lemma 4.6.
For every q < , the following ˜ P − a.s. convergence holds: (˜ ̺ ε − → in L ∞ (0 , T ; L min { ,γ } ( R )) , (4.9) P (˜ ̺ ε ˜ u ε ) → ˜ u in L (0 , T ; W − , ( R )) , (4.10) P ˜ u ε → ˜ u in L (0 , T ; L q loc ( R )) . (4.11) Proof.
See [3]. (cid:3)
Remark . Henceforth, we write ‘ . ’ for ‘ ≤ c ’ and ‘ h ’ for ‘= c ’ where c , whichmay varies from line to line is some universal constant that is independent of ε butmay depend on other variables. Proposition 4.8.
The strong convergence below holds. Q (˜ ̺ ε ˜ u ε ) → in L (0 , T ; L γγ +1 loc ( R )) ˜ P − a.s. XISTENCE AND LOW-MACH LIMIT FOR STOCHASTIC COMPRESSIBLE FLOWS 21
Proof.
Let define the function ˜Ψ ε = ∆ − div(˜ ̺ ε ˜ u ε ) such that ∇ ˜Ψ ε = Q (˜ ̺ ε ˜ u ε ).Then equation (4.3) becomes(4.12) ε d( ˜ ϕ ε ) + ∆ ˜Ψ ε d t = 0 ,ε d ∇ ˜Ψ ε + γ ∇ ˜ ϕ ε d t = ε ˜ F ε d t + ε Q Φ(˜ ̺ ε , ˜ ̺ ε ˜ u ε )d ˜ W ε . We however observe that Eq. (4.12) is equivalent to(4.13) ε d " ˜ ϕ ε ∇ ˜Ψ ε = A " ˜ ϕ ε ∇ ˜Ψ ε d t + ε " F ε d t + ε " Q Φ d ˜ W ε where the usual wave operator A = " − div − γ ∇ (4.14)is an infinitesimal generator of a strongly continuous semigroup S ( · ) = exp( A· ).See for example [8]. Also since Φ := Φ(˜ ̺, ˜ ̺ ˜ u ) is the Hilbert–Schmidt operator andequation (4.12) is satisfied weakly in the probabilistic sense, it follows that thisweak solution is also a mild solution. See for example [7, Theorem 6.5]. As suchafter rescaling, we obtain the mild equation(4.15) " ˜ ϕ ε ∇ ˜Ψ ε ( t ) = S (cid:18) tε (cid:19) " ˜ ϕ ε (0) ∇ ˜Ψ ε (0) + ˆ t S (cid:18) t − sε (cid:19) " F ε d s + ˆ t S (cid:18) t − sε (cid:19) " Q ˜Φ ε d ˜ W s,ε where the semigroup S ( t ) is such that(4.16) S ( t ) " ˜ ϕ ∇ ˜Ψ = " ˜ ϕ ∇ ˜Ψ ( t )is the solution to the homogeneous problem(4.17) d( ˜ ϕ ) + ∆ ˜Ψ d t = 0 , d ∇ ˜Ψ + γ ∇ ˜ ϕ d t = 0 , ˜ ϕ (0) = ˜ ϕ ; ∇ ˜Ψ(0) = ∇ ˜Ψ . Using Fourier transforms (in space), we obtain solution of Eq. (4.17) which is givenby the pair(4.18) ∇ ˜Ψ( t, x ) = e i √− γ ∆ t (cid:18) ∇ ˜Ψ ( x ) − i √ γ √− ∆ ˜ ϕ ( x ) (cid:19) + e − i √− γ ∆ t (cid:18) ∇ ˜Ψ ( x ) + i √ γ √− ∆ ˜ ϕ ( x ) (cid:19) , ˜ ϕ ( t, x ) = e i √− γ ∆ t (cid:18) i √− ∆ √ γ ∇ ˜Ψ ( x ) + ˜ ϕ ( x ) (cid:19) − e − i √− γ ∆ t (cid:18) i √− ∆ √ γ ∇ ˜Ψ ( x ) − ˜ ϕ ( x ) (cid:19) . The lemma below is crucial to the proof of Proposition 4.8 and is an adaptationof [31, Lemma 2.2] to our setting. cf. [12, Lemma 3.1].
Lemma 4.9.
Let φ ( x ) ∈ C ∞ c ( R ) , we have ˆ R k e i √− γ ∆ t [ v φ ] k L ( R ) d t ≤ c ( φ ) k v k L ( R ) for any v ∈ L ( R ) .Proof. For simplicity, we assume that γ = 1. General γ > δ below.Using Plancherels theorem in t and x , we have that ˆ R k e i √− ∆ t [ v φ ] k L ( R ) d t = c ( π ) ˆ R ˆ R (cid:12)(cid:12)(cid:12) ˆ R b φ ( ξ − η ) δ ( τ − | η | ) b v ( η ) d η (cid:12)(cid:12)(cid:12) d ξ d τ = c ( π ) ˆ R ˆ R (cid:12)(cid:12)(cid:12) ˆ { τ = | η |} b φ ( ξ − η ) b v ( η ) d S η (cid:12)(cid:12)(cid:12) d ξ d τ ≤ c ( π ) ˆ R ˆ R (cid:16) ˆ { τ = | η |} | b φ ( ξ − η ) | d S η (cid:17)(cid:16) ˆ { τ = | η |} | b φ ( ξ − η ) || b v ( η ) | d S η (cid:17) d ξ d τ ≤ c ( π, φ ) ˆ R ˆ R ˆ { τ = | η |} | b φ ( ξ − η ) || b v ( η ) | d S η d τ d ξ ≤ c ( π, φ ) ˆ R ˆ R | b φ ( ξ − η ) || b v ( η ) | d η d ξ ≤ c ( π, φ ) k v k L ( R ) where we have used the Cauchy-Schwartz inequality. (cid:3) Moving on, we now consider a smooth cut-off function (with expanding support) η r ∈ C ∞ ( B r ) with η r ≡ B r for r > v is one of the functions in(4.12), we set v κ = ( η r v ) ∗ ϕ κ where ϕ κ is the standard mollifier. This we do to ensure that the regularizedfunctions are globally integrable. First off, we note that since (4.8) holds uniformlyin ε , for an arbitrary small δ >
0, we can find a κ ( δ ) such that˜ E sup t ∈ [0 ,T ] k (˜ ̺ ε ˜ u ε ) κ − ˜ ̺ ε ˜ u ε k pL γγ +1 ( B ) ≤ δ (4.19)for any 1 ≤ p < ∞ and an arbitrary ball B ⊂⊂ B r for r >
0. Then using (4.16) ,(4.18) and Lemma 4.9, we obtain(4.20) ˜ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) S ( t ) " ˜ ϕ κ ∇ ˜Ψ κ L ( R × B ) ≤ c h,γ ˜ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)" ˜ ϕ κ ∇ ˜Ψ κ L ( R ) , for any ball B ⊂ R and where in particular, the constant is independent of κ . Soby rescaling in time, i.e, setting s = tε so that d s = d tε , we get(4.21) ˜ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) S (cid:18) tε (cid:19) " ˜ ϕ κε (0) ∇ ˜Ψ κε (0) L ((0 ,T ) × B ) ≤ ˜ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) S (cid:18) tε (cid:19) " ˜ ϕ κε (0) ∇ ˜Ψ κε (0) L ( R × B ) . ε ˜ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)" ˜ ϕ κε (0) ∇ ˜Ψ κε (0) L ( R )XISTENCE AND LOW-MACH LIMIT FOR STOCHASTIC COMPRESSIBLE FLOWS 23 with a constant that is independent of ε . Now by the continuity of Q , (4.19), andthe initial law defined in the statement of Theorem 2.4, we conclude that(4.22) ˜ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) S (cid:18) tε (cid:19) " ˜ ϕ κε (0) ∇ ˜Ψ κε (0) L ((0 ,T ) × B ) . ε ˜ E k ˜ ϕ κε (0) k L min { ,γ } ( R ) + k ˜ q κε (0) k L γγ +1 ( R ) ! ≤ ε c κ,M . Similarly we have that for any ball B ⊂ R ,(4.23) ˜ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) t ˆ S (cid:18) t − sε (cid:19) ˜ F κε d s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ((0 ,T ) × B ) ≤ ˜ E (cid:13)(cid:13)(cid:13)(cid:13) S (cid:18) t − sε (cid:19) ˜ F κε (cid:13)(cid:13)(cid:13)(cid:13) L ((0 ,t ) × (0 ,T ) × B ) ≤ ˜ E (cid:13)(cid:13)(cid:13)(cid:13) S (cid:18) tε (cid:19) S (cid:18) − sε (cid:19) ˜ F κε (cid:13)(cid:13)(cid:13)(cid:13) L ( R × (0 ,T ) × B ) ≤ ε c γ ˜ E (cid:13)(cid:13)(cid:13)(cid:13) S (cid:18) − sε (cid:19) ˜ F κε (cid:13)(cid:13)(cid:13)(cid:13) L ((0 ,T ) × B ) h ε ˜ E (cid:13)(cid:13)(cid:13) ˜ F κε (cid:13)(cid:13)(cid:13) L ((0 ,T ) × R ) ≤ ε c γ,κ Where we have used Jensen’s inequality and Fubini’s theorem in the first inequality,extended (0 , t ) to R and used the semigroup property in the second inequality,applied similar reasoning as in (4.21) in the third inequality and then used that( S ( t )) t is a group of isometries on L (extended by zero outside of the ball) in thelast line above.We have therefore obtained the following bounds(4.24) ˜ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) S (cid:18) tε (cid:19) " ˜ ϕ κε (0) ∇ ˜Ψ κε (0) L (0 ,T ; L ( B )) . ε, ˜ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) t ˆ S (cid:18) t − sε (cid:19) " F κε d s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L (0 ,T ; L ( B )) . ε for any ball B ⊂ R . Now let make the notation ˜Φ κε ( e i ) := g i ( · , ˜ ̺ ε ( · ) , (˜ q ε )( · )) κ =:˜ g ε,κi . We notice that for a continuous function S ( t ) and a continuous operator Q ,the quantity S ( t ) Q Φ is Hilbert–Schmidt if Φ is Hilbert–Schmidt. As such, it follows from It´o isometry that˜ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) t ˆ S (cid:18) t − sε (cid:19) " Q ˜Φ κε d ˜ W ε ( s ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ((0 ,T ) × B ) = ˜ E t ˆ (cid:13)(cid:13)(cid:13)(cid:13) S (cid:18) t − sε (cid:19) Q ˜Φ κε (cid:13)(cid:13)(cid:13)(cid:13) L ( U ; L ((0 ,T ) × B )) d s = ˜ E t ˆ X i ∈ N (cid:13)(cid:13)(cid:13)(cid:13) S (cid:18) t − sε (cid:19) Q ˜ g ε,κi (cid:13)(cid:13)(cid:13)(cid:13) L ((0 ,T ) × B ) d s . T ˆ X i ∈ N ˆ R ˜ E (cid:13)(cid:13)(cid:13)(cid:13) S (cid:18) t − sε (cid:19) Q ˜ g ε,κi (cid:13)(cid:13)(cid:13)(cid:13) L ( B ) d s d t where the above involved extending s from (0 , t ) to R as well as Fubini’s theorem.Now using the semigroup property and similar estimate as in equation (4.20)and (4.21), followed by the fact that the semigroup is an isometry with respect tothe L -norm, we get that T ˆ X i ∈ N ˆ R ˜ E (cid:13)(cid:13)(cid:13) S (cid:18) t − sε (cid:19) Q ˜ g ε,κi (cid:13)(cid:13)(cid:13) L ( B ) d s d t = T ˆ X i ∈ N ˜ E (cid:13)(cid:13)(cid:13)(cid:13) S (cid:18) tε (cid:19) S (cid:18) − sε (cid:19) Q ˜ g ε,κi (cid:13)(cid:13)(cid:13)(cid:13) L ( R × B ) d t . ε T ˆ X i ∈ N ˜ E (cid:13)(cid:13)(cid:13)(cid:13) S (cid:18) − sε (cid:19) Q ˜ g ε,κi (cid:13)(cid:13)(cid:13)(cid:13) L ( B ) d t h ε T ˆ X i ∈ N ˜ E kQ ˜ g ε,κi k L ( R ) d t . ε T ˆ X i ∈ N ˜ E k ˜ g ε,κi k L ( R ) d t . ε ˜ E T ˆ X i ∈ N k ˜ g εi k L ( R ) d t . ε. The last inequality follows because the noise term is assumed to be compactlysupported in R . See (2.1). We have therefore shown that˜ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) t ˆ S (cid:18) t − sε (cid:19) Q ˜Φ κε d ˜ W ε ( s ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ((0 ,T ) × B ) ≤ ε c h,γ,κ XISTENCE AND LOW-MACH LIMIT FOR STOCHASTIC COMPRESSIBLE FLOWS 25 where the constant is independent of ε . Combining this with the estimates from(4.24), we get from (4.15) that˜ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)" ˜ ϕ ε ( t ) ∇ ˜Ψ ε ( t ) L ((0 ,T ) × B ) = ˜ E k ˜ ϕ ε ( t ) k L ((0 ,T ) × B ) + ˜ E k∇ ˜Ψ ε ( t ) k L ((0 ,T ) × B ) . I + I + I ≤ ε c h,γ,κ . where we have set I := ˜ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) S (cid:18) tε (cid:19) " ˜ ϕ ε (0) ∇ ˜Ψ ε (0) L ((0 ,T ) × B ) I := ˜ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) t ˆ S (cid:18) t − sε (cid:19) ˜ F ε d s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ((0 ,T ) × B ) I := ˜ E (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) t ˆ S (cid:18) t − sε (cid:19) Q Φd W s,ε (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ((0 ,T ) × B ) So in particular, ˜ E k∇ ˜Ψ κε ( t ) k L ((0 ,T ) × B ) ≤ ε c h,γ,κ (4.25)holds for any ball B ⊂ R . We also deduce from Eq. (4.19) together with theembedding L ∞ (0 , T ; L r ( B )) ֒ → L (0 , T ; L r ( B )) where r = γγ +1 , and the continuityof Q that(4.26) ˜ E k∇ ˜Ψ κε − ∇ ˜Ψ ε k L (0 ,T ; L r ( B )) ≤ c δ,t , ˜ E k ˜ q κε − ˜ q ε k L (0 ,T ; L r ( B )) ≤ c δ,t where δ is the arbitrarily constant from (4.19) which is independent of κ and ε . Assuch, the constant c δ,t can be made arbitrarily small for an arbitrary choice of δ sothat lim κ ↓ ˜ E k∇ ˜Ψ κε − ∇ ˜Ψ ε k L (0 ,T ; L r ( B )) = 0 , r = 2 γγ + 1 . Thus, it follows from (4.25) and the uniform bound (4.26) that we may exchangethe order of taking limits in (4.26). As such for any ball B ⊂ R , we have that(4.27) 0 ≤ lim ε ↓ ˜ E k∇ ˜Ψ ε k L (0 ,T ; L r ( B )) = lim κ ↓ lim ε ↓ ˜ E k∇ ˜Ψ ε k L (0 ,T ; L r ( B )) ≤ ε ↓ lim κ ↓ ˜ E k∇ ˜Ψ κε − ∇ ˜Ψ ε k L (0 ,T ; L r ( B )) + 2 lim κ ↓ lim ε ↓ ˜ E k∇ ˜Ψ κε k L (0 ,T ; L r ( B )) ≤ c (cid:18) lim κ ↓ ˜ E k∇ ˜Ψ κε − ∇ ˜Ψ ε k L (0 ,T ; L r ( B )) + lim ε ↓ ˜ E k∇ ˜Ψ κε k L ((0 ,T ) × B ) (cid:19) = 0hence our claim. (cid:3) Remark . We observe that by combining (4.10) and Proposition 4.8, we canonly conclude that ˜ ̺ ε ˜ u ε → ˜ u in L (0 , T ; W − , ( R ))(4.28)˜ P − a.s.However, we can improve this spatial regularity. We give this as part of thelemma below. Lemma 4.11.
Let γ > , q < and l > . Then for all r ∈ ( , , we have that div(˜ ̺ ε ˜ u ε ⊗ ˜ u ε ) ⇀ div(˜ u ⊗ ˜ u ) in L (0 , T ; W − l, ( B )) , (4.29) ˜ ̺ ε ˜ u ε → ˜ u in L (0 , T ; L r ( B ))(4.30)˜ P − a.s. for any ball B ⊂ R .Proof. To avoid repetition, we refer the reader to [3, Proposition 3.13] for the proofof (4.29). However we proof (4.30) below.By using the identity P (˜ ̺ ε ˜ u ε ) = P (˜ ̺ ε − u ε + P ˜ u ε , the reverse triangle inequalityand then the triangle inequality, we have that for any ball B ⊂ R , (cid:12)(cid:12)(cid:12) kP (˜ ̺ ε ˜ u ε ) k L (0 ,T ; L r ( B )) − k ˜ u k L (0 ,T ; L r ( B )) (cid:12)(cid:12)(cid:12) ≤ kP (˜ ̺ ε − u ε + P ˜ u ε − ˜ u k L (0 ,T ; L r ( B )) ≤ kP (˜ ̺ ε − u ε k L (0 ,T ; L r ( B )) + kP ˜ u ε − ˜ u k L (0 ,T ; L r ( B )) ≤ c n k ˜ ̺ ε − k L ∞ (0 ,T ; L min { ,γ } ( R )) k ˜ u ε k L (0 ,T ; L rγγ − r ( B )) + kP ˜ u ε − ˜ u k L (0 ,T ; L q ( B )) o → , (4.9), (4.11) and the continuity of P .Combining this with Proposition 4.8 finishes the proof. (cid:3) By combining (4.9) with Lemma 4.11 we finish the proof of Lemma 4.5. (cid:3)
The following lemma now completes the proof of Proposition 4.4.
Lemma 4.12.
For all t ∈ [0 , T ] and φ ∈ C ∞ c ( R ) , we define N ( ̺, q ) t = X k ∈ N ˆ t h g k ( ̺, q ) , φ i d s, N k ( ̺, q ) t = ˆ t h g k ( ̺, q ) , φ i d s. Then we have that for ε ∈ (0 , N (˜ ̺ ε , ˜ ̺ ε ˜ u ε ) t → N (1 , ˜ u ) t ˜ P − a.s.,N k (˜ ̺ ε , ˜ ̺ ε ˜ u ε ) t → N k (1 , ˜ u ) t ˜ P − a.s. as ε → .Proof. By Minkowski’s inequality, we have that kh Φ(˜ ̺ ε , ˜ ̺ ε ˜ u ε ) · , φ i − h Φ(1 , ˜ u ) · , φ ik L ( U ; R ) = X k ∈ N |h (Φ(˜ ̺ ε , ˜ ̺ ε ˜ u ε ) − Φ(1 , ˜ u )) ( e k ) , φ i| ! ≤ c ( φ ) X k ∈ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ supp( φ ) ( g k (˜ ̺ ε , ˜ ̺ ε ˜ u ε ) − g k (1 , ˜ u )) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ˆ supp( φ ) X k ∈ N | g k (˜ ̺ ε , ˜ ̺ ε ˜ u ε ) − g k (1 , ˜ u ) | ! d x where ´ supp( φ ) f d x is the restriction of the integral of f to the support of φ . XISTENCE AND LOW-MACH LIMIT FOR STOCHASTIC COMPRESSIBLE FLOWS 27
Now let x := (˜ ̺ ε , ˜ ̺ ε ˜ u ε ) and y := (1 , ˜ u ) be vectors in R and define the linesegment joining them by L ( x , y ) = { t x + (1 + t ) y : 0 ≤ t ≤ } . Then by the Mean value inequality, we can find ( ̺ ε , q ε ) ∈ L ( x , y ) such that ˆ supp( φ ) X k ∈ N | g k (˜ ̺ ε , ˜ ̺ ε ˜ u ε ) − g k (1 , ˜ u ) | ! d x ≤ ˆ supp( φ ) | (˜ ̺ ε , ˜ ̺ ε ˜ u ε ) − (1 , ˜ u ) | X k ∈ N (cid:12)(cid:12)(cid:12) ∇ ̺ ε , q ε g k ( ̺ ε , q ε ) (cid:12)(cid:12)(cid:12) ! d x ≤ c ˆ supp( φ ) | ˜ ̺ ε − | d x + ˆ supp( φ ) | ˜ ̺ ε ˜ u ε − ˜ u | d x ! =: I + I where we have used (2.2) and [19, Eq. 6.13.6] in the last inequality.Hence by using the embeddings L min { ,γ } ֒ → L and L r ֒ → L , which holds truefor any compact set or ball in R and where r is as defined in Lemma 4.11, we getthat I → I → ω, t ) in ˜Ω × (0 , T ). This is due to (4.9) and (4.30).Hence h Φ(˜ ̺ ε , ˜ ̺ ε ˜ u ε ) · , φ i → h Φ(1 , ˜ u ) · , φ i in L ( U ; R ) ˜ P × L − a.e. which implies that N (˜ ̺ ε , ˜ ̺ ε ˜ u ε ) t → N (1 , ˜ u ) t in L ( U ; R ) ˜ P × L − a.e. Similar argument holds for N k (˜ ̺ ε , ˜ ̺ ε ˜ u ε ) t → N k (1 , ˜ u ) t ˜ P − a.s. (cid:3) Using Lemmata 4.5 and Lemma 4.12, we can now pass to the limit in equation[3, Eq. 3.14-3.16] to get that :(4.31) ˜ E h ( r s ˜ u , r s ˜ W ) [ M (1 , ˜ u , ˜ u ) s,t ] = 0 , ˜ E h ( r s ˜ u , r s ˜ W ) h(cid:2) M (1 , ˜ u , ˜ u ) (cid:3) s,t − N (1 , ˜ u ) s,t i = 0 , ˜ E h ( r s ˜ u , r s ˜ W ) (cid:20)h M (1 , ˜ u , ˜ u ) ˜ β k i s,t − N (1 , ˜ u ) s,t (cid:21) = 0 . Equation (4.31) means that M (1 , ˜ u , ˜ u ) t is an ( F t ) − martingale. Moreover, using(4.31) , we get the quadratic and cross-variation of M (1 , ˜ u , ˜ u ) t as (cid:10)(cid:10) M (1 , ˜ u , ˜ u ) t (cid:11)(cid:11) = N (1 , ˜ u ) , (cid:10)(cid:10) M (1 , ˜ u , ˜ u ) t , ˜ β k (cid:11)(cid:11) = N k (1 , ˜ u )which yields DD M (1 , ˜ u , ˜ u ) t − ˆ t h Φ(1 , ˜ u ) d ˜ W , φ i EE = 0 . That is, for φ ∈ C ∞ c, div ( R ) and t ∈ [0 , T ], we have that h ˜ u ( t ) , φ i = h ˜ u (0) , φ i + ˆ t h ˜ u ⊗ ˜ u , ∇ φ i d s − ν ˆ t h∇ ˜ u , ∇ φ i d s + ˆ t h Φ(1 , ˜ u ) d ˜ W , φ i ˜ P − a.s. keeping in mind that div φ = 0. (cid:3) References [1]
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Department of Mathematics, Heriot-Watt University, Edinburgh, EH14 4AS, UnitedKingdom
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