In this paper, we study the stochastic three-dimensional modified Leray-alpha model arising from the turbulent flows of fluids. We prove the existence of the probabilistic weak solution under the non-Lipschitz condition for the nonlinear forcing terms. We also discuss its uniqueness.
In this paper, we focus on the study of the probabilistic weak solution to the following three-dimensional modified Leray-alpha model in the periodic box T=[0,2πL]3, L>00$]]>:
∂t(uα−α2Δuα)−νΔ(uα−α2Δuα)+((uα−α2Δuα)·∇)uα=−∇pα+f(t,uα)+g(t,uα)dWdt,inT×[0,T]∇·uα=0,inT×[0,T]uα(x,0)=u0(x),inTuα=uα(x,t)is periodic inT,∫Tuαdx=0,
where the vector uα=(u1α,u2α,u3α) is the unknown fluid velocity vector, the scalar pα is the unknown pressure, the constant ν>00$]]> is the kinematic viscosity and the real number α>00$]]> is a given parameter. The vectors f(t,uα) and g(t,uα)dWdt are external forces, where W is an Rm-valued standard Wiener process. The initial velocity is given by u0. The time T>00$]]> is the final time.
Alpha models are known to be a good regularisation of the three-dimensional Navier-Stokes equations. For a physical motivations, model derivation, and analytical and numerical aspects of these models, readers can consult [6–8, 21, 24]. Specifically, the deterministic three-dimensional modified Leray-alpha equations were studied in [12], where the global well-posedness was shown, and it was explained that this model can provide a reliable, computationally sound analytical subgrid large eddy simulation model of turbulence. The study of the stochastic incompressible Navier-Stokes equations driven by a white noise dates back to the early 1970s with the pioneering work [3]. For the stochastic versions of the alpha models, we refer the reader to [5, 10, 11]. In [5], the authors proved the existence and uniqueness of the variational solution to the three-dimensional stochastic Navier-Stokes-alpha (NS-α) model equations in a bounded domain, with Lipschitz assumptions on the nonlinear forcing terms. In [10], the authors ameliorated the result of [5] by avoiding the Lipschitz conditions on the nonlinear forcing terms when proving existence. Furthermore, the authors of [11] obtained the existence of a probabilistic weak solution for the stochastic version of the three-dimensional Bardina model arising from the turbulent flows of fluids with non-Lipschitz conditions. A stochastic three-dimensional inviscid simplified Bardina turbulence model was recently investigated, in [17], and the existence of a global strong and pathwise solution was proven in a periodic domain.
In [23], we established the exponential mixing and ergodic theorems for a stochastic damped nonlinear quintic wave equation driven by a localised space-time noise. Our aim in this paper is to show that the deterministic model in the case of a periodic domain introduced in [12] is reasonable, in the sense that, when some stochastic terms are present in the model, we can suggest a stochastic version with an accurate mathematical setting, yielding the existence and uniqueness of weak solutions to the problem. As is known in the fluid mechanics literature, the challenge is to handle the nonlinear term. The main difference with the publications cited above is the structure of the nonlinearity we consider, especially in a stochastic framework. Our nonlinear term generates new difficulties to be overcome and estimates to be established. To the best of our knowledge, this paper is the first work dealing with the existence and uniqueness of solution to a three-dimensional stochastic modified Leray-alpha subgrid scale model of fluid turbulence. To prove the existence of a weak solution in a periodic domain, in Theorem 2.2, we put the system in an abstract form. Then, using the Galerkin method, we construct an approximating solution. Next, we establish a priori estimates on the approximating solution that allow us to prove the compactness properties of the corresponding probability measures. Thereafter, we use Prokhorov’s criterion and Skorokhod’s theorem to prove the existence of a subsequence that converges strongly as the approximating parameter goes to infinity. For the proof of the existence of the pressure, we use a generalisation of Rham’s theorem [15]. Finally, we will prove the pathwise uniqueness of the weak solution to establish the uniqueness of the weak solution in Theorem 2.3. Naturally, the uniqueness property needs hold with the Lipschitz condition of the external nonlinear terms.
The paper is organised as follows. Section 2 is devoted to stating our problem, the functional settings and the main results. The proofs of our main results are included in Section 3. The convergence results of the unique weak solution of the three-dimensional modified Leray-alpha model as the regularising parameter alpha vanishes was addressed in [22].
Functional setting and main resultsFunctional setting
Before presenting our main results, we introduce some functional settings.
We set V={ϕ,ϕto be a trigonometric polynomial with period2πL,such that∇·ϕ=0and∫Tϕ(x)dx=0}.
Let H and V be the closure of the set V in the space (L2(T))3 and (H1(T))3, respectively. As mentioned in [9], we denote by |·| and (·,·) the associated norm and the inner product, respectively, of (L2(T))3 and by ‖·‖ and ((·,·))=(∇·,∇·) the (H1(T))3 associated norm and inner product, respectively. We note that H is a Hilbert space equipped with the inner product of (L2(T))3 and V is a Hilbert space for the scalar product ((·,·))V=(·,·)+α2(∇·,∇·). Its associate norm, denoted by ‖·‖V, is equivalent to the usual H1-norm. Indeed, we can deduce from the definition of ‖·‖V and Poincaré inequality [16, Chapter 5] that
(L−2+α2)−1‖v‖V≤‖v‖≤α−2‖v‖V,v∈V.
The orthogonal projection of (L2(T))3 onto H (called the Helmholtz-Leray projection) is denoted by P:(L2(T))3→H.
Let the operator A=−PΔ be the Stokes operator with the domain D(A)=(H2(T))3∩V. In the space-periodic case, for all u∈D(A), we have Au=−PΔu=−Δu. Throughout, we will denote by Y′ the dual of any topological space Y and by ⟨·,·⟩Y′ the duality between Y and Y′. Moreover, as mentioned in [9, 26], the operator A is an isomorphism from V to V′, and it is a self-adjoint, positive, and compact operator on H. Hence, the space H has an orthonormal basis {ej}j≥1 of eigenfunctions of A, such that Aej=λjej, where L−2=λ1≤λ2≤⋯λj… are the eigenvalues of A, repeated according to their multiplicities. Furthermore, from [9, 26], we have D(Aρ)′=D(A−ρ), for any ρ>00$]]> and
‖v‖D(Aρ)=|Aρv|,v∈V,ρ∈R.
By the Riesz representation, we will identify H with its dual, and we will consider the chain of inclusions
D(A)⊂V⊂H≡H′⊂V′⊂D(A)′,
where each space is densely and compactly embedded into the next one. Next, let I denote the identity operator in H. Recall that (I+α2A)−1 is an isomorphism from H onto D(A), and for all ϕ∈H, φ∈V, we have (((I+α2A)−1ϕ,φ))V=(ϕ,φ),‖(I+α2A)−1ϕ‖V≤|ϕ|.
It follows from the Poincaré inequality [16] that
|v|2≤λ1−1‖v‖2,v∈V.
Hereafter, C denotes some uniform constant that is independent of the parameter α in the equations.
For any v1,v2∈V, we define the standard bilinear form associated with the Navier-Stokes equation.
B(v1,v2)=P((v1·∇)v2).
In particular, for every v1,v2,v3∈V, the bilinear form B:V×V→V′ is continuous and satisfies the following estimates [9, 26].
|⟨B(v1,v2),v3⟩V′|≤c|v1|1/2‖v1‖1/2‖v2‖‖v3‖.
Moreover,
⟨B(v1,v2),v3⟩V′=−⟨B(v1,v3),v2⟩V′,
and in particular,
⟨B(v1,v2),v2⟩V′=0.
For any v1∈H, v2∈V and v3∈D(A), we have
|⟨B(v1,v2),v3⟩D(A)′|≤c|v1|‖v2‖‖v3‖1/2|Av3|1/2,
and therefore
‖B(v1,v2)‖D(A)′≤c|v1|‖v2‖.
Applying the Helmholtz-Leray orthogonal projection P to equation (1), we obtain the following equivalent abstract stochastic evolution equation:
dvα+[νAvα+B(vα,uα)]dt=f(t,uα)dt+g(t,uα)dWvα=uα+α2Auαuα(0)=u0,
where ∇·uα=0, ∫Tuαdx=0 and f and g are two nonlinear operators such that
f:(0,T)×H→H is measurable, almost every (a.e.) t. Moreover, u↦f(t,u) is continuous from H to H, and we have a.e.
|f(t,u)|≤C(1+|u|).
g:(0,T)×H→(L2(T))3m is measurable, a.e. t. Moreover, u↦g(t,u) is continuous from H to (L2(T))3m, and we have a.e.
|g(t,u)|(L2(T))3m≤C(1+|u|).
Furthermore, we assume that Pf=f. We always can do so due to the modification of the pressure p in such a way that it includes the gradient part of f. Using the Poincaré inequality and the second equation of (14), we have
|vα|≤(λ1−1+α2)|Auα|=(L2+α2)|Auα|.
Now, we introduce some probabilistic functional spaces.
Let (Ω,F,P) be a complete probability space and {Ft}0≤t≤T be an increasing and right-continuous family of sub-σ-algebras of F such that F0 contains all the P-null sets of F. Let W be an Rm-valued Wiener process on (Ω,F,{Ft}0≤t≤T,P).
Let X be a Banach space. For any r,p≥1, we denote by the functional space Lp(Ω,F,{Ft}0≤t≤T,P,Lr((0,T),X)) the set of functions u=u(x,t,ω) with values in X defined on T×(0,T)×Ω, such that u is measurable with respect to (t,ω) and for almost all t, u is Ft-measurable. Moreover, we have
‖u‖Lp(Ω,F,{Ft}0≤t≤T,P,Lr((0,T),X))=[E(∫0T‖u‖Xrdt)p/r]1/r<∞,
where E is the mathematical expectation with respect to the probability measure P. The space Lp(Ω,F,{Ft}0≤t≤T,P,Lr((0,T),X)) is a Banach space. If r=∞, then Lp(Ω,F,{Ft}0≤t≤T,P,L∞((0,T),X)) is endowed with the norm
‖u‖Lp(Ω,F,{Ft}0≤t≤T,P,L∞((0,T),X))=(Esup ess0≤t≤T‖u‖Xp)1/p.
Finally, we recall Vitali’s convergence theorem [20], which is crucial in the proof of our existence result.
Suppose that(φk)k≥1is a sequence of real integrable functions on Ω. Let φ be a real function on Ω, such thatφk⟶φa.e.ask→∞. Then, the following insertions are equivalent
(φk)is uniformly integrable.
φ∈L1(Ω,F,P,R)andφk⟶φinL1(Ω,F,P,R)ask→∞.
Main results
The meaning of a weak solution to (1) is understood as follows.
A weak solution to (1) is a system (Ω,F,{Ft},P,W,uα), where
(Ω,F,{Ft},P) is a filtered probability space, where {Ft}0≤t≤T is an increasing and right-continuous family of sub σ-algebra of F.
W is an m-dimensional Ft-standard Wiener process.
uα(t) is Ft-adapted for all t∈[0,T], and for all 1≤p<∞, uα belongs to Lp(Ω,F,{Ft},P;L2(0,T;D(A)))∩Lp(Ω,F,{Ft},P;L∞(0,T;V)).
For any t∈[0,T], the following equation holds P-a.s.
((uα(t),φ))V−((uα(0),φ))V+ν∫0t(vα(s),Aφ)ds+∫0t⟨B(vα(s),uα(s)),φ⟩V′ds=∫0t⟨f(s,uα(s)),φ⟩V′ds+(∫0tg(s,uα(s))dW(s),φ),
for all φ∈D(A).
Our first result is the following existence theorem.
Letu0∈V, and suppose that (15) and (16) hold. Then, there exists a weak solution(Ω,F,{Ft}0≤t≤T,P,W,uα)of (1) in the sense of Definition2.1, and we haveuα∈Lp(Ω,F,P,C(0,T;V)).
Our second result is the following uniqueness theorem.
If f and g are Lipschitz with respect to the variableuα, then the solution given by Theorem2.2is a unique weak solution of (1) in the sense of Definition2.1.
We note that uniqueness is the main target behind regularisations of three-dimensional stochastic Navier-Stokes equations. As stated by Theorem 2.3, in the case of the three-dimensional stochastic modified Leray-alpha model, a pathwise uniqueness holds, in contrast with the case of the original Navier-Stokes equations.
Proofs of main resultsProof of Theorem <xref rid="j_vmsta175_stat_003">2.2</xref>
We will perform the proof of Theorem 2.2 in five steps.
Step 1: Construction of an approximating sequence
Let {ej}j≥1 be an orthonormal basis of H, such that {ej} are eigenfunctions of the operator A. Denote by HN=span{e1,…,eN} and by PN the L2-orthogonal projection from H onto HN. We look for a sequence {uNα} in HN that is a solution to the following system of stochastic differential equations:
dvNα+νAvNα+PNB(vNα,uNα)dt=PNf(t,uNα(t))dt+PNg(t,uNα(t))dW¯vNα=uNα+α2AuNαuNα(0)=PNu0,
defined on a fixed stochastic basis (Ω¯,F¯,{F¯t}0≤t≤T,P¯,W¯). By classical existence results of solutions to stochastic differential equations, see i.e., [18, Theorem 3.1.1], there exists a local probabilistic weak solution to (19), denoted by (ΩN,FN,{FtN}0≤t≤TN,PN,WN,uNα) and defined on [0,TN], TN>00$]]>. We denote by EN the mathematical expectation with respect to (ΩN,FN,PN). The following a priori estimates will allow us to show that TN=T.
Step 2: Estimates for the approximating sequence
Throughout this step, C, Ci, i=1,… denote positive constants that are independent of N and α that may need change from line to line.
The approximating sequenceuNαsatisfies the a priori estimates:
ENsup0≤t≤T‖uNα(t)‖V+4να2EN∫0T‖uNα(t)‖D(A)2dt≤C1,
ENsup0≤t≤T‖uNα(t)‖Vp≤C2,p,p∈[1,∞),
EN(∫0T‖uNα(t)‖D(A)2dt)p≤C3,pα2p,p∈[1,∞).
Applying (I+α2A)−1 to the first equation of (19), it follows that
duNα+[νAuNα+(I+α2A)−1PNB(vNα,uNα)]dt=(I+α2A)−1PNf(t,uNα)dt+(I+α2A)−1PNg(t,uNα)dWN.
By Itô’s formula for ‖uNα‖V2, we have
d‖uNα‖V2+2[ν((AuNα,uNα))V+(((I+α2A)−1PNB(vNα,uNα),uNα))V]dt=2(((I+α2A)−1PNf(t,uNα),uNα))Vdt+‖(I+α2A)−1PNg(t,uNα)‖V2dt+2(((I+α2A)−1PNg(t,uNα(t)),uNα))VdWN.
From (5) and (11), we obtain that
((AuNα,uNα))V=(AuNα,uNα)+α2(AuNα,AuNα)=‖uNα‖2+α2|AuNα|2,(((I+α2A)−1PNB(vNα,uNα),uNα))V=(PNB(vNα,uNα),uNα)=0,(((I+α2A)−1PNf(t,uNα),uNα))V=(PNf(t,uNα),uNα)=(f(t,uNα),uNα),(((I+α2A)−1PNg(t,uNα),uNα))V=(PNg(t,uNα),uNα)=(g(t,uNα),uNα).
Thus, we infer that
d‖uNα‖V2+2ν(‖uNα‖2+α2|AuNα|2)dt=2(f(t,uNα),uNα)dt+‖(I+α2A)−1g(t,uNα)‖V2dt+2(g(t,uNα),uNα)dWN.
By (6), we have ‖(I+α2A)−1g(t,uNα)‖V2≤|g(t,uNα)|2. It follows that
d‖uNα‖V2+2ν(‖uNα‖2+α2|AuNα|2)dt≤2(f(t,uNα),uNα)dt+|g(t,uNα)|2dt+2(g(t,uNα),uNα)dWN.
Since |·|≤‖·‖V, inequalities (15) and (16) imply that
(f(t,uNα),uNα)≤C(1+‖uNα‖V2)and(g(t,uNα),uNα)≤C(1+‖uNα‖V2).
Hence, (22) and (23) yield
d‖uNα‖V2+2ν(‖uNα‖V2+α2|AuNα|2)dt≤C(1+‖uNα‖V2)dt+2(g(t,uNα(t)),uNα)dWN.
Dropping the term 2ν‖uNα‖V2 and integrating with respect to time, we obtain
‖uNα(t)‖V2+2να2∫0t‖uNα(s)‖D(A)2dt≤‖uNα(0)‖V2+CT+C∫0t‖uNα(s)‖V2dt+2∫0t(g(t,uNα(s)),uNα(s))dWN(s).
For each integer n>00$]]>, we consider the FtN-stopping time τNn defined by
τNn=inf{t∈[0,T],‖uNα(t)‖V2+2να2∫0t‖uNα(s)‖D(A)2dt≥n2}∧T,
where a∧b=min(a,b). This stopping time will be useful later on to apply Burkhölder-Davis-Gundy’s inequality. Moreover, the sequence (τNn)n increases to T when n goes to ∞. Taking the supremum and the mathematical expectation of (25), we obtain that
ENsup0≤s≤t∧τNn‖uNα(s)‖V2+2να2EN∫0t∧τNn‖uNα(s)‖D(A)2ds≤‖uNα(0)‖V2+CT+CEN∫0t∧τNn‖uNα(s)‖V2ds+2ENsup0≤s≤t∧τNn∫0s(g(r,uNα(r)),uNα(r))dWN(r),
for all t∈[0,T] and all n,N≥1. Let us estimate the last term of the right-hand side of (26). Using Burkhölder-Davis Gundy’s inequality [13, Chapter 3, Theorem 3.28], we have
2ENsup0≤s≤t∧τNn|∫0s(g(r,uNα(r)),uNα(r))dWN(r)|≤6CEN(∫0t∧τNn(g(t,uNα(s)),uNα(s))2ds)1/2.
Using Young’s inequality and (23), we deduce that
2ENsup0≤s≤t∧τNn|∫0s(g(r,uNα(r)),uNα(r))WN(r)|≤C1EN(∫0t∧τNn(1+‖uNα(s)‖V2)‖uNα(s)‖V2ds)1/2≤12ENsup0≤s≤t∧τNn‖uNα(s)‖V2+C2T+C2EN∫0t∧τNn‖uNα(s)‖V2ds.
We obtain from (26) that
ENsup0≤s≤t∧τNn‖uNα(s)‖V2+4να2EN∫0t∧τNn‖uNα(s)‖D(A)2ds≤2‖uNα(0)‖V2+C3T+C3EN∫0t∧τNn‖uNα(s)‖V2ds.
Dropping the viscous term from the left-hand side of (27), we obtain
ENsup0≤s≤t∧τNn‖uNα(s)‖V2≤2‖uNα(0)‖V2+C3T+C3EN∫0t∧τNn‖uNα(s)‖V2ds.
Since τNn increases to T as n goes to ∞, Gronwall’s lemma implies that
ENsup0≤s≤T‖uNα(s)‖V2≤C.
Thus, it follows from (27) that
ENsup0≤t≤T‖uNα(s)‖V2+4να2EN∫0T‖uNα(s)‖D(A)2ds≤C.
By Itô’s formula and (21), it follows, for any p≥4, that
d‖uNα‖Vp2=p2‖uNα‖Vp2−2(−ν(‖uNα‖V2+α2|AuNα|2)dt+2(f(t,uNα),uNα)+p−44(g(t,uNα),uNα)2‖uNα‖V2)dt+p2‖uNα‖Vp2−2(g(t,uNα),uNα)WN.
Young’s inequality and (23), yield
‖uNα‖Vp2−2(f(t,uNα),uNα)≤C(1+‖uNα‖Vp2).
Thanks to Cauchy-Schwarz’s and Young’s inequalities and (23), we obtain that
(g(t,uNα),uNα)2‖uNα‖V2≤C(1+‖uNα‖V2).
Next, we drop −ν(‖uNα‖2+α2|AuNα|2) from the right-hand side of (30). Using (31) and (32) and integrating with respect to t∈[0,T], we have
‖uNα(t)‖Vp2≤‖uNα(0)‖Vp2+C∫0t(1+‖uNα(s)‖Vp2)ds+p2∫0t‖uNα‖Vp2−1(g(t,uNα),uNα)WN.
For each integer n>00$]]>, we consider the FtN-stopping time τNn defined by
τ˜Nn=inf{t∈[0,T],∫0t‖uNα(s)‖Vpdt≥n2}∧T.
Taking the supremum and squaring both sides of (33), Young’s inequality leads to
(sup0≤s≤t∧τ˜Nn‖uNα(s)‖Vp2)2≤2(‖uNα(0)‖Vp2+C∫0t∧τ˜Nn(1+‖uNα(s)‖Vp2)ds)2+p22sup0≤s≤t∧τ˜Nn|∫0s‖uNα‖Vp2−1(g(r,uNα),uNα)WN|2.
Using Cauchy-Schwarz’s and Young’s inequalities, we deduce that
(‖uNα(0)‖Vp2+C∫0t∧τ˜Nn(1+‖uNα(s)‖Vp2)ds)2≤2‖uNα(0)‖Vp+2C(T2+T∫0t∧τ˜Nn‖uNα(s)‖Vpds).
Taking the mathematical expectation in (34), we obtain
ENsup0≤s≤t∧τ˜Nn‖uNα(s)‖Vp≤C(‖u0,Nα‖Vp+T+EN∫0t∧τ˜Nn‖uNα(s)‖Vpds)+p22ENsup0≤s≤t∧τ˜Nn|∫0s‖uNα(r)‖Vp2−1g(r,uNα(r)),uNα(r))WN(r)|2.
The Burkhölder-Davis-Gundy inequality implies that
ENsup0≤s≤t∧τ˜Nn|∫0s‖uNα(r)‖Vp2−1(g(r,uNα(r)),uNα(r))WN(r)|2≤CpEN∫0t∧τ˜Nn‖uNα(s)‖Vp−2(g(s,uNα(s)),uNα(s))ds≤CpEN∫0t∧τ˜Nn‖uNα(s)‖Vp−2(1+‖uNα(s)‖V2)ds≤Cp+EN∫0t∧τ˜Nn‖uNα(s)‖Vpds,
where we use Young’s inequality in the last step. Thus, it follows that
ENsup0≤t≤t∧τ˜Nn‖uNα(s)‖Vp≤Cp+CEN∫0t∧τ˜Nn‖uNα(s)‖Vpds.
Since τNn increases to T, as n goes to ∞, Gronwall’s inequality implies that
ENsup0≤t≤T‖uNα(t)‖Vp≤Cp,p≥4.
As (35) is proven for any p≥4, it is consequently true for any p∈[1,∞). From (26), we obtain
(2να2)p(∫0T‖uNα(t)‖D(A)2dt)p≤Cp(‖uNα(0)‖V2p+Tp+(∫0T‖uNα(t)‖V2dt)p)+cpsup0≤t≤T|∫0t(g(s,uNα(s)),uNα(s))WN(s)|p.
To handle the last term on the right-hand side of inequality (36), we take the mathematical expectation. Then, by the stopping time techniques used in the proof of (29), we apply Burkhölder-Davis-Gundy’s inequality. Finally, we obtain
EN(∫0T‖uNα(t)‖D(A)2dt)p≤Cpα2p,p∈[1,∞).
□
Estimate (38) given in the lemma below is crucial in the proof of the tightness of the law to the Galerkin solution uNα.
There exists a positive constantC(α)such thatENsup0≤|θ|≤η∫0T‖uNα(t+θ)−uNα(t)‖D(A)′2dt≤C(α)η,where0<η≤1is a fixed constant.
For θ>00$]]>, we obtain from (19)
‖vNα(t+θ)−vNα(t)‖D(A)′2≤2‖∫tt+θPN(νAvNα(s)+B(vNα(s),uNα(s))−f(s,uNα(s)))ds‖D(A)′2+2|∫tt+θg(s,uNα(s))WN(s)|2.
This implies that
‖vNα(t+θ)−vNα(t)‖D(A)′2≤6θ[ν2∫tt+θ‖AvNα(s)‖D(A)′2ds+∫tt+θ‖PNB(vNα(s),uNα(s))‖D(A)′2ds+∫tt+θ|f(s,uNα(s))|2ds]+2|∫tt+θg(s,uNα(s))WN(s)|2.
Since
‖AvNα‖D(A)′=|vN|≤|uN|+α2|AuN|
and
‖PNB(vNα(s),uNα(s))‖D(A)′2≤c|vNα(s)|2‖uNα(t)‖2,
Lemma 3.1 implies that
ENsup0≤θ≤η∫0T∫tt+θ‖AvNα(s)‖D(A)′2dsdt≤EN∫0T∫tt+η|vNα(s)|dsdt≤EN∫0T∫tt+ηsup0≤t≤T‖uNα(t)‖V2dsdt+α4EN∫0T∫tt+η‖uNα(s)‖D(A)2dsdt≤CTη+α2C.
Moreover, we have
ENsup0≤θ≤η∫0T∫tt+θ‖PNB(vNα(s),uNα(s))‖D(A)′2dsdt≤cEN∫0T∫tt+η|vNα(s)|2‖uNα(t)‖2dsdt≤Cα−4ηENsup0≤t≤T‖uNα(t)‖V4+CηEN(∫0T(|uN|2+α4|AuN|2)dt)2dt≤Cα−4η+CηENsup0≤t≤T‖uNα(t)‖V4+Cα4ηEN(α2∫0T‖uNα(s)‖D(A)dt)2dt≤Cα−4η+Cη+Cα4η.
It follows from (15) that
ENsup0≤θ≤η∫0T∫tt+θ|f(s,uNα(s))|2ds≤CEN∫0T∫tt+η(1+‖uNα(t)‖V2)dsdt≤Cη+EN∫0T∫tt+ηsup0≤t≤T‖uNα(t)‖V2≤Cη.
As previously, using Burkhölder-Davis-Gundy’s inequality and (16), we obtain
EN∫0Tsup0≤θ≤η|∫tt+θg(s,uNα(s))WN(s)|2dt≤EN∫0T∫tt+η|g(s,uNα(s))|2dsdt≤Cη.
Collecting the estimates above, we obtain that
ENsup0≤θ≤η∫0T‖vNα(t+θ)−vNα(t)‖D(A)′2dt≤C(α2η+1+α−4+α4)η2≤C(α)η.
In fact, as 0≤η≤1, we have η2≤η. Therefore, from (4) and the fact that
α4|uNα(t+θ)−uNα(t)|2≤|A−1(vNα(t+θ)−vNα(t))|2≤‖vNα(t+θ)−vNα(t)‖D(A)′2,
estimate (39) implies that
ENsup0≤θ≤η∫0T‖uNα(t+θ)−uNα(t)‖D(A)′2dt≤C(α)η.
To achieve the proof of (38), we can use similar discussions to prove a similar estimate for the case θ<0. □
Step 3: Tightness Property
First, we introduce the concept of tightness of probability measure (see, e.g., [13]). Let X be a separable Banach space, equipped with the Borel σ-algebra B(X).
A set Γ of probability measures defined on (X,B(X)) is called tight if, for each ε>00$]]>, there is a compact set A such that P(A)>1−ε1-\varepsilon $]]>, for all P∈Γ.
Next, we will prove the tightness property of the Galerkin solution. Similar to the proof in [2, Section 4.3.3], we have the following lemma.
For any sequences of positive real numbersμn,νnthat tend to 0 asn→∞, the injection ofχμn,νn={z∈L2(0,T;D(A)∩L∞(0,T;V),supn1νnsup|τ|≤μn(∫0T‖z(t+τ)−z(t)‖D(A)′2dt)1/2<∞},inL2(0,T;V)is compact.
The space χμn,νn is a Banach space with the norm
‖z‖χμn,νn=sup0≤t≤T‖z(t)‖V+(∫0T‖z(t)‖D(A)2dt)1/2+supn1νnsup|τ|≤μn(∫0T‖z(t+τ)−z(t)‖D(A)′2dt)1/2.
We consider a Banach space Bp,μn,νn (1≤p≤∞) of random variables z defined on some probability space. If we denote the expectation on Bp,μn,νn by Eˆ, then we have
Eˆsup0≤t≤T‖z(t)‖Vp<∞,Eˆ(∫0T‖z(t)‖D(A)2dt)p/2<∞
and
Eˆsupn1νnsup|τ|≤μn∫0T‖z(t+τ)−z(t)‖D(A)′2dt<∞.
The space Bp,μ,ν is endowed with the norm
‖z‖Bp,μn,νn=(Eˆsup0≤t≤T‖z(t)‖Vp)1/p+(Eˆα(∫0T‖z(t)‖D(A)2dt)p/2)2/p+Eˆsupn1νn(sup|τ|≤μn∫0T‖z(t+τ)−z(t)‖D(A)′2dt)1/2.
According to a priori estimates in Lemmas 3.1 and 3.2, we deduce that for every μn, νn such that the series ∑n=1∞μnνn converges, the approximate solutions {uNα}N∈N remain bounded in Bp,μn,νn, for any 1≤p<∞.
Now, we consider the set S=C((0,T),Rm)×L2((0,T),V), equipped with its Borel σ-algebra B(S). We denote by ψ the measurable S-valued map defined on (ΩN,FN,PN) by ψ(ω)=WN(ω),uNα(ω)). For each N, we introduce a probability measure ΓN defined on (S,B(S)) by ΓN(Λ)=PN(ψ−1(Λ)), for Λ∈B(S).
The family of probability measures{ΓN;N∈N}is tight.
For any ε>00$]]>, we shall find two compact subsets
Θε⊂C(0,T;Rm),Zε⊂L2(0,T;V),
such thatPN(ω:WN(ω,.)∉Θε)≤ε2PN(ω:uNα(ω,.)∉Zε)≤ε2. For Θε, we use the following classical results about the Wiener process
EN|WN(t2)−WN(t1)|2j=(2j−1)!(t2−t1)j,j∈N.
For a constant ℓε to be chosen depending on ε, we consider the set
Θε={WN(.)∈C(0,T;Rm):supt1,t2∈[0,T],|t2−t1|≤1/n6n|WN(t2)−WN(t1)|≤ℓε}.
Using an extended version for monotonically increasing functions of Markov’s inequality [4, Chapter 1] (for a nonnegative random variable ζ, β>00$]]> and k∈N, we have P(|ζ|≥β)≤E(|ζ|k)βk), we obtain that
PN(ω:WN(ω,.)∉Θε)≤c∑n=1∞(nℓε)4(Tn−6)2n6=cℓε4∑n=1∞1n2.
To obtain (40), we take ℓε4=2Cεε∑n=1∞1n2. Next, we choose Zε as a ball in χμn,νn of radius Mε centred at zero, where μn and νn are independent of ε, the series ∑n=1∞μnνn converges and μn, νn⟶0 as n⟶∞. From Lemma 3.4, we know that Zε is a compact subset of L2(0,T;V), and
PN(ω:uNα(ω,.)∉Zε)≤PN(ω:‖uNα‖χμn,νn>Mε)≤1MεEN‖uNα‖χμn,νn≤cMε.{M_{\varepsilon }}\big)\\ {} & \le \frac{1}{{M_{\varepsilon }}}{\mathbb{E}_{N}}\| {u_{N}^{\alpha }}{\| _{{\chi _{{\mu _{n}},{\nu _{n}}}}}}\le \frac{c}{{M_{\varepsilon }}}.\end{aligned}\]]]>
Choosing Mε=2cε, (41) holds. Finally, using (40) and (41), we obtain that
PN(ω:WN(ω,.)∈Θε,uNα(ω,.)∈Zε)≥1−ε.
This completes the proof of Proposition 3.5. □
Step 4: Applications of Prokhorov’s and Skorokhod’s Theorems
In this step, we recall two lemmas due to Prokhorov [19] and Skorokhod [25], which will be crucial in the following.
If a set Γ of probability measures on(X,B(X))is tight, then for each sequenceΓn⊂Γ, there exists a subsequence{Γnk}that converges weakly to a probability measure Γ:∫ϕdΓnk⟶∫ϕdΓ,asnk⟶∞,for all bounded continuous integrands ϕ.
For an arbitrary sequence of probability measures{Γk}on(X,B(X))weakly convergent to a probability measure Γ, there exists a probability space(Ω,F,P)and random variablesξn,n∈N, and ξ with values in X, such that the probability law ofξnisΓn, the probability law of ξ is Γ andlimn→∞ξn=ξ,P-a.s.
Using the tightness property of {ΓN}N and Prokhorov’s theorem, we obtain the existence of a subsequence {ΓNk} and a probability measure Γ such that ΓNk converges weakly to Γ. Hence, by Skorokhod’s theorem, there exist a probability space (Ω,F,P) and random variables (WNk,uNkα), (W,uα) with values in S, such that the law of (WNk,uNkα) is ΓNk and the law of (W,uα) is Γ. Moreover, we have
(WNk,uNkα)⟶(W,uα)strongly, inSP−a.s.,
where (WNk) is a sequence of an m-dimensional standard Wiener process. Let Ft=σ{(W(s),uα(s)),0≤s≤t}. As in [1], we can show that W(t) is an m-dimensional Ft standard Wiener process. Using the same arguments as in [2], we can prove that the pair (WNk,uNkα) satisfies the following equation, dP⊗dt-almost everywhere:
vNkα(t)+∫0tPNk(νAvNkα(s)+B(vNkα(s),uNkα(s))−f(s,uNkα(s)))ds=vNkα(0)+∫0tPNkg(s,uNkα(s))dWNk(s),
where vNkα(t)=uNkα(t)+α2AuNkα(t).
Step 5: Convergence
By (43), uNkα satisfies the same estimates of uNα, that is ENsup0≤t≤T‖uNkα(t)‖Vp≤Cp,p∈[1,∞)EN(∫0T‖uNkα(t)‖D(A)2dt)p≤Cp,p∈[1,∞)ENsup0≤|θ|≤η∫0T‖uNkα(t+θ)−uNkα(t)‖D(A)′2dt≤C(α)η, where 0<η≤1. Thus, we can extract a subsequence, denoted again by uNkα, such that uNkα⇀uαweakly*, inLp(Ω,F,P;L∞(0,T;V))uNkα⇀uαweakly, inLp(Ω,F,P;L2(0,T;D(A))). Using (42), it follows that
uNkα⟶uαstrongly, inL2(0,T;V),P−a.s.
Then, estimates (44)–(46) and Vitali’s theorem yield
uNkα⟶uαstrongly, inL2(Ω,F,P;L2(0,T;V)).
Thus, we can extract from (uNkα) a subsequence denoted again by uNkα, such that
uNkα⟶uαinV,
for almost all (ω,t) with respect to the measure dP⊗dt.
Using (48), we obtain, for p=2, that
uNkα⇀uαweakly, inL2(Ω,F,P,L2(0,T;D(A))).
Hence, we deduce that
vNkα⇀vαweakly, inL2(Ω,F,P,L2(0,T;H)),
for almost all (ω,t) with respect to the measure dP⊗dt. Since A is a linear bounded operator, it follows that
∫0tPNkAvNkα(s)ds⇀∫0tAvα(s)dsweakly, inL2(Ω,F,P,L2(0,T;D(A)′)).
The strong convergence (49) and the weak convergence (51) allow us to pass to the limit in the nonlinear terms of (43).
Thanks to (50), the continuity of f and g and Vitali’s theorem, we obtain PNkf(s,uNkα(s))⟶f(s,uα(s))strongly, inL2(Ω,F,P,L2(0,T;H))PNkg(s,uNkα(s))⟶g(s,uα(s))strongly, inL2(Ω,F,P,L2(0,T;H)). In view of (53), we use the same arguments, as in [2] to prove that
∫0tPNkg(s,uNkα(s))dWNk(s)⇀∫0tg(s,uα(s))dW(s)
weakly in L2(Ω,F,P,L2(0,T;H)).
Now, it remains to prove that
∫0tPNkB(vNkα(s),uNkα(s))ds⇀∫0tB(vα(s),uα(s))ds
weakly in L2(Ω,F,P,L2(0,T;D(A)′)). Then, we introduce MFt∞(0,T,D(A)) the space of all processes w∈L∞(Ω,(0,T),dP⊗dt,D(A)) that are Ft-progressively measurable. The space MFt∞(0,T,D(A)) is a Banach subspace of L∞(Ω,(0,T),dP⊗dt,D(A)). For any w∈MFt∞(0,T,D(A)), we have
|E∫0t⟨PNkB(vNkα(s),uNkα(s))−B(vα(s),uα(s)),w⟩D(A)′ds|≤INk(1)+INk(2)+INk(3),
where
INk(1)=|E∫0t⟨B(vNkα(s),uNkα(s)),PNkw−w⟩D(A)′ds|INk(2)=|E∫0t⟨B(vNkα(s)−vα(s),uNkα(s)),w⟩D(A)′ds|INk(3)=|E∫0t⟨B(v,uNkα(s)−uα(s)),w⟩D(A)′ds|.
Recall that for any φ∈H, we have |PNkφ−φ|⟶0, as Nk→∞. Additionally, we obtain from (4) that
‖PNkφ−φ‖D(A)⟶0,asNk→∞.
Using (12), (4) and Young’s inequality, we have
INk(1)≤cE∫0t|vNkα|‖uNkα‖‖PNkw−w‖1/2|A(PNkw−w)|1/2ds≤CE∫0t|A(PNkw−w)||vNkα|‖uNkα‖ds≤C‖PNkw−w‖MFt∞[Esup0≤t≤T‖uNk‖V2+E∫0T|vNkα|2ds].
Since
E∫0T|vNkα|2ds≤α−2Esup0≤t≤T‖uNk‖V2+α4E∫0T|AuNkα|2ds<∞,
(54), (44) and (45) imply that INk(1)⟶0, as Nk→∞.
Next, we investigate INk(2). For any χ∈L2(Ω,F,P,L2(0,T;H)), we define the function L(χ)=E∫0t⟨B(χ(s),uNkα(s)),w⟩D(A)′ds. By (12) and (4), we have
|L(χ)|≤cE∫0t|χ|‖uNkα‖‖w‖1/2|Aw|1/2ds≤CE∫0t|Aw||χ|‖uNkα‖ds.
Using (2) and Cauchy-Schwarz’s inequality, we infer that
|L(χ)|≤C‖w‖MFt∞(E∫0T‖uNk‖V2ds)1/2(E∫0T|χ|2ds)1/2.
Therefore, (44) and (56) imply that L is a linear continuous form. By (51), we deduce that L(vNkα)⟶L(vα), as Nk→∞. As INk(2)=L(vNkα−vα), we obtain that INk(2)⟶0, as Nk→∞.
Finally, we turn to INk(3). Using (12), (4) and Cauchy-Schwarz’s inequality, it follows that
INk(3)≤cE∫0t|vNkα|‖uNkα−uα‖‖w‖1/2|Aw|1/2ds≤C‖w‖MFt∞(E∫0T|vNkα|2ds)1/2(E∫0T‖uNkα−uα‖2ds)1/2.
As a consequence of (2), (49) and (55), we deduce that INk(3)⟶0, as Nk→∞.
Gathering all convergence results, we obtain that
vα(t)+∫0tAvα(s)ds+∫0tB(vα(s),uα(s))ds=vα(0)+∫0tf(s,uα(s))ds+∫0tg(s,uα(s))dW(s),P-a. s.for a. e.t. Moreover, we note that Avα∈L2(Ω,F,P;L2(0,T;D(A)′)), B(vα,uα)∈L2(Ω,F,P;L2(0,T;V′)), f(t,uα)∈L2(Ω,F,P;L2(0,T;H)) and g(t,uα)∈L2(Ω,F,P;L2(0,T;(L2(T))3m)).
To prove the continuity of uα with respect to time, we note that (57) allows us to apply [14, Theorem 3.2, Chapter I]. By this theorem, we deduce that there exists Ω˜∈F, such that P(Ω˜)=1 and vα is continuous in H, P-a.s. with respect to t, for any ω∈Ω˜. Therefore, uα is P-a. s. continuous with values in V. This concludes the proof of Theorem 2.2.
To prove the existence and the uniqueness of the pressure, we may use a generalisation of Rham’s theorem for processes [15, Theorem 4.1, Remark 4.3].
Proof of Theorem <xref rid="j_vmsta175_stat_004">2.3</xref>
In the following, we will prove the pathwise uniqueness to (1). Let u1α and u2α be two weak solutions of (1) that have in D(A) almost surely continuous trajectories with the same initial data u0 and are defined on the same probability space with the same standard Wiener process. Since f and g are Lipschitz with respect to uα, there exist Lf>00$]]> and Lg>00$]]>, such that
|f(t,u1α)−f(t,u2α)|≤Lf|u1α−u2α|
and
|g(t,u1α)−g(t,u2α)|(L2(T))3m≤Lg|u1α−u2α|.
We denote by v1α=u1α+α2Au1α and v2α=u2α+α2Au2α. Let δuα=u1α−u2α and δvα=v1α−v2α. It follows from (14) that
dδvα+[νAδvα+B(v1α,u1α)−B(v2α,u2α)]dt=(f(t,u1α)−f(t,u2α))dt+(g(t,u1α)−g(t,u2α))dW.
Applying (I+α2A)−1 and using Itô’s formula, we obtain that
d‖δuα(t)‖V2+2ν((Aδuα,δuα))V+2(((I+α2A)−1(B(v1α,u1α)−B(v2α,u2α)),δuα))V=2(((I+α2A)−1(f(s,u1α)−f(s,u2α)),δuα))V+‖(I+α2A)−1(g(s,u1α)−g(s,u2α))‖V2+2(((I+α2A)−1(g(s,u1α)−g(s,u2α)),δuα))VdW,
for any t∈[0,T]. Integrating with respect to t and using (5), we obtain
‖δuα(t)‖V2+2ν∫0t((Aδuα(s),δuα(s)))Vds+2∫0t⟨B(v1α(s),u1α(s))−B(v2α(s),u2α(s)),δuα(s)⟩D(A)′ds=2∫0t(f(s,u1α(s))−f(s,u2α(s)),δuα(s))ds+∫0t‖(I+α2A)−1(g(s,u1α(s))−g(s,u2α(s)))‖V2ds+2∫0t(g(s,u1α(s))−g(s,u2α(s)),δuα(s))dW(s).
Next, we define ϑα(t)=exp(−∫0tγ(s)‖u1α(s)‖2ds), 0≤t≤T, where γ is a real-valued function that will be fixed below. Applying Itô’s formula for the real process ϑα(t)‖δuα(t)‖V2, we deduce from (60) that
ϑα(t)‖δuα(t)‖V2+2ν∫0tϑα(s)(‖δuα(s)‖+α2|Aδuα(s)|)ds≤2∫0tϑα(s)|⟨B(v1α(s),u1α(s))−B(v2α(s),u2α(s)),δuα(s)⟩D(A)′|ds+2∫0tϑα(s)|(f(s,u1α(s))−f(s,u2α(s)),δuα(s))|ds+∫0tϑα(s)|g(s,u1α(s))−g(s,u2α(s))|(L2(T))3m2ds+2∫0tϑα(s)|(g(s,u1α(s))−g(s,u2α(s)),δuα(s))|dW(s)−∫0tϑα(s)|γ(s)|‖u1α(s)‖V2‖δuα(s)‖V2ds.
From (11) and (4), we have
|⟨B(v1α,u1α)−B(v2α,u2α),δuα⟩D(A)′|=|⟨B(δvα,u1α),δuα⟩D(A)′+⟨B(v2α,δuα),δuα⟩D(A)′|=|⟨B(δvα,u1α),δuα⟩D(A)′|.
Using (12), (17), Young’s inequality, (7) and (2), we infer that
|⟨B(δvα,u1α),δuα⟩D(A)′|≤c|δvα|‖u1α‖‖δuα‖1/2|Aδuα(s)|1/2≤c(λ1−2+α2)|Aδuα|3/2‖u1α‖‖δuα‖1/2≤c(λ1−2+α2)2|δuα|‖u1α‖2+c2|Aδuα|3≤c2((λ1−2+α2)α4+|Aδuα|3)‖u1α‖2‖δuα‖V2.
Since |·|≤‖·‖V, (58) and (59) imply that
|(f(s,u1α(s))−f(s,u2α(s)),δuα(s))|≤Lf|δuα(s)|2≤c2λ1−2α4‖δuα(s)‖V2
and
|g(s,u1α(s))−g(s,u2α(s))|(L2(T))3m≤Lg|δuα(s)|≤cλ1−1α2‖δuα(s)‖V2.
It follows from (61) that
ϑα(t)‖δuα(s)‖V2+2ν∫0tϑα(s)(‖δuα(s)‖+α2|Aδuα(s)|)ds≤c∫0tϑα(s)((λ1−1+α2)α4+|Aδuα(s)|3)‖u1α(s)‖2‖δuα(s)‖V2ds+c2λ1−2α4(2Lf+Lg2)∫0tϑα(s)‖δuα(s)‖V2ds+2∫0tϑα(s)|(g(s,u1α(s))−g(s,u2α(s)),δuα(s))|dW(s)−∫0tγ(s)ϑα(s)‖u1α(s)‖2‖δuα(s)‖V2ds,
for any t∈[0,T]. Taking γ(s)=c((λ1−1+α2)α4+|Aδuα(s)|3) in (62), we obtain that
ϑα(t)‖δuα(s)‖V2+2ν∫0tϑα(s)(‖δuα(s)‖+α2|Aδuα(s)|)ds≤c2λ1−2α4(2Lf+Lg2)∫0tϑα(s)‖δuα(s)‖V2ds+2∫0tϑα(s)|(g(s,u1α(s))−g(s,u2α(s)),δuα(s))|dW(s).
Since 0<ϑα(t)≤1, t∈[0,T], we have
E∫0tϑα(s)|(g(s,u1α(s))−g(s,u2α(s)),δuα(s))|dW(s)≤E∫0t|(g(s,u1α(s))−g(s,u2α(s)),δuα(s))|dW(s).
Then, using the property of the stochastic integral, the expectation of the stochastic integral in (63) vanishes. Dropping the viscous term from the left-hand side of (63), we deduce that for all t∈[0,T]E(ϑα(t)‖δuα(s)‖V2)≤c2λ1−2α4(2Lf+Lg2)E(∫0tϑα(s)‖δuα(s)‖V2ds).
By Gronwall’s lemma, we obtain that ‖δuα(s)‖V=0, P-a.s. for all t∈[0,T]. Hence, we deduce that u1α(t)=u2α(t)P-a.s. for all t∈[0,T]. This concludes the proof of Theorem 2.3.
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