Abstract
We consider in a smooth bounded and simply connected two dimensional domain the convergence in the \(L^2\) norm, uniformly in time, of the solution of the stochastic second-grade fluid equations with transport noise and no-slip boundary conditions to the solution of the corresponding Euler equations. We prove, that assuming proper regularity of the initial conditions of the Euler equations and a proper behavior of the parameters \(\nu \) and \(\alpha \), then the inviscid limit holds without requiring a particular dissipation of the energy of the solutions of the second-grade fluid equations in the boundary layer.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The second-grade fluid equations are a model for viscoelastic fluids, with two parameters: \(\alpha >0\), corresponding to the elastic response, and \(\nu >0\), corresponding to viscosity. Considering a constant density, \(\rho =1\), their stress tensor is given by
where
being p the pressure and u the velocity field. Given this stress tensor, the equations of motion for an incompressible homogeneous fluid of grade 2 are given by
where f describes some external forces, possibly stochastic, acting on the fluid, see [10, 36] for further details on the physics behind this system. The analysis of the deterministic system started with [7]. They proved global existence and uniqueness without restricting the problem to the two dimensional case. Setting, formally, \(\alpha =0\) in Eq. (1) we can reduce the system to the well-known Navier–Stokes one:
Thus (1) can be seen as a generalization of (2). Moreover, in [19], it has been shown that second-grade fluid equations are a good approximation of the Navier–Stokes system. Due to these good properties of the system it is a legitimate question trying to understand if the second-grade fluid equations behave better than the Navier–Stokes ones in problems related to turbulence, like the inviscid limit for domain with boundary and no-slip boundary conditions. In fact, such question is far for being solved for system (2) also in the deterministic framework. Partial results are available:
-
1.
Unconditioned results. They are based on strong assumptions about the flows. For example flows with radial symmetry [25, 26], or flows with analytic boundary layers [29, 37].
-
2.
Conditioned results. They are based on stating some criteria about the behavior of the solutions of the Navier–Stokes equations in the boundary layer in order to prove the inviscid limit. This line of research started with the famous work by Kato [21], see [8, 43, 44] for other results. For what concerns the Stochastic framework few results are available, see for example [28] for a generalization of the Kato’s results to the additive noise case and a wider set of initial conditions and [3] for some analysis on the validity of a Large Deviation Principle for the inviscid limit of the Navier–Stokes equations in two-dimensional bounded domains perturbed by additive noise.
The analysis of the inviscid limit for the deterministic second-grade fluid equations is a partially well-understood topic. In particular, in [27], the authors show that the behavior of the system changes considering different scaling between \(\nu \) and \(\alpha ^2\).
If we set, formally, \(\nu =0\) in system (1) second-grade fluid equations reduce to the so-called Euler-\(\alpha \) equations:
This system models the averaged motion of an ideal incompressible fluid when filtering over spatial scales smaller than \(\alpha \) and its well-posedness has been treated in [30, 39]. Euler-\(\alpha \) equations, formally, satisfies the condition of [27, Theorem 3]. Therefore we can expect that the inviscid limit holds also in this framework. Indeed, this is true as has been showed in [24].
In this work, we will consider stochastic second-grade fluid equations and stochastic Euler-\(\alpha \) equations with transport noise which scales with respect to the elasticity. We want to understand if the good behavior proved in [27] if \(\nu =O(\alpha ^2)\) and in [24] if \(\nu =0\) is preserved also in this case. There are several motivations to consider transport noise, as the effect of small scales on large scales in fluid dynamics problems, see [9, 13, 14, 18] for several discussions on this topic. A first issue related to the analysis of the inviscid limit in the case of the transport noise is the well-posedness of the systems. In fact the existence of strong probabilistic solutions of such systems is outside the framework treated in [32, 34], thus we need to improve slightly these results thanks to the properties of the transport noise. In the following \(\nu \ge 0\) and we will always speak of second-grade fluid equations even if \(\nu =0.\)
The paper is organized as follows. In Sect. 2 we introduce the mathematical problem, we state our main theorems and we give some well-known results for the Euler equations and the analysis of the stochastic second-grade fluid equations. In Sect. 3 we prove that the stochastic second-grade fluid equations with transport noise and no-slip boundary conditions are well posed. In Sect. 4, thanks to the already proven well-posedness and Hypothesis 6 below we improve the energy estimates obtained in Sect. 3 in order to get some estimates crucial for the proof of Theorem 9. The proof of our main theorem on the inviscid limit occupies Sect. 5. Lastly in Sect. 6 we add some remarks for the analysis of the additive noise case.
2 Main results
Let us start this section introducing some general assumptions which will be always adopted under our analysis even if not recalled.
Hypothesis 1
The following hold:
-
\(0<T<+\infty \).
-
D is a bounded, smooth, simply connected domain.
-
\(\left( \Omega ,\mathcal {F},\mathcal {F}_t,\mathbb {P}\right) \) is a filtered probability space such that \((\Omega , \mathcal {F},\mathbb {P})\) is a complete probability space, \((\mathcal {F}_t)_{t\in [0,T]}\) is a right continuous filtration and \(\mathcal {F}_0\) contains every \(\mathbb {P}\) null subset of \(\Omega \).
For square integrable semimartingales taking value in separable Hilbert spaces \(U_1,\ U_2\) we will denote by \([M, N]_t\) the quadratic covariation process. If M, N take values in the same separable Hilbert space U with orthonormal basis \(u_i\), we will denote by \(\langle \langle M,N\rangle \rangle _t=\sum _{i\in \mathbb N} [\langle M,u_i\rangle _U, \langle N,u_i\rangle _U]_t\). For each \(k\in \mathbb {N},\ 1\le p\le \infty \) we will denote by \(L^p(D)\) and \(W^{k,p}(D)\) the well-known Lebesgue and Sobolev spaces. We will denote by \(C_{c}^{\infty }(D)\) the space of smooth functions with compact support and by \(W^{k,p}_0(D)\) their closure with respect to the \(W^{k,p}(D)\) topology. If \(p=2\), we will write \(H^k(D)\) (resp. \(H^k_0(D)\) ) instead of \(W^{k,2}(D)\) (resp. \(W^{k,2}_0(D)\)). Let X be a separable Hilbert space, denote by \(L^p(\mathcal {F}_{t_0},X)\) the space of p integrable random variables with values in X, measurable with respect to \(\mathcal {F}_{t_0}\). We will denote by \(L^p(0,T;X)\) the space of measurable functions from [0, T] to X such that
and obvious generalization for \(p=\infty .\) For any \(r,\ p\ge 1\), we will denote by \(L^p(\Omega ,\mathcal {F},\mathbb {P};L^r(0,T;X))\) the space of processes with values in X such that
-
1.
\(u(\cdot ,t)\) is progressively measurable.
-
2.
\(u(\omega ,t)\in X\) for almost all \((\omega ,t)\) and
$$\begin{aligned} {\mathbb {E}}\left[ \Vert u(\omega ,\cdot )\Vert _{L^r(0,T;X)}^p\right] <+\infty . \end{aligned}$$Obvious generalizations for \(p=\infty \) or \(r=\infty \).
Set
Moreover we introduce the vector space
with norm \(\Vert {u}\Vert _W^2=\Vert u\Vert ^2+\alpha ^2\Vert \nabla u\Vert _{L^2(D;\mathbb {R}^2)}^2+\Vert {\text {curl}}(u-\Delta u)\Vert _{L^2(D)}^2.\) It is well-known, see for example [7], that we can identify W with the space
Moreover there exists a constant such that
We denote by \(\langle \cdot ,\cdot \rangle \) and \(\Vert \cdot \Vert \) the inner product and the norm in H respectively. Other norms and scalar products will be denoted with the proper subscript. On V we introduce the norm \(\Vert u\Vert _V^2=\Vert u\Vert ^2+\alpha ^2\Vert \nabla u\Vert _{L^2(D;\mathbb {R}^2)}^2.\) We will shortly denote by \(\Vert u\Vert _*=\Vert {\text {curl}}(u-\alpha ^2\Delta u)\Vert _{L^2(D)}.\) Obviously the following inequality holds for \(u\in V\), where \(C_p\) is the Poincarè constant associated to D,
Denote by P the linear projector of \(L^2\left( D;\mathbb {R}^2\right) \) on H and define the unbounded linear operator \(A:D(A)\subseteq H\rightarrow H\) by the identity
for all \(v \in D(A),\ w \in H\). A will be called the Stokes operator. It is well-known (see for example [42]) that A is self-adjoint, generates an analytic semigroup of negative type on H and moreover \(V=D\left( \left( -A)^{1/2}\right) \right) .\) Denote by \(\mathbb {L}^{4}\) the space \(L^{4}\left( D,\mathbb {R}^{2}\right) \cap H\), with the usual topology of \(L^{4}\left( D,\mathbb {R}^{2}\right) \). Define the trilinear, continuous form \(b:\mathbb {L}^{4}\times V\times \mathbb {L} ^{4}\rightarrow \mathbb {R}\) as
Now we introduce some assumptions on the stochastic part of the system.
Hypothesis 2
The following hold:
-
K is a (possibly countable) set of indexes.
-
\(\sigma _k\in W^{1,\infty }(D;\mathbb {R}^2)\cap V\) satisfying
$$\begin{aligned} \sum _{k\in K}\Vert \sigma _k\Vert _{W^{1,\infty }}^2<+\infty . \end{aligned}$$ -
\(u_0\in \cap _{p\ge 2}L^p(\mathcal {F}_0,W)\).
-
\(\{W^k_t\}_{k\in K}\) is a sequence of real, independent Brownian motions adapted to \(\mathcal {F}_t\).
Let us consider the stochastic second-grade fluid equations below. Some physical motivations for the introduction of transport noise in fluid dynamic models can be found in [13, 18].
We need to add the additional pressure term \(\sum _{k\in K} \nabla \tilde{p}_k \circ dW^k_t\), the so-called turbulent pressure, in the system above in order to deal with the fact that \(\sum _{k\in K} \sigma _k\cdot \nabla u \circ dW^k_t\) is not divergence free, therefore an additional martingale term orthogonal to H must be added to make the system feasible.
Introducing the Stokes operator, the previous equation can be rewritten as
or the corresponding Itô form
Indeed each of the Stratonovich integrals in Eq. (8) can be rewritten, thanks to the Stratonovich–Itô corrector associated to previous equation, in the following form:
We denote by \(F(u)=\frac{1}{2}\sum _{k\in K} P(\sigma _k\cdot \nabla ((I-\alpha ^2A)^{-1}P(\sigma _k\cdot \nabla u)))\) and \(G^k(u)=P(\sigma _k\cdot \nabla u)\). By Corollary 22 below
Definition 3
A stochastic process with weakly continuous trajectories with values in W is a weak solution of Eq. (9) if
and \(\mathbb {P}-a.s.\) for every \(t\in [0,T]\) and \(\phi \in W\) we have
Theorem 4
Under Hypothesis 1–2, Eq. (9) has a unique solution in the sense of Definition 3. Moreover, almost surely, the stochastic process u has V continuous paths.
Remark 5
Actually we can weaken the integrability assumption of \(u_0\) with respect to \(\mathbb {P}\) in order to get less integrable solution, but regular enough to prove that the inviscid limit holds. Indeed \(u_0\in L^4(\mathcal {F}_0,W)\) is the minimal assumption to prove either the well-posedness, see [2] and Sect. 3.2 below, and the inviscid limit, see Sects. 4 and 5. However, we prefer to not stress this assumption in order to make our results comparable to [34].
As stated in Sect. 1, the proof of Theorem 4 will be the object of Sect. 3. Usually, in stochastic analysis, the well-posedness of a stochastic partial differential equation is obtained considering some approximating sequence, \(\{u^N\}_{N\in \mathbb {N}}\), which solves an approximate equation in the original probability space and showing the tightness of their law in some spaces of functions. Then, by Prokhorov’s theorem and Skorokhod’s representation theorem, one can find an auxiliary probability space and a solution of the limit equation in this auxiliary probability space, u. Lastly, by a Gyongy–Krylov argument, one can recover that the limit process belongs to the original probability space and that the approximating sequence converge in probability to u. See [1, 4, 13] for some examples of this method. Here, we follow a different, perhaps, more direct approach introduced by Breckner in [2] for Navier–Stokes equations with multiplicative noise with particular regularity properties, but well-suited to treat transport noise, which a priori does not satisfy the general assumptions of [2, Section 2]. This approach uses, in particular, the properties of stopping times, some basic convergence principles from functional analysis and some properties of fluid dynamic non-linearities. Therefore, even if the results of [2] were related to Navier–Stokes equations, this approach can be applied also to other fluid dynamic models, see [5, 34] for some examples to different fluid dynamic systems. An important byproduct of this way of proceed is that the approximations converge in mean square to the solution of the second-grade fluid equations, see Theorem 33 below. This fact will be crucial in order to obtain some apriori estimates on the solution, see Lemma 35 below.
Now we move to consider the inviscid limit problem and introduce a new set of hypotheses.
Hypothesis 6
The following hold:
-
\(\nu =O(\alpha ^2),\ \tilde{\nu }=O(\alpha ^2)\).
-
\(\overline{u}_0\in H^s(D;\mathbb {R}^2)\cap H\) for some \(s\ge 3.\)
-
$$\begin{aligned}&{\mathbb {E}}\left[ \Vert u^{\alpha }_0-\bar{u}_0\Vert ^2\right] \rightarrow 0 ; \end{aligned}$$(11)$$\begin{aligned}&{\mathbb {E}}\left[ \alpha ^2\Vert \nabla u^{\alpha }_0\Vert _{L^2(D;\mathbb {R}^2)}^2\right] =o(1) ;\end{aligned}$$(12)$$\begin{aligned}&{\mathbb {E}}\left[ \alpha ^6\Vert u^{\alpha }_0\Vert _{H^3(D;\mathbb {R}^2)}^2\right] =O(1) . \end{aligned}$$(13)
Let us consider the family of equations
where \(\sigma _k\) are independent from \(\nu ,\ \tilde{\nu },\ \alpha \) and \(u^{\alpha }_0\) are random variable satisfying the assumptions of Theorem 4. Energy relations and the behavior of the \(H^3\) norm of \(u^{\alpha }\) play a crucial role in the analysis of the inviscid limit in the deterministic framework, see Eqs. (3.2) and (3.7) in [27]. If we want to have some hope of replicating the approach of [27] we need some estimates in that direction. This is exactly what happens. Indeed, under Hypothesis 6, Eqs. (3.2) and (3.7) in [27] continue to hold in the stochastic framework, see Lemma 35 below. Therefore there is some hope to generalize the results of [24, 27] to our stochastic framework. Now, let us consider the Euler equations
Definition 7
Given \(\bar{u}_0\in H,\) we say that \(\bar{u}\in C(0,T;H)\) is a weak solution of Eq. (15) if for every \(\phi \in C^{\infty }_0([0,T]\times D)\cap C^1([0,T];H)\)
for every \(t \in [0,T]\) and the energy inequality
holds.
For what concerns the well posedness of the Euler equations the following results hold true, see [22, 41].
Theorem 8
Fix \(T>0,\ s\ge 3\). Let \(\bar{u}_0\in H^s(D;\mathbb {R}^2)\cap H\). Then there exist a unique weak solution of (15) with initial condition \(\bar{u}_0\) such that
and \(\Vert {\bar{u}(t)}\Vert =\Vert \bar{u}_0\Vert ,\ \forall t\in [0,T].\)
Now we can state our main Theorem. According to the analysis started in [16] and continued, recently, in [11, 12] the influence of the transport noise on the averaged solution is related to the \(\ell ^2\) norm of its coefficients, therefore we expect that the solution of Eq. (14) converges to the solution of the Euler equations with null forcing term.
Theorem 9
Under Hypotheses 1–6, calling \(u^{\alpha }\) the solution of (14) and \(\bar{u}\) the solution of (15), it holds
Remark 10
If \(\overline{u}_0\in H\cap H^1(D;\mathbb {R}^2)\), the existence of a family \(u_0^{\alpha }\) satisfying Eqs. (11), (12), (13) is guaranteed by Proposition 1 of [24].
Remark 11
Due to the poor regularity of the coefficients F and \(G^k\), Eq. (9) are not guaranteed to be well-posed from the results of [34]. Indeed, neither F nor \(G^k\) satisfy the assumptions of [33] or [34]. However, due to relation (47) and the good estimates of Corollary 22, we will be able to prove in Lemmas 25 and 26 the same, actually stronger, energy estimates that are available in [34]. These and Lemma 28 are the main ingredients in order to prove the well-posedness of system (9). On the contrary the well-posedness in the case of additive noise is completely solved by the results of [34], thus in Sect. 6 we will only explain some remarks about the inviscid limit and the well-posedness in the additive noise framework.
Remark 12
Both Theorems 4 and 9 continue to hold for \(\nu =0\). We will give the proof of all the statements below in full details considering the case \(\nu >0\). However if something in the proof changes considering \(\nu =0\) we will explain in a remark at the end of each proof what we need to change in order to deal with the other case.
Remark 13
The arbitrariness in the choice of the parameters \(\nu \) and \(\tilde{\nu }\) allows us to generalize to this stochastic framework, via Theorem 9, some results of [24, 27]. As a byproduct of its proof we obtain that under Hypotheses 1–2–6
Moreover, considering \(\tilde{\nu }=\nu >0\) we recover the scaling introduced by Kuksin [23] which is relevant for the inviscid limit at the level of invariant measures. The scaling above has been proved of being of interest also for the evolution of the solutions of the stochastic Navier–Stokes equations in a Kato-type regime, see [17, Corollary 2.5.1].
Remark 14
The results of these notes are in a certain sense complementary to what we obtained in [28]. In [28] we required poor regularity on the initial conditions of the Euler equations and the Navier–Stokes equations but we got a conditioned result. On the contrary, in these notes we require strong regularity on the initial conditions of the two problems and a special type of convergence of the initial conditions but we arrive at a not conditioned result.
Remark 15
The assumption on \(\nu =O(\alpha ^2)\) is hidden in Eq. (14). For high frequencies \(\Delta u\) is a damping term in Eq. (14). In fact, for high frequencies \(v\approx -\alpha ^2 \Delta u\), thus the equation becomes, formally,
Asking \(\nu =O(\alpha ^2)\), means requiring that the damping coefficient does not blow-up.
We conclude this section with few notations that will be adopted: by C we will denote several constant independent from \(\nu \), \(\alpha ^2\) and \(\sigma _k\), perhaps changing value line by line. In the case C depends by \(\nu ,\alpha \) or \(\sigma _k\) we will add the dependence as a subscript. Sometimes we will use the notation \(a \lesssim b\), if it exists a constant independent from \(\nu \) and \(\alpha ^2\) such that \(a \le C b\). In order to simplify the notation we will denote Sobolev spaces by \(H^s\), forgetting domain and range.
3 Well-posedness
3.1 Preliminaries
Before starting with the analysis of Eq. (9), we need to recall some preliminaries results on the nonlinear term in the second-grade fluid equations, the Stokes operator A and the embedding between W and V. We will consider the Hilbert triple
We start recalling in a single lemma some classical facts on the nonlinear part of Eq. (1). We refer to [7, 33, Lemma 2.4], [34, Lemma 2.4] for the proof of the various statements.
Lemma 16
For any smooth, divergence free \(\phi ,\ v,\ w\) the following relation holds
Moreover for \(u,\ v,\ w\) the following inequalities hold
Therefore there exists a bilinear operator \(\hat{B}:W\times V\rightarrow W^*\) such that
which satisfies for \(u\in V,\ v\in W\)
Lastly, for \(u\in W,\ v\in V, \ w \in W\)
We need a basis orthonormal either in W and in V in order to deal with the Galerkin approximation of Eq. (9). The existence of such basis is guaranteed by the lemma below. The first part is a consequence of the spectral theorem for self-adjoint compact operators stated in [35], we refer to [6, Lemma 4.1] for the proof of the second part.
Lemma 17
The injection of W into V is compact. Let I be the isomorphism of \(W^*\) onto W, then the restriction of I to V is a continuous compact operator into itself. Thus, there exists a sequence \(e_i\) of elements of W which forms an orthonormal basis in W, and an orthogonal basis in V. This sequence verifies:
where \(\lambda _{i+1}> \lambda _i> 0,\ i= 1,2,\ldots \). Thus \(\sqrt{\lambda _i}e_i\) is an orthonormal basis of V. Moreover \(e_i\) belong to \(H^4(D;\mathbb {R}^2)\).
We will use also some properties of the projection operator P and the solution map of the Stokes operator. We refer to [42] for the proof of the lemmas below.
Lemma 18
The restriction of the projection operator \(P:L^2(D;\mathbb {R}^2)\rightarrow H\) to \(H^r(D;\mathbb {R}^2)\) is a continuous and linear map between \(H^r(D;\mathbb {R}^2)\) and itself.
Lemma 19
Let \(f\in H^m(D;\mathbb {R}^2)\). Then, there exists a unique couple (u, p), with p defined up to an additive constant, solution of
Moreover \(u=(I-\alpha ^2A)^{-1}f\in H^{m+2}(D;\mathbb {R}^2),\ p\in H^{m+1}(D)\),
Lemma 20
The injection of V in H is compact. Thus there exists a sequence \(\tilde{e}_i\) of elements of H which forms an orthonormal basis in H and an orthogonal basis in V. This sequence verifies
where \(\tilde{\lambda }_{i+1}>\tilde{\lambda }_{i}>0,\ i=1,2,\ldots \). Moreover \(\tilde{\lambda }_i\rightarrow +\infty \). Lastly \(\tilde{e}_i\in C^{\infty }(\overline{D};\mathbb {R}^2)\) under our assumptions on D
Combining Lemmas 17 and 19 above, it follows that for each \(f\in H^1\), \(i \in \mathbb {N}\)
Moreover, Lemmas 18, 19, 20 above allow us to prove some useful estimates that will be exploited along the paper. We will need Corollary 22 in order to evaluate the regularity of the linear operators appearing in Eq. (9). Instead we will need Lemma 21 in order to quantify explicitly the dependence from \(\alpha \) in several embeddings and operators. This will be crucial in Sects. 4 and 5.
We recall first that by Poincaré inequality, Eq. (4), triangle inequality and Eq. (5) the following relations hold:
Before going on with the statements of Lemma 21 and Corollary 22 we recall the definitions of the linear operators, \(F,\ \{G^{k}\}_{k\in K},\) appearing in Eq. (9):
Lemma 21
Let \(h\in H,\ u\in V,\ w\in V\cap H^2\), then
Therefore, if \(u\in V\) the following inequalities hold true
Proof
Inequalities (27), (28) are trivial. Indeed, by Lemma 18 it holds
In order to prove inequalities (29), (30), (31) we exploit the Fourier decomposition \(h=\sum _{i\in \mathbb {N}}\langle h,\tilde{e}_i\rangle \tilde{e}_i\). Therefore it holds
For what concerns inequality (32), by definition of the norm in the space W it holds
From Lemma 19, we know that
Combining (37) and (38), inequality (32) follows.
Combining relation (27) with relations (29) and (30), inequalities (33) and (34) follow immediately. Let us now prove Eq. (35). By Hölder’s inequality and relation (28) we have
For what concerns the last one, by Lemma 18, 19 and relations (31) it holds
Corollary 22
It holds
In particular
Lastly we recall two technical tools used in the proof of Theorem 9. We refer to [15] for the proof of the interpolation inequality and to [38] for the proof of the stochastic Grönwall’s Lemma.
Theorem 23
Each function \(f\in H^2\) satisfies the following inequality:
Theorem 24
Let Z(t) and H(t) be continuous, nonnegative, adapted processes, \(\psi (t) \) a nonnegative deterministic function and M(t) a continuous local martingale such that
Then Z(t) satisfies the following inequality
3.2 Galerkin approximation and limit equations
Let \(W^N={\text {span}}\{e_1,\ldots ,\ e_N\}\subseteq W\) and \(P^N:W\rightarrow W^N\) the orthogonal projector. We start looking for a finite dimension approximation of the solution of Eq. (9). We define
The \(c_{i,N}\) have been chosen in order to satisfy \(\forall e_i,\ \ 1\le i\le N\)
where \(u_0^N=\sum _{i=1}^N\langle u_0,e_i\rangle _W e_i\), \(F^N(s)=F(u^N(s))\) and \(G^{k,N}(s)=G^k(u^N(s))\). The local well-posedness of this equation follows from classical results about stochastic differential equations with locally Lipshitz coefficients, see for example [20, 40]. The global well-posedness follows from the a priori estimates in Lemmas 25, 26.
Lemma 25
Assuming Hypothesis 2, the following relations hold:
-
The Itô’s formula
$$\begin{aligned} d \Vert u^N\Vert _V^2&=-2\nu \Vert \nabla u^N\Vert _{L^2}^2dt\nonumber \\&\quad -\sum _{k\in K} b(\sigma _k, u^N,(I-P^N)(I-\alpha ^2 A)^{-1}P(\sigma _k\cdot \nabla u^N))dt. \end{aligned}$$(44) -
The inequality below holds uniformly in N
$$\begin{aligned} {\mathbb {E}}\left[ {\text {sup}}_{t\in [0,T]}\Vert u^N(t)\Vert _V^p\right] \le C_{p,\alpha , u_0,\{\sigma _k\}_{k\in K}},\ \ \forall p\ge 1. \end{aligned}$$(45)
Proof
If we apply the Itô’s formula to \(\sum _{i=1}^N \lambda _i\langle u^N(t),e_i\rangle _V^2\), we get
In the last relation we exploited the fact that \(b(u^N(s), u^N(s), u^N(s))=b(u^N(s),\Delta u^N(s), u^N(s))=0\). Now we observe that for each k, \(\langle G^{k,N}(s), u^N(s)\rangle =0\). In fact
Moreover we have
In fact,
It remains to show that
Thus it is enough to show that \(\langle \sum _{i=1}^N \lambda _i e_i b(\sigma _k,u^N(s),e_i),v\rangle _V=\langle (I-\alpha ^2A)^{-1}P(\sigma _k\cdot \nabla u^N(s))),v\rangle _V\) for all \(v\in V_N\), where \(V_N={\text {span}}\{e_i\}_{i=1}^N\). The last claim is true, in fact
Therefore, combining Eqs. (46), (47) and (48) we obtain
Thus, by Grönwall,
Taking the expected value of Eq. (49) we get the thesis for \(p\le 2\). If \(p>2\), raising to the power p/2 both sides of Eq. (49) the thesis follows easily.
Lemma 26
Assuming Hypothesis 2, the following relation holds:
where \(C_{p,\nu , \alpha ,\{\sigma _k\}_{k\in K}}\) is a constant independent from N.
Proof
This proof is similar to Lemmas 2.4–2.5 of [33]. We will need some changes due to the poor regularity of the coefficients F and \(G^k\). In the part where we will not need any changes, we will refer to the equations in [33]. Let
and \(\tilde{G}^{k,N}=(I-\alpha ^2A)^{-1}G^{k,N}\) the solution of Stokes problem defined in Lemma 19. From the regularity of the eigenvectors \(e_i\), \(G^{k,N}\in H^1\), thus \(\tilde{G}^{k,N}\in W\) and by Eqs. (24) and (32) the following relations hold true
Let us call
From the regularity of the \(e_i\), we have that \(\phi ^N\in H^1\). Thus we can find a \(v^N\in W\) such that \(v^N=(I-\alpha ^2A)^{-1}\phi ^N\). We rewrite shortly the weak formulation satisfied by \(u^N\)
Multiplying each equation by \(\lambda _i\) we get
Now we apply the Itô’s formula to \(\sum _{i=1}^N \langle u^N, e_i\rangle _W^2\) and we obtain
Exploiting the definition of \(v^N,\ \tilde{G}^{k,N}\), Eq. (47) and the classical fact that \({\text {curl}}\nabla =0\) we get
From Lemma 25 we already know that
Substituting this relation in the Itô’s formula (53) we get
Analogously to Eq. (4.48) in [33], the relation below holds true
Using this relation in the Itô’s formula (54) and integrating between 0 and \(t\le \tau _M^N\) we get
Taking the supremum between 0 and \(r\wedge \tau _M^N\) in relation (55) and, then, the expected value we get
Choosing \(\epsilon _1=\frac{1}{4}\) and \(\epsilon _2=\frac{\nu }{4\alpha ^2}\) we arrive at
From Eqs. (48) and (33) we know that
Thanks to Eqs. (51), (52), the interpolation estimate (42) and relation (25) we have
Thanks to Burkholder–Davis–Gundy inequality, Eq. (28), the interpolation inequality (42) and relation (25) we get
Lastly, thanks to Eq. (41) we have
Combining estimates (58),(59),(60),(61) above we obtain
Therefore, choosing \(\epsilon \) small enough, by Eq. (49) we have
Last inequality proves the Lemma for \(p=2\), letting M to \(+\infty \) thanks to monotone convergence Theorem. Now we consider \(p\ge 4\) and we restart from Eq. (4.79) in [33].
Let us consider all the terms, one by one. Arguing as before we have
Exploiting the relations above and the continuous embedding \(W\hookrightarrow V\) we get
Thus taking the supremum in time for \(t\le r\) and the expected value of this we get the thesis via Grönwall’s Lemma arguing exactly as in the proof of Lemma 4.3 in [33] and exploiting previous estimate (60) on \(\langle {\text {curl}}G^{k,N}(s),{\text {curl}}(u^N(s)-\alpha ^2\Delta u^N(s))\rangle _{L^2}\).
Remark 27
In case of \(\nu =0 \), arguing as above we get
Therefore, thanks to Lemma 25 and estimates (58),(59),(60),(61) we obtain
Since \({\mathbb {E}}\left[ {\text {sup}}_{t\le r\wedge \tau _M^N}\Vert u^N(t)\Vert _*^2\right] ={\mathbb {E}}\left[ {\text {sup}}_{t\le r}\Vert u^N(t)\Vert _*^21_{[0,\tau _M^N]}(t)\right] ,\) by Grönwall’s Lemma
Last inequality proves the Lemma for \(p=2\), letting M to \(\infty \) thanks to monotone convergence Theorem. The case \(p\ge 4\) can be treated as in the case \(\nu >0\), therefore we do not add other details.
Let us now introduce the operator \(\hat{A}=(I-\alpha ^2A)^{-1}A\). By Lemmas 16 and 19 the weak formulation satisfied by the Galerkin approximations can be rewritten as
Thanks to relations (45),(50) and the continuity of \(B,\ F\) and \(G^k\), we know that exists a subsequence of the Galerkin approximations, which we will denote again by \(u^N\) just for simplicity, and processes u and \(\hat{B}^*\) such that
The next step will be showing that \(\hat{B}^*=\hat{B}(u,u)\). In this way the existence of a solution of Eq. (9) will follow. In fact, we know that \(\mathbb {P}-a.s.\) for each \(i\in \mathbb {N}\), for each \(t\in [0,T]\)
For what concerns the continuity in V we can argue in the following way via Itô’s formula and Kolmogorov continuity Theorem. From the weak formulation above we get the weak continuity in V of u applying the Kolmogorov continuity Theorem for the SDE satisfied by \(\langle u(t), e_i\rangle _V\), applying the Itô’s formula to \(\Vert u\Vert _V^2\) we get
From this, we get the continuity of \(\Vert u\Vert _V^2\) thanks to the integrability properties of u. Weak continuity and continuity of the norm implies strong continuity, thus we have the strong continuity of u as a process taking values in V. Weak continuity of u as a process taking values in W follows from Lemma 1.4, p. 263 in [42]. Alternatively the strong continuity in V of u follows arguing as in [2] or [31].
3.3 Existence, uniqueness and further results
To prove the existence of the solutions of Eq. (9) we need the following Lemma. As stated in Sect. 2, this way of proceed has been introduced in [2] for Navier–Stokes equations.
Lemma 28
Let
then
Proof
Let \(P^N\) be the projection of W on \(W^N={\text {span}}\{e_1,\ldots , e_N\}\). Thanks to dominated convergence Theorem,
Consequently we have also convergence in \(L^r(\Omega ,L^q(0,T;V))\). Moreover, if \(w\in W,\ i\le N,\ \langle P^Nw,e_i\rangle _V=\langle w,e_i\rangle _V\). Let \(\hat{F}(u)=(I-\alpha ^2 A)^{-1}F(u)\), \(\hat{G}^k(u)=(I-\alpha ^2 A)^{-1}G^k(u)\). From the weak formulation satisfied by u, for each \(i\le N\), we get
Exploiting the relation satisfied by \(u^N\), we get
Thanks to (67), applying the Itô’s formula to \(\sigma (t)\Vert P^N u(t)-u^N(t)\Vert _V^2\), where \(\sigma (t)=exp(-\eta _1 t-\eta _2\int _0^t \Vert u(s)\Vert _W^2 ds)\), we obtain
Let us analyze the terms in (68) one by one. We will not add details where the computations are analogous to Lemma 3.9 in [34].
Inserting these relations in equality (68) we obtain
Taking \(\eta _1=2C^G_{\alpha ,\{\sigma _k\}_{k\in K}}+4C^F_{\alpha ,\{\sigma _k\}_{k\in K}}+1\), \(\eta _2=C_B^2\) we get
Considering the expected value of (69) for \(t=\tau _M\wedge r\), \(r\in [0,T]\), the stochastic integral cancel out, thus we arrive at
We want understand the behavior of the last term in the inequality above. From Lemma 26 and relation (66) we have
Instead we have
In fact thanks to relation (66) and the boundedness properties of \(\hat{B}\) (20),(21), \(\mathbb {P}-a.s.\) for each \(t\in [0,T]\) it holds
Moreover
By dominated convergence Theorem we have the validity of relation (72). Combing the weak convergence guaranteed by relation (71) and the strong convergence guaranteed by (72) we obtain
From this relation, by triangle inequality, we can analyze easily the last term in (70)
Thanks to the boundedness of \(u^N\) and relation (66)
Combining (73) and (74) in relation (70) we obtain
From relation (75), \(\sigma (t)\ge C_M>0\ \forall t\le \tau _M \) and the properties of \(P^N\) via triangle inequality the thesis follows considering \(r=T\).
Remark 29
The proof presented above works only in the case \(\nu >0.\) In order to treat the case \(\nu =0\) we start from relation (75). Then, triangle inequality allows to prove
By dominated convergence theorem we can improve the pointwise convergence of relation (76) in order to obtain Lemma 28. We omit the easy details at this stage, since this argument will be described in full details in the proof of Corollary 30 below.
Combing Lemma 28 and the moment estimates for u and \(u^N\) we get the following Corollary.
Corollary 30
The subsequence \(u^N\) satisfies
Proof
By relation (75) and triangle inequality we already know that
We start proving convergence (77). By definition of \(\tau _M\), Lemma 26 and the weak-\(*\) convergence of \(u^N\) to u described by relation (65) and Markov’s inequality it follows that
where \(C_{\nu , \alpha ,u_0,\{\sigma _k\}_{k\in K}}\) is a constant independent from M and N. If we fix \(\epsilon >0\) and choose M large enough such that \(\frac{C_{\nu , \alpha ,u_0,\{\sigma _k\}_{k\in K}}}{M^2}\le \epsilon \) then by relation (79) we have
From the arbitrariness of \(\epsilon \), the first thesis follows. In order to obtain the other convergence we apply dominated convergence Theorem. Indeed, by relation (77) we already know that for each \(t\in [0,T]\)
Moreover, by Lemma 25, for each N
Therefore convergence (78) follows.
From Lemma 28, without any change with respect to the proof of Lemma 3.8 in [34], we have that the Lemma below holds, thus u is a solution of problem (9) in the sense of Definition 3.
Lemma 31
\(\hat{B}^*=\hat{B}(u,u) \textit{\ in\ } L^2(\Omega ,\mathcal {F},\mathbb {P};L^2(0,T;W^*))\)
Now we can prove the uniqueness.
Theorem 32
The solution of problem (9) in the sense of Definition 3 is unique.
Proof
Let \(u_1\) and \(u_2\) be two solutions. Let w be their difference, then for each \(\phi \in W\) and \(t>0\)
Now we apply the Itô’s formula to compute \(\Vert w\Vert _V^2\). Arguing as in the first part of the proof of Lemma 35 we obtain
Let us consider \(exp(-\int _0^t \Vert u_2(s)\Vert _W^2ds)\Vert w(t)\Vert _V^2:=\sigma (t)\Vert w(t)\Vert _V^2\), via Itô’s formula we get
Combining relations (16) and (18) it follows that
Therefore
where in the last step we applied Young’s inequality. From the last chain of inequalities, via Grönwall’s Lemma we get the thesis.
Theorem 33
The entire Galerkin’s sequence \(u^N\) satisfies
Proof
Each subsequence \(u^{N_k}\) has a converging sub-subsequence \(u^{N_{k,k}}\) which satisfies all previous Lemmas. By uniqueness of the solution of Eq. (9) and Corollary 30 then the thesis follows.
Remark 34
Theorem 33 plays no role concerning the well-posedness of Eq. (9), but it will be crucial for obtaining the energy estimates of Sect. 4, and thus for proving Theorem 9.
4 Energy estimates
Now we start considering Eq. (14) and assuming also Hypothesis 6. The goal of this section is to prove the following lemma:
Lemma 35
Under Hypothesis 2–6, if \(u^{\alpha }\) is the solution of problem (14) in the sense of Definition 3, then
Proof
For the sake of simplicity we write u and \(u_0\) instead of \(u^{\alpha },\ u^{\alpha }_0\) since \(\alpha \) is fixed in this proof. Therefore all the asymptotic expansions and limits will be considering \(N\rightarrow +\infty .\)
-
Let \(\tilde{e}_i\) be the eigenfunctions of the Stokes operator \(-A\), and \(\tilde{\lambda }_i\) the corresponding eigenvalues introduced in Lemma 20. Let, moreover, \(\tilde{u}^N=\sum _{i=1}^N\langle u,\tilde{e}_i\rangle \tilde{e}_i=\tilde{P}^N u\). Exploiting the weak formulation with test functions \(\tilde{e}_i\) we get
$$\begin{aligned}&\langle u(t), \tilde{e}_i\rangle -\alpha ^2\langle u(t), A\tilde{e}_i\rangle - \langle u_0, \tilde{e}_i\rangle +\alpha ^2 \langle u_0, A\tilde{e}_i\rangle \\&\quad =\nu \int _0^t \langle u(s), A\tilde{e}_i\rangle ds- \int _0^t b(u(s), u(s)-\alpha ^2 \Delta u(s),\tilde{e}_i)ds\\&\qquad -\alpha ^2 \int _0^t b(\tilde{e}_i, \Delta u(s),u(s))ds +\tilde{\nu }\int _0^t \langle F(u),\tilde{e}_i \rangle ds\\&\qquad + \sqrt{\tilde{\nu }}\sum _{k\in K} \int _0^t \langle G^k(u(s)),\tilde{e}_i \rangle dW^k_s \ \ \mathbb {P}-a.s. \end{aligned}$$Multiplying each equation by \(\tilde{e}_i\) and summing up, we get
$$\begin{aligned} d(\tilde{u}^N-\alpha ^2A\tilde{u}^N)&=\nu A\tilde{u}^N dt-\sum _{i=1}^N b(u, u-\alpha ^2 \Delta u,\tilde{e}_i)\ dt\\&\quad -\alpha ^2 \sum _{i=1}^N\int _0^t b(\tilde{e}_i, \Delta u,u) \tilde{e}_i\ dt+ \tilde{\nu }\sum _{i=1}^N\langle F(u),\tilde{e}_i \rangle \tilde{e}_i \ dt \\&\quad + \sqrt{\tilde{\nu }}\sum _{k\in K} \sum _{i=1}^N \langle G^k(u),\tilde{e}_i \rangle \tilde{e}_i \ dW^k_t. \end{aligned}$$Now we can apply the Itô’s formula to the process
$$\begin{aligned} \frac{1}{2} (\Vert \tilde{u}^N(t)\Vert ^2+\alpha ^2 \Vert \nabla \tilde{u}^N(t)\Vert _{L^2}^2)=\frac{1}{2} \langle (I-\alpha ^2 A)\tilde{u}^N(t),\tilde{u}^N(t)\rangle \end{aligned}$$obtaining
$$\begin{aligned}&\frac{\Vert \tilde{u}^N(t)\Vert ^2+\alpha ^2 \Vert \nabla \tilde{u}^N(t)\Vert }{2}=\frac{\Vert \tilde{u}^N_0\Vert ^2+\alpha ^2 \Vert \nabla \tilde{u}^N_0\Vert }{2} -\nu \int _0^t \langle \nabla \tilde{u}^N(s), \nabla u^N(s)\rangle _{L^2} ds\\&\quad -\int _0^t b(u(s),u(s) -\alpha ^2\Delta u(s),\tilde{u}^N(s))-\alpha ^2\int _0^t b(\tilde{u}^N(s),\Delta u(s), u(s))ds\\&\quad +\frac{\tilde{\nu }}{2}\int _0^t \sum _{k\in K} \langle P(\sigma _k\cdot \nabla ((I-\alpha ^2 A)^{-1}P(\sigma _k\cdot \nabla u(s)))),\tilde{u}^N(s)\rangle ds\\&\quad + \sqrt{\tilde{\nu }}\sum _{k\in K}\int _0^t \langle P(\sigma _k\cdot \nabla u(s)),\tilde{u}^N\rangle dW^K_s\\&\quad +\frac{\tilde{\nu }}{2}\sum _{k\in K}\int _0^t \sum _{i=1}^N\langle P(\sigma _k\cdot \nabla u(s)),\tilde{e}_i\rangle ^2\langle \tilde{e}_i, (I-\alpha ^2A)^{-1}\tilde{e}_i\rangle ds. \end{aligned}$$Thanks to the properties of the projector \(\tilde{P}^N\) we get easily the first relation. The only thing we need to prove is that
$$\begin{aligned}&\sum _{i=1}^N\langle P(\sigma _k\cdot \nabla u),\tilde{e}_i\rangle ^2\langle \tilde{e}_i, (I-\alpha ^2A)^{-1}\tilde{e}_i\rangle \\ {}&\quad +\langle P(\sigma _k\cdot \nabla ((I-\alpha ^2 A)^{-1}P(\sigma _k\cdot \nabla u))),\tilde{u}^N\rangle \rightarrow 0. \end{aligned}$$The last relation is true, in fact
$$\begin{aligned}&\sum _{i=1}^N\langle P(\sigma _k\cdot \nabla u),\tilde{e}_i\rangle ^2\langle \tilde{e}_i, (I-\alpha ^2A)^{-1}\tilde{e}_i\rangle \\&\qquad +\langle P(\sigma _k\cdot \nabla ((I-\alpha ^2 A)^{-1}P(\sigma _k\cdot \nabla u))),\tilde{u}^N\rangle \\&\quad = \sum _{i=1}^N\langle P(\sigma _k\cdot \nabla u),(I-\alpha ^2 A)^{-1/2}\tilde{e}_i\rangle ^2\\&\qquad +\langle P(\sigma _k\cdot \nabla ((I-\alpha ^2 A)^{-1}P(\sigma _k\cdot \nabla u))),\tilde{u}^N\rangle \\&\quad \rightarrow \langle (I-\alpha ^2 A)^{-1}P(\sigma _k\cdot \nabla u),P(\sigma _k\cdot \nabla u)\rangle \\&\qquad +\langle P(\sigma _k\cdot \nabla ((I-\alpha ^2 A)^{-1}P(\sigma _k\cdot \nabla u))),u\rangle \\&\quad =0. \end{aligned}$$ -
From Theorem 33 and Eq. (5), we know that
$$\begin{aligned} \int _0^T{\mathbb {E}}\left[ \Vert \nabla u^N(s)\Vert _{L^2}^2\right] ds&\le \frac{1}{\alpha ^2}\int _0^T {\mathbb {E}}\left[ \Vert u^N(s)\Vert _V^2\right] ds\\&=\frac{1}{\alpha ^2}\int _0^T{\mathbb {E}}\left[ \Vert u(s)\Vert _V^2\right] ds+o(1). \end{aligned}$$Thus, from the Itô formula (81) the following relations hold true:
$$\begin{aligned}&\int _0^T {\mathbb {E}}\left[ \Vert \nabla u^N(s)\Vert _{L^2}^2 \right] ds\le \frac{C}{\alpha ^2}{\mathbb {E}}\left[ \Vert u_0\Vert ^2\right] +C{\mathbb {E}}\left[ \Vert \nabla u_0\Vert _{L^2}^2\right] +o(1)\end{aligned}$$(83)$$\begin{aligned}&{\mathbb {E}}\left[ {\text {sup}}_{t\in [0,T]}\Vert \nabla u(t)\Vert _{L^2}^2 \right] \le \frac{1}{\alpha ^2}{\mathbb {E}}\left[ \Vert u_0\Vert ^2\right] +{\mathbb {E}}\left[ \Vert \nabla u_0\Vert _{L^2}^2\right] \end{aligned}$$(84)According to inequality (25), in order to prove relation (82), it remains to study
$$\begin{aligned} {\mathbb {E}}\left[ {\text {sup}}_{t\in [0,T]}\Vert u^N(t)\Vert _*^2 \right] . \end{aligned}$$Before going on we recall some notation. For each \(N\in \mathbb {N}\)
$$\begin{aligned} \tau _M^N=\inf \{t:\ \Vert u^N(t)\Vert _V+\Vert u^N(t)\Vert _*\ge M \}\wedge T, \end{aligned}$$Thanks to the scaling factor \(\sqrt{\tilde{\nu }}\) appearing in front of the noise and exploiting the asymptotic relation between \(\nu ,\ \tilde{\nu }\) and \(\alpha ^2\) described by Hypothesis 6, if we choose
$$\begin{aligned} \epsilon =\frac{1}{2\sum _{k\in K}\Vert \sigma _k\Vert _{W^{1,\infty }}^2}, \end{aligned}$$Equation (62) in Lemma 26 becomes
$$\begin{aligned}&{\mathbb {E}}\left[ {\text {sup}}_{t\le r\wedge \tau _M^N}\Vert u^N(t)\Vert _*^2\right] +{\mathbb {E}}\left[ \int _0^{r\wedge \tau _M^N} \Vert u^N(s)\Vert _*^2ds\right] \nonumber \\&\quad \le C\left( {\mathbb {E}}\left[ \Vert u^N_0\Vert _*^2\right] +\left( \alpha ^2+1\right) {\mathbb {E}}\left[ \int _0^{r\wedge \tau _M^N} \Vert \nabla u^{N}(s)\Vert _{L^2}^2\right] \right) . \end{aligned}$$(85)Therefore, thanks to Eq. (83), we have
$$\begin{aligned}&{\mathbb {E}}\left[ {\text {sup}}_{t\le r\wedge \tau _M^N}\Vert u^N(t)\Vert _*^2\right] +{\mathbb {E}}\left[ \int _0^{r\wedge \tau _M^N} \Vert u^N(s)\Vert _*^2ds\right] \nonumber \\&\quad \le C\left( {\mathbb {E}}\left[ \Vert u^N_0\Vert _*^2\right] +\left( \alpha ^2+1\right) \left( {\mathbb {E}}\left[ \frac{\Vert u_0\Vert ^2}{\alpha ^2}\right] +{\mathbb {E}}\left[ \Vert \nabla u_0\Vert _{L^2}^2\right] \right) \right) +o(1). \end{aligned}$$(86)So far we showed that \(u^N\in L^2(\Omega ;L^2([0,T];H^1))\), \({\text {curl}}(u^N-\alpha ^2\Delta u^N)\in L^2(\Omega , L^{\infty }([0,T];L^2))\). By monotone convergence Theorem, we can remove the dependence from M in relation (86). Therefore
$$\begin{aligned} {\mathbb {E}}\left[ {\text {sup}}_{t\le T}\Vert u^N(t)\Vert _*^2\right]&\le C\left( {\mathbb {E}}\left[ \Vert u^N_0\Vert _*^2\right] \right. \nonumber \\ {}&\quad \left. +\left( \alpha ^2+1\right) \left( {\mathbb {E}}\left[ \frac{\Vert u_0\Vert ^2}{\alpha ^2}\right] +{\mathbb {E}}\left[ \Vert \nabla u_0\Vert _{L^2}^2\right] \right) \right) \nonumber \\&\quad +o(1). \end{aligned}$$(87)Thus, by Theorem 33 and the uniform bound (87) there exists a subsequence \(N_k\) such that
$$\begin{aligned}&u^{N_k}\rightarrow u\text { in } L^2(\Omega ;L^2([0,T];H^1))\\&{\text {curl}}(u^{N_k}-\alpha ^2\Delta u^{N_k}){\mathop {\rightharpoonup }\limits ^{*}} g \text { in } L^2(\Omega , L^{\infty }([0,T];L^2)). \end{aligned}$$If we take a test function \(\phi \in L^2(\Omega ;L^2(0,T;C^{\infty }_c(D)))\), we get easily
$$\begin{aligned}&{\mathbb {E}}\left[ \int _0^T \langle \phi (s),g(s)\rangle _{L^2} ds \right] \\ {}&=\lim _{k\rightarrow +\infty }{\mathbb {E}}\left[ \int _0^T \langle \phi (s),{\text {curl}}(u^{N_k}(s)-\alpha ^2\Delta u^{N_k}(s))\rangle _{L^2} ds\right] \\&= \lim _{k\rightarrow +\infty }{\mathbb {E}}\left[ \int _0^T \langle (I-\alpha ^2\Delta )\nabla ^{\perp }\phi (s),u^{N_k})(s)\rangle _{L^2} ds \right] \\&={\mathbb {E}}\left[ \int _0^T \langle (I-\alpha ^2\Delta )\nabla ^{\perp }\phi (s),u(s))\rangle _{L^2} ds\right] . \end{aligned}$$Therefore \(g={\text {curl}}(u-\alpha ^2\Delta u)\in L^2(\Omega , L^{\infty }([0,T];L^2)) \) and the following inequality holds true
$$\begin{aligned} {\mathbb {E}}\left[ {\text {sup}}_{t\le T}\Vert u(t)\Vert _{*}^2\right]&\le C\left( \liminf _{k\rightarrow +\infty }{\mathbb {E}}\left[ \Vert u^{N_k}_0\Vert _*^2\right] \right. \nonumber \\&\quad \left. +\left( \alpha ^2+1\right) \left( {\mathbb {E}}\left[ \frac{\Vert u_0\Vert ^2}{\alpha ^2}\right] +{\mathbb {E}}\left[ \Vert \nabla u_0\Vert _{L^2}^2\right] \right) \right) . \end{aligned}$$(88)Let us analyze better the first term. We denote by \(u_0^{N,\infty }=u_0-u_0^N\).
$$\begin{aligned} \Vert u^{N_k}_0\Vert _*^2&=\Vert u^{N_k}_0\Vert _W^2-\Vert u^{N_k}_0\Vert _V^2\\&\le \Vert u_0\Vert _W^2-\Vert u^{N_k}_0\Vert _V^2\\&\le \Vert u_0\Vert _*^2+\Vert u^{{N_k},\infty }_0\Vert _V^2\\&\le \Vert u_0\Vert _*^2+\Vert u_0\Vert _V^2\\&\le C( \Vert \nabla u_0\Vert _{L^2}^2+\alpha ^4\Vert {\text {curl}}\Delta u_0\Vert _{L^2}^2+\Vert u_0\Vert ^2+\alpha ^2\Vert \nabla u_0\Vert _{L^2}^2)\\&\le C( \Vert u_0\Vert ^2+(1+\alpha ^2)\Vert \nabla u_0\Vert _{L^2}^2+\alpha ^4\Vert u_0\Vert _{H^3}^2). \end{aligned}$$In conclusion, combining the observation above, relations (25), (84) and (88) we get
$$\begin{aligned} {\mathbb {E}}\left[ {\text {sup}}_{t\le T}\Vert u(t)\Vert _{H^3}^2\right]&\le C\left( \frac{\alpha ^4+\alpha ^2+1}{\alpha ^6}{\mathbb {E}}\left[ \Vert u_0\Vert ^2\right] +\frac{1+\alpha ^2+\alpha ^4}{\alpha ^4}{\mathbb {E}}\left[ \Vert \nabla u_0\Vert _{L^2}^2\right] \right. \nonumber \\&\quad \left. + {\mathbb {E}}\left[ \Vert u_0\Vert _{H^3}^2\right] \right) . \end{aligned}$$(89)
Thanks to the assumptions on \(u_0^{\alpha }\), see Hypothesis 6, the thesis follows.
Remark 36
In the case \(\nu =0\), relation (81) follows without any change with respect to the main proof. For what concerns relation (82), Eq. (85) above is false in this framework. However, introducing the proper scaling in front of the noise we can restart from relation (63) obtaining
Therefore, combining estimates (58),(59),(60),(61), exploiting the asymptotic relation between \(\tilde{\nu }\) and \(\alpha ^2\) described by Hypothesis 6 and choosing \(\epsilon =\frac{1}{\sum _{k\in K}\Vert \sigma _k\Vert _{W^{1,\infty }}^2}\), we obtain
Therefore, thanks to Eq. (83), we have
Arguing as in Remark 27, we can apply Grönwall’s Lemma in inequality (92) obtaining
Relation (93) is completely analogous to relation (86) above. Therefore we can follow the same argument of the main proof in order to obtain estimate (82) and we omit the details.
5 Proof of Theorem 9
In order to prove Theorem 9, we will follow the ideas of [27, 28]. We will start with a weaker result with the supremum in time outside the expected value and then we will move to the stronger one with the supremum in time inside the expected value.
Proof of Theorem 9
Let \(W^{\alpha }=u^{\alpha }-\bar{u}\), it satisfies \(\mathbb {P}-a.s.\) for each \(\phi \in H\) and \(t\in [0,T]\)
Following the idea of [21], let v the corrector of the boundary layer of width \(\delta \), i.e. a divergence free vector field with support in a strip of the boundary of width \(\delta \) such that \(\bar{u}-v\in V\) and
Let \(\delta =\delta (\alpha )\) such that
We want to write the Itô’s formula for \(\Vert W^{\alpha }(t)\Vert ^2\). Let us take an orthonormal basis of H, \(\{\tilde{e}_i\}\) made by eigenvectors of A, let \(\{-\tilde{\lambda }_i\}\) the corresponding eigenvalues. Let us consider the weak formulation with test functions \(\phi =\tilde{e}_i\), let us call \(W^{\alpha ,n}=\sum _{i=1}^n \langle W^{\alpha },\tilde{e}_i\rangle \tilde{e}_i\), \(u^{\alpha ,n}=\sum _{i=1}^n \langle u^{\alpha },\tilde{e}_i\rangle \tilde{e}_i\), \(\bar{u}^n=\sum _{i=1}^n \langle \bar{u},\tilde{e}_i\rangle \tilde{e}_i\) e \(v^n=\sum _{i=1}^n\langle v,\tilde{e}_i\rangle \tilde{e}_i\), then, arguing as in the proof of Lemma 35, we get
Therefore
In the same way, considering the weak formulation satisfied by \(u^{\alpha }\), we get
Combining relation (96), (97), (98) we obtain
Let us rewrite \(\langle W^{\alpha ,n},dAu^{\alpha ,n}\rangle \) in a different way
Therefore, we arrive to this final expression
Now, letting \(n\rightarrow +\infty \), exploiting the regularity of \(u^{\alpha },\ \bar{u},\ v\) and the continuity of the trilinear form b we arrive to the formula below
where:
Our approach is almost completely pathwise. Therefore we need to estimate the terms \(I_i(t),\ t\in \{1,\ldots ,6\}\). The analysis of \(I_1(t)\) follows by Young’s inequality, the estimates on the boundary layer corrector (94) and the interpolation estimate (42)
The analysis of \(I_2(t)\) follows by Young’s inequality and the results of Lemma 21, Corollary 22. Indeed it holds
The analysis of \(I_3(t)\) follows by Young’s inequality, the estimates on the boundary layer corrector (94) and the interpolation estimate (42)
The analysis of \(I_4(t)\) is analogous to Eqs. (3.20)–(3.22) in [27], it implies:
Therefore by the interpolation inequality (42) and Young’s inequality we have
The analysis of \(I_5(t)\) follows immediately by Hölder’s inequality:
For what concerns the analysis of \(I_6(t)\), preliminary we observe that
Arguing as in [24], Equations (4.18)–(4.19) we get
Combining Eqs. (99), (100), (101), (102), (103), (104) and exploiting our assumptions on the behavior of \(\nu ,\, \tilde{\nu }, \alpha ^2\), see Hypothesis 6, we have the integral relation below:
By the stochastic Grönwall’s Lemma 24 above we have:
Thanks to Hypothesis 6 and our assumptions on \(\delta \), see Eq. (95), we have that
Thanks to Lemma 35, we have that
Therefore
Restarting from Eq. (105) and considering the expected value of the supremum of both the terms in the left hand side we have
We already proved that almost all the terms in the right hand side of Eq. (110) go to 0. Therefore in order to complete the proof we left to show that
By the weaker convergence described by Eq. (109) and Fubini Theorem
For what concerns the other, the convergence follows by Burkholder–Davis–Gundy inequality, Hypothesis 6, Eq. (109), Fubini Theorem and relation (27). Indeed
Now the proof is complete.
Remark 37
Combining Lemma 35 and Theorem 9 we understand that, if \(\nu =O(\alpha ^2)\) and \(\tilde{\nu }=O(\alpha ^2)\), the assumptions on the behavior of the initial conditions \(u_0^{\alpha }\) in norm \(H,\ H^1\) and \(H^3\) are satisfied also for \(t\in [0,T]\).
6 The case of additive noise
For what concerns the case with additive noise, as stated in Sect. 2, the well-posedness is a well-known fact in case of \(\nu >0\) and we can prove a result completely analogous to Theorem 9, following exactly the same argument. However, the restriction \(\nu >0\) can be omitted modifying slightly the proof of [34] as described in Remarks 27, 29 and 36. However, we do not stress this assumption in this section, therefore \(\nu >0\) in what follows. What was crucial for the proof of Theorem 9 were the energy estimates of Lemma 35. Thus in this section we want to explain a different approach to prove these energy estimates in the case of additive noise. These computations are more similar to what happens in the deterministic framework. We keep previous assumptions on the coefficients \(\sigma _k\) and the Brownian motions \(W^k\). For generality reasons we consider the equations without any scaling factor on the noise. Thus we consider
Before going on, we need to recall a result of [24].
Lemma 38
Let \(q\in L^2(D)\), there exists a unique \(\phi \in H^2_0(D)\) solution of
which satisfies
Moreover, the solution map is continuous from \(L^2(D)\) to \(H^2_0(D)\cap H^4(D).\)
Thanks to this Lemma, we can define an operator \(\mathbb {K}:L^2(D)\rightarrow H^3(D)\cap W^{1,\infty }_0(D)\) which associates to each \(q\in L^2(D)\) the vector field \(u=\nabla ^{\perp } \phi \), where \(\phi \) is the solution of the equation of Lemma 38.
Definition 39
A stochastic process u weakly continuous with values in W and continuous with values in V is a weak solution of Eq. (111) if
and \(\mathbb {P}-a.s.\) for every \(t\in [0,T]\) and \(\phi \in D(A)\) we have
Arguing as in the first part of the proof of Lemma 35 we can prove the following result.
Lemma 40
Let u be a weak solution of problem (111) in the sense of Definition 39, then the following relations hold true
-
1.
$$\begin{aligned} d\Vert u\Vert ^2+\alpha ^2d\Vert \nabla u\Vert _{L^2}^2= & {} \left( -2\nu \Vert \nabla u\Vert _{L^2}^2+\sum _{k\in K} \langle \sigma _k,(I-\alpha ^2A)^{-1}\sigma _k\rangle \right) dt\\{} & {} + 2\sum _{k\in K} \langle \sigma _k,u \rangle dW^k_t \end{aligned}$$
-
2.
$$\begin{aligned}&{\mathbb {E}}\left[ \Vert u(t)\Vert ^2\right] +\alpha ^2{\mathbb {E}}\left[ \Vert \nabla u(t)\Vert _{L^2}^2\right] +2\nu \int _0^t {\mathbb {E}}\left[ \Vert \nabla u(s)\Vert _{L^2}^2\right] ds\\&\quad = {\mathbb {E}}\left[ \Vert u_0\Vert ^2\right] +\alpha ^2{\mathbb {E}}\left[ \Vert \nabla u_0\Vert _{L^2}^2\right] +t\sum _{k\in K} \langle \sigma _k,(I-\alpha ^2A)^{-1}\sigma _k\rangle \end{aligned}$$
-
3.
$$\begin{aligned}&{\mathbb {E}}\left[ {\text {sup}}_{t\in [0,T]}\Vert u(t)\Vert ^2\right] +\alpha ^2{\mathbb {E}}\left[ {\text {sup}}_{t\in [0,T]}\Vert \nabla u(t)\Vert _{L^2}^2\right] +2\nu \int _0^T {\mathbb {E}}\left[ \Vert \nabla u(s)\Vert _{L^2}^2\right] ds \\&\quad \le C\left( {\mathbb {E}}\left[ \Vert u_0\Vert ^2\right] +\alpha ^2{\mathbb {E}}\left[ \Vert \nabla u_0\Vert _{L^2}^2\right] +T\sum _{k\in K} \langle \sigma _k,(I-\alpha ^2A)^{-1}\sigma _k\rangle \right. \\&\qquad \left. +{\mathbb {E}}\left[ \left( \int _0^T \sum _{k\in K}\langle \sigma _k,u(s) \rangle ^2ds\right) ^{1/2}\right] \right) \end{aligned}$$
Let us introduce the vorticity formulation of (111), we denote \(s_k={\text {curl}}\sigma _k\)
Definition 41
A stochastic process q, which is weakly continuous with values in \(L^2(D)\) and continuous with values in \(H^{-1}(D)\), is a weak solution of Eq. (112) if
and \(\mathbb {P}-a.s.\) for every \(t\in [0,T]\) and \(\phi \in H^2_0(D)\) we have
\(u=\nabla ^{\perp }\varphi ,\) \(\varphi \) obtained by Lemma 38, \( u\in W.\)
Let us obtain a result about the equivalence between the solutions of these two problems. Since we know from the results of [34] that problem (111) is well-posed, then problem (112) is well-posed as well.
Proposition 42
Let u be a solution of (111) in the sense of Definition 39, then \(q:={\text {curl}}(u-\alpha ^2\Delta u)\) is a solution of (112) in the sense of Definition 41. Conversely, if q is a solution of (112) in the sense of Definition 41 then \(u=\nabla ^{\perp }\varphi \), \(\varphi \) obtained by Lemma 38, is a solution of (111) in the sense of Definition 39.
Proof
Definition 39\(\implies \) Definition 41 is immediate taking \(\phi =-\nabla ^{\perp } \tilde{\phi }\), \(\tilde{\phi }\in H^2_0(D)\) as test function for problem (111).
Therefore it remains to show that Definition 41\(\implies \) Definition 39. We take \(u=\nabla ^{\perp }\varphi ,\ v=u-\alpha ^{2}\Delta u,\) where \(\varphi \) is obtained by Lemma 38 and \(\phi =-\nabla ^{\perp }\tilde{\phi },\) where \(\tilde{\phi }\in H^2_0(D)\). Then integrating by parts and exploiting that \({\text {curl}}\nabla ^{\perp }=\Delta \), \(\Delta \varphi -\alpha ^2\Delta ^2\varphi =q\) and q is a solution of (112) in the sense of Definition 41 we get
From the last relation the thesis follows if we are able to prove the continuity properties of u. The weak continuity of u with values in W follows immediately from the regularity of q and Lemma 38. Again by Lemma 38 we get the strong continuity of u with values in V. In fact, via Lax–Milgram Lemma we get the regularity of the solution mapping of the problem described in Lemma 38 between \(H^{-2}(D)\) and \(H^2_0(D)\). Via interpolation techniques we recover the regularity of the solution mapping between \(H^{-1}(D)\) and \(H^3(D)\cap H^2_0(D)\), therefore the required regularity for u.
Approximating the process q(t) solution of (112) by the eigenvectors of the Laplacian with Dirichlet boundary conditions and then arguing as in the first part of the proof of Lemma 35, we can obtain some Itô’s formula and energy estimates. Moreover, if \(u\in V\) we have \(\Vert \nabla u\Vert _{L^2}^2=\Vert {\text {curl}}u\Vert _{L^2}^2\). Thanks to Proposition 42, we know that u appearing in problem (112) is a solution of problem (111). Therefore, thanks to Lemma 40 we know that
and we can obtain the following energy relations.
Lemma 43
Let q be a weak solution of problem (112) in the sense of Definition 41, then the following relations hold true
-
1.
$$\begin{aligned} d\Vert q\Vert ^2=-\frac{2\nu }{\alpha ^2}\langle q-{\text {curl}}u,q\rangle dt+\sum _{k\in K}\Vert s_k\Vert ^2\ dt +2\sum _{k\in K} \langle s_k,q\rangle dW^k_t \end{aligned}$$
-
2.
$$\begin{aligned} {\mathbb {E}}\left[ \Vert q(t)\Vert ^2\right]&\le e^{-\frac{\nu }{\alpha ^2}t} {\mathbb {E}}\left[ \Vert q_0\Vert ^2\right] +\frac{\alpha ^2}{\nu }(1-e^{-\frac{\nu t}{\alpha ^2}})\sum _{k\in K}\Vert s_k\Vert ^2 \\&\quad +\frac{1}{2\nu }\left( {\mathbb {E}}\left[ \Vert u_0\Vert ^2\right] +\alpha ^2{\mathbb {E}}\left[ \Vert \nabla u_0\Vert _{L^2}^2\right] \right. \\&\quad \left. +T\sum _{k\in K} \langle \sigma _k,(I-\alpha ^2A)^{-1}\sigma _k\rangle \right) \end{aligned}$$
-
3.
$$\begin{aligned} {\mathbb {E}}\left[ {\text {sup}}_{t\in [0,T]}\Vert q(t)\Vert ^2\right]&\le {\mathbb {E}}\left[ \Vert q_0\Vert ^2\right] +\sum _{k\in K} \Vert s_k\Vert ^2 T\\&\quad + C{\mathbb {E}}\left[ \left( \sum _{k\in K}\int _0^T\langle s_k,q(s) \rangle ^2ds\right) ^{1/2}\right] \\&\quad +\frac{1}{2\alpha ^2}\left( {\mathbb {E}}\left[ \Vert u_0\Vert ^2\right] +\alpha ^2{\mathbb {E}}\left[ \Vert \nabla u_0\Vert _{L^2}^2\right] \right. \\&\quad \left. +T\sum _{k\in K} \langle \sigma _k,(I-\alpha ^2A)^{-1}\sigma _k\rangle \right) . \end{aligned}$$
Remark 44
We can control the \(H^3\) norm of u via the \(H^1\) norm of u and the \(L^2\) norm of q in the following way
Therefore, thanks to Lemma 43 it holds
Remark 45
If we consider the scaled equations with \(\sqrt{\tilde{\nu }}\) in front of the noise, then each \(\sigma _k\) and \(s_k\) is multiplied by \(\sqrt{\tilde{\nu }}\) in Lemmas 40, 43 and Remark 44.
Thanks to Remark 44, if we consider the scaled equation with additive noise and initial condition \(u_0^{\alpha }\) satisfying Hypothesis 6, then the following result follows immediately.
Lemma 46
If we consider the stochastic second-grade fluid equations with additive noise (111) scaled by \(\sqrt{\tilde{\nu }}\), under Hypothesis 2–6, if \(u^{\alpha }\) is the solution in the sense of Definition 39 of the problem with initial condition \(u_0^{\alpha }\), then
Looking carefully at the proof of Theorem 9, Lemma 46 contains the crucial bounds on the norm of the solutions to obtain the inviscid limit. Therefore, following the same ideas of Sect. 5, one can prove that the inviscid limit holds:
Theorem 47
Under Hypotheses 1–6, calling \(u^{\alpha }\) the solution of the stochastic second-grade fluid equations with additive noise (111) scaled by \(\sqrt{\tilde{\nu }}\) and \(\bar{u}\) the solution of (15), then
References
Bensoussan, A.: Stochastic Navier–Stokes equations. Acta Appl. Math. 38(3), 267–304 (1995)
Breckner, H.: Galerkin approximation and the strong solution of the Navier–Stokes equation. J. Appl. Math. Stoch. Anal. 13(3), 239–259 (2000)
Butori, F., Luongo, E.: Large Deviations principle for the inviscid limit of fluid dynamic systems in 2D bounded domains. arXiv:2305.11148 (2023)
Capinski, M., Gatarek, D.: Stochastic equations in Hilbert space with application to Navier–Stokes equations in any dimension. J. Funct. Anal. 126(1), 26–35 (1994)
Carigi, G., Luongo, E.: Dissipation properties of transport noise in the two-layer quasi-geostrophic model. J. Math. Fluid Mech. 25(2), 28 (2023)
Cioranescu, D., Girault, V.: Weak and classical solutions of a family of second grade fluids. Int. J. Non-Linear Mech. 32(2), 317–335 (1997)
Cioranescu, D., Ouazar, E.H.: Existence and uniqueness for fluids of second grade. Nonlinear Partial Differ. Equ. 109, 178–197 (1984)
Constantin, P., Kukavica, I., Vicol, V.: On the inviscid limit of the Navier–Stokes equations. Proc. Am. Math. Soc. 143(7), 3075–3090 (2015)
Debussche, A., Pappalettera, U.: Second order perturbation theory of two-scale systems in fluid dynamics. arXiv:2206.07775 (2022)
Dunn, J.E., Fosdick, R.L.: Thermodynamics, stability, and boundedness of fluids of complexity 2 and fluids of second grade. Arch. Ration. Mech. Anal. 56(3), 191–252 (1974)
Flandoli, F., Galeati, L., Luo, D.: Quantitative convergence rates for scaling limit of SPDEs with transport noise. arXiv:2104.01740 (2021)
Flandoli, F., Galeati, L., Luo, D.: Scaling limit of stochastic 2D Euler equations with transport noises to the deterministic Navier–Stokes equations. J. Evol. Equ. 21(1), 567–600 (2021)
Flandoli, F., Luongo, E.: Stochastic Partial Differential Equations in Fluid Mechanics, vol. 2328. Springer Nature, Berlin (2023)
Flandoli, F., Pappalettera, U.: From additive to transport noise in 2D fluid dynamics. In: Stochastics and Partial Differential Equations: Analysis and Computations, pp. 1–41 (2022)
Galdi, G.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations: Steady-State Problems. Springer Science & Business Media, Berlin (2011)
Galeati, L.: On the convergence of stochastic transport equations to a deterministic parabolic one. Stoch. Partial Differ. Equ. Anal. Comput. 8(4), 833–868 (2020)
Goodair, D., Crisan, D.: The zero viscosity limit of stochastic Navier–Stokes flows. arXiv:2305.18836 (2023)
Holm, D.D.: Variational principles for stochastic fluid dynamics. Proc. R. Soc. A Math. Phys. Eng. Sci. 471(2176), 20140963 (2015)
Iftimie, D.: Remarques sur la limite \(\alpha \rightarrow 0\) pour les fluides de grade 2. In: Studies in Mathematics and its Applications, vol. 31, pp. 457–468. Elsevier (2002)
Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus, vol. 113. Springer Science & Business Media, Berlin (2012)
Kato, T.: Remarks on zero viscosity limit for nonstationary Navier–Stokes flows with boundary. In: Seminar on Nonlinear Partial Differential Equations, pp. 85–98. Springer (1984)
Kato, T., Lai, C.Y.: Nonlinear evolution equations and the Euler flow. J. Funct. Anal. 56(1), 15–28 (1984)
Kuksin, S.B.: The Eulerian limit for 2D statistical hydrodynamics. J. Stat. Phys. 115(1), 469–492 (2004)
Lopes-Filho, M.C., Lopes, H.J.N., Titi, E.S., Zang, A.: Convergence of the 2D Euler-\(\alpha \) to Euler equations in the Dirichlet case: indifference to boundary layers. Phys. D: Nonlinear Phenom. 292, 51–61 (2015)
Lopes Filho, M.C., Mazzucato, A.L., Lopes, H.N.: Vanishing viscosity limit for incompressible flow inside a rotating circle. Physica D 237(10–12), 1324–1333 (2008)
Lopes Filho, M.C., Mazzucato, A.L., Lopes, H.N., Taylor, M.: Vanishing viscosity limits and boundary layers for circularly symmetric 2D flows. Bull. Braz. Math. Soc. New Ser. 39(4), 471–513 (2008)
Lopes Filho, M.C., Nussenzveig Lopes, H.J., Titi, E.S., Zang, A.: Approximation of 2D Euler equations by the second-grade fluid equations with Dirichlet boundary conditions. J. Math. Fluid Mech. 17(2), 327–340 (2015)
Luongo, E.: Inviscid limit for stochastic Navier–Stokes equations under general initial conditions. arXiv:2111.14189 (2021)
Maekawa, Y.: On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half-plane. Commun. Pure Appl. Math. 67(7), 1045–1128 (2014)
Marsden, J.E., Ratiu, T.S., Shkoller, S.: The geometry and analysis of the averaged Euler equations and a new diffeomorphism group. Geom. Funct. Anal. GAFA 10(3), 582–599 (2000)
Pardoux, E.: Equations aux dérivées partielles stochastiques monotones. University, These (1975)
Razafimandimby, P.A.: Grade-two fluids on non-smooth domain driven by multiplicative noise: existence, uniqueness and regularity. J. Differ. Equ. 263(5), 3027–3089 (2017)
Razafimandimby, P.A., Sango, M.: Weak solutions of a stochastic model for two-dimensional second grade fluids. Bound. Value Probl. 1–47, 2010 (2010)
Razafimandimby, P.A., Sango, M.: Strong solution for a stochastic model of two-dimensional second grade fluids: existence, uniqueness and asymptotic behavior. Nonlinear Anal. Theory Methods Appl. 75(11), 4251–4270 (2012)
Reed, M.: Methods of Modern Mathematical Physics: Functional Analysis. Elsevier, Amsterdam (2012)
Rivlin, R. S., Ericksen, J. L.: Stress-deformation relations for isotropic materials. In: Collected Papers of RS Rivlin, pp. 911–1013 (1997)
Sammartino, M., Caflisch, R. E.: Zero viscosity limit for analytic solutions of the Navier–Stokes equation on a half-space. P II. Construction of the Navier–Stokes Solution. Commun. Math. Phys. 192(2), 463–491 (1998)
Scheutzow, M.: A stochastic Gronwall lemma. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16(02), 1350019 (2013)
Shkoller, S.: Analysis on groups of diffeomorphisms of manifolds with boundary and the averaged motion of a fluid. J. Differ. Geom. 55(1), 145–191 (2000)
Skorokhod, A.V.: Studies in the Theory of Random Processes, vol. 7021. Courier Dover Publications, Mineola (1982)
Temam, R.: On the Euler equations of incompressible perfect fluids. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi" Séminaire Goulaouic-Schwartz", pp. 1–14 (1974)
Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis, vol. 343. American Mathematical Society (2001)
Temam, R., Wang, X.: On the behavior of the solutions of the Navier–Stokes equations at vanishing viscosity. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 25(3–4), 807–828 (1997)
Wang, X.: A Kato type theorem on zero viscosity limit of Navier–Stokes flows. Indiana Univ. Math. J. 223–241 (2001)
Acknowledgements
I want to thank Professor Franco Flandoli and Professor Edriss Titi for useful discussions and valuable insights into the subject. I also would like to express my sincere gratitude to the referees and the editor for their careful reading and helpful suggestions which improved drastically the presentation of the paper from its initial version.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declares no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Luongo, E. Inviscid limit for stochastic second-grade fluid equations. Stoch PDE: Anal Comp 12, 1046–1099 (2024). https://doi.org/10.1007/s40072-023-00303-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40072-023-00303-y