数理解析研究所講究録
第 1905 巻 2014 年 5-17

5

UNCONDITIONAL EXISTENCE OF DENSITIES FOR THE NAVIER-STOKESEQUATIONS WITH NOISEMARCO ROMITO

ABSTRACT. The first part of the paper contains a short review of recent results about theexistence of densities for finite dimensional functionals of weak solutions of the NavierStokes equations forced by Gaussian noise. Such results are obtained for solutions limitof spectral Galerkin approximations.
In the second part of the paper we prove via a ”transfer principl that existenceof densities is universal, in the sense that it does not depend on how the solution has
been obtained, given some minimal and reasonable conditions of consistence underconditional probabilities and weak-strong uniqueness. A quantitative version of thetransfer principle is also available for stationary solutions.
$e’$1. INTRODUCTION

When dealing with a stochastic evolution PDE, the solution depends not only on thetime and space independent variables, but also on th chanc variable, which plays
a completely diﬀerent role. The existence of a density for the probability distributionof the solution is thus a form of regularity with respect to this new variable.
In this paper we detail some results related to the existence of densities of finite
dimensional projections of any solution of the Navier-Stokes equations$e^{}$$e’$$\dot{u}+(u\cdot\nabla)u+\nabla p=v\Delta u+\eta,$(1.1)

divu $=0,$

with Dirichlet boundary conditions on a bounded domain, or with periodic boundary
conditions on the torus. Here is Gaussian noise. Most of the results have appeared in[DR14, Rom13], some additional results are in progress $[Roml4b, Roml4c, Roml4a].$
To be more precise, our result concerns the existence of densities for finite dimensional functionals of the solution, and one reason for this is that there is no canonicalreference measure in infinite dimension, as is the Lebesgue measure in finite dimension. To understand the right reference measure is an open problem even in dimension
two and for any suitable choice of the driving noise.
Our interest in the existence of densities stems from a series of mathematical mo$\eta$tivations. The first and foremost is the investigation of the regularity properties ofsolutions of the Navier-Stokes equations.
On the other hand regularity is not the only open problem in the mathematical theory of the Navier-Stokes equations (either with random forcing, or without). The firstobvious choice is the related problem of uniqueness. In the probabilistic framework we
can deal with diﬀerent notions of uniqueness, the weaker being the statistical uniqueness, that is the uniqueness of distributions. Although the results detailed in this paper2010 Mathematics Subject Classification. Primary $76M35,\cdot$ Secondary $60H15,$ $60G30,$ $35Q30.$
Key words and phrases. Density of laws, Navier-Stokes equations, stochastic partial diﬀerential equations, weak solutions.

6
are very far from any uniqueness result, we mention that the law of an infinite dimensional random variable can be characterized by the laws of its finite dimensional
projections. By the results of [FR08], it is then suﬃcient to show that the laws of two
solutions agree at every time. An even easier condition, following from the results of[Rom08], requires that we show agreement between the laws of the corresponding invariant measures, that is, if the processes agree at time $t=\infty$ , then they agree for every
time, including their time correlations.

An additional rather vague though folkloristi motivation for the interest in finite dimensional projections is that most of the real-life experiments to evaluate thevelocity of a fluid are based on a finite number of samples in a finite number of points(Eulerian point of view), or by tracing some particles (smoke, etc ) moving according
to the fluid velocity (Lagrangian point of view). The literature on experimental fluid
dynamics is huge. Here we refer for instance to [Tav05] for some examples of design$c’$$)^{}$$($$\cdots$of experiments. Let us focus on the Eulerian point of view. To simplify, consider atorus, then sampling the velocity field means measuring the velocity in some spacepoints , .. . , $v_{d}:urightarrow(u(t,y_{1}), \ldots,u(t_{Vd}))$ , and a bit of Fourier series manipulations
shows that this is $a”projection”.$
An interesting diﬃculty in proving regularity of the density emerges as a by-productof the (more general and fundamental!) problem of proving uniqueness and regularityof solutions of the Navier-Stokes equations. Indeed, a fundamental and classical tool
is the Malliavin calculus, a diﬀerential calculus where the diﬀerentiating variable is the, the derivative
underlying noise driving the system. The Malliavin derivativewith respect to the variations of the noise perturbation, is given as$y_{1}$$\mathcal{D}_{H}u(t)$$\mathcal{D}_{H}u=\lim_{\epsilon\downarrow 0}\frac{u(W+\epsilon\int Hds)-u(W)}{\epsilon},$where we have written the solution as $u(W)$ to show the explicit dependence offrom the noise forcing. We point, for instance, to [Nua06] for further details andof the solution ofdefinitions, and we only notice that the Malliavin derivative
solution,variation,the(1.1), as anamely,satisfies the linearization around
$u$

$u$

$\mathcal{D}_{H}u$$u$$\frac{d}{dt}\mathcal{D}_{H}u-v\Delta \mathcal{D}_{H}u+(u\cdot\nabla)\mathcal{D}_{H}u+((\mathcal{D}_{H}u)\cdot\nabla)u=SH,$and good estimates on
originate only from good estimates on the linearizationof (1.1), which are not available so far. This settles the need of methods to prove existence and regularity of the density that do not rely on this calculus, as done in [DR14].
In this paper we tackle the problem of universality of the result obtained in [DR14],
which are valid only for limits of Galerkin approximations. At the present time we
do not know if the Navier-Stokes equations admit a unique $distribution_{r}$ therefore it
might happen that solutions obtained by diﬀerent means may have diﬀerent properties. In a way this is reminiscent of the problem of suitable weak solutions introduced
by [Sch77]. Only much later it has been proved that solutions obtained by the spectral
Galerkin methods are suitable [Gue06] (under some $non\dashv$ rivial conditions though),and hence results of partial regularity are true for those solutions.
Our main theorem is transfer principl (Theorem 4.1), that states that as long as
we can prove existence of a density for a finite dimensional functional of the solution
and for a weak solution that satisfies weak strong uniqueness, then existence of a density$\mathcal{D}_{H}u(t)$$a^{ノ/}$$e’$7
holds for any other solution satisfying weak-strong uniqueness and a closure property
with respect to conditional probabilities.
An important limitation of our transfer principle is that it applies only on’ instantaneou properties, namely to random variables depending only on one time, in particular, the results on time continuity of densities in $[Roml4c]$ are still out of reach.
The transfer principle is qualitative in nature, as it may transfer only the existence. In
general no quantitative information can be inherited. This seems to be mainly an artefact of the proof, that in turns depends on good moments of the solution in smootherspaces. Indeed, in the case of stationary solutions, we can prove a quantitative versionof the principle (Theorem 4.2).
$s’$

2. WEAK SOLUTIONS
We consider problem (1.1) with either periodic boundary conditions on the threedimensional torus
or Dirichlet boundary conditions on a smooth domain. We will understand weak martingale solutions of (1.1) as probability measureson the path space. We will then define legit families of solutions as classes of solutions that are closed by conditional probability and for which weak-strong uniquenessholds.
$\mathbb{T}_{3}=[0,$ $2\pi|^{3}$$0\subset R^{3}$

2.1. Preliminaries. Let be the standard space of square summable divergence freevector fields, defined as the closure of divergence free smooth vector fields satisfying
the boundary condition, with inner productand norm. Define likewise
V as the closure with respect to thenorm. Letbe the Leray projector, $A=$
the Stokes operator, and denote byandthe eigenvalues and theorthonormal
corresponding
basis of eigenvectors of A. Define the bi-linear operator: $V\cross Varrow V’$ as $B(u,\nu)=\Pi_{L}(u\cdot\nabla\nu)$ , $u,\nu\in$ V. We recall that
We refer to Temam [Tem95] for a detailed account of all the above
definitions.
The noisein (1.1) is coloured in space by a covariance operator S @(H),where $W$ is a cylindrical Wiener process (see [DPZ92] for further details). We assume
that @ @ is trace-class and we denote byits trace. Finally, consider thesequenceof eigenvalues of@ @, and let
be the orthonormal basis inof eigenvectors of @ S. For simplicity we may assume that the Stokes operator A andthe covariance commute, so that$H$$\rangle_{H}$$H^{1}$$-\Pi_{L}\Delta$$\Vert_{H}$$\Vert$$\Pi_{L}$$(\lambda_{k})_{1c\geq 1}$$(e_{k})_{k\geq 1}$$B$$-\langle u_{3},$$\langle u_{1},$$B(u_{2},u_{3})\rangle=$$B(u_{2},u_{1}$$\dot{\eta}=S\dot{W}$$\star$$\in \mathscr{L}$$\star$$\sigma^{2}=R(@^{\star}@)$$\star$$(\sigma_{k}^{2})_{k\geq 1}$$H$$(q_{k})_{k\geq 1}$$\star$$\dot{\eta}(t,y)=SdW=\sum_{k\in Z_{\star}^{3}}\sigma_{k}\dot{\beta}_{k}(t)e_{k}(v)$.

2.2. Weak and strong solutions. With the above notations, we can recast problem (1.1)
as an abstract stochastic equation,(2.1)$du+(vAu+B(u))dt=SdW,$
with initial condition $u(0)=x\in$ H. It is well-known [Fla08] that for every $x\in H$
there exist a martingale solution of this equation, that is a filtered probability space, a cylindrical Wiener process
and a process with trajectories insuch that the above
adapted to$W.$equation is satisfied with replacing
We will describe, equivalently, a martingale solution as a measure on the path space(in other words via a martingale problem). Let $\Omega_{NS}=C([0, \infty);D(A)’)$ and let$\overline{W}$$(\tilde{\Omega},\overline{\mathscr{F},}\tilde{\mathbb{P}},\{\overline{\mathscr{F}_{t}}\}_{t\geq 0})$$u$$C([0, \infty);D(A)’)\cap L_{1oc}^{\infty}([0, \infty), H)\cap L_{1oc}^{2}([0, \infty);V)$$(\overline{\mathscr{F}_{t}})_{t\geq 0}$$\overline{W}$$\mathscr{F}^{NS}$8the -algebra generated by the restrictions of
be its Borel -algebra. Denote by$[0,t]$(roughly speaking, this is the same as the Borel
to the interval
elements of,-algebra of $C([0, t];D(A)’)$ ). Let be the canonical process, defined byforis a solution of the martingale
Definition 2.1 ([FR08]). A probability measure on
problem associated to (2.1) with initial distribution if$\mathscr{F}_{t}^{NS}$$\sigma$$\sigma$$\Omega_{NS}$$\xi_{t}(\omega)=\omega(t)$$\xi$$\sigma$$\omega\in\Omega_{NS}$$\mathbb{P}$$\Omega_{NS}$$\mu$$\blacksquare \mathbb{P}[L_{1\circ c}^{\infty}(R^{+}, H)\cap L_{1oc}^{2}(R^{+};V)]=1,$$\blacksquare$for each $\phi\in D(A)$ , the process$\langle\xi_{t}-\xi_{0}, \phi\rangle+\int_{0}^{t}\langle\xi_{s}, A\phi\rangle-\langle B(\xi_{s}, \phi) , \xi_{s}\rangle ds$$\blacksquare$is a continuous square summable martingale with quadratic variation(hence a Brownian motion),
the marginal of at time is$\mathbb{P}$$0$$t\Vert S\phi\Vert_{H}^{2}$$\mu.$The second condition in the definition above has a twofold meaning. On the one
hand it states that the canonical process is a weak (in terms of PDEs) solution, on theother hand it identifies the driving Wiener process, and hence is a weak (in terms ofstochastic analysis) solution.

2.2.1. Strong solutions. It is also well-known that (2.1) admits local smooth solutions
that corresponds to the (posdefined up to a random time (a stopping time, in fact)
sible) time of blow-up in higher norms. To consider a quantitative version of the local
can be approximated monotonically by a sequence ofsmooth solutions, notice thattimes
stopping$\tau_{\infty}$$\tau_{\infty}$$\tau_{R}=\inf\{t>0:\Vert Au_{R}(t)\Vert_{H}\geq R\},$

where$u_{R}$

is a solution of the following problem,$du_{R}+(vAu_{R}+\chi(\Vert Au_{R}\Vert_{H}^{2}/R^{2})B(u_{R},u_{R}))dt=@dW,$) $arrow[0$ , 1 is a suitable $cut-0ﬀ$ funcwith initial condition in $D(A)$ , and where :ontion, namely a non-increasing
on , 1 andfunction such that[2,
. The process
is a strong (in PDE sense) solution of the cut-oﬀ equation.
$\chi$$C^{\infty}$$\infty)$$[0, \infty$$]$$\chi\equiv 1$

$[0$

$]$

$\chi_{R}\equiv 0$$u_{R}$

Moreover it is a strong solution also in terms of stochastic analysis, so it can be realized
uniquely on any probability space, given the noise perturbation.
As it is well-known in the theory of Navier-Stokes equations, the regular solution is
unique in the class of weak solutions that satisfy some form of the energy inequality.
We will give two examples of such classes for the equations with noise.
Remark 2.2. The analysis of strong (PDE meaning) solutions can be done on larger
spaces, up to $D(A^{1/4})$ , which is a critical space with respect to the Navier-Stokes scaling. The extension is a bit technical though, see [Romll].

92.2.2. Solutions satisfying the almost sure energy inequality. An almost sure version of theenergy inequality has been introduced in [Rom08, RomlO]. Given a weak solution
choose $\phi=e_{k}$ as a test function in the second property of Definition 2.1, to get a one
dimensional standard Brownian motion . Since
is an orthonormal basis, theare a sequence of independent standard Brownian motions. Then the process
be the solution to theis a cylindrical Wiener process1 on H. Let,(2.1),
linearization at ofwith initial condition $z(0)=0.$
namely. It turns out that is a solution of
Finally, set$\mathbb{P},$$(e_{k})_{k\geq 1}$$\beta^{k}$$(\beta^{k})_{k\geq 1}$$W_{\mathbb{P}}= \sum_{k}\beta^{k}e_{k}$$z_{\mathbb{P}}$$0$$dz_{\mathbb{P}}+Az_{\mathbb{P}}=SdW_{\mathbb{P}}$$\nu_{\mathbb{P}}=\xi-z_{\mathbb{P}}$$\nu_{\mathbb{P}}$$\dot{\nu}+vA\nu+B(\nu+z_{\mathbb{P}},\nu+z_{\mathbb{P}})=0, \mathbb{P}-a.s.,$with initial condition $\nu(0)=\xi_{0}$ . An energy balance functional can be associated to$\mathcal{E}_{t}(\nu, z)=\frac{1}{2}\Vert\nu(t)\Vert_{H}^{2}+v\int_{0}^{t}\Vert\nu(r)\Vert_{V}^{2}dr-\int_{0}^{t}\langle z(r)$$\nu_{\mathbb{P}},$, $B(\nu(r)+z(r),\nu(r))\rangle_{H}$ dr.

We say that a solution of the martingale problem associated to (2.1) (as in Definition
2.1) satisfies the almost sure energy inequality if there is a setof null Lebesgue
measure such that for all
and all $t\geq s,$$\mathbb{P}$$T_{P}\subset(0, \infty)$$s\not\in T_{P}$$P[\mathcal{E}_{t}(\nu, z)\leq \mathcal{E}_{s}(\nu, z)]=1.$

It is not diﬃcult to check that$\mathcal{E}$is measurable and finite almost surely.

2.2.3. A martingale version of the energy inequality. An alternative formulation of theenergy inequality that, on the one hand is compatible with conditional probabilities,
and on the other hand does not involve additional quantities (such as the processes
andcan be given in terms of super-martingales. The additional advantage is that
this definition is keen to generalization to state-dependent noise.
Define, for every, the process$z_{\mathbb{P}}$$\nu_{\mathbb{P}})$$\mathfrak{n}\geq 1$

ds-2Tr(@ S),$\star$$\mathscr{E}_{\iota^{1}}=\Vert\xi_{t}\Vert_{H}^{2}+2v\int_{0}^{t}\Vert\xi_{s}\Vert_{V}^{2}$

and, more generally, for every$\mathfrak{n}\geq 1,$$\mathscr{E}_{t}^{\mathfrak{n}}=\Vert\xi_{t}\Vert_{H}^{2\mathfrak{n}}+2\mathfrak{n}v\int_{0}^{t}\Vert\xi_{s}\Vert_{H}^{2\mathfrak{n}-2}\Vert\xi_{s}\Vert_{V}^{2}ds-\mathfrak{n}(2\mathfrak{n}-1)R(S^{\star}S)\int_{0}^{t}\Vert\xi_{s}\Vert_{H}^{2\mathfrak{n}-2}$

ds,

, and
when
elsewhere.
We say that a solution of the martingale problem associated to (2.1) (as in Definition
2.1) satisfies the super-martingale energy inequality if for each, the process
defined above is -integrable and for almost every $s\geq 0$ (including $s=0$ ) and all$\xi\in L_{1oc}^{\infty}([0, \infty);H)\cap L_{1\circ c}^{2}([0, \infty);V)$$\infty$$\mathbb{P}$$\mathfrak{n}\geq 1$$\mathbb{P}$$t\geq s,$$\mathbb{E}[\mathscr{E}_{\iota^{\mathfrak{n}}}|\mathscr{F}_{s}^{NS}]\leq \mathscr{E}_{s}^{\mathfrak{n}},$or, in other words, if each
$1_{Notice}$$\mathscr{E}^{\mathfrak{n}}$is an almost sure supermartingale.

that $W$ is measurable with respect to the solution process.

$\mathscr{E}_{t}^{\mathfrak{n}}$102.3. Legit families of weak solutions. Following the spirit of [FR08], given $x\in H,$
denote byany family of non-empty sets of probability measures onthat are solutions of (1.1) with initial condition , as specified by Definition 2.1, and$(\Omega_{NS}, \mathscr{F}^{NS})$$\mathscr{C}(x)$$x$such that the following properties hold,andthe sets
are close under conditioning, namely for every
is the regular conditional probability distribution of
every $t>0$, if
, for
,
then
given
,
weak-strong uniqueness holds, namely for every $x\in D(A)$ and every$\xi(t)=u_{R}(t,x)$ for every $t<\tau_{R},$x) is the local smooth solution, wherewith initial condition
We will call each family
satisfying the two above property a legit family.
It is clear that the classes defined in [FR08] (detailed in section 2.2.3) and in [Rom08,
RomlO] (detailed in section 2.2.2) are of this kind, as they actually satisfy the more
restrictive condition called reconstruction in the above-mentioned papers.2 It is alsostraightforward that the -wise set union of two legit families is again a legit family. A less obvious fact is that the family of sets of solutions obtained as limits ofGalerkin approximations is legit. This is remarkable as limits of Galerkin approximations do not satisfy the reconstruction property. To see this fact, we first observethat limit of Galerkin approximatio\’{n} satisfy the energy inequality, and hence fall inthe same class defined in [Rom08, RomlO]. In particular, due to the energy inequality,
weak-strong uniqueness holds. Moreover, once the sub-sequence of Galerkin approximations is identified, the regular conditional probability distributions of the approximations, along the sub-sequence, converge to the corresponding regular conditional
probability distributions of the limit (uniquely identified by the sub-sequence).
$(\mathscr{C}(x))_{x\in H}$$\blacksquare$$\mathbb{P}\in \mathscr{C}(x)$$(\mathbb{P}|_{\mathscr{F}_{t}^{NS}}^{\omega})_{\omega\in\Omega_{NS}}$$\mathbb{P}$$\mathscr{F}_{t}^{NS}$$\mathbb{P}|_{\mathscr{T}_{t}^{NS}}^{\omega}\in \mathscr{C}(\omega(t))$$\mathbb{P}-a.e.$$\omega\in\Omega_{NS},$$\mathbb{P}\in \mathscr{C}(x)$$\blacksquare$$\mathbb{P}-a.s$$u_{R}$$x.$$(\mathscr{C}(x))_{x\in H}$$x$

3. EXISTENCE OF DENSITIESIn this section we give a short review of the results contained in the papers [DR14,
To this end we recall the definition of Besov
spaces. The general definition is based on the Littlewood-Paley decomposition, but it
is not the best suited for our purposes. We shall use an altemative equivalent definition(see [Tri83, Tri92]) in terms of diﬀerences. Define$Roml4c,$ $Roml4a]$ (see also [Rom13]).

$(\Delta_{b}^{1}f)(x)=f(x+b)-f(x)$ ,$( \Delta_{b}^{\mathfrak{n}}f)(x)=\Delta_{b}^{1}(\Delta_{h}^{\mathfrak{n}-1}f)(x)=\sum_{\mathfrak{j}=0}^{\mathfrak{n}}(-1)^{\mathfrak{n}-j}(\begin{array}{l}\mathfrak{n}\mathfrak{j}\end{array})f(x+\mathfrak{j}b)$then the following norms, for $s>0,$ $1\leq p\leq\infty,$$1\leq q<\infty\backslash$,,$\Vert f\Vert_{B_{p,q}^{s}}=\Vert f\Vert_{Lp}+(\int_{\{|b|\leq 1\}}\frac{\Vert\Delta_{b}^{\mathfrak{n}}f||_{L}^{q_{p}}}{|b|^{sq}}\frac{dh}{|h|^{d}})^{\frac{1}{q}}$and for $q=\infty,$$\Vert f\Vert_{B_{p,\infty}^{s}}=\Vert f\Vert_{Lp}+\sup_{|b|\leq 1}\frac{\Vert\Delta_{b}^{\mathfrak{n}}f\Vert_{Lp}}{|h|^{\alpha}},$measurable map
roughly speaking, requires that if one has awith, then the probability measure given by on $[0, t]$ , and, conditionaly on,.
by the values of . , is an element of$2_{Reconstruction}$ ,$\mathbb{Q}_{x}\in \mathscr{C}(x)$, and$\mathscr{F}_{t}^{NS}$$\mathbb{P}$$\mathbb{P}\in \mathscr{C}(x_{0})$$\mathbb{Q}$$\mathscr{C}(x_{0})$$x\mapsto \mathbb{Q}_{x/}$$\omega(t)$

11
where

$\mathfrak{n}$is any integer such that$s<\mathfrak{n}$, are equivalent norms of$B_{p,q}^{s}(R^{d})$for the given

range of parameters.
The technique introduced in [DR14] is based on two ideas. The first is the following
analytic lemma, which provides a quantitative integration by parts. The lemma is
implicitly given in [DR14] and explicitly stated and proved in $[Roml4c].$
Lemma 3.1 (smoothing lemma). If is a finite measure on
and there are an integer$m\geq 1$ , two real numbers $s>0,$ $\alpha\in(0,1)$ , with $\alpha<s<m$, and a constant $c_{1}>0$ such that$R^{d}$$\mu$

for every$\phi\in C_{b}^{\alpha}(R^{d})$and $b\in R^{d},$$| \int_{R^{d}}\Delta_{h}^{m}\phi(x)\mu(dx)|\leq c_{1}|b|^{S}\Vert\phi\Vert_{C_{b}^{\alpha}},$has a density with respect to the Lebesgue measure on$r<s-\alpha$. Moreover, there is $c_{2}=c_{2}(r)$ such thatthen$R^{d}$$f_{\mu}$

$\mu$

and

$f_{\mu}\in B_{1,\infty}^{r}$for every$\Vert f_{\mu}\Vert_{B_{1,\infty}^{r}}\leq c_{2}c_{1}.$The second idea is to use the random perturbation to perform th fractiona integration by parts along the noise to be used in the above lemma. The bulk of this idea
can be found in [FP10]. Our method is based on the one hand on the idea that theNavier-Stokes dynamics i goo $d’$ for short times, and on the other hand that Gaussian processes have smooth densities. When trying to estimate the Besov norm of thedensity, we approximate the solution by splitting the time interval in two parts,$l’$$e^{}$$s^{}$

time

On the first part the approximate solution is the same as the original solution, on thesecond part the non-linearity is killed. By Gaussianity this is enough to estimate theincrements of the density of . Since
is the one-step explicit Euler approximationof , the error in replacing by
can be estimated in terms of . By optimizing theincrement versus we have an estimate on the derivatives of the density.
The final result is given in the proposition below. In comparison with Theorem 5.1
of [DR14], here we give an explicit dependence of the Besov norm of the density withrespect to time. The estimate looks not optimal though.
The regularity of the density can be slightly improved from
if is thetosolution,
the
from
time.
solution
whose
statistics
arestationary
namelyindependent$u_{\epsilon}$$u_{e}$$u$$u$$u_{\epsilon}$$e$$u_{\epsilon}$

$e$

$u$

$B_{1,\infty}^{2-}$$B_{1,\infty}^{1-}$Proposition 3.2. Given $x\in H$ and a finite dimensional subspace of $D(A)$ generated by
the eigenvectors of , namely
for some arbitrary indices , ,assume thatis invertible on F. Thenfor every $t>0$ the projection
has an almost
with respect to the Lebesgue measure on , where is any
everywhere positive density
solution of (2.1) which is limit point of the spectral Galerkin approximations.
Moreover, for every $\alpha\in(0,1)$ ,and for every small $e>0$, there exists$c_{3}=c_{3}(e)>0$ such that
$F$

$A$

$F=span[e_{\mathfrak{n}_{1}}, \cdots, e_{\mathfrak{n}_{F}}]$$\mathfrak{n}_{1}$$\cdots$$\mathfrak{n}_{F},$$\pi_{F}u(t)$$\pi_{F}S$$F$$f_{F,t}$$u$$f_{F,t}\in B_{1,\infty}^{\alpha}(R^{d})$$\Vert f_{F,t}\Vert_{B_{1,\infty}^{\alpha}}\leq\frac{c_{3}}{(1\wedge t)^{\alpha+e}}(1+\Vert x\Vert_{H}^{2})$.

Proof. Given a finite dimensional space as in the statement of the proposition, fix$F$$t>0$ , and let $\alpha\in(0,1)$ ,$\phi\in C_{b}^{\alpha}$, and $h\in F$ , with $|b|\leq 1$ . For $m\geq 1$ , consider two

12cases.

$If|b|^{2\mathfrak{n}/(2\alpha+\mathfrak{n})}<t$, then we use the same estimate in [DR14] to get$|\mathbb{E}[\Delta_{\dagger\iota}^{m}\phi(\pi_{F}u(t))]|\leq c_{4}(1+\Vert x\Vert_{H}^{2\alpha})\Vert\phi\Vert_{C_{b}^{\alpha}}|b|^{\frac{2\mathfrak{n}\alpha}{2\infty+\mathfrak{n}}}.$

If on the other hand
. As in [DR14],$t\leq|b|^{2\mathfrak{n}/(2\infty+\mathfrak{n})}$, we introduce the process$u_{\epsilon}$as above, but with$\epsilon=t$$\mathbb{E}[\Delta_{h}^{m}\phi(\pi_{F}u(t))]=\mathbb{E}[\Delta_{h}^{m}\phi(\pi_{F}u_{\epsilon}(t))]+\mathbb{E}[\Delta_{b}^{m}\phi(\pi_{F}u(t))-\Delta_{b}^{m}\phi(\pi_{F}u_{\epsilon}(t))]$and$|\mathbb{E}[\Delta_{b}^{m}\phi(\pi_{F}u(t))-\Delta_{b}^{m}\phi(\pi_{F}u_{\epsilon}(t))]|\leq c_{5}(1+\Vert x\Vert_{H}^{2\propto})\Vert\phi\Vert_{C_{b}^{\alpha}}t^{\infty}.$

For the probabilistic error we use the fact that$u_{\epsilon}(t)$is Gaussian, hence$| \mathbb{E}[\Delta_{b}^{m}\phi(\pi_{F}u_{\epsilon}(t))]|\leq c_{6}\Vert\phi\Vert_{\infty}(\frac{|h|}{\sqrt{t}})^{\frac{2\mathfrak{n}\alpha}{2\alpha+\mathfrak{n}}}$In conclusion, from both cases we finally have$|\mathbb{E}[\Delta_{b}^{m}\phi(\pi_{F}u(t))]|\leq C_{7(1+\Vert x\Vert_{H}^{2})\Vert\phi\Vert_{C_{b}^{\alpha}}|b|^{\frac{2\mathfrak{n}\alpha}{2\alpha+\mathfrak{n}}(1\wedge t)^{-\frac{\mathfrak{n}\alpha}{2\infty+\mathfrak{n}}}}}.$

The choice of$\mathfrak{n}$and$\alpha$yields the final result.

$\square$Remark 3.3. In [DR14] we introduced three diﬀerent methods to prove existence ofdensities. The first method is based on the Markov machinery developed in [FR08](see also [DPD03]), while the third one is the one on Besov bounds detailed above.
second possibility is to use an appropriate version of the Girsanov change of measure.
It tums out that, together, the Girsanov change of measure and the Besov bounds yieldtime regularity of the densities of finite dimensional projections $[Roml4c].$
As it may be expected, the time regularity obtained is ”hal the space regularity,, for $\alpha<1.$
and the density is at most H\"older in time with values in
$A$

$f’$

$\frac{1}{2}$$B_{1,\infty}^{\alpha}$

Remark 3.4. An apparent drawback of the method is that it can only handle finite dimensional projections. There are interesting functions of the solution, the energy for
instance, that cannot be seen in any way as finite dimensional projections. On the other
hand, one can use the same ideas (fractional integration by parts and smoothing eﬀectof the noise) directly on such quantities.
andFollowing this idea, in $[Roml4b]$ it is shown that the two quantities, have a density. Unfortunately, there is a regularity issue thatwith
, unless $s=0$ . The special quantity
prevents getting densities when$\Vert u(t)\Vert_{H^{-s}}^{2}$$\int_{0}^{t}\Vert u(t)\Vert_{H^{1-s/}}^{2}$$s<\frac{3}{4}$$s\geq\frac{3}{4}$$\Vert u(t)\Vert_{H}^{2}+2v\int_{0}^{t}\Vert u(s)\Vert_{V}^{2}$

$ds$

,

which represents the energy balance and is quite relevant in the theory, admits a density. This is possible due to $tHe$ fundamental cancellation property of the Navier-Stokes

non-linearity.
Remark 3.5. An interesting question, that has been completely answered for the twodimensional case in [MP06], concems the existence of densities when the covarianceof the driving noise is essentially non-invertible. The typical perturbation in (1.1) we
consider here is$\dot{\eta}(t,v)=\sum_{k\in X}\sigma_{k}\dot{\beta}_{k}(t)e_{k}(V)$

,

13

$0$

$t$

$\epsilon$$t$

FIGURE 1. The strategy for the transfer principle: we only look at thesmooth solution immediately before the evaluation time.
where
and is usually much smaller (finite, for instance). The idea is that thenoise influence is spread, by the non-linearity, to all Fourier components. The condition that should ensure this has been already well understood [Rom04], and corresponds to the fundamental algebraic property that SC should generate the whole group$\mathcal{Z}\neq Z_{\star}^{3}$$Z^{3}.$

It is clear that the method we have used to obtain Besov bounds cannot work in thiscase, because the non-linearity plays a major role. On the other hand in $[Roml4a]$ weprove, using ideas similar to those leading to the transfer principle (Theorem 4.1), theexistence of a density No regularity properties are possible, though.
4. THE TRANSFER PRINCIPLE
In this final section we present two results in the direction of extending results
proved only for a special class of solutions (limits of spectral Galerkin approximationsin [DR14]) to every legit solution of (1.1). As already mentioned, the transfer principle
allows the extension of instantaneous properties, namely properties that depend on asingle time.
Given $t_{0}>0$, consider the following event in$\Omega_{NS},$$L(t_{0})=${$\omega$: there is $e>0$ such that

From $[Roml4b]$ we know that, if$\sup_{t\in[t_{0}-e,t_{0}]}\Vert A\omega(t)\Vert_{H}<\infty$}.

, thenis a legit family, if $x\in H$ andfor a.e. $t_{0}>0$ . To be more precise, the proof is given in $[Roml4b]$ only for
those legit families introduced in [FR08] and [RomlO], but the two crucial properties
used in the proof of the probability one statement are exactly those defining a legit$(\mathscr{C}(x))_{x\in H}$$\mathbb{P}\in \mathscr{C}(x)$$\mathbb{P}[L(t_{0})]=1$

family.
Our main theorem is given below. The intuitive idea is that if we are able to proveexistence of a density (with respect to a suitable Lebesgue measure) for a finite dimensional functional of a solution, then the same holds for any other solution, regardlessof the way we were able to produce it.
In other words, we can prove existence of a density for solutions obtained fromGalerkin approximation, and this result will extend straight away to any other solutions, for instance those produced by the Leray regularization (see for instance [Lio96]).
Or we can use the special properties of Markov solutions given in [FR08, RomlO] toprove existence of densities of a large class of finite dimensional functionals, as donein the first part of [DR14], and again this extends immediately to any (legit) solution.

14
Theorem 4.1 (Transfer principle). Let $d\geq 1$ and let : $D(A)arrow R^{d}$ be a measurable
and afamily
junction. Assume that we are given a legit classof solutions
(1.1)
weak
uniqueness.
(only)
strongsatisfyingofonIﬀor every $x\in D(A)$ and almost every $t_{0}>0$ the random variable, then for every
has a density with respect to the Lebesgue measure on$x\in H$ , everyon
and almost every $t_{0}>0$, the random variabletomeasureontheLebesguehas a density with respect$F$$(\mathscr{C}(x))_{x\in H}$$(\mathbb{Q}_{x})_{x\in H}$$\omega\mapsto F(\omega(t_{0}))$$R^{d}$$(\Omega_{NS}, \mathscr{F}^{NS}, \mathbb{Q}_{X})$$\omega\mapsto F(\omega(t_{0}))$$\mathbb{P}\in \mathscr{C}(x)$$R^{d}.$$(\Omega_{NS}, \mathscr{F}^{NS},\mathbb{P})$Proof. Following $[Roml4b]$ , consider for every$e\leq 1$

and every $R\geq 1$ the event$L_{\epsilon,R}(t_{0})$defined as$L_{\epsilon,R}(t_{0})=\{\sup_{t\in[t_{0}-\epsilon,t_{0}]}\Vert A\omega(t)\Vert_{H}\leq R\}.$

Clearly $L(t_{0})=\cup L_{e,R}(t_{0})$ . Given a measurable function as in the standing assump,, a state $x\in H$ and a solution
tions, a Lebesgue null set$F$$E\subset R^{d}$$\mathbb{P}\in \mathscr{C}(x)$$\mathbb{P}[F(\omega(t_{0}))\in E]=\sup_{e\leq 1,R\geq 1}\mathbb{P}[\{F(\omega(t_{0}))\in E\}\cap L_{e,R}(t_{0})].$

Given $e\leq 1$ and $R\geq 1$ , we condition at time $t_{0}-e$ and we know that
onditional probability distribution of given
where$niqueness\mathscr{C}(\omega(t_{0}-\epsilon)),$Hence, using w
ustrong$\mathbb{P}$$\mathbb{P}|_{\mathscr{F}_{t}^{NS}}^{\omega}.$$\in$$\mathbb{P}$$\mathbb{P}|_{\mathscr{F}_{t}^{N}}^{\omega}\underline{s}istheregu1arc$$e^{0}ak^{\epsilon}-$$\mathscr{F}_{t_{0}-\epsilon}^{NS}.$$\mathbb{P}[\{F(\omega(t_{0}))\in E\}\cap L_{\epsilon,R}(t_{0})]=\mathbb{E}^{\mathbb{P}}[\mathbb{P}[\{F(\omega(t_{0}))\in E\}\cap L_{\epsilon,R}(t_{0})|\mathscr{F}_{t_{0}-\in}^{NS}]]$$=\mathbb{E}^{\mathbb{P}}[\mathbb{P}|_{\mathscr{F}_{t_{0}\epsilon}^{N\underline{S}}}^{\omega}[F(\omega’(e))\in E, \tau_{R}\geq e]1_{A_{\epsilon,R}}]$$\leq \mathbb{E}^{\mathbb{P}}[\mathbb{P}|_{\mathscr{F}_{\iota_{0}}^{N}}^{\omega}\underline{s}_{\epsilon}[F(\omega’(e))\in E, \tau_{2R}>e]1_{A_{\epsilon,R}}]$$=\mathbb{E}^{\mathbb{P}}[\mathbb{P}_{\omega(t_{0}-\epsilon)}^{2R}[F(u_{2R}(e))\in E, \tau_{2R}>e]1_{A_{\epsilon,R}}],$

where. Again by weak-strong uniqueness,of positive probability for every with

$A_{\epsilon,R}=\{\Vert A\omega(t_{0}-e)\Vert_{H}\leq R\}$agree on the event$\{\tau_{2R}>e\}$$V$$\mathbb{P}_{v}^{2R}$and, hence$\mathbb{Q}_{v}$$\Vert Av\Vert_{H}\leq R$

for all such $v,$$\mathbb{P}_{v}^{2R}[F(u_{2R}(e))\in E, \tau_{2R}>e]=0.$

Therefore$\mathbb{P}[\{F(\omega(t_{0}))\in E\}\cap L_{\epsilon,R}(t_{0})]=0$

for every $e\leq 1$ and every $R\geq 1$ . In conclusion$\mathbb{P}[F(\omega(t_{0}))\in E]=0.$$\square$

The previous theorem has two crucial drawbacks. The first is that it deals only withinstantaneous properties, namely properties depending only on one single time, andit looks hardly possible, by the nature of the proof, that the principle might ever be
extended, at this level of generality, to multi-time statements, such as the existence of
ajoint density for multiple times (see Remark 4.3 in [DR14]).
The second drawback is that the result is qualitative in nature. Whenever one canfind quantitative bounds on the density, such as the Besov bounds in [DR14], it is againarivial task, one that the present author is not able to figure out in general, toprove that the bounds ar universal hence true for any solution.
If we try to repeat the proof of our main theorem above, with the purpose of extending the Besov bound in a quantitative way, in general we are doomed to failure.
Proposition 3.2 above shows that the control of the Besov norm of the density becomessingular for short times. This is clearly expected when the initial condition is deterministic.
$non\dashv$$e^{}$

15
Let us try to understand what is preventing us from getting a quantitative estimate,for instance in the case the finite dimensional map of the Theorem above is a finitedimensional projection as in Proposition 3.2. We proceed in a slightly diﬀerent way,following loosely the idea of Lemma 3.7 in [Rom08]. Let be a weak solution of (1.1)$b\in F$ with $|h|\leq 1$ , and $m\geq 1$
from a legit class, and fix
withlarge. For $e\in(0,1)$ and $R\geq 1$ set$\mathbb{P}$$\phi\in L^{\infty}(F)$$\Vert\phi\Vert_{\infty}\leq 1,$$A_{\epsilon,R}=\{\Vert A\omega(t-\epsilon)\Vert_{H}\leq R\}, B_{e,R}=\{\sup_{[t-\epsilon,t]}\Vert A\omega(s)\Vert_{H}\leq 2R\}.$

We have$\mathbb{E}^{\mathbb{P}}[\Delta_{b}^{m}\phi(\pi_{F}\omega(t))]=\mathbb{E}[\Delta_{b}^{m}\phi(\pi_{F}\omega(t))1_{B_{\epsilon,R}\cap A_{\epsilon,R}}]+$

error,

where the error can be simply estimated as
error$\prod$ $\leq \mathbb{P}[B_{\epsilon,R}^{c}\cup A_{\epsilon,R}^{c}]\leq \mathbb{P}[B_{\epsilon,R}^{c}\cap A_{\epsilon,R}]+\mathbb{P}[A_{\epsilon,R}^{c}].$For the second term of the error, there is not much we can do, so we keep it unchanged.
As it regards the first term, we exploit the legit class assumption on and use Proposition 3.5 in [FR07] (or [Romll, Proposition 5.7]) to get,$\mathbb{P}$

(4.1)

if

$\mathbb{P}[B_{\epsilon,R}^{c}\cap A_{\epsilon,R}]=\mathbb{E}^{\mathbb{P}}[\mathbb{P}|^{\omega}[\tau_{2R}\leq\epsilon]1_{A_{\epsilon,R}}]\leq ce^{-c_{9^{\frac{R^{2}}{e}}}}$$R^{2}e\leq c_{10}$, for some constants$c_{8}$, C9, $c_{10}>0$ . Finally, again by the legit class condition,$\mathbb{E}^{\mathbb{P}}[\Delta_{b}^{m}\phi(\pi_{F}\omega(t))1_{B_{\epsilon,R}\cap A_{\epsilon,R}}]=\mathbb{E}^{\mathbb{P}}[\mathbb{E}^{\mathbb{P}1_{J_{t\epsilon}^{\underline{N}S}}^{\omega}}[\Delta_{b}^{m}\phi(\pi_{F}\omega(e))1_{\{\tau_{2R}\geq\epsilon\}}]1_{A_{\epsilon_{)}R}}].$, by weak-strong uniqueness, the inner expectation does not
the eventdepend on , but only on the smooth solution starting from $\omega(t-e)$ , in particular the
Besov estimate holds and for $\alpha\in(0,1)$ , by Proposition 3.2,$\circ n$$\{\tau_{2R}\geq e\}$$\mathbb{P}$$\mathbb{E}^{\mathbb{P}1_{\mathscr{T}_{t\epsilon}^{N\underline{S}}}^{\omega}}[\Delta_{\dagger\iota}^{\mathfrak{n}\iota}\phi(\pi_{F}\omega(e))1_{\{\tau_{2R}\geq\epsilon\}}]\leq\frac{c_{3}}{e^{\alpha+\delta}}(1+\Vert\omega(t-e)\Vert_{H}^{2})|b|^{\alpha}+\mathbb{P}|_{\mathscr{F}_{te}^{\underline{N}S}}^{\omega}[\tau_{2R}\leq e].$Using again [FR07, Proposition 3.5] as in (4.1),$\mathbb{E}^{\mathbb{P}}[\Delta_{b}^{m}\phi(\pi_{F}\omega(t))1_{B_{\epsilon,R}\cap A_{e,R}}]\leq\frac{c_{3}}{\epsilon^{\alpha+\delta}}(1+\mathbb{E}[\Vert\omega(t-e)\Vert_{H}^{2}])|b|^{\alpha}+c_{8}e^{-c_{9^{\frac{R^{2}}{\epsilon}}}},$

with 6 small (so that $\alpha+6<1$ ). In conclusion,$| \mathbb{E}^{\mathbb{P}}[\Delta_{b}^{m}\phi(\pi_{F}\omega(t))]|\leq\frac{c_{3}}{\epsilon^{\alpha+6}}(1+\mathbb{E}[\Vert\omega(t-\epsilon)\Vert_{H}^{2}])|b|^{\alpha}+2c_{8}e^{-c_{9^{\frac{R^{2}}{\epsilon}}}}+$$+\mathbb{P}[A_{e,R}^{c}]$Choose $R\approx e^{-1/2}$ , so that the constraint
is satisfied. Integrate the above, and use the moment in $D(A)$ proved in $[Roml4a]$ toinequality over $e\in(0, e_{0})$ ,get$eR^{2}\leq c_{10}$$6_{0}\leq 1$(4.2)$| \mathbb{E}^{\mathbb{P}}[\Delta_{b}^{m}\phi(\pi_{F}\omega(t))]|\leq c_{11}(1+\Vert x\Vert_{H}^{2})(\frac{|\dagger\iota|^{\alpha}}{\epsilon_{0}^{\alpha+6}}+e^{--2}ce)+\frac{1}{\epsilon_{0}}|_{0}^{\epsilon_{0}}\mathbb{P}[A_{\epsilon,R}^{C}]$de.

Neglect, only for a moment, the last term. The choice $e=-c_{12}/\log|h|$ would finallygive the same result as Proposition 3.2 (up to a logarithmic correction). Unfortunately

16
the last term prevents this computation. There is no much we can do here, our best
option seems Chebychev’s inequality and the -moment in $D(A)$ proved in $[Roml4a],$$\frac{2}{3}$$\frac{1}{\epsilon_{0}}\int_{0}^{\epsilon_{0}}\mathbb{P}[A_{\epsilon,R}^{C}]$$de$$\leq\frac{1}{\epsilon_{0}}\mathbb{E}^{\mathbb{P}}[\int_{0}^{\epsilon_{0}}\frac{1}{R^{\frac{2}{3}}}\Vert A\omega(t-e)\Vert^{\frac{2}{H3}}de]$(4.3)$\leq e_{0}^{\frac{2}{3}}\mathbb{E}^{\mathbb{P}}[\int_{0}^{t}\Vert A\omega\Vert^{\frac{2}{H3}}ds],$

and no quantitative counterpart of the transfer principle can be proved in this way.
Notice that the same technique is successful in [Rom08]. The reason is that in thementioned paper (a diﬀerent form of) the transfer principle was used for moments ofrivialthe solution. Here we are estimating the size of an increment. It makes adiﬀerence, since here the crucial mechanism is the smoothing eﬀect of the random
perturbation, as it can be seen in a simple example with a one dimensional Brownis. indeed, it is elementary to compute that
ian motion
and, where is the total variation distance between the laws ofbounded by
much
hand,
is
. On the otherthe seemingly similar quantity$non\dashv$$|\mathbb{E}[\phi(B_{t+h})-\phi(B_{t})]|$$(B_{t})_{t\geq 0}$$B_{t}$$\frac{h}{t}$$\Vert\phi\Vert_{\infty}\frac{\dagger\iota}{t}$$\mathbb{E}[|\phi(B_{t+b})-\phi(B_{t})|]$$B_{t+b}$

worse.
There is one case though where our computations above can be carried on. If weassume that is a stationary solution, with time marginal , the quantity in (4.3) can),be estimated as (recall that
$u$

$\mu$

$R\approx e^{-\gamma}$$\frac{1}{\epsilon_{0}}\int_{0}^{\epsilon_{0}}\mathbb{P}[A_{\epsilon,R}^{c}]$$de$$\leq\frac{1}{e_{0}}\mathbb{E}^{\mathbb{P}}[\int_{0}^{\epsilon_{0}}\frac{1}{R^{\frac{2}{3}}}\Vert A\omega(t-e)\Vert^{\frac{2}{H3}}de]\leq \mathbb{E}^{\mu}[\Vert Ax\Vert^{\frac{2}{H3}}]\epsilon^{\frac{1}{0^{3}}},$and (4.2) this time reads,$| \mathbb{E}^{\mathbb{P}}[\Delta_{h}^{m}\phi(\pi_{F}\omega(t))]|\leq c_{13}(e^{\frac{1}{0^{3}}}+\frac{|h|^{\alpha}}{\epsilon_{0}^{\alpha+\delta}})$.

by optimization and Lemma 3.1 show that the density is Besov.
Since can run over all values in $(0,2)$ by Theorem 5.2 in [DR14], we obtain the following result.

A suitable choice of$e_{0}$$\alpha$Theorem 4.2 (Quantitative transfer principle). Let $d\geq 1$ and consider a dimensional
sub space of $D(A)$ spanned by a finite number of eigenvectors of the Stokes operator.
, and leta stationary solution whose
Assume that we are given a legit classconditional probabilities at time are elements of the legit class
has a density with respect the Lebesgue
Denote bya process with law
measure on F. Moreover, the density belongs to the Besov spacefor every$d$$F$$(\mathscr{C}(x))_{x\in H}$$(\mathscr{C}(x))_{x\in H}.$$0$$u_{\star}$$\mathbb{P}_{\star}$$\mathbb{P}_{\star/}then$$\pi_{F}u_{\star}(t)$$B_{1,\infty}^{\alpha}(F)$One can slightly improve the exponent
than $D(A)$ , see Remark 2.2.

$\frac{2}{7}$$\alpha<\frac{2}{7}.$by using moments in a diﬀerent topology

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DIPARTIMENTO DI MATEMATICA, UNIVERSITA DI PISA, LARGO BRUNO PONTECORVO 5, I-56127
PISA, ITALIA$E$-mail address: romito@dm.unipi.it: http: $//www$ . dm. unipi. $it/pages/$ romito$URL$