$\rho$-White noise solution to 2D stochastic Euler equations

Abstract

A stochastic version of 2D Euler equations with transport type noise in the vorticity is considered, in the framework of Albeverio–Cruzeiro theory (Commun Math Phys 129:431–444, 1990) where the equation is considered with random initial conditions related to the so called enstrophy measure. The equation is studied by an approximation scheme based on random point vortices. Stochastic processes solving the Euler equations are constructed and their density with respect to the enstrophy measure is proved to satisfy a Fokker–Planck equation in weak form. Relevant in comparison with the case without noise is the fact that here we prove a gradient type estimate for the density. Although we cannot prove uniqueness for the Fokker–Planck equation, we discuss how the gradient type estimate may be related to this open problem.

Appendix

The notion of uniqueness for a stochastic equation has several faces: beyond the most classical ones, namely pathwise uniqueness and uniqueness in law, there are others, like uniqueness of the Lagrangian flow and uniqueness of the solution of the Fokker–Planck equation. Due to the outstanding difficulty of the particular case of stochastic Euler equations, in the following discussion let us concentrate on the weakest one of the previous concepts, namely uniqueness of the solution of the Fokker–Planck equation; when it is true, in some cases one can prove more (see for instance [3] for Lagrangian flows and [29] for uniqueness in law).

We outline below a few approaches to uniqueness of a Fokker–Planck equation of the form

$$\begin{aligned} \partial _{t}\nu _{t}=\mathcal {L}^{*}\nu _{t},\quad \nu |_{t=0}=\nu _{0}, \end{aligned}$$

(5.1)

where $\mathcal {L}^{*}$ is an operator acting on measures, which is the formal dual of a certain linear operator $\mathcal {L}$ acting on functions f, namely, $\left\langle \mathcal {L}^{*}\nu ,f\right\rangle =\left\langle \nu ,\mathcal {L} f\right\rangle$ (we write $\left\langle \nu ,g\right\rangle$ for $\int g(x) \nu (\mathrm{d}x)$). The particular case considered in this paper has the generator defined as

$$\begin{aligned} \left( \mathcal {L}f\right) \left( \omega \right) =\left\langle b\left( \omega \right) ,D_{\omega }f\left( \omega \right) \right\rangle +\frac{1}{2}\sum _{k=1}^{\infty } \big \langle \sigma _{k}\cdot \nabla \omega ,D_{\omega } \langle \sigma _{k}\cdot \nabla \omega ,D_{\omega }f\left( \omega \right) \rangle \big \rangle \end{aligned}$$

with the notations $b\left( \omega \right)$, $D_{\omega }f\left( \omega \right)$ etc. introduced in the previous sections; the function f is in $\mathcal {FC}_{P}$.

As we have already remarked in the introduction, at present we are unable to prove uniqueness for Eq. (5.1). The aim of the following sections is only to show that gradient estimates are at the core of this question.

Before we continue, let us notice that, in our case, if we denote by $\mu$ the enstrophy measure and by $\mathcal {L}_{\mu }^{*}$ the formal dual of $\mathcal {L}$ in $L^{2}( H^{-1-},\mu )$, then

$$\begin{aligned} \left( \mathcal {L}_{\mu }^{*}f\right) \left( \omega \right) =-\,\left\langle b\left( \omega \right) , D_{\omega }f\left( \omega \right) \right\rangle +\frac{1}{2}\sum _{k=1}^{\infty } \big \langle \sigma _{k} \cdot \nabla \omega , D_{\omega }\langle \sigma _{k}\cdot \nabla \omega ,D_{\omega }f\left( \omega \right) \rangle \big \rangle , \end{aligned}$$

hence the structure of $\mathcal {L}$ and $\mathcal {L}_{\mu }^{*}$ is similar, up to a sign in the drift term. If $\nu _t$ has a density $\rho _t$ w.r.t. $\mu$, we write the Eq. (5.1) as

$$\begin{aligned} \partial _{t} \rho _{t}=\mathcal {L}^{*}_\mu \rho _{t}, \quad \rho |_{t=0}=\rho _{0}. \end{aligned}$$

(5.2)

1.1 Lions theorem

We first describe the difficulties in the attempt to apply a classical variational theorem of J. L. Lions [23, Chap. 4]. Probably this is one of the most restrictive approaches, compared for instance to duality or semigroup theory. However, it clarifies in the easiest possible way some of the difficulties.

We need a Gelfand triple $\mathcal {V}\subset \mathcal {H}\subset \mathcal {V} ^{\prime }$ and a bilinear map $a:\mathcal {V}\times \mathcal {V}\rightarrow \mathbb {R}$, such that Eq. (5.2) is interpreted in the form

$$\begin{aligned} \left\langle \rho _{t},f\right\rangle +\int _{0}^{t} a( \rho _{s},f)\, \mathrm{d}s= \langle \rho _{0},f \rangle , \end{aligned}$$

where $\langle \,,\rangle$ is the inner product in $L^2(H^{-1-},\mu )$. When a is continuous and coercive on $\mathcal {V}$, existence and uniqueness is true. In our case the natural choice for $\mathcal {H}$ is $L^{2}( H^{-1-},\mu )$, with $\left\| f\right\| _{\mathcal {H}}^{2} = \langle f,f\rangle$; for $\mathcal {V}$, it is the space of all $f\in \mathcal {H}$ such that the derivatives $\left\langle \sigma _{k}\cdot \nabla \omega ,D_{\omega }f\left( \omega \right) \right\rangle$ exist in $\mathcal {H}$ for all $k\in \mathbb {N}$ in the distributional sense described in the paper and

$$\begin{aligned} \left\| f\right\| _{\mathcal {V}}^{2}:=\left\| f\right\| _{\mathcal {H}}^{2}+\sum _{k=1}^{\infty }\int _{H^{-1-}}\left\langle \sigma _{k}\cdot \nabla \omega ,D_{\omega }f\left( \omega \right) \right\rangle ^{2} \mu ( \mathrm{d}\omega ) <\infty ; \end{aligned}$$

and the bilinear form is given by

$$\begin{aligned} a\left( f,g\right)&=a_{0}\left( f,g\right) +a_{1}\left( f,g\right) , \\ a_{0}\left( f,g\right)&=\frac{1}{2}\sum _{k=1}^{\infty }\int _{H^{-1-} }\left\langle \sigma _{k}\cdot \nabla \omega ,D_{\omega }f\left( \omega \right) \right\rangle \left\langle \sigma _{k}\cdot \nabla \omega ,D_{\omega }g\left( \omega \right) \right\rangle \mu ( \mathrm{d}\omega ), \\ a_{1}\left( f,g\right)&=-\,\int _{H^{-1-}}f\left( \omega \right) \left\langle b\left( \omega \right) ,D_{\omega }g\left( \omega \right) \right\rangle \mu ( \mathrm{d}\omega ). \end{aligned}$$

Coercivity is true, since

$$\begin{aligned} a_{0}\left( f,f\right)&=\frac{1}{2} \big ( \Vert f\Vert _{\mathcal {V}}^{2} -\Vert f\Vert _{\mathcal {H}}^{2} \big ), \\ a_{1}\left( f,f\right)&=0. \end{aligned}$$

What is not clear is continuity (and even the fact that the term $a_{1}\left( f,g\right)$ is well defined on $\mathcal {V}\times \mathcal {V}$), which requires

$$\begin{aligned} \left| a_{1}\left( f,g\right) \right| \le C\left\| f\right\| _{\mathcal {V}}\left\| g\right\| _{\mathcal {V}} \end{aligned}$$

(5.3)

for all $f,g\in \mathcal {V}$.

We do not claim that property (5.3) is not true; simply that our understanding is still too poor. At present, we do not know how to use efficiently the fact that f can be estimated in the $\mathcal {V}$ topology, when dealing with property (5.3). Using the non-optimal inequality $\int _{H^{-1-}}f^{2}\left( \omega \right) \mu ( \mathrm{d}\omega ) \le \left\| f\right\| _{\mathcal {V}}^{2}$ we have

$$\begin{aligned} \left| a_{1}\left( f,g\right) \right| \le \left\| f\right\| _{\mathcal {V}}\left( \int _{H^{-1-}}\left\langle b\left( \omega \right) ,D_{\omega }g\left( \omega \right) \right\rangle ^{2} \mu ( \mathrm{d}\omega ) \right) ^{1/2}, \end{aligned}$$

and thus we are faced to prove

$$\begin{aligned} \int _{H^{-1-}}\left\langle b\left( \omega \right) ,D_{\omega }g\left( \omega \right) \right\rangle ^{2} \mu ( \mathrm{d}\omega ) \le C\left\| g\right\| _{\mathcal {V}}^{2}. \end{aligned}$$

If $\sigma _{k}\left( x\right) =e^{2\pi \mathrm{i} k\cdot x}\frac{k^{\perp }}{\left| k\right| ^{\gamma }}$ for $k\in \mathbb {Z}_{0}^{2}:=\mathbb {Z}^{2} \backslash \left\{ 0\right\}$ (for $\gamma >2$, since $\sigma _{k}\cdot \nabla \sigma _{k}=0$, these vector fields satisfy our assumptions) we may write

$$\begin{aligned} \left\langle b\left( \omega \right) ,D_{\omega }g\left( \omega \right) \right\rangle =2\pi \mathrm{i} \sum _{k\in \mathbb {Z}_{0}^{2}}\left| k\right| ^{\gamma -2} \big \langle \omega ,e^{2\pi \mathrm{i}k\cdot x} \big \rangle \left\langle \sigma _{k}\cdot \nabla \omega ,D_{\omega }g\left( \omega \right) \right\rangle . \end{aligned}$$

Even the trivial $L^{1}(\mu )$-estimate

$$\begin{aligned} \int _{H^{-1-}}\left| \left\langle b\left( \omega \right) ,D_{\omega }g\left( \omega \right) \right\rangle \right| \mu \left( d\omega \right)&\le 2\pi \left( \sum _{k\in \mathbb {Z}_{0}^{2}}\left| k\right| ^{2\gamma -4}\int _{H^{-1-}} \big \vert \big \langle \omega ,e^{2\pi ik\cdot x} \big \rangle \big \vert ^{2}\mu \left( d\omega \right) \right) ^{1/2}\left\| g\right\| _{\mathcal {V}}\\&=2\pi \left( \sum _{k\in \mathbb {Z}_{0}^{2}}\left| k\right| ^{2\gamma -4}\right) ^{1/2}\left\| g\right\| _{\mathcal {V}} \end{aligned}$$

requires too much on $\gamma$, namely $\gamma <1$. The previous elementary computation fails to prove (5.3).

1.2 Duality

A widely used general idea to prove uniqueness of a linear equation is to prove existence for a suitable dual equation; a reference in Probability theory is [29]. With our notations, an example of dual equation is

$$\begin{aligned} \partial _{t}u+\mathcal {L}u=0\text { on }\left[ 0,t_{0}\right] ,\quad u|_{t=t_{0}}=\phi \end{aligned}$$

(5.4)

where $t_{0}$ is arbitrary in $\left[ 0,T\right]$. The heuristic argument is that, by ordinary calculus and the duality between $\mathcal {L}$ and $\mathcal {L}^{*}$,

$$\begin{aligned} \left\langle \nu _{t_{0}},\phi \right\rangle -\left\langle \nu _{0} ,u_{0}\right\rangle =\int _{0}^{t_{0}}\left( \left\langle \mathcal {L}^{*}\nu _{s},u_{s}\right\rangle -\left\langle \nu _{s},\mathcal {L}u_{s} \right\rangle \right) \mathrm{d}s=0. \end{aligned}$$

Assume that $\phi$ may vary in a determining class (a class with the property that two measures coinciding over it are the same measure). Then the action of $\nu _{t_{0}}$ on test functions $\phi$ is identified by the initial condition $\nu _{0}$ and by the solution u:

$$\begin{aligned} \left\langle \nu _{t_{0}},\phi \right\rangle = \langle \nu _{0},u_{0} \rangle . \end{aligned}$$

(5.5)

The existence of u identifies the measure $\nu _{t_{0}}$, providing uniqueness for (5.1).

Having described the general idea, let us see now the technical problems arising in its implementation in our particular case. Let us recall we have proved the existence of a solution of (5.1) in the sense of the weak identity ($\nu _{t}=\rho _{t}\mu$ and $\langle \, ,\rangle$ now denotes the inner product in $L^2 (H^{-1-},\mu )$)

$$\begin{aligned} \left\langle \rho _{t},F_{t}\right\rangle -\left\langle \rho _{0},F_{0} \right\rangle =\int _{0}^{t}\left( \int _{H^{-1-}}\rho _{s}\left( \mathcal {L}F_{s}+\partial _{s}F_{s}\right) \mathrm{d}\mu \right) \mathrm{d}s \end{aligned}$$

(5.6)

for every function F of class $\mathcal {FC}_{P,T}$; solution satisfying the gradient estimate. Exactly with the same proof (due to the analogy between $\mathcal {L}$ and $\mathcal {L}^{*}= \mathcal {L}_{\mu }^{*}$) one can prove existence of a solution of (5.4) in the sense of the weak identity

$$\begin{aligned} \left\langle u_{t_{0}},F_{t_{0}}\right\rangle -\left\langle u_{t} ,F_{t}\right\rangle +\int _{t}^{t_{0}}\left( \int _{H^{-1-}}u_{s}\left( \mathcal {L}^{*}F_{s} -\partial _{s}F_{s}\right) \mathrm{d}\mu \right) \mathrm{d}s=0 \end{aligned}$$

for every $F\in \mathcal {FC}_{P,t_{0}}$. Now the question is how to prove rigorously (5.5) starting from the previous two identities. No one of the functions $\rho$ or u has sufficient regularity to be used as a test function in the weak formulation of the other function. Using the gradient estimate known for $\rho$ we may rewrite the weak formulation (5.6) as

$$\begin{aligned}&\left\langle \rho _{t},F_{t}\right\rangle +\int _{0}^{t}a_{0}\left( \rho _{s},F_{s}\right) \mathrm{d}s =\left\langle \rho _{0},F_{0}\right\rangle +\! \int _{0}^{t}\left( \int _{H^{-1-}}\rho _{s}\left( \partial _{s}F_{s}+\left\langle b,DF_{s}\right\rangle \right) \mathrm{d}\mu \right) \mathrm{d}s. \end{aligned}$$

This formulation is less demanding in terms of regularity of u and F since only first order differential operators are applied to them. But, in order to take $F=u$, even taking advantage of the additional property that $\rho$ is bounded when $\rho _{0}\in L^{\infty }\left( H^{-1-},\mu \right)$, we need

$$\begin{aligned} \left\langle b,Du\right\rangle \in L^{1}\left( 0,T;L^{1}\left( H^{-1-},\mu \right) \right) . \end{aligned}$$

(5.7)

This requirement is weaker than that in the Lions approach but, as described in the previous subsection, we do not know if it holds true under our assumptions.

Thus a direct substitution of u as a test function in the equation of $\rho$ meets problems similar to (although weaker than) those of the Lions approach. This is not the end of the story: a general idea, with plenty of possible implementations, consists in introducing a sequence $\left( u_{t}^{N}\right)$ with the following properties:

$$\begin{aligned} u_{t}^{N}\text { is an admissible test function for the }\rho _{t} \text {-equation }\quad {(5.6)}, \end{aligned}$$

$$\begin{aligned} u_{t}^{N}\text { converges to }u_{t} \text { and} \end{aligned}$$

$$\begin{aligned} \partial _{t}u_{t}^{N}+\mathcal {L}u_{t}^{N}\text { converges to zero} \end{aligned}$$

in suitable topologies. At present we have not found a solution to this question. It is also strongly related to the next two strategies: for instance, one may construct $u_{t}^{N}$ by space-mollifiers which lead to the fact that $\partial _{t}u_{t}^{N}+\mathcal {L}u_{t}^{N}$ is equal to a commutator, as in Sect. 5.4 below.

Remark 5.1

A well known method to prove uniqueness in law for a stochastic differential equation based on the existence of a solution to Kolmogorov equation is closely related to the topic of this section. In our case we should take a solution $\omega _{t}$ of the stochastic Euler equation and a solution $u_{t}$ of the backward Kolmogorov equation (5.4) and compute

$$\begin{aligned} \mathrm{d}u( t,\omega _{t}), \end{aligned}$$

realizing that it is given by the local martingale (in differential notation)

$$\begin{aligned} \sum _{k=1}^{\infty }\left\langle \sigma _{k}\cdot \nabla \omega _{t},D_{\omega }u\left( t,\omega _{t}\right) \right\rangle \mathrm{d}W_{t}^{k}. \end{aligned}$$

One can easily recognize the structure of the duality argument and the role of a gradient type estimate. The difficulty to make this argument rigorous is similar to what described above: the regularity of u is not sufficient to apply the classical Itô formula, hence a regularization is needed, leading to the same difficulties mentioned above, and below in Sect. 5.4.

1.3 Dense range conditions

In semigroup theory a powerful theorem providing existence and uniqueness of solutions to a linear differential equation is Lumer–Phillips theorem. It is based on two assumptions: dissipativity, which is natural for Fokker–Planck type equations; plus a dense range condition (see [26] for several versions and details). A version of this argument for Fokker–Planck equations in infinite dimensions, interpretable also as a result of duality, is given by [8].

The version of this strategy we invoke here is simply based on identity (5.6), hence it is strongly related to the previous section. But the question here is: define a domain $\mathcal {D}$ for the operator

$$\begin{aligned} \left( \partial _{t}+\mathcal {L}\right) :\mathcal {D}\rightarrow L^{1}\left( 0,T;L^{1} ( H^{-1-},\mu ) \right) \end{aligned}$$

which includes the final time condition $u|_{t=T}=0$ and prove that

$$\begin{aligned} \left( \partial _{t}+\mathcal {L}\right) \left( \mathcal {D}\right) \text { is dense in }L^{1}\left( 0,T;L^{1} ( H^{-1-},\mu ) \right) . \end{aligned}$$

(5.8)

If we succeed, then we use the additional property that $\rho$ is bounded when $\rho _{0}\in L^{\infty } ( H^{-1-},\mu )$, from identity (5.6) we deduce that $\rho$ is identified.

One way of proving the dense range condition is by replacing the operator $\mathcal {L}$ with a simpler one $\mathcal {L}^{N}$ such that

$$\begin{aligned} \left( \partial _{t}+\mathcal {L}^{N}\right) \left( \mathcal {D}\right) \text { is dense in }L^{1}\left( 0,T;L^{1} ( H^{-1-},\mu ) \right) . \end{aligned}$$

Assume this happens (in itself this can be done in various ways). Take $G\in L^{1}\left( 0,T;L^{1} ( H^{-1-},\mu ) \right)$, $\epsilon >0$ and $G_{N}\in L^{1}\left( 0,T;L^{1} ( H^{-1-},\mu ) \right)$ with distance less than $\epsilon$ from G and equal to $\left( \partial _{t}+\mathcal {L}^{N}\right) F_{N}$ for a certain $F_{N}\in \mathcal {D}$. Then

$$\begin{aligned} \left( \partial _{t}+\mathcal {L}\right) F_{N}=G_{N}+\left( \mathcal {L} F_{N}-\mathcal {L}^{N}F_{N}\right) . \end{aligned}$$

If we prove

$$\begin{aligned} \underset{N\rightarrow \infty }{\lim \inf }\int _{0}^{t}\int _{H^{-1-}}\left| \mathcal {L}F_{N}-\mathcal {L}^{N}F_{N}\right| \mathrm{d}\mu \mathrm{d}s=0 \end{aligned}$$

then we have proved the dense range condition (5.8) and thus uniqueness for Eq. (5.1). A strategy is to take $\mathcal {L} ^{N}$ of the form

$$\begin{aligned} \left( \mathcal {L}^{N}f\right) \left( \omega \right) =\left\langle b^{N}\left( \omega \right) ,D_{\omega }f\left( \omega \right) \right\rangle +\frac{1}{2}\sum _{k=1}^{\infty } \big \langle \sigma _{k}\cdot \nabla \omega ,D_{\omega } \langle \sigma _{k}\cdot \nabla \omega ,D_{\omega }f\left( \omega \right) \rangle \big \rangle , \end{aligned}$$

so that we have only to prove

$$\begin{aligned} \underset{N\rightarrow \infty }{\lim \inf }\int _{0}^{t}\int _{H^{-1-}}\left| \left\langle b\left( \omega \right) -b^{N}\left( \omega \right) ,D_{\omega }F_{N}\left( \omega \right) \right\rangle \right| \mathrm{d}\mu \mathrm{d}s=0. \end{aligned}$$

We see again that the key property is a gradient type condition, similar to (5.7).

1.4 Renormalized solutions

Instead of using duality, heuristically one could use the following energy type argument. Let $\rho _{t}^{\left( i\right) }$, $i=1,2$, be two solutions of Eq. (5.6). Then $\rho _{t}=\rho _{t}^{\left( 1\right) }-\rho _{t}^{\left( 2\right) }$ satisfies identity (5.6) with $\rho _{0}=0$. If we may choose $F=\rho$, we get (using also $\int _{0}^{t}\rho _{s}\partial _{s}\rho _{s} \mathrm{d}s=\frac{1}{2} \int _{0}^{t}\partial _{s}\rho _{s}^{2} \,\mathrm{d}s=\frac{1}{2}\rho _{t}^{2}$)

$$\begin{aligned} \frac{1}{2}\int _{H^{-1-}}\rho _{t}^{2} \,\mathrm{d}\mu +\int _{0}^{t} a_0(\rho _s, \rho _s)\, \mathrm{d}s=0, \end{aligned}$$

hence

$$\begin{aligned} \int _{H^{-1-}}\rho _{t}^{2}\,\mathrm{d}\mu \le 0 \end{aligned}$$

(5.9)

which implies $\rho _{t}=0$.

Rigorously speaking, the problem is to pass from (5.6) to (5.9). Notice that an analog of (5.9) is known for each $\rho _{t}^{\left( i\right) }$, $i=1,2$, but not for the difference.

A classical way (see [2, 14]) to go from (5.6) to (5.9) is to mollify $\rho _{t}$ so that computations can be performed rigorously. Call generically

$$\begin{aligned} \rho _{t}^{\epsilon }:=P_{\epsilon }\rho _{t} \end{aligned}$$

a smoothed version of $\rho _{t}$ obtained by the application of a smoothing linear operator (see examples in [4, 12, 15]). We have (taking $P_{\epsilon }^{*}\rho _{t}^{\epsilon }$ as test function)

$$\begin{aligned} \left\langle \rho _{t}^{\epsilon },\rho _{t}^{\epsilon }\right\rangle&=\int _{0}^{t}\left( \int _{H^{-1-}}\rho _{s}^{\epsilon }\left( \mathcal {L} \rho _{s}^{\epsilon }+\partial _{s}\rho _{s}^{\epsilon }\right) \mathrm{d}\mu \right) \mathrm{d}s \\&\quad +\int _{0}^{t}\left( \int _{H^{-1-}}\rho _{s}\left( \mathcal {L}P_{\epsilon }^{*}-P_{\epsilon }^{*}\mathcal {L}\right) \rho _{s}^{\epsilon }\, \mathrm{d}\mu \right) \mathrm{d}s. \end{aligned}$$

The commutator $\left[ \mathcal {L},P_{\epsilon }^{*}\right] :=\mathcal {L} P_{\epsilon }^{*}-P_{\epsilon }^{*}\mathcal {L}$ arises. We deduce

$$\begin{aligned} \frac{1}{2}\int _{H^{-1-}}\left( \rho _{t}^{\epsilon }\right) ^{2} \mathrm{d}\mu +\int _{0}^{t} a_0( \rho _{s}^{\epsilon },\rho _{s}^{\epsilon }) \, \mathrm{d}s=\int _{0}^{t}\left( \int _{H^{-1-}}\rho _{s}\left( \mathcal {L}P_{\epsilon }^{*}-P_{\epsilon }^{*}\mathcal {L}\right) \rho _{s}^{\epsilon }\, \mathrm{d}\mu \right) \mathrm{d}s. \end{aligned}$$

If we can prove that the r.h.s. converges to zero, then we deduce $\rho _{t}=0$. Convergence to zero of commutators is a very technical subject, well understood in finite dimensions ([2, 14] and several subsequent references) with a number of results in the infinite dimensional case ([4, 5, 12, 15, 20]). The present known conditions in infinite dimensions cannot be applied to our case; however, some degree of differentiability of either the solution or the drift is needed in the estimates and here, in our case, we have a mild form of differentiability provided by the gradient estimates for $\rho _{t}^{\left( i\right) }$, $i=1,2$. This direction, as the previous ones, may deserve further study.