A fluid-particle system related to Vlasov-Navier-Stokes equations (Mathematical Analysis of Viscous Incompressible Fluid)

A

_{fluid‐particle}

_system

related

to

Vlasov‐Navier‐Stokes

equations

Franco Flandoli

_(Universita

di

_Pisa)

Abstract

Preliminaryresultson_{the convergence}totheVlasov‐Navier‐Stokes_equationsofa

systemofparticles interactingwithafluid areannounced. Mainemphasisis_givento

thedifficulties that arise and hints forsolutionsare_given.

1

Introduction

Theaim of this workis to

_investigate

a

fluid‐particle

_systemwhichseems_{to converge, in}

the hmit of

_infinitely

many

particles,

toaVlasov‐Navier‐Stokes

(VNS)

_system. We restrict

this

preliminary investigation mainly

to dimension d=2 and\mathrm{W}shall assumeto beon a

torus

$\Gamma$^{2}

with

_{periodic boundary}

conditions. The facts described in thisnotehave

_only

the

character ofa

preliminary investigation

andannouncement of

_partial

results.

Let $\epsilon$\in

(0,1)

and N\in \mathrm{N} be

given,

where N is the number of

_particles;

consider the

system:

\displaystyle \frac{\partial u^{N}}{\partial t}= $\Delta$ u^{N}-u^{N}

.

\displaystyle \nabla u^{N}-\nabla$\pi$^{N}-\frac{c_{0}}{N}\sum_{i=1}^{N}(u_{ $\epsilon$}^{N}(t, X_{t}^{i})-V_{t}^{i})$\delta$_{X_{\mathrm{t}}^{i}}^{ $\epsilon$}

\displaystyle \frac{d}{dt}X_{t}^{i}=V_{t}^{i}

\displaystyle \frac{1}{N}dV_{t}^{i}=\frac{c_{0}}{N}(u_{ $\epsilon$}^{N}(t, X_{\mathrm{t}}^{i})-V_{t}^{i})dt+\frac{1}{N^{2}}\sum_{j=1}^{N}K(X_{t}^{i}-X_{t}^{j})dt+\frac{$\sigma$_{p}}{N}dW_{t}^{i}.

The first

_equation

isthe usualNavier‐Stokessystemfor the

_velocity

andpressure

(u^{N}, $\pi$^{N})

ofa

fluid,

“forced”

by

the_presenceof N

particles;

a

_precise

description

of the interaction

between

particles

and fluid is adifficult

_topic

(just

as an

instance,

see

[2], [5],

[8], [9], [16]),

outside the _scopeof this

preliminary

note, hence we

adopt

a

partially phenomenological

description,

where

_particles

act as delta Dirac

forces,

with

intensity

proportional

to the

velocity

difference between fluid and

particle.

For technical_reasons, but alsoasatrace of

force;

and

_analogously

the

_velocity

differenceis

_computed

between the

particle velocity

V_{\mathrm{t}}^{i}

andalocal_{average at}

particle

center

X_{t}^{i}

of the fluid

velocity,

_{u_{ $\epsilon$}^{N}(t, X_{t}^{i})}

.

The

_smoothings

usedinthe first

_equation

above are

given by

classical mollifiers of the

form

_{$\theta$_{ $\epsilon$}^{0}(x)=$\epsilon$^{-d}$\theta$^{0}($\epsilon$^{-1}x)}

,where

$\theta$^{0}

isasmooth

prObability density

withcompact support

which includesa

_neighbor

of the

_origin,

andaredefinedas

$\delta$_{X_{t}^{i}}^{ $\epsilon$}(x)=($\theta$_{ $\epsilon$}^{0}*$\delta$_{X_{t}^{i}})(x)=$\theta$_{ $\epsilon$}^{0}(x-X_{t}^{i}) , u_{ $\epsilon$}^{N}=$\theta$_{ $\epsilon$}^{0}*u^{N}.

The lasttwo

_equations

of the_systemabove describe the Newtonian

dynamics

of

_parti‐

cles andweassumethe

velocity

V_{t}^{i}

satisfies astochasticdifferential

equation

driven

by

the

Brownianmotion

_{W_{t}^{i}}

in

\mathbb{R}^{d}

; the Brownianmotions

W_{t}^{i},

i=

1,

N are

independent

and

defined on a

probability

_space

( $\Omega$, \mathcal{F}, P)

.

Weassumethe

particles

havemass

\displaystyle \frac{1}{N}

; theforce

acting

on

particle

ihas threecompo‐

nents: the Stokes

drug

force duetothe

_fluid,

aninteractionforce

_{given by}

the interaction

kernel K andanoise

perturbation.

Remark 1 Recall that Stokes

_{drag force}

is

given

by

6 $\pi$ r $\mu$ \mathrm{v}

wherer is

particle radius,

\mathrm{v} is

the relative

_velocity

_of

_particle

and $\mu$ is

viscosity.

Hence the

interpretation

of

the

scalings

in N chosen above is: the

_particle

mass is

of

order

\displaystyle \frac{1}{N}

_; the

particle

radius is

of

order

\displaystyle \frac{1}{N}

and

_{c_{0}\sim 6 $\pi \mu$}

. Particles with a mass

density

similar to the

fluid

should have mass

of

the

order

_{\displaystyle \frac{1}{N^{\mathrm{d}}}}

, while here we assumeit

of

order

\displaystyle \frac{1}{N}

, much

bigger.

This

corresponds

to a

regime

of

sparse

heavy partides.

Example

2 Theinteractionkernelis

usually

absentinclassical

formulations

of

VNSsys‐ tem. We include it here since it may be

interesting

in some

applications.

For

instance,

thinktometastaticcancercells

flowing

inthe blood_stream, an

example

where the condition

of

sparse

heavy

particles

may be realistic. These cells do not

_only

interact with the

_fluid

but also between themselves and

_possibly

with other

_special

cells.

Example

3

_Having

in mind

_applications

to

_{biological fluids,}

an

_interesting

variations

could betointroduceadeath‐rate

of

the

form

$\lambda$_{t}^{i}=g(u_{ $\epsilon$}^{N}(t,X_{t}^{i})-V_{t}^{i})

, motivated

by

the

fact

thatstress mayinduce cell death. Such additionaltermleadtothe

_{term-g(u(t,x)-v)F(t, x, v)}

inthe limit. PDE.

Aswesaid

above,

ouraimis

_proving

_{convergence to}aVlasov‐Navier‐Stokes

system.We

would hke to_prove

_that,

as N\rightarrow\infty and $\epsilon$\rightarrow 0 with

_appropriate

link between

_them,

the

pair

u^{N}(t,x) , S_{t}^{N}=\displaystyle \frac{1}{N}\sum_{i=1}^{N}$\delta$_{X_{\mathrm{t}}^{i},V_{\mathrm{r}^{i}}}

(S_{\mathrm{t}}^{N}

(

dx,

dv)

isa

time‐dependent

random

probability

_measure, called

empirical

measureof

the

_particle

_system)

_converges tothe solution

_{(u, F)}

of

\displaystyle \frac{\partial F}{\partial t}+v\cdot\nabla_{x}F+\mathrm{d}\mathrm{i}\mathrm{v}_{v}((u-v)F+(K*F)F)=\frac{$\sigma$_{p}^{2}}{2}$\Delta$_{v}F

(2)

Remark 4 The term

-\displaystyle \int(u(t,x)-v)F(x,v)dv=\int vF(x,v)

dv‐

u

(t,x)\displaystyle \int F(x,\prime v)dv

is the so called Brinkman’s

force, usually

denotedinthe

physical

literature

by

”

j-u $\rho$

”

We shall see that

_proving

this hmit result is adifficult

problem.

Let us

preliminarily

describeatechnical

difficulty

withthe Navier‐Stokes_part.

Concerning

the

_hterature,

thereareresultson

particle

_systemsrelatedtoVNS

_equations

but

_only

under

_{special conditions,}

caused

_by

thefact that atrue

_{fluid‐particle}

interaction is

_imposed,

see

[1], [2], [6], [15];

and there are resultson _convergence of PDEs to

_PDEs,

although

motivated

_{by particle}

_arguments,see

[7], [10], [11].

1.1

_Difficulty

with the Navier‐Stokes

_forcing

Letus restricthereto d=2. The

equation

\displaystyle \frac{\partial u^{N}}{\partial t}= $\Delta$ u^{N}-u^{N}

.

\displaystyle \nabla u^{N}-\nabla$\pi$^{N}-\frac{c_{0}}{N}\sum_{i=1}^{N}(u_{ $\epsilon$}^{N}(t, X_{t}^{i})-V_{t}^{i})$\delta$_{X_{\mathrm{t}}^{i}}^{ $\epsilon$}

còntainsasubtle

difficulty.

Ifwe_put $\epsilon$=0,weforceNavier‐Stokes

equations

withan

input

which isworsethan

H^{-1}

(recall

that intwodimensions the delta Dirac is

only

in

H^{-1- $\gamma$}

forevery

$\gamma$>0

)

andwe

pretend

to

_speak

of

_{u^{N}(t, X_{t}^{i})}

_(for

$\epsilon$=0

₎

which

_{requires u^{N}}

tobe

continuous.

For $\epsilon$>0 wedo not seethis

regularity

issue;

butweneed uniform estimates in

( $\epsilon$, N)

to pass to the

_limit,

and

thus,

sooner or

later,

wemeet the

_difficulty

_just

described.

Remark 5 The need

for

continuous‐in‐space

velocity field

in this areahas been

recognized

also

_dealing

with other

_questions,

see

l141

who assumes

u\in L^{2}(0, T;C(D))

.

Letus

explain

this

difficulty

also with the

following

_argument.

To

simplify,

assumewe

have the heat

_equation

in

place

of the Navier‐Stokesone andwehave

_only

one

fixed

_point

particle

at

_{position X_{0}}

:

\displaystyle \frac{\partial u}{\partial t}=\frac{1}{2} $\Delta$ u+(u_{ $\epsilon$}(t, X_{0})-V_{0})$\delta$_{X_{0}}^{ $\epsilon$}

The solution with

_{u_{0}^{N}=0}

is

where

_{p_{t}(x)}

isthe heat kernel. Ifwetake the hmit as $\epsilon$\rightarrow 0wehave

u(t, x)=\displaystyle \int_{0}^{t}\frac{1}{(2 $\pi$(t-s))^{d/2}}e^{-\frac{|x-X_{\mathrm{f}1}|^{2}}{2(t- $\epsilon$)}}(u(s, X_{0})-V_{0})ds.

Already

ind=2, for

x=X_{0}

, wesee that

|u(t, X_{0})|=+\infty!

1.2

_Why

the

_problem

should be solvable

Notice however that the

_conjectured

limit

_equation,

the Valsov‐Navier‐Stokes system is better: nodelta Dirac_appearthere.

What is

_meaningless,

as remarked inthe

_{previous section,}

is the model with a finite

number

N_{0}

of

_point

_particles

ofmass

\displaystyle \frac{1}{N_{0}}

, ifwetake the limit $\epsilon$\rightarrow 0. But this is notwhat

wewant to do: we want totake the limit of

infinitely

many

particles,

with infinitesimal

interaction

_strength.

We_may

_{hope that,}

inthe limitas $\epsilon$\rightarrow 0,wemaycontrol the

quantities

\displaystyle \frac{\mathrm{b}\mathrm{e}1}{N}

.cause

we also take N\rightarrow\infty and the

_intensity

of

_{fluid‐particle}

interaction isrescaled

by

However,

torealize this_{program, it}isessentialto provethat

_particles

donot concentrate too

_much,

otherwisewe take the riskto have

_again,

in the

_{limit, cơńcentrated}

masses of

particles

withfinite interaction

_{strength. Therefore,}

amain_purposeof the estimates below

is

_proving

aform ofnonconcentration.

2

_Energy

balance

Lemma 6

_Setting

\displaystyle \mathcal{E}_{t}=\frac{1}{2}\int|u^{N}(t, x)|^{2}dx+\frac{1}{2N}\sum_{i=1}^{N}|V_{t}^{i}|^{2}

if

u^{N}

is a

regular

solution thenwehave

d\displaystyle \mathcal{E}_{t}+(\int|\nabla u^{N}(t, x)|^{2}dx+\frac{1}{N}\sum_{i=1}^{N}(u_{ $\epsilon$}^{N}(t, X_{t}^{i})-V_{t}^{i})^{2})dt

= (\displaystyle \frac{1}{N^{2}}\sum_{i,j=1}^{N}V_{t}^{i}K(X_{t}^{i}-X_{t}^{j})+\frac{$\sigma$_{p}^{2}}{2}) dt+\frac{$\sigma$_{p}}{N}\sum_{i=1}^{N}V_{t}^{i}dW_{\mathrm{t}}^{i}.

The

_proof

is

_elementary

_by

Itôformula. Notice that the

_previous

result also

gives

us a

controlon

because it is bounded

_(up

to

_constants)

_by

\displaystyle \frac{1}{N}\sum_{i=1}^{N}|V_{t}^{i}|^{2}

_plus

\displaystyle \frac{1}{N}\sum_{i=1}^{N}(u_{ $\epsilon$}^{N}(t, X_{t}^{i})-V_{t}^{i})^{2}

that areboth controlled

(the

secondone

integrated

in

time).

Using

the

_previous

a

_priori

estimatesonecan_prove, under the

_assumptions

u_{0}\in \mathrm{L}_{ $\sigma$}^{2}($\Gamma$^{2})

E[\displaystyle \frac{1}{N}\sum_{i=1}^{N}(|X_{0}^{i}|^{2}+|V_{0}^{i}|^{2})] \leq C

(

_{\mathrm{L}_{ $\sigma$}^{2}($\Gamma$^{2})}

is the usual space of

divergence

free

periodic

zero mean vector fields on

$\Gamma$^{2}

)

existence and

_uniqueness

of solutions

_(for

finiteN

₎

such that

E[\displaystyle \sup_{t\in[0,T]}\int|u^{N}(t, x)|^{2}dx] \leq C

E[\displaystyle \int_{0}^{T}\int|\nabla u^{N}(t, x)|^{2}dxdt] \leq C

E[\displaystyle \sup_{t\in[0,T]}\frac{1}{N}\sum_{i=1}^{N}(|X_{t}^{i}|^{2}+|V_{t}^{i}|^{2})] \leq C

E[\displaystyle \int_{0}^{T}\frac{1}{N}\sum_{i=1}^{N}(u_{ $\epsilon$}^{N}(t, X_{\mathrm{t}}^{i})-V_{t}^{i})^{2}dt] \leq C

E[\displaystyle \int_{0}^{T}\frac{1}{N}\sum_{i=1}^{N}u_{ $\epsilon$}^{N}(t,X_{t}^{i})^{2}dt] \leq C.

From these

_bounds,

with

_relatively

classicalcompactness

theorems,

onecanshow that

the

_family

of laws of

_{(u^{N}, S^{N})}

are

tight

and thus there exist

subsequences

which_converge

in

_law;

_changing

_probability

spaceitis

possible

to assume a.s. _convergencein

appropriate

topologies.

In the

sequel,

to understand the

difficulties,

we assume for

simplicity

such

a.s. _convergence. We do not want to

_give

the details

_here,

which will be included in a

forthcoming

technical work.

Let

us

only

mention that

subsequences

(u^{N_{k}}, S^{N_{k}})

, on the

new

probability

_space, will have the_propertythat

\bullet

u^{N_{k}}

_converges

strongly

in

L^{2}(0, T;L^{2}($\Gamma$^{2}))

\bullet

u^{N_{k}}

_converges

_weakly

_in

L^{2}(0, T;W^{1,2}($\Gamma$^{2}))

_{and weakstar in}

L^{\infty}(0,T;L^{2}($\Gamma$^{2}))

\bullet

S^{N_{k}}

_converges inthe weak

topology

ofmeasures

uniformly

in time.

In the

sequel,

when we

informally

discuss

questions

of convergence, we

replace

the

3

A

_difficulty

about

_{passage to}

the

limit

in the Navier‐Stokes

system

Letusstressthat the existenceofa_convergent

_subsequence

(u^{N_{k}}, S^{N_{k}})

(denoted

below

by

(u^{N},

S^{N}\backslash

inthe

_topologies

indicated at the endof the

_{previous section,}

is trUe both if

we

keep

$\epsilon$>0

_unchanged

with N, orifwehnkit to N

by choosing $\epsilon$= $\epsilon$ N\rightarrow 0

.

However,

in the first case we can_{pass to}the

hmit,

in the secondonewe meet arelevant techmical

difficulty,

that we now

explain.

In weak formon

divergence

free smoothtest vectorfields

$\phi$

, theNavier‐Stokessystem

reads

\displaystyle \langle u^{N}(t) , $\phi$\rangle_{x}-\{u_{0}, $\phi$\rangle_{x}+\int_{0}^{t}\langle\nabla u^{N}, \nabla $\phi$\rangle_{x}ds

=\displaystyle \int_{0}^{t}\langle u^{N}\cdot\nabla $\phi$, u^{N}\rangle_{x}ds-\{\frac{1}{N}\sum_{i=1}^{N}(u_{ $\epsilon$}^{N}(t,X_{t}^{i})-V_{t}^{i})$\delta$_{X_{t}^{i}}^{ $\epsilon$} \mathrm{V}\mathrm{q}1), \mathrm{X}, $\phi$(\cdot)\}_{x}

where

_\{f,

_{g\displaystyle \rangle_{x}=\int_{$\Gamma$^{d}}f(x)\cdot g(x)dx}

for suitablevectorfields

_f,

g,and

\cdotdenotes scalar

product

in

\mathbb{R}^{d}

. The

difficulty

is

only

inthe convergenceof the lastterm, when

u^{N}

converges

only

inthe usual

_topologies

of weak solutions mentionedattheend of lastsection. What about

the_convergenceof

u_{ $\epsilon$}^{N}(t, X_{t}^{i})=($\theta$_{ $\epsilon$}^{0}*u_{t}^{N})(X_{t}^{i})

?

Itseems_{necessary to prove}some_convergenceof

u^{N}

inthe

uniform topology.

Butuniform

estimatesare_{not among}thea

_priori

bounds.

Although

not

_being

the

_only

one,anatural_{way to prove}bounds in theuniform

topology

for

_{u_{t}^{N}}

is

_by

Sobolev

_embedding,

hence

_{investigating}

bounds on derivatives of

u_{t}^{N}

. Since

we are on a torus and werestrict to d=2, we use

vorticity.

The

question

then is: can

we _prove

enstrophy

_type bounds? Consider thenthe

vorticity

equation,

which in thecase

d=2,for the

vorticity

function

$\omega$^{N}=\nabla^{\perp}\cdot u^{N}

, is

\displaystyle \frac{\partial$\omega$^{N}}{\partial t}= $\Delta \omega$^{N}-u^{N}

.

\displaystyle \nabla$\omega$^{N}-\frac{\mathrm{c}_{0}}{N}\sum_{i=1}^{N}\nabla^{\perp}.

((u_{ $\epsilon$}^{N}(t,X_{t}^{i})-V_{t}^{i})$\delta$_{x_{t^{\dot{\mathrm{a}}}}^{N}}^{ $\epsilon$})

.

(3)

A main

_conceptual

remark is that

_particles

create

_vorticity.

Terms hke

_{\partial_{i}$\delta$_{X_{{\$}}^{i}}^{$\epsilon$_{N}}}

contribute

diverging

termsinNfor _{$\epsilon$= $\epsilon$ N\rightarrow 0}and thus

_vorticity

doesnot seemtobe under control.

4

_Summary

of results

\bullet First we

develop

a “two_steps

approach

which consists intwo_separate hmit theo‐

rems.

1. The first oneis

_only

the hmit as N\rightarrow\infty,

given

aconstant value of

$\epsilon$\in(0,1)

; it

identifies ahmit

\mathrm{P}\mathrm{D}\mathrm{E}_{ $\epsilon$}

;

2. Thesecondoneisthe limit

_{PDE_{ $\epsilon$}\rightarrow PDE}

as $\epsilon$\rightarrow 0.

Linking artificialy

( $\epsilon$, N)

, therearesequences

( $\epsilon$ k, N_{k})

where Particles

($\epsilon$_{k},N_{k})\rightarrow PDE

as k\rightarrow\infty. Thisis notthe solutionwe were

looking

for,

butitis

important

toknow

that atleast this

_{relatively simple}

two‐step

approach

works.

\bullet

Second,

we

_conjecture

a local in _{time result of the form Particles ($\epsilon$_{N},N)} \rightarrow PDE as

N \rightarrow \infty. It is based on local‐in‐time uniform‐in‐x estimates on

u^{N}

_,

jointly

with

estimatesonno concentrationof

particles

(these

twofacts

_proceed

_together).

A full

proof

still

_requires

to solve

\mathrm{t}\mathrm{e}\mathrm{c}\mathrm{b}\mathrm{u}\mathrm{c}\mathrm{s}^{\mathrm{R}}\mathrm{J}

_problems,

so we limit ourselves _{to express} a

reasonable

_conjecture.

5

_{Two‐steps approach}

5.1

Preliminaries

The

_advantage

ofthe

“two‐steps”

or”’_separatelimit”

_strategy

isthat itworks with minimal

ingredients:

wedonotneedto proveno‐concentrationof

_particles;

wedonotneed the noise

to

_regularize;

wecan_say

something

alsointhecased=3

(always

on atorus

$\Gamma$^{3}

,to

simplify)

Weassume,for

simplicity

of

_notations,

_K=0,

_{$\sigma$_{p}=0,}

_{c_{0}=1}

. However the result remains

true _{when $\sigma$_{p}}

_\neq

0 and when Kis bounded

Lipschitz

continuous and

_presumably

also in

some caseswhen

K=K_{N}

isrescaled in a_{proper way.}

For sake of

_clarity

_(also

because here there is no _average over the

randomness),

we

restatethe well

_posedness

mentioned above forfinite N and the_energybounds.

Lemma 7 For_every

_{$\epsilon$\in(0,1)}

and N\in \mathrm{N}, the

system

\displaystyle \frac{\partial u^{N}}{\partial t}= $\Delta$ u^{N}-u^{N}\cdot\nabla u^{N}-\nabla$\pi$^{N}-\frac{1}{N}\sum_{i=1}^{N}(u_{ $\epsilon$}^{N}(t, X_{\mathrm{t}}^{i})-V_{t}^{i}).$\delta$_{X_{\mathrm{t}}^{i}}^{ $\epsilon$}

\displaystyle \frac{d}{dt}X_{t}^{i}=V_{t}^{i}, \frac{d}{dt}V_{\mathrm{t}}^{i}=u_{ $\epsilon$}^{N}(t, X_{t}^{i})-V_{t}^{i}

has a

unique

solution such that

\displaystyle \sup_{t\in[0,T]}\int|u^{N}(t,x)|^{2}dx+\int_{0}^{T}\int|\nabla u^{N}(t,x)|^{2}dxdt\leq C.

5.2 First limit:

_{N\rightarrow\infty,}

_{$\epsilon$\in(0,1)}

_given

Wenowconsider the

following mollified

VNS_system

\displaystyle \frac{\partial u}{\partial t}= $\Delta$ u-u\cdot\nabla u-\nabla $\pi-\theta$_{ $\epsilon$}^{0}*\int(u_{ $\epsilon$}(t, \cdot)-v)F(\cdot, dv)

(4)

\displaystyle \frac{\partial F}{\partial t}+v\cdot\nabla_{x}F+\mathrm{d}\mathrm{i}\mathrm{v}_{v}((u_{ $\epsilon$}-v)F)=0

(5)

where

u_{ $\epsilon$}=$\theta$_{ $\epsilon$}^{0}*u.

Thanks to the

_{mollification,}

we are allowed to

_investigate

this_system and _convergenceof the

_particle

_system when the

density

of

_particles

is treated

_just

as a _{measure, not} as a

density

function. Letus

_give

the

_appropriate

definition. Denote

by

\mathrm{P}\mathrm{r}_{1}($\Gamma$^{d}\times \mathbb{R}^{d})

theset

of all Borel

probability

measures _$\mu$on

$\Gamma$^{d}\times \mathbb{R}^{d}

such that

\displaystyle \int_{$\Gamma$^{d}}\prime\int_{\mathbb{R}^{d}}|v| $\mu$

(

dx,

dv)<\infty

we endow

\mathrm{P}\mathrm{r}_{1}($\Gamma$^{d}\times \mathbb{R}^{d})

with the weak

_topology,

withconvergenceof firstmoment. The notation

_{$\theta$_{ $\epsilon$}^{0}*\displaystyle \int(u_{ $\epsilon$}(t, \cdot)-v)}

F.

_(dv),

when F \in

C([0, T];\mathrm{P}\mathrm{r}_{1}($\Gamma$^{d}\times \mathbb{R}^{d}))

and

_{\mathrm{u}_{ $\epsilon$}(t, \cdot)}

is measurable and

bounded,

standsfor

($\theta$_{ $\epsilon$}^{0}*\displaystyle \int(u_{ $\epsilon$}(t, \cdot)-v)F(\cdot, dv))(x)=\int_{$\Gamma$^{\mathrm{d}}}\int_{\mathbb{R}^{d}}$\theta$_{ $\epsilon$}^{0}(x-x')(u_{ $\epsilon$}(t, x')-v')F(t, dx', dv')

.

Beside the notation

_{\langle f,}

_{g\}_{x}}

_already

introduced

above,

herewealsowrite

( $\mu$,

f\}_{x,v}

for

\displaystyle \{ $\mu$, f\}_{x,v}=\int_{$\Gamma$^{d}}^{\backslash }\int_{\mathbb{R}^{\mathrm{d}}}f(x, v) $\mu$

(dx

,

dv)

.

Definition8 Let _{u_{0}} \in

\mathrm{L}_{ $\sigma$}^{2}($\Gamma$^{2})

and

_{F_{0}}

\in

\mathrm{P}\mathrm{r}_{1}($\Gamma$^{d}\times \mathbb{R}^{d})

be

_given.

A

_pair

(u, F)

is a

solution

_of

_system

_(4)‐(5)

with initial condition

_{(u_{0}, F_{0})}

_if

u\in L^{\infty}(0, T;L^{2}($\Gamma$^{d}))\cap L^{2}(0, T;W^{1,2}($\Gamma$^{d}))

F\in C([0, T] ; Pr_{1}($\Gamma$^{d}\times \mathbb{R}^{d}))

\displaystyle \{u(t), $\phi$\}_{x,\backslash }-\{u_{0}, $\phi$)_{x}+\int_{0}^{t}\langle\nabla u, \nabla $\phi$\}_{x}ds

=\displaystyle \int_{0}^{t}\{u\cdot\nabla $\phi$, u\rangle_{x}ds-\int_{0}^{t}\langle$\theta$_{ $\epsilon$}^{0}*\int(u_{ $\epsilon$}(s, \cdot)-v)F(s, \cdot, dv) , $\phi$\rangle_{x}ds

\{F(t)

,

$\varphi$\rangle_{x,v}=\{F_{0},

$\varphi$\displaystyle \rangle_{x,v}+\int_{0}^{t}\{F(s)

,

v\displaystyle \cdot\nabla_{x} $\varphi$\rangle_{x,v}ds+\int_{0}^{t}\{F(s), (u_{ $\epsilon$}(s)-v)\cdot\nabla_{v} $\varphi$\}_{x,v}ds

Theorem 9 Let

_{$\epsilon$\in(0,1)}

be

_given.

Assume

_{u_{0}\in \mathrm{L}_{ $\sigma$}^{2}}

_(T2),

\displaystyle \frac{1}{N}\sum_{i=1}^{N}(|X_{0}^{i}|^{2}+|V_{0}^{i}|^{2})

\leq C,

and

_{S_{0}^{N}\rightarrow F_{0}}

inthe weaksense

of probability

measures.

1)

Let d=3. Given $\epsilon$\in

(0

,1

)

, there exists a

subsequence N_{k}\rightarrow\infty

such that the

pair

(u^{N_{k}}, S^{N_{k}})

converges

(in

the sense described at the end

_of

section

₂₎

to a solution

(u, F)

of

system

(4)‐(5)

with initial condition

_{(u_{0}, F_{0})}

.

2)

Let d = 2.

System

(4)‐(5),

with initial condition

(u0, F_{0})

, has a

unique

solution

(u, F)

and,

as N\rightarrow\infty, the

pair

(u^{N}, S^{N})

converges to

(u, F)

.

Remark 10 In d=3

obviously

we donotknow

uniqueness

duetotheNavier‐Stokespart;

hence the convergence holds

only for

certain

subsequences

(for

every

subsequence

there is

a

sub‐subsequence

which

converges).

Remark 11 Existence

of

a solution

(u, F)

of

the limit

_system

(with

given

$\epsilon$ \in

_(0,1))

eithercan be

proved

directly

orit

follows from

the_convergence result

itself,,

being

based on a _compactness

_{argument. Uniqueness}

of

(u, F) (

for

d=2)

has to be

_{proved directly.}

Letus

give

afew elementsofthe

proof.

From the estimates

\displaystyle \sup_{t\in[0,T]}\int|u^{N}(t,x)|^{2}dx+\int_{0}^{T}\int|\nabla u^{N}(t,x)|^{2}dxdt\leq.C

it isclassical

_{(cf. [17])}

to

_apply

the_compactnessAubin‐Lions lemma. Duetothe estimate

\displaystyle \sup_{t\in[0,T]}\frac{1}{N}\sum_{i=1}^{N}(|X_{t}^{i}|^{2}+|V_{t}^{i}|^{2}) \leq C

one can use a criterion based on Wasserstein distance _{to prove}

_compactness

of

S^{N_{k}^{(0)}}

in

_{C([0,T];\mathrm{P}\mathrm{r}_{1}($\Gamma$^{d}\times \mathbb{R}^{d}}

From these fact one has the existence of a

subsequence

(u^{N_{k}}, S^{N_{k}})

which _converges as described at the end of section 2. Call

_{(u, F)}

the limit

of such

_subsequence.

Taking

the limitinthe first fourtermsof the weak formulation ofNavier‐Stokesequa‐

tions

_(see

Definition

₈₎

is classical

_{(cf. [17]).}

_Concerning

the last term, wehaveto prove

that

k\displaystyle \rightarrow\infty \mathrm{h}\mathrm{m}\{\frac{1}{N_{k}}\sum_{i=1}^{N_{k}}(u_{ $\epsilon$}^{N_{k}}(t,X_{t}^{i,N_{k}})-V_{t}^{i,N_{k}})$\delta$_{X_{t}^{i,N_{h}}}^{ $\epsilon$}, $\phi$\}_{x}

Thetermon theleft‐hand sideis

equal

to

\displaystyle \int_{$\Gamma$^{d}}\frac{1}{N_{k}}\sum_{i=1}^{N_{k}}(u_{ $\epsilon$}^{N_{k}}(t, X_{t}^{i,N_{k}})-V_{t}^{i,N_{k}})$\theta$_{ $\epsilon$}^{0}(x-X_{t}^{i,N_{k}}) $\phi$(x)dx

=\displaystyle \int_{$\Gamma$^{d}}\int_{$\Gamma$^{d}}\int_{\mathbb{R}^{d}}(u_{ $\epsilon$}^{N_{k}}(t, x')-v')$\theta$_{ $\epsilon$}^{0}(x-x') $\phi$(x)S^{N_{k}}

(dx

’,

dv')dx

=\displaystyle \int_{$\Gamma$^{d}}\int_{\mathbb{R}^{d}}(($\theta$_{ $\epsilon$}^{0}*u^{N_{k}}(t))(x')-v')($\theta$_{ $\epsilon$}^{0-}* $\phi$)(x')S^{N_{k}}

(dx

’,

_dv')

where

_{$\theta$_{ $\epsilon$}^{0-}(x) =$\theta$_{ $\epsilon$}^{0}(-x)}

. The term

$\theta$_{ $\epsilon$}^{0}*u^{N_{k}}(t)

converges‐

uniformly

to

$\theta$_{ $\epsilon$}^{0}*u(t)

fora.e. t

_(passing

to a

subsequence).

With httleadditionalcarewe cantake the limitas k\rightarrow\infty

and get

\displaystyle \int_{$\Gamma$^{d}}\int_{\mathbb{R}^{d}}(($\theta$_{ $\epsilon$}^{0}*u(t))(x')-v')($\theta$_{ $\epsilon$}^{0-}* $\phi$)(x')F(t, dx', dv')

=\displaystyle \langle\int_{$\Gamma$^{d}}\int_{\mathbb{R}^{d}}$\theta$_{ $\epsilon$}^{0}(\cdot-x')(u_{ $\epsilon$}(t, x')-v')F(t, dx', dv') , $\phi$\rangle_{x}

Toprove that F satisfies the weak

identity

inDefinition 8 we have first to derive an

identity

for

S^{N_{k}}

.

By

chain rule

applied

to

$\varphi$(X_{t}^{i,N_{k}}, V_{t}^{i,N_{k}})

we_get

\langle S_{t}^{N_{k}},

_{$\varphi$\rangle_{x,v}=\langle S_{0}^{N_{h}},}

$\varphi$\displaystyle \rangle_{x,v}+\int_{0}^{t}\langle S_{8}^{N_{k}},

v\displaystyle \cdot\nabla_{x} $\varphi$\rangle_{x,v}ds+\int_{0}^{t}\langle S_{8}^{N_{k}},

(u_{ $\epsilon$}(s)-v)\cdot\nabla_{v} $\varphi$\rangle_{x,v}ds

(in

thedeterministic caseit is a well knownfact that the

empirical

measure is

already

a

solution of the limit

_PDE;

inthe stochasticcase,

$\sigma$_{p}\neq 0

,onehasto

apply

Itô formula and

an additional

_martingale

_{term appears,}which

_however,

_{converges to}

_zero).

Thenonecan

pass to the limit.

Finaly,

for d=2wehaveto prove

uniqueness

for the limit_system

(4)-(5)

. In

principle,

oneof the difficultiesisthatwedeal with solutions F whichare

only

measures.

However,

we

mayuse awellknown method

_(see

for instance

_[3])

basedonWasserstein distance

d_{1}

(

_$\mu$,

iỷ)

between $\mu$,

\mathrm{v}\in \mathrm{P}\mathrm{r}_{1}($\Gamma$^{d}\times \mathbb{R}^{d})

. Assume that

(u, F)

,

(u', F')

aretwosolutions. One has

d_{1}(F(t), F'(t))\leq E[|X_{t}-X_{t}'|+|V_{t}-V_{t}

where

_{(X_{t}, V_{t})}

satisfies

\displaystyle \frac{d}{dt}X_{t}=V_{t}

whereuisthefirst_componentof the solution

(u, F)

, and

similarly

for

(Xt’, V_{t}'

) (with

respect

to

₍

_u',

F Theinitial conditions for thesetwo

_problems

arethe_same,

(X_{0},

V_{0}

with law

F_{0}

. One can

easily

provethat

E[|X_{t}-X_{t}^{l}|+|V_{t}-V_{t} \displaystyle \leq C\int_{0}^{t}E[|X_{s}-X_{s}|+|V_{s}-\acute{V}_{s}'|]ds

+C\displaystyle \int_{0}^{t}|u_{ $\epsilon$}(s, X_{s})-u_{ $\epsilon$}'(s, X_{s})|ds.

Then one has to _repeat classical _{energy type}

_computations

of the 2‐dimensionaì

_theory

of Navier‐Stokes

_equations

_{(cf. [17])}

to control u-u' inthe norms

L^{\infty}(0, T;L^{2}($\Gamma$^{d}))\cap

L^{2}(0, T;W^{1,2}($\Gamma$^{d}))

, boundsto be used

jointly

with the

previous

one. The

only

nonclas‐

sicaltermis

|\displaystyle \langle$\theta$_{ $\epsilon$}^{0}*\int(u_{ $\epsilon$}(t, \cdot)-v)F(\cdot, dv)

,

u(t)\displaystyle \rangle_{x}-\langle$\theta$_{ $\epsilon$}^{0}*\int(u_{ $\epsilon$}'(t, \cdot)-v)F'(\cdot, dv)

,

u'(t)\rangle_{x}|

whichis controlled

_by

the

_previous

normsofu-u'

plus

theterm

|\displaystyle \langle$\theta$_{ $\epsilon$}^{0}*\int(u_{ $\epsilon$}(t, \cdot)-v)(F(\cdot, dv)-F'(\cdot, dv)) , u(t)\rangle_{x}|

=|\displaystyle \int_{$\Gamma$^{d}}\int_{\mathbb{R}^{d}}($\theta$_{ $\epsilon$}^{0-}*u(t))(x)(u_{ $\epsilon$}(t, x)-v)(F (dx, dv)-F' (dx, dv))|

\leq C\cdot d_{1}(F(t), F'(t))

(the

last bound

_requires

some

work,

omitted

here).These

recursive estimates aJlow oneto

apply

Gronwall lemma andprove that

(u, F)=(u', F')

.

5.3 Second limit: $\epsilon$\rightarrow 0

Untilnowwehave_proventhat the

fluid,

coupled

with the

_particles,

converges, as N\rightarrow\infty

to _system

_(4)-(5)

, where the molhfication with $\epsilon$ \in

(0,1)

survives. Called

(u^{ $\epsilon$}, F^{ $\epsilon$})

the

solution in d=2

_(or

asolutionind=3

₎

of

_(4)-(5)

, it remains to

investigate

thehmit as

$\epsilon$\rightarrow 0.

The hmit cannot be taken at the’levelofmeasure solutions F, orat least this looks

verydifficult. Ońecan

give

a

meaning

tothe weak formulation of the VNS

_(1)-(2)

when F is

_{only measure‐valued,}

butatthe

_price

of

_imposing

a

_priori

thatuiscontinuous bounded. This direction could be

investigated

but

_requires

a

fully

original

approach

toVNS _system which is

beyond

the _scope of this note. And in

_addition,

as remarked

below,

oneshould

expect

only

local‐in‐time solutions.

Therefore let us consider the modified VNS system and the true one when F is a

fmction.

_First,

onehas_{to prove} that themodified VNS_systemhas asolution

(u^{ $\epsilon$}, F^{ $\epsilon$})

in

be

_umique

also inthe weaker class ofmeasure valued solutions. Then one should prove

convergenceof

(u^{ $\epsilon$}, F^{ $\epsilon$})

to asolution

(u, F)

of_system

(1)-(2)

. We have checked that both

these _steps are

plausible following

the

approach

of

[18];

see also

[4];

however there are

several details and results willappearina

forthcoming

work.

When this is

_done,

it is

_possible

to extract suitable _sequences

_{( $\epsilon$ N)}

such that

(u^{N_{k}}, S^{N_{k}})

converges to

(u, F)

as k\rightarrow\infty.

Here, by

(u^{N_{k}}, S^{N_{k}})

_, wemean thoseobtained

by

the

_{fluid‐particle}

_systemwith $\epsilon$= $\epsilon$ k.

6

Joint

limit

6.1

Introduction

As described inSections 1.1 and

3,

uniform estimatesonthe

velocity

are

required

to pass

to the hmit

_{simultaneously}

in N\rightarrow\infty and $\epsilon$\rightarrow 0. A natural

approach

to proveuniform

bounds on

u^{N}

is to _get

W^{ $\epsilon$,2}

(for

$\epsilon$ > 0

₎

bounds onthe

vorticity $\omega$^{N}

=\nabla^{\perp}\cdot \mathrm{u}^{N}

, which

satisfies

_equation

_(3):

Bounds on the

enstrophy

arenot

_sufficient,

since

_they

are bounds onthe

W^{1,2}

‐normof

u^{N}

which donot

_imply

uniform boundson

u^{N}

. Hence wework with

semigroups

and look for boundsinmore

regular topologies.

Notice however that

enstrophy

boundsmeet thesamedifficultieswehave with

semigroups.

Denoting by e^{t $\Delta$}

the

_semigroup

associatedtothe

_Laplacian

_operatorinƯorC^{ $\alpha$}_spaces

onthe _torus, wehave

\displaystyle \frac{\partial$\omega$^{N}}{\partial t}= $\Delta \omega$^{N}-u^{N}

.

\displaystyle \nabla$\omega$^{N}-\frac{c_{0}}{N}\sum_{i=1}^{N}\nabla^{\perp}.

((u_{ $\epsilon$}^{N}N(t, X_{t}^{i})-V_{t}^{i})$\delta$_{X_{t}^{i}}^{$\epsilon$_{N}})

.

$\omega$^{N}(t)=e^{t $\Delta$}$\omega$^{N}(0)-\displaystyle \int_{0}^{t}e^{(t-s) $\Delta$}u^{N}(s) \nabla$\omega$^{N}(s)ds

-\displaystyle \int_{0}^{t}e^{(t- $\epsilon$) $\Delta$}\nabla^{\perp}. \frac{c_{0}}{N}\sum_{i=1}^{N}(u_{$\epsilon$_{N}}^{N}(s, X_{s}^{i})-\dot{ $\psi$}_{s})$\delta$_{X_{s}^{i}}^{ $\epsilon$}ds.

Wewant to estimate

_{$\omega$^{N}(t)}

in

W^{2 $\alpha$,2}

_(Td),

hencewe usethe

inequality

\Vert(I- $\Delta$)^{a}$\omega$^{N}(t)\Vert_{L^{2}($\Gamma$^{d})} \leq \Vert(I- $\Delta$)^{ $\alpha$}$\omega$^{N}(0)\Vert_{L^{2}($\Gamma$^{d})}

+\displaystyle \int_{0}^{t}\Vert(I- $\Delta$)^{ $\alpha$}e^{(t-s) $\Delta$}u^{N}(s)

.

\nabla$\omega$^{N}(s)\Vert_{L^{2}($\Gamma$^{d})}ds

+\displaystyle \int_{0}^{t}\Vert(I- $\Delta$)^{\frac{1}{2}+ $\alpha$}e^{(t-s) $\Delta$}\nabla^{\perp}(I- $\Delta$)^{-\frac{1}{2}}\cdot\frac{\mathrm{c}_{0}}{N}\sum_{i=1}^{N}(u_{$\epsilon$_{N}}^{N}(s, X_{s}^{i})-V_{8}^{i})$\delta$_{X_{8}^{i}}^{ $\epsilon$}\Vert_{L^{2}('$\Gamma$^{d})^{ds}}

.

(6)

Let us

only

concentrate on the term

₍₆₎

without

_{V_{8}^{i}}

, which is the sóurce of the main

difficulty.

For every T> 0, denote

by

\Vert\cdot\Vert_{T,\infty}

the supremumnorm over

[0, T]

\times$\Gamma$^{d}

. We

have

|\displaystyle \frac{1}{N}\sum_{i=1}^{N}u_{$\epsilon$_{N}}^{N}(s, X_{s}^{i})$\delta$_{X_{s}^{i}}^{ $\epsilon$}(x)|\leq\Vert u_{$\epsilon$_{N}}^{N}\Vert_{T,\infty}F_{S}^{0,N}(x)

where

F_{t}^{0,N}=$\theta$_{$\epsilon$_{N}}^{0}*(\displaystyle \frac{1}{N}\sum_{i=1}^{N}$\delta$_{X_{t}^{i}})

and therefore theterm

_(ô)

without

_{V_{ $\epsilon$}^{i}}

isbounded above

by

(

\nabla^{\perp}(I- $\Delta$)^{-\frac{1}{2}}

is aboundedoperatorin

L^{2}(.$\Gamma$^{d})

)

\displaystyle \Vert\nabla^{\perp}(I- $\Delta$)^{-\frac{1}{2}}\Vert_{L^{2}($\Gamma$^{d})\rightarrow L^{2}($\Gamma$^{d})}\Vert u_{$\epsilon$_{N}}^{N}\Vert_{T,\infty}\int_{0}^{t})

\leq C\Vert u_{$\epsilon$_{N}}^{N}\Vert_{T,\infty}T^{\frac{1}{2}- $\alpha$}

\displaystyle \sup

\Vert F^{0,N}\Vert_{L^{2}($\Gamma$^{d})}

(7)

t\in[0,T] because

\displaystyle \Vert(I- $\Delta$)^{\frac{1}{2}+ $\alpha$}e^{(t-s) $\Delta$}\Vert_{L^{2}('$\Gamma$^{d})\rightarrow L^{2}($\Gamma$^{d})} \leq\frac{C}{(t-s)^{\frac{1}{2}+ $\alpha$}}

by

well known

analytic semigroup

estimates.

We need an estimate on

\Vert F^{0,N}\Vert_{L^{2}( $\Gamma$)^{d}}

: this is the _property of no concentration of

particles,

asannouncedinSection 1.2.

6.2 No concentration of

_particles

Set

F_{t}^{N}=$\theta$_{$\epsilon$_{N}}*(\displaystyle \frac{1}{N}\sum_{i=1}^{N}$\delta$_{X_{\mathrm{t}}^{l},V_{\mathrm{r}^{i)}}}

, wherenow

$\theta$_{ $\epsilon$}=$\theta$_{ $\epsilon$}(x, v)

are sm

\cdot

tablemolhfiers inboth

variables,

related to

_{$\theta$_{ $\epsilon$}^{0}}

. Hereweneed

$\sigma$_{p}\neq 0

. One has

dF_{t}^{N}= (\displaystyle \frac{$\sigma$_{p}^{2}}{2}$\Delta$_{v}F_{t}^{N}-\nabla_{x} . $\theta$_{$\epsilon$_{N}}*(vS_{\mathrm{t}}^{N}))dt

-(\nabla_{v}\cdot$\theta$_{$\epsilon$_{N}}*((u_{$\epsilon$_{N}}^{N}(t, x)-v)S_{t}^{N}))dt+dM_{t}^{N}

where

M_{t}^{N}(x, v)=\displaystyle \frac{1}{N}\sum_{i=1}^{N}\int_{0}^{t}\nabla_{v}$\theta$_{$\epsilon$_{N}}(x-X_{s}^{i}, v-V_{s}^{i})$\sigma$_{p}dB_{s}^{i}.

\displaystyle \frac{1}{2}d\int_{$\Gamma$^{\mathrm{d}}}\int_{\mathbb{R}^{d}}(F_{t}^{N})^{2}dxdv+\frac{$\sigma$_{p}^{2}}{2}\int_{$\Gamma$^{\mathrm{d}}}\int_{\mathbb{R}^{d}}|\nabla_{v}F_{t}^{N}|^{2}

dxdvdt

=\displaystyle \int_{$\Gamma$^{d}}\int_{\mathbb{R}^{d}}($\theta$_{$\epsilon$_{N}}*(vS_{t}^{N}))\nabla_{x}F_{t}^{N}

dxdvdt

+\displaystyle \int_{$\Gamma$^{d}}\int_{\mathbb{R}^{d}}($\theta$_{$\epsilon$_{N}}*((u_{$\epsilon$_{N}}^{N}(t, x)-v)S_{t}^{N}))\nabla_{v}F_{t}^{N}

dxdvdt

plus

termsrelatedtothe

_martingale

partthat wedonotdiscuss

_explicitly

here. Letus see

howto treatthe mostdifficult term: since

|($\theta$_{$\epsilon$_{N}}*(u_{$\epsilon$_{N}}^{N}(t, x)S_{t}^{N}))(x, v)|

=|\displaystyle \int_{$\Gamma$^{\mathrm{d}}}\int_{\mathbb{R}^{d}}$\theta$_{$\epsilon$_{N}}(x-x', v-v')u_{$\epsilon$_{N}}^{N}(t, x')S_{t}^{N}

(dx

’,

dv')|

\displaystyle \leq\int_{\mathrm{N}^{d}}\'{I}_{\mathbb{R}^{d}}$\theta$_{$\epsilon$_{N}}(x-x', v-v')|u_{$\epsilon$_{N}}^{N}(t, x')|S_{t}^{N}(dx', dv')

\leq\Vert u_{$\epsilon$_{N}}^{N}\Vert_{T,\infty}($\theta$_{$\epsilon$_{N}}*S_{t}^{N})(x, v)

wehave

|\displaystyle \int_{$\Gamma$^{d}}\int_{\mathbb{R}^{\mathrm{d}}}($\theta$_{$\epsilon$_{N}}*(u_{$\epsilon$_{N}}^{N}(t, x)S_{t}^{N}))\nabla_{v}F_{t}^{N}dxdv|

\displaystyle \leq\Vert u_{$\epsilon$_{N}}^{N}\Vert_{T,\infty}\int_{$\Gamma$^{d}}\int_{\mathbb{R}^{d}}F_{t}^{N}|\nabla_{v}F_{t}^{N}|dxdv

\displaystyle \leq $\epsilon$\int_{\mathrm{T}^{d}}\int_{\mathbb{R}^{d}}|\nabla_{v}F_{t}^{N}|^{2}dxdv+\frac{\Vert u_{ $\epsilon$}^{N}N\Vert_{T,\infty}^{2}}{ $\epsilon$}\int_{$\Gamma$^{d}}\int_{\mathbb{R}^{d}}(F_{t}^{N})^{2}dxdv.

Summarizing,

\displaystyle \frac{1}{2}d\int_{$\Gamma$^{d}}\int_{\mathbb{R}^{\mathrm{d}}}(F_{t}^{N})^{2}dxdv+\frac{$\sigma$_{p}^{2}}{4}\int_{$\Gamma$^{\mathrm{d}}}\int_{\mathbb{R}^{d}}|\nabla_{v}F_{t}^{N}|^{2}

dxdvdt

=\displaystyle \int_{$\Gamma$^{d}}\int_{\mathbb{R}^{\mathrm{d}}}($\theta$_{$\epsilon$_{N}}*(vS_{t}^{N}))\nabla_{x}F_{\mathrm{t}}^{N}

dxdvdt

+\displaystyle \frac{\Vert u_{$\epsilon$_{N}}^{N}\Vert_{T,\infty}^{2}}{ $\epsilon$}\int_{$\Gamma$^{d}}\int_{\mathrm{N}^{d}}(F_{t}^{N})^{2}dxdv

plus

termsrelatedto the

_martingale.

Heuristically,

itseems that for small T,

using

(7),

the

previous

estimates “close” and

give

aboundon

\Vert u^{N}\Vert_{T,\infty}

and

However,

therearestill several nontrivial technical

problems

tobeovercome. In the

previ‐

oussectionwe needed anestimateon

\Vert F_{t}^{0,N}\Vert_{L^{2}($\Gamma$^{d})}

.

Here,

inthis

section,

wehaveshown

a controlon

F_{t}^{N}

,not

F_{t}^{0,N}

. Onecanprovethe estimate

\displaystyle \int F_{t}^{0,N}(x)^{2}dx\leq C\int\int|v|^{3}F_{t}^{N}(x, v)dxdv+C\int\int F_{t}^{N}(x, v)^{4}dxdv

(this

is avariant of Lemma 1 of

_[4],

which avoids

_{\Vert F_{t}^{N}\Vert_{\infty}}

, sinceit looks too difficultto

estimate

_{\Vert F_{t}^{N}\Vert_{\infty}}

_).

But then thetwo

_quantities

onthe

right‐hand‐side

of this

inequality

have tobe controlled. We_presume that all these_steps can be done but dueto thecom‐

plexity

of these

_estimates,

instead of

_formulating

a

result,

we

prefer

to limit ourselvesto

statea

conjecture.

Conjecture

12 Assume

d=2,

_{u_{0}\in W^{2 $\alpha$,2}($\Gamma$^{d})}

,

\displaystyle \int_{$\Gamma$^{d}}\int_{\mathbb{R}^{d}}(F_{0}^{N})^{2}dxdv\leq C

. Let

(u^{N}, X_{t}^{i}, V_{$\iota$^{i}})

be the solution

_of

the

_{fluid‐particle}

_interacting

_system, with

_{$\epsilon$= $\epsilon$ N\rightarrow 0}

as N\rightarrow\infty. Set

F_{t}^{N}=$\theta$_{$\epsilon$_{N}}*(\displaystyle \frac{1}{N}\sum_{i=1}^{N}$\delta$_{X_{t}^{i},V_{$\iota$^{i}}})

.

Then,

for

small

T,

(u^{N}, F^{N})

converges tothe

unique

solution

(u, F)

of

VNSsystem.

The _convergence should holds in several

_{topologies, including}

the _strong

_topology

of

L^{2}(0, T;L^{2}($\Gamma$^{d}))

for

u^{N}

, and of

L^{2}(0, T;L^{2}($\Gamma$^{d}\times \mathbb{R}^{2}))

for

F^{N}.

6.3

_{Open questions}

A first main hmitation of the results described here is the

_{phenomenological}

description

of the

_fluid‐body

interaction. We have

already

remarked in the Introduction about the difficulties met

_by

morereaJistic models.

Thé

two‐step

approach

is

_complete

and extendible to stochastic

_dynamics.

Ón

the contrary, themore

_{interesting joint}

limit

approach,

evenif_true, containstwo restrictions: the short time and the _presence of noise in the

_particles

‐

viscosity

in the PDE. The

short timeis

_{due, conceptually,}

tothe

_vorticity

_{produced by}

the immersed

_particles,

which increases both with fluid

_velocity

and

_{particle density,}

therefore

_introducing

a

quadratic

terminthe

_equations.

_Blow‐up

dueto

_quadratic

terms is

_{prevented by}

suitable conservation laws andwehavean _energy

_inequality

butwemissaconservation law for

vorticity,

dueto

the

_vorticity

_{production by particles.}

Following

[18],

section

_4.1,

itcould be that uniform‐in‐z estimateson

u^{N}

canbe

replaced

by

estimates in

L^{4}

,whichare

global;

correspondingly,

an

L^{4}

‐controlon

F^{N}

isneeded. Here

andfor otherpurposes, weseethe

_importance

ofa

major

problem: F^{N}

doesnot

_satisfy

a

continuity equation,

butan

identity

with weaker

_{geometric properties.}

Asa

remark,

as soon aswerestricttolocalintime

_results,

itseems

possible

toextend the result of the

joint

limit to the 3\mathrm{D} case,

by working

in spaces of

_sufficiently

_regular

solutionsu_;see arelated

problem

in

[12].

Thisstressesoncemorethefact that

presumably

wehave nottaken in full

_advantage

the

_properties

of 2\mathrm{D} fluids.

The

_viscosity

has been used in our

approach

to obtain an

L^{2}

‐estimate on

F_{t}^{N}

_, con‐

ceptually

fundamentalas a mean_{to prove} noconcentration of

particles.

However,

for the

limit

_equation

for F, Ư‐estimates are

easily

obtained in terms of the Ư‐norm ofinitial

conditions,

without need ofany

Laplacian

(see

forinstance

[13]).

This could bea

signature

of the fact that noise is notneededto prove

L^{2}

‐estimateon

F_{t}^{N}.

Finally,

thea

priori

estimate on

\displaystyle \int_{0}^{T}\frac{1}{N}\sum_{i=1}^{N}(u_{ $\epsilon$}^{N}(t, X_{t}^{i})-V_{t}^{i})^{2}dt=\int_{0}^{T}\int(u_{ $\epsilon$}^{N}(t,x)-v)^{2}S_{t}^{N}

(dx

,

dv)dt

obtained

_by

the_energybound looks

_promising

tocontrol thedifficult

_quadratic

_terms,but

inall

_{computations they}

seemto be

_coupled

withotherterms notunder control.

Acknowledgement

13 The author thanks Christian Olivera

_for

several

_preliminary

dis‐

cussionsand the

_organizers

_of

_workshops

atKomaba

_University

an\grave{d}

Waseda

University

in

Tokyo

and at RIMS

_Kyoto

in November 2016

_for

their

_hospitality

and the

_opportunity

to

discuss the

topics

of

this note.

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