prague sum · awell-posednessresultfortheregularizedsystem theorem(moussa-s.2012) existence and...

On the motion of rigid bodies immersedin a two dimensional incompressible perfect fluid

Franck Sueur

Laboratoire Jacques-Louis Lions, Université Paris 6 (for 4 more days);Institut de Mathématiques de Bordeaux, Université de Bordeaux (from Sept. 1st)

Summer School and Workshop "Particles in Flows",Prague, August 2014

Franck Sueur Rigid bodies in 2d perfect flows Prague, August 2014 1 / 145

Part 1. On the motion of a rigid body immersed in a two dimensionalirrotational and incompressible perfect fluid.

In the first part we revisit the complex-analytic approach of the motion of a rigidbody immersed in a two dimensional irrotational and incompressible perfect fluidinitiated by Blasius, Kutta, Joukowski, Chaplygin and Sedov.


The equations

Fluid equations :

∂u∂t

+ (u · ∇)u +∇π = 0 for t ∈ (0,∞), x ∈ F(t),

div u = 0 for t ∈ [0,∞), x ∈ F(t),

Solid equations :

m(h)′′(t) =

∫∂S(t)

πn ds for t ∈ (0,∞),

J r ′(t) =

∫∂S(t)

(x − h(t))⊥ · πn ds for t ∈ (0,∞),


Boundary and initial conditions

Boundary conditions :

u · n =(

(h)′(t) + r(t)(x − h(t))⊥)· n for t ∈ [0,∞), x ∈ ∂S(t),

lim|x|→∞

|u(t, x)| = 0 for t ∈ [0,∞),

Initial data :

u|t=0 = u0 for x ∈ F0,

h(0) = 0, (h)′(0) = `0, θ(0) = 0, r(0) = r0.


Rotation

There exists a rotation matrix

Rθ(t) :=

(cos θ(t) − sin θ(t)sin θ(t) cos θ(t)

)such that

S(t) := h(t) + Rθ(t)x , x ∈ S0.

Furthermore, the angle satisfies

(θ)′(t) = r(t).


Change of frame

We apply the following isometric change of variable :v(t, x) = RT

θ(t) u(t,Rθ(t)x + h(t)),

π(t, x) = π(t,Rθ(t)x + h(t)),

`(t) = RTθ(t) (h)′(t).


Recasting of the equations

The equations become

∂v∂t

+[(v − `− rx⊥) · ∇

]v + rv⊥ +∇π = 0 x ∈ F0,

div v = 0 x ∈ F0,

m`′(t) =

∫∂S0

πn ds −mr`⊥

J r ′(t) =

∫∂S0

x⊥ · πn ds

v · n =(`+ rx⊥

)· n x ∈ ∂S0,

v(0, x) = v0(x) x ∈ F0,

`(0) = `0, r(0) = r0.


The Kirchoff decomposition

−∆Φi = 0 in F0,∂Φi

∂n= Ki on ∂F0, Φi −→ 0 when x →∞,

where(K1, K2, K3) := (n1, n2, x⊥ · n).

One then assumes that the velocity is of the form

v := `1∇Φ1 + `2∇Φ2 + r∇Φ3.


Added inertia

We introduceMa :=

[mi,j

]i,j∈1,2,3 .

where, for i , j ∈ 1, 2, 3,

mi,j :=

∫F0∇Φi · ∇Φj dx .

The index a refers to “added”, by opposition to the genuine inertia :

Mg :=

m 0 00 m 00 0 J

,


Reformulation of the solid equations

We can first reformulate the main solid equations as follows :

(Mg +Ma)

(`r

)′= −(Ai )i=1,2,3 −

(mr`⊥

0

),

where for i = 1, 2, 3,

Ai :=12

∫∂S0|v |2Ki ds −

∫∂S0

(`+ rx⊥) · vKi ds.


A complex-analytic approach

We identify C and R2 through (x1, x2) = x1 + ix2 = z .

We also use the notation f = f1 − if2 for any f = (f1, f2).

If f is divergence and curl free if and only if f is holomorphic.

Lemma (Blasius)

Let C be a smooth Jordan curve, f := (f1, f2) and g := (g1, g2) two smoothtangent vector fields on C. Then∫

C(f · g)n ds = i

(∫Cf g dz

)∗,∫

C(f · g)(x⊥ · n) ds = Re

(∫Czf g dz

).


Laurent expansion of the Kirchhoff potentials

Lemma

Let C be a smooth Jordan curve, f := (f1, f2) a smooth vector field on C :∫Cf dz =

∫Cf · τ ds − i

∫Cf · n ds.

We infer from this lemma that ∫∂S0∇Φi dz = 0.


Laurent expansion of the Kirchhoff potentials

Corollary

Let C be a smooth Jordan curve, f := (f1, f2) a smooth vector field on C :∫Czf dz =

∫C

(x1 + ix2)(f · τ) ds − i∫C

(x1 + ix2)(f · n) ds.

Lemma

We infer from this corollary that∫∂S0

z∇Φi dz = −(mi,2 + |S0|δi,2) + i(mi,1 + |S0|δi,1), for i = 1, 2;∫∂S0

z∇Φ3 dz = −(m3,2 + |S0|xG ,1) + i(m3,1 − |S0|xG ,2).


Decomposition of the pressure

We start with the following observation :

Ai =12

∫∂S0|v − (`+ rx⊥)|2Ki ds −

12

∫∂S0|`+ rx⊥|2Ki ds,

and replace v by the decomposition

v = v# + r∇Φ3,

to get

Ai =12

∫∂S0|v# − `|2Ki ds +

12

∫∂S0|r(∇Φ3 − x⊥)|2Ki ds

+

∫∂S0

r(v# − `) · (∇Φ3 − x⊥)Ki ds −12

∫∂S0|`+ rx⊥|2Ki ds

=: Ai,a + Ai,b + Ai,c + Ai,d .


Computation of Ai ,d

Using Stokes formula we easily check that(Ai,d

)i=1,2

= −r`⊥|S0|+ r2xG |S0| and A3,d = −r(` · xG )|S0|,

where |S0| is the Lebesgue measure of S0 and

xG :=1|S0|

∫S0

x dx .


Computation of Ai ,a, i = 1, 2

We go on with the computation of

Ai,a =12

∫∂S0|v# − `|2Ki ds.

As v# − ` is tangent to the boundary, we can apply the Blasius formula,

v# − `(z) = −`1 + i`2 + `1∇Φ1 + `2∇Φ2,

and Cauchy’s residue theorem, to obtain(Ai,a

)i=1,2

= 0.


Computation of A3,a

We proceed in the same way for i = 3 :

A3,a =12

∫∂S0|v#,I − `|2K3 ds

=12Re

(∫∂S0

z( v#,I − `)2 dz)

= Re

(∫∂S0

z(−`1 + i`2)(`1∇Φ1 + `2∇Φ2

)dz

)= (−`1)

[− `1m1,2 − `2(m2,2 + |S0|)

]+(−`2)

[`1(m1,1 + |S0|) + `2m2,1

]= `⊥M[`,

whereM[ is the following 2× 2 restriction ofMa :

M[ :=[mi,j

]i,j∈1,2 .


Force and torque in the moving frame

We thus obtain : (A1A2

)= r2

(−m3,2m3,1

)+ r(M[`

)⊥and

A3 = `⊥M[`− r` ·(−m3,2m3,1

).


Solid motion

Let

p :=

(`r

)and µ :=

m1,3m2,30

,

Let Γg : R3 × R3 → R3 and Γa : R3 × R3 → R3 be the bilinear symmetricmappings defined as follows :

∀p =

(`r

)∈ R3, 〈Γg , p, p〉 = mr

(`⊥

0

)and 〈Γa, p, p〉 =

(r(M[`)

⊥

`⊥ · M[`

)+ rp × µ,

Thus the solid equations read :[Mg +Ma

](p)′ + 〈Γg , p, p〉+ 〈Γa, p, p〉 = 0.


Harmonic field

Let H the unique solution vanishing at infinity of

div H = 0 in F0, curlH = 0 in F0, H · n = 0 on ∂S0,

∫∂S0

H · τ ds = 1.

the vector field H admits a harmonic stream function ΨH(x) :

H = ∇⊥ΨH ,

which vanishes on the boundary ∂S0, and behaves like 12π ln |x | as x goes to

infinity.


Harmonic field

the function H is holomorphic (as a function of z = x1 + ix2), and can bedecomposed in Laurent Series :

H(z) =1

2iπz+O(

1z2 ) as z →∞.

Coming back to the variable x ∈ R2, the previous decomposition implies

H(x) = O(

1|x |

)and ∇H = O

(1|x |2

).


Harmonic field

The so-called conformal center of S0 is :

ξ1 + iξ2 :=

∫∂S0

zH dz .

In the case of a disk, we have H = HR2 where we denote

HR2(x) :=12π

x⊥

|x |2.


Decomposition of the velocity field

For ` in R2, r and γ in R given, there exists a unique vector field v verifying :

div v = 0 and curl v = 0 in F0,

v · n =(`+ rx⊥

)· n on ∂S0,

limx→∞

v = 0,∫∂S0

v · τ ds = γ,

and it is given by the law :

v = γH + `1∇Φ1 + `2∇Φ2 + r∇Φ3.


Decomposition of the velocity field

We will denote by

v := v − γH= v# + r∇Φ3,

where, as previously,v# := `1∇Φ1 + `2∇Φ2.


Reformulation of the solid equations

The solid equations can be rewritten in the form

(Mg +Ma)

(`r

)′= −(Ai + Bi + Ci )i=1,2,3 −

(mr`⊥

0

),


Ai :=12

∫∂S0|v |2Ki ds −

∫∂S0

(`+ rx⊥) · vKi ds,

Bi := γ

∫∂S0

(v − (`+ rx⊥)) · HKi ds,

Ci :=γ2

2

∫∂S0|H|2Ki ds.


Computation of Bi

Let us start with the term Bi . Let us prove the following.

Lemma

One has (B1B2

)= −γ(`)⊥ + γrξ,

andB3 = −γ ξ · `.


Computation of Ci

From Blasius lemma 1, the Laurent Series of H and Cauchy’s Residue Theorem,we deduce that

C1 = C2 = C3 = 0.


Conclusion

• Therefore the solid equations can be recast as follows :[Mg +Ma

](p)′ + 〈Γg , p, p〉+ 〈Γa, p, p〉 = γp × B,

where

B :=

(ξ⊥

−1

).

• Let us define, for any p ∈ R3, the energies

Eg (p) =12p · Mgp and Ea(p) =

12p · Map.

Then the total energyEg (p) + Ea(p)

is conserved along time.


Part 2. On the motion of a rigid body immersed in a two dimensionalincompressible perfect fluid with vorticity.

In the second part we consider the motion of a rigid body immersed in a twodimensional incompressible perfect fluid with vorticity.

We prove a result of global in time existence and uniqueness similar to thecelebrated result by Yudovich about a fluid alone.


Equations in the body frame

The equations become

∂v∂t

+[(v − `− rx⊥) · ∇

]v + rv⊥ +∇π = 0 x ∈ F0,

div v = 0 x ∈ F0,

m`′(t) =

∫∂S0

πn ds −mr`⊥

J r ′(t) =

∫∂S0

x⊥ · πn ds

v · n =(`+ rx⊥

)· n x ∈ ∂S0,

v(0, x) = v0(x) x ∈ F0,

`(0) = `0, r(0) = r0.


Vorticity in the body frame

The vorticity, in the body frame, is given by

ω(t, x) := curl v(t, x).

Taking the curl of the Euler equation we get

∂tω +[(v − `− rx⊥) · ∇

]ω = 0 for x ∈ F0

which yields the following conservation laws, at least for smooth solutions : forany t > 0, for any p in [1,+∞],

‖ω(t, ·)‖Lp(F0) = ‖ω0‖Lp(F0).


Global weak formulation

We introduce the following space

H :=

Ψ ∈ L2loc(R2)

/div Ψ = 0 in R2 and ∇Ψ + (∇Ψ)T = 0 in S0

.

It is classical that the space H can be recast thanks to the property :

∃(`Ψ, rΨ) ∈ R2 × R, ∀x ∈ S0, Ψ(x) = `Ψ + rΨx⊥. (1)

More precisely,

H =

Ψ ∈ L2loc(R2)

/div Ψ = 0 in R2 and satisfies (1)

,

and the ordered pair (`Ψ, rΨ) above is unique.



Let us also introduce

H :=

Ψ ∈ H/

Ψ|F0 ∈ C 1c (F0)

and HT := C 1([0,T ]; H).

When (u, v) ∈ H × H, we denote

< u, v >= m `u · `v + J ru rv +

∫F0

u · v dx .



Definition (Weak Solution)

Let us be given v0 ∈ H and T > 0. We say that v ∈ C ([0,T ];H− w) is a weaksolution in [0,T ] if for any test function Ψ ∈ HT ,

< Ψ(T , ·), v(T , ·) > − < Ψ(0, ·), v0 >=

∫ T

0<∂Ψ

∂t, v > dt

+

∫ T

0

∫F0

v ·((v − `v − rvx⊥

)· ∇)

Ψ dx dt −∫ T

0

∫F0

rv v⊥ ·Ψ dx dt

−∫ T

0mrv `⊥v · `Ψ dt.

We say that v ∈ C ([0,+∞);H− w) is a weak solution on [0,+∞) if it is a weaksolution in [0,T ] for all T > 0.


Log-Lipschitz functions

The notation LL(F0) refers to the space of log-Lipschitz functions on F0, that isthe set of functions f ∈ L∞(F0) such that

‖f ‖LL(F0) := ‖f ‖L∞(F0) + supx 6=y

|f (x)− f (y)||(x − y)(1 + ln− |x − y |)|

< +∞.


Result

Theorem (Glass-S. 2012)

For any u0 ∈ C 0(F0;R2), (`0, r0) ∈ R2 × R, such that :

div u0 = 0 in F0 and u0 · n = (`0 + r0x⊥) · n on ∂S0,

ω0 := curl u0 ∈ L∞c (F0),

lim|x|→+∞

u0(x) = 0,

there exists a unique solution (`, r , u) in

C 1(R+;R2 × R)× L∞(R+,LL(F0)).

Moreover for all t > 0,ω0 := curl u(t) ∈ L∞c (F0).


Green’s function and Biot-Savart operator

LetG (x , y) be the Green’s function of F0 with Dirichlet boundary conditions,K (x , y) = ∇⊥G (x , y),K [ω] be the so-called Biot-Savart operator acting on ω ∈ L∞c (F0) throughthe formula

K [ω](x) =

∫F0

K (x , y)ω(y) dy .


Green’s function and Biot-Savart operator

Proposition

Let ω ∈ L∞c (F0). Then K [ω] is in LL(F0), divergence-free, tangent to theboundary and such that curlK [ω] = ω.Moreover, it satisfies

K [ω](x) = O(

1|x |2

)as x →∞,

and its circulation around ∂S0 is given by∫∂S0

K [ω] · τ ds = −∫F0ω dx .


Velocity decomposition

Proposition

Let be given ω in L∞c (F0), ` in R2, r and γ in R. Then there is a unique solutionv in LL(F0)

div v = 0, for x ∈ F0,curl v = ω for x ∈ F0,v · n =

(`+ rx⊥

)· n for x ∈ ∂S0,

v −→ 0 as x →∞,∫∂S0 v · τ ds = γ.

Moreover v is given by

v = K [ω] + (γ + α)H + `1∇Φ1 + `2∇Φ2 + r∇Φ3,

withα :=

∫F0ω dx .


Hydrodynamic Biot-Savart operator and Green function

Let us also introduce the so-called hydrodynamic Biot-Savart operator

KH := K + αH,

which satisfies

div KH [ω] = 0, and curlKH [ω] = ω, for x ∈ F0,

KH [ω] · n = 0, for x ∈ ∂S0,∫∂S0

KH [ω] · τ ds = 0.


Hydrodynamic Biot-Savart operator and Green function

Then v can be decomposed as

v = KH [ω] + γH + `1∇Φ1 + `2∇Φ2 + r∇Φ3.

We also introduce the hydrodynamic Green function GH as

GH(x , y) := G (x , y) + ΨH(x) + ΨH(y).

Consequently one has

KH [ω](x) =

∫F0∇⊥GH(x , y)ω(y) dy .


A priori estimates. Vorticity transport and circulationconservation

In addition to the conservation of Lp norms of vorticity, Kelvin’s theorem alsoholds true, at least for smooth solutions, and the total vorticity is conserved aswell :

γ :=

∫∂S0

v(t, ·) · τ ds =

∫∂S0

v0 · τ ds.,

α =

∫F0ω(t, x) dx =

∫F0ω0(x) dx .

We introducev := v − βH,

whereβ = α + γ.


A priori estimates. Energy-like estimate

Proposition

There exists a constant C > 0 (depending only on S0, m and J ) such that forany solution (`, r , v) of the problem on the time interval [0,T ], with

(`, r) ∈ C 1([0,T ];R3) and v ∈ C 1([0,T ]; (L2 ⊗ RH) ∩ C∞(F0)),

and with compactly supported vorticity,

E (t) :=12

(m|`(t)|2 + J r(t)2 +

∫F0

v(t, ·)2dx),

satisfies the inequalityE (t) 6 E (0)eC |β|t .

In the case where β = 0, that is, when the solution is of finite energy, the energyis conserved.


A priori estimates. Bound of the body acceleration

Proposition

There exists a constant C > 0 depending only on S0, m, J , β and E (0) such thatany classical solution satisfies the estimate

‖(`′, r ′)‖L∞(0,T ) ≤ C .


Reformulation

The system can be recast as follows :

∂tω +[(v − `− rx⊥) · ∇

]ω = 0,

(Mg +Ma)

[`r

]′=[∫F0

(v ·[((v − `− rx⊥) · ∇)∇Φi

]− rv⊥ · ∇Φi

)dx]i∈1,2,3

+

[−mr`⊥

0

],

wherev = KH [ω] + γH + `1∇Φ1 + `2∇Φ2 + r∇Φ3.


Existence

Theorem (Schauder)

Let E denotes a Banach space and let C be a nonempty closed convex set in E .Let F : C 7→ C be a continuous map such that F (C ) ⊂ K, where K is a compactsubset of C . Then F has a fixed point in K.

F : (ω, `, r) 7→ (ω, ˜, r) with

∂t ω +[(v − `− rx⊥) · ∇

]ω = 0,

(Mg +Ma)

[˜

r

]′=[∫F0

(v ·[((v − `− rx⊥) · ∇)∇Φi

]− rv⊥ · ∇Φi

)dx]i∈1,2,3

+

[−mr`⊥

0

].


Energy conservation

Let us define

E(ω) := −12

∫F0×F0

GH(x , y)ω(x)ω(y) dx dy − γ∫F0ω(x)ΨH(x) dx ,

andH := Eg (p) + Ea(p) + E(ω).

Proposition

The quantity H is conserved along the motion.


Short-range and longe-range interactions

Lemma

Let f in L1(R2) ∩ L∞(R2). We denote by

ρf := inf d > 1 / Supp(f ) ⊂ B(0, d).

Then there exists C > 0 such that∫R2

∣∣∣ln |x − y |f (x)∣∣∣ dx ≤ C‖f ‖L∞ + ln(2ρf )‖f ‖L1 ,

for any y ∈ B(0, ρf ).


Short/long-range interactions

Proposition

For some constant C, depending only on m,J , ‖ω0‖L1∩L∞ , |`0|, |r0|, |γ|, ρω0 andthe geometry,

|`(t)|+ |r(t)| ≤ C [1 + ln(ρ(t))],

whereρ(t) := ρω(t,·) = infd > 1 / Supp(ω(t, ·)) ⊂ B(0, d).


Part 3. A vanishingly small body in an rotational 2d flow.

In the third part we still consider the motion of a rigid body immersed in a twodimensional incompressible perfect fluid with vorticity but we are now interested inthe limit where the body shrinks to a pointwise particle.

We will consider two cases depending on whether the body shrinks to a massive ora massless particle.

In both cases the circulation is assumed to be fixed.


A vanishingly small body

Sε0 := εS0,

The body moves rigidly so that at times t it occupies a domain Sε(t) which isisometric to Sε0 . We denote

Fε(t) := R2 \ Sε(t)

the domain occupied by the fluid at time t starting from the initial domain

Fε0 := R2 \ Sε0 .


The equations

∂uε

∂t+ (uε · ∇)uε +∇πε = 0 for t ∈ (0,∞), x ∈ Fε(t),

div uε = 0 for t ∈ [0,∞), x ∈ Fε(t),

mε(hε)′′(t) =

∫∂Sε(t)

πεn ds for t ∈ (0,∞),

J ε(rε)′(t) =

∫∂Sε(t)

(x − hε(t))⊥ · πεn ds for t ∈ (0,∞),


Indeed, since Sε(t) is isometric to Sε0 there exists a rotation matrix

Rθε(t) :=

(cos θε(t) − sin θε(t)sin θε(t) cos θε(t)

)such that

Sε(t) := hε(t) + Rθε(t)x , x ∈ Sε0.

Furthermore, the angle satisfies

(θε)′(t) = rε(t).


Two pointwise limits

Case (i) : the solid occupying the domain Sε(q) is assumed to have a massand a moment of inertia of the form

mε = m and J ε = ε2J ,

where m > 0 and J > 0 are fixed, so that the solid tends to a massivepointwise particle.Case (ii) : the solid occupying the domain Sε(q) is assumed to have a massand a moment of inertia of the form

mε = εαm and J ε = εα+2 J ,

where α > 0 and m > 0 and J > 0 are fixed, so that the solid tends to amassless pointwise particle.


Vorticity and circulation

We will consider an initial fluid vorticity wε0 = w0 ∈ L∞c (R2 \ 0) independent of

ε and an initial circulationγ :=

∫∂Sε

0

uε0 · τ ds

around the solid independent of ε as well.


Initial solid velocity

We will consider an initial solid velocity (`ε0, rε0 ) independent of ε :

(`ε0, rε0 ) = (`0, r0) ∈ R2 × R.


Initial fluid velocity

The initial fluid velocity uε0 is then defined as the unique log-Lipschitz solution ofthe div-curl type system :

div uε0 = 0, curl uε0 = wε0 in Fε0 ,

uε0 · n = (`0 + r0x⊥) · n on ∂Sε0 ,lim|x|→∞ |uε0(x)| = 0,

∫∂Sε

0uε0 · τ ds = γ,

where wε0 := w0|Fε

0, hence, for ε small enough (depending on dist(supp w0; h0)

and the size of S0), we havewε

0 := w0.


Biot-Savart operator in R2

We will use the following notation for the Biot-Savart operator in R2 :

KR2 [w ] :=

∫R2

HR2(x − y)w(t, y) dy ,

which is the convolution operator with the vector field HR2 (already) defined by

HR2(x) :=12π

x⊥

|x |2.


Case (i). The massive limit

Theorem (Glass-Lacave-S. 2012)

Consider T > 0. Then, up to a subsequence, one has the following :hε converges to h weakly-∗ in W 2,∞(0,T ;R2),εθε converges to 0 weakly-∗ in W 2,∞(0,T ;R),wε converges to w in C 0([0,T ]; L∞(R2)− w∗),(h,w) satisfy

∂w∂t

+ div([

u +γ

2π(x − h(t))⊥

|x − h(t)|2

]w)

= 0,

mh′′(t) = γ(h′(t)− u(t, h(t))

)⊥,

w |t=0 = w0, h(0) = 0, h′(0) = `0.

with u(t, x) := KR2 [w(t, ·)](x).


Case (ii). The massless limit

Theorem (Glass-Lacave-S. Soon)

For any T > 0, as ε→ 0+,hε converges to h weakly-∗ in W 1,∞(0,T ;R2),wε converges to w in C 0([0,T ]; L∞(R2)− w∗),(h,w) satisfy

∂w∂t

+ div([

u +γ

2π(x − h(t))⊥

|x − h(t)|2

]w)

= 0,

h′(t) = u(t, h(t)) in [0,T ],

w |t=0 = w0, h(0) = 0.


Change of frame

We apply again the following isometric change of variable :vε(t, x) = RT

θε(t) u(t,Rθε(t)x + hε(t)),

ωε(t, x) = wε(t,Rθε(t)x + hε(t)) = curl vε(t, x),

`ε(t) = RTθε(t) (hε)′(t).


A few basic remarks on scaling laws

Hε(x) =1εH1(xε

),

(∇Φεi )(x) = (∇Φ1

i )(xε

) for i = 1, 2,

(∇Φε3(x) = ε (∇Φ1

3)(xε

),


A few basic remarks on scaling laws

LetMε

a :=[mε

i,j]i,j∈1,2,3 ,

withmε

i,j :=

∫Fε0

∇Φεi · ∇Φε

j = ε2+δi≥3+δj≥3

∫F0∇Φ1

i · ∇Φ1j .

Therefore

Mεa = ε2 IεMaIε, with Iε :=

1 0 00 1 00 0 ε

.

This has to compared with

Mεg :=

[mε Id2 0

0 J ε]

= εα IεMg Iε.


Energy estimates

We have seen that the following quantity is conserved along the motion :

Hε =12

(Iεpε)T(εαMg + ε2Ma

)(Iεpε)

−12

∫Fε0×Fε

0

G εH(x , y)ωε(x)ωε(y) dx dy − γ∫Fε0

ωε(x)ΨHε(x) dx ,

where

pε :=

(`ε

rε

)and Iεpε :=

(`ε

εrε

).


Energy estimates

Using the short/long-range decomposition of the interactions we deduce that

|ε`ε(t)|+ |ε2rε(t)| ≤ C [1 + ln(ρε(t))],

whereρε(t) := ρωε(t,·) = infd > 1 / Supp(ωε(t, ·)) ⊂ B(0, d),

for some constant C = C (m,J0, ‖w0‖L1∩L∞ , |`0|, |r0|, |γ|, ρw0), depending only onthese values and the geometry for ε = 1.


Decomposition of the pressure

In the presence of vorticity the solid equations can be rewritten in the form

Mε

(`ε

rε

)′= −(Aεi + Bεi + C εi )i=1,2,3 − (Dε

i )i=1,2,3 −(mrε(`ε)⊥

0

),


Aεi :=12

∫∂Sε

0

|vε|2Ki ds −∫∂Sε

0

(`ε + rεx⊥) · vεKi ds,

Bεi := γ

∫∂Sε

0

(vε − (`ε + rεx⊥)) · HεKi ds,

C εi :=γ2

2

∫∂Sε

0

|Hε|2Ki ds,

andDε

i :=

∫Fε0

ωε[vε − `ε − rεx⊥]⊥ · ∇Φεi (x) dx .


Obstacle to the complex-analytical strategy

In the previous slide,vε := vε − γHε

is not curl free.


Two extra Kirchhoff potentials

−∆Φεi = 0 in Fε0 , Φε

i −→ 0 when x →∞, ∂Φεi

∂n= Ki on ∂Fε0 ,

where

(K4, K5) :=

((−x1x2

)· n,(x2x1

)· n).


Approximation of the velocity

We have

(∇KR2 [ωε]|x=0)x = aε(−x1x2

)+ bε

(x2x1

).

We introduce

vε# := KR2 [ωε]|x=0 + (∇KR2 [ωε]|x=0)x

−2∑

i=1

(KR2 [ωε]|x=0)i∇Φεi − aε∇Φε

4 − bε∇Φε5.

which is a good approximation of

K εH [ωε].


An exercise

Letn be an integerA be a real n × n skew-symmetric matrix,B be a real n × n matrix,y0, y1 ∈ Rn,ε ∈ (0, 1) and yε(t) the solution to

εy ′′ε = A(y ′ε − Byε),yε(0) = y0, y ′ε(0) = y1.

Prove that there exists T > 0 such that (yε)ε∈(0,1) is bounded in C 1([0,T ];Rn).


Modulation

Let

˜ε(t) := `ε(t)− KR2 [ωε(t, ·)](0)− ε∇KR2 [ωε(t, ·)](0) · ξ,

where

`ε(t) := RTθε(t) (hε)′(t).

Let

pε := (˜ε, εrε).


Normal Form

Proposition

Let us fix ρ > 0. If for a given T > 0 and an ε ∈ (0, 1) one has for all t ∈ [0,T ] :

d(hε(t), Supp (ωε(t))) ≥ 1/ρ and Supp (ωε(t)) ⊂ B(hε(t), ρ),

then on [0,T ] : [εαMg + ε2Ma

](pε)′ + 〈εα−1Γg + εΓa, pε, pε〉

= γ pε × B+εγG (ε, t) + εmin(α,2)F (ε, t),

where G is weakly gyroscopic and F is weakly nonlinear.


The function G = G (ε, t) : (0, 1)× [0,T ]→ R3 satisfies∣∣∣∣∫ t

0pε(s) · G (ε, s) ds

∣∣∣∣ ≤ εC (1 + t +

∫ t

0|pε(s)|2 ds

),

and the function F = F (ε, t) : (0, 1)× [0,T ]→ R3 satisfies

|F (ε, t)| ≤ C(1 + |pε(t)|+ ε|pε(t)|2

).


Part 1.Bis. A new look at the irrotational case

We revisit the irrotational case of the first part following this time a real-analyticapproach after Lamb.


We have seen that the solid equations can be rewritten in the form

(Mg +Ma)(p)′ + 〈Γg , p, p〉 = −(12

∫∂S0|v |2Ki ds −

∫∂S0

(`+ rx⊥) · vKi ds)i ,

where i runs over the integers 1, 2, 3.

Let us recall that(K1, K2, K3) := (n1, n2, x⊥ · n).


Lamb’s lemma

Letζ1(x) := e1 , ζ2(x) := e2 and ζ3(x) := x⊥

denote the elementary rigid velocities.

Lemma

For any pair of vector fields (u, v) in C∞(R2 \ S0;R2) satisfyingdiv u = div v = curl u = curl v = 0,u(x) = O(1/|x |) and v(x) = O(1/|x |) as |x | → +∞,

one has, for any i = 1, 2, 3,∫∂S0

(u · v)Kids =

∫∂S0

ζi ·(

(u · n)v + (v · n)u)ds.


As a consequence, using Lamb’s lemma and the boundary conditions, we obtain :

12

∫∂S0|v |2Ki ds =

∫∂S0

(v · n)(v · ζi ) ds

=

∫∂S0

((`+ rx⊥) · n

)(v · ζi

)ds

=

∫∂S0

((`+ rx⊥) · n

)(v · n

)Ki ds

+

∫∂S0

((`+ rx⊥) · n

)(v · τ

)(ζi · τ

)ds,

so that

12

∫∂S0|v |2Ki ds −

∫∂S0

(`+ rx⊥) · vKi ds

= −∫∂S0

((`+ rx⊥) · τ

)(v · τ

)Ki ds +

∫∂S0

((`+ rx⊥) · n

)(v · τ

)(ζi · τ

)ds

=∑k

pk

∫∂S0

(v · τ

)[(ζi · τ

)Kk −

(ζk · τ

)Ki ] ds


Computation of the brackets

For instance,

(ζ1 · τ

)K2 −

(ζ2 · τ

)K1 =

(ζ1 · τ

)(ζ2 · n

)−(ζ2 · τ

)(ζ1 · n

)=

(ζ2 · n

)2+(ζ2 · τ

)2= 1

and (ζ1 · τ

)(ζ3 · n

)−(ζ3 · τ

)(ζ1 · n

)=

(ζ2 · n

)(ζ3 · n

)+(ζ3 · τ

)(ζ2 · τ

)= ζ2 · ζ3.


Conclusion

One obtains exactly the same kind of formulation than previously, that is :[Mg +Ma

](p)′ + 〈Γg , p, p〉+ 〈Γa, p, p〉 = γp × B,

with

B :=

(ξ⊥

−1

),

whereξ :=

∫∂S0

x(H · τ

)ds,

(instead of

ξ1 + iξ2 :=

∫∂S0

zH dz

in the first part).


Link with the complex-analytic approach

Lemma

Denote ξ := (ξ1, ξ2). Then ξ = ξ1 + i ξ2.

The proof relies on the formula :∫∂S0

(f1 − if2) dz =

∫∂S0

(f · τ − if · n

)ds. (2)


Back to the original frame

Going back to the original frame the equations read

q′ = p, (Mg +Ma, ϑ) p′ + 〈Γϑ, p, p〉 = Fϑ(p),

where

Ma,ϑ := R(ϑ)MaR(ϑ)t ,

〈Γϑ, p, p〉 := −(Pa, ϑ0

)× p − rMa, ϑ

(0`⊥

),

Fϑ(p) := γ

(`⊥ − rζϑζϑ · `

)= γp × Bϑ,

with

R(ϑ) :=

(R(ϑ) 00 1

)∈ SO(3),

Pa, ϑ for the two first coordinates ofMa, ϑ p,

Bϑ :=

(−1ζ⊥ϑ

).


A geodesic interpretation

Let

〈Γϑ, p, p〉 =:

∑1≤i,j≤3

(Γϑ)ki,jpipj

1≤k≤3

∈ R3.

Then for every i , j , k ∈ 1, 2, 3,

(Γϑ)ki,j(q) :=

12

((Ma,ϑ)i

k,j + (Ma,ϑ)jk,i − (Ma,ϑ)k

i,j

)(q),

where(Ma,ϑ)k

i,j :=∂(Ma,ϑ)i,j

∂qk.


Part 4. A rigid body immersed in a bounded domain

We will make use of Lamb’s approach to tackle the case where the fluid-solidsystem occupies a bounded domain, still in the irrotational case.


An immersed body

Let now S(t) be a rigid body moving in Ω.

S(t)F(t) Ω


Setting

Ω : bounded open regular connected and simply connected domain of R2

occupied by the system fluid-solid,S0 ⊂ Ω : domain initially occupied by the solid,F0 := Ω \ S0 : domain initially occupied by the fluid,at time t, the solid occupies S(t) := R(ϑ(t))S0 + h(t),

at time t, he fluid occupies F(t) := Ω \ S(t).


Motion of the rigid body

The solid can translate : its center of gravity h(t) evolves in Ω,The solid can rotate of an angle ϑ(t),

The evolutions of h and ϑ are given by Newton’s laws :

mh′′ =

∫∂S(t)

πn ds,

where n denotes the normal,There’s a similar equation for ϑ :

J ϑ′′ =

∫∂S(t)

π(x − h(t))⊥ · n ds,

The walls (of the cavity and of the body) are impermeable so that the normalcomponent of the velocity is continuous at the fluid-solid interface.


Helmholtz and Kelvin

For any t,curl u(t, ·) = 0 in F(t),

and ∫∂S(t)

u(t) · τds = γ :=

∫∂S0

u0 · τds.


Leth := (h1, h2), q := (h1, h2, ϑ) ∈ R3,

and their time derivatives

` := (`1, `2), p := (`1, `2, r) ∈ R3.

S(t), F(t) depend on q only, we shall rather denote them S(q) and F(q),

Q := q ∈ R3 : d(S(q), ∂Ω) > 0,


Kirchhoff potentials

Let, for any q in Q, ζj(q, ·), Kj(q, ·) and Φj(q, ·)/

ζj(q, x) := ej−1, for j = 1, 2 and ζ3(q, x) := (x − h)⊥,

Kj(q, ·) := n · ζj(q, ·) on ∂Ω ∪ ∂S(q).

∆Φj = 0 in F(q),

∂Φj

∂n(q, ·) = Kj(q, ·) on ∂S(q),

∂Φj

∂n(q, ·) = 0 on ∂Ω.


Kirchhoff potentials

We also denote

K (q, ·) := (K1(q, ·),K2(q, ·),K3(q, ·))t ,

and

Φ(q, ·) := (Φ1(q, ·),Φ2(q, ·),Φ3(q, ·))t .


Inertia

Let

Mg :=

J 0 00 m 00 0 m

,

Ma(q) :=

∫∂S(q)

Φ(q, ·)⊗ ∂Φ

∂n(q, ·)ds =

(∫F(q)

∇Φi · ∇Φjdx)

16i,j63,

and

M(q) :=Mg +Ma(q).


Christoffel symbols

Let

〈Γ(q), p, p〉 :=

∑1≤i,j≤3

Γki,j(q)pipj

1≤k≤3

∈ R3,

where, for every i , j , k ∈ 1, 2, 3, we set

Γki,j(q) :=

12

((Ma)i

k,j + (Ma)jk,i − (Ma)k

i,j

)(q),

where(Ma)k

i,j :=∂(Ma)i,j

∂qk.


Stream function for the circulation term

For every q ∈ Q, there exists a unique C (q) ∈ R such that the unique solutionψ(q, ·) of the Dirichlet problem :

∆ψ(q, ·) = 0 in F(q),

ψ(q, ·) = C (q) on ∂S(q),

ψ(q, ·) = 0 on ∂Ω,

satisfies ∫∂S(q)

∂ψ

∂n(q, ·)ds = −1.


Force term

Eventually, we also define :

B(q) :=

∫∂S(q)

(∂ψ

∂n

(∂Φ

∂n× ∂Φ

∂τ

))(q, ·) ds,

E (q) := −12

∫∂S(q)

(∣∣∣∣∂ψ∂n∣∣∣∣2 ∂Φ∂n

)(q, ·) ds,

and the force termF (q, p) := γ2E (q) + γ p × B(q).


Reformulation as an ODE

Theorem (Glass-Munnier-S. 2014)

Up to the first collision, the fluid-body system is equivalent to the second orderODE

q′ = p,M(q)p′ + 〈Γ(q), p, p〉 = F (q, p),

with Cauchy data

q(0) = 0 ∈ Q, p(0) = (r0, `0) ∈ R× R2.


Scheme of proof

We will use the weak formulation :

m`′ · `∗ + J r ′r∗ +

∫F(q)

(∂u∂t

+12∇|u|2

)· u∗dx = 0,

for all p∗ = (`∗, r∗) ∈ R3, with

u∗ := ∇(Φ(q, ·) · p∗),

which is defined on F(q).


Scheme of proof

We decomposeu(q, ·) = u1(q, ·) + u2(q, ·),

with

u1(q, ·) := ∇(Φ(q, ·) · p) = ∇

3∑j=1

Φj(q, ·)pj

and

u2(q, ·) := γ∇⊥ψ(q, ·).


Scheme of proof

This leads to

m`′ · `∗ + J r ′r∗ +

∫F(q)

(∂u1

∂t+

12∇|u1|2

)· u∗dx = −

∫F(q)

(12∇|u2|2

)· u∗dx

−∫F(q)

(∂u2

∂t+

12∇(u1 · u2)

)· u∗dx .


Scheme of proof

The proof then reduces to proving that

m`′ · `∗ + J r ′r∗ = Mg (q)p′ · p∗,∫F(q)

(∂u1

∂t+

12∇|u1|2

)· u∗dx = Ma(q)p′ · p∗ + 〈Γ(q), p, p〉 · p∗,

−∫F(q)

(12∇|u2|2

)· u∗dx = γ2E (q) · p∗,

−∫F(q)

(∂u2

∂t+∇(u1 · u2)

)· u∗dx = γ

(p × B(q)

)· p∗.


Scheme of proof of the second identity

First we prove that∫F(q)

(∂u1

∂t+

12∇|u1|2

)· u∗dx =

(ddt∂E1∂p− ∂E1

∂q

)· p∗,

withE1(q, p) :=

12

∫F(q)

|u1|2dx .

Then we observeE1(q, p) =

12Ma(q)p · p.

and deduce that(ddt∂E1∂p− ∂E1

∂q

)· p∗ =Ma(q)p′ · p∗ + 〈Γ(q), p, p〉 · p∗.


Energy conservation

Proposition

For any solution (q, p) ∈ C∞([0,T ];Q× R3),

ddtE(q, p) = 0, where E(q, p) :=

12M(q)p · p + U(q),

where the potential energy U(q) is given by

U(q) := −12γ2C (q).

Moreover∀q ∈ Q, E (q) =

12DC (q).


No collision implies that the velocity is bounded

Let, for δ > 0,

Qδ := q ∈ R3 : d(S(q), ∂Ω) > δ.

Then there exists K > 0 (depending only on S0,Ω, p0, γ,m,J , δ) such that|p|R3 ≤ K on [0,T ].


First proof of the energy conservation : with the PDEformulation

First we prove the conservation of

E :=12

∫F(q)

u2 dx +12m`2 +

12J r2

thanks to the weak formulation of the PDE formulation.



Then we prove that E coincides with E in the following way. We first decomposeagain u into

u(q, ·) = u1(q, ·) + u2(q, ·),

and use again that ∫F(q)

u21 dx =Ma(q)p,

to obtain that

E =12

∫F(q)

u22 dx +

∫F(q)

u1 · u2 dx +12M(q)p · p,



Moreover, by some integration by parts, we have that

12

∫F(q)

u22 dx =

12γ2∫F(q)

∇⊥ψ · ∇⊥ψ dx

=12γ2∫S(q)

∂ψ

∂n(q, ·)ψ(q, ·)ds

= −12γ2C (q),

since ψ(q, ·) is constant equal to C (q) on ∂S(q) and satisfies∫∂S(q)

∂ψ

∂n(q, ·)ds = −1.



Finally by another integral by parts,∫F(q)

u1 · u2 dx = 0.


Second proof : with the ODE formulation

First we observe that(E(q, p)

)′=M(q)p′ · p +

12

(DM(q) · p)p · p − 12γ2DC (q) · p.

Using the ODE we have

M(q)p′ · p = −〈Γ(q), p, p〉 · p + F (q, p) · p,

andF (q, p) · p = γ2E (q) · p,

so that (E(q, p)

)′= −〈Γ(q), p, p〉 · p +

12

(DM(q) · p)p · p

+γ2(E (q) · p − 1

2DC (q) · p

).



The proof then follows from the two following identities :

−〈Γ(q), p, p〉 · p +12

(DM(q) · p)p · p = 0,

E (q) · p − 12DC (q) · p = 0.



The second identity follows from the fact that the matrix

S(q, p) :=

∑1≤i≤3

Γki,j(q)pi

1≤k,j≤3

,

is such that〈Γ(q), p, p〉 = S(q, p)p.

and such that12DM(q) · p − S(q, p)

is skew-symmetric.


Hamiltonian structure and third proof of the energyconservation

We introduce the impulses :(PgΠg

):=Mgp,

(PaΠa

):=Ma(q)p,

(PΠ

)=

(Pg + PaΠg + Πa

).

LetI := (P,Π) andM(q) :=Mg +Ma(q),

such that (PΠ

):=M(q)p.


Poisson manifold

Definition

We say that the manifold P of the smooth functions (q, I ) has a Poisson structureif there exists a a bracket ·, · acting on C∞ functionals f : P → R, bilinear andskew-symmetric, satisfying the Jacobi and the Leibniz identities.

Jacobi : f 1, f 2, f 3+ f 2, f 3, f 1+ f 3, f 1, f 2 = 0,

Leibniz : f 1f 2, f 3 = f 1f 2, f 3+ f 2f 1, f 3.


Non-canonical Hamiltonian structure

We set

f 1, f 2 :=∂f 2

∂q· ∂f

1

∂I− ∂f 1

∂q· ∂f

2

∂I− γB(q) ·

(∂f 1

∂I× ∂f 2

∂I

),

and consider

H = −E = −12M(q)p · p − U(q) = −1

2I · M(q)−1I − U(q),

as a functional on P.


Hamiltonian structure

Proposition

If (q, p) satisfies the previous ODE then, for any smooth functional f on P,

ddt

f = f ,H.

(Above f and H stand respectively for f (q, I ) and H(q, I )).

As a trivial consequence of the skew-symmetry of the bracket ·, · we get that Hand therefore E is conserved by the solutions.


Part 5. A vanishingly small body in a bounded domain

Let us now turn our attention to the limit of the dynamics in the setting of Part 4when the size of the solid goes to 0.

As above, we will distinguish the two cases :Case (i) : when the mass of the solid is fixed, and then the solid tends to amassive pointwise particle, andCase (ii) : when the mass tends to 0 along with the size, and then the solidtends to a massless pointwise particle.


A vortex point in a bounded domain

If initiallyω|t=0 = γδh0

then at time t > 0,ω = γδh(t)

with

(h)′(t) = γuΩ(h),

where uΩ is a smooth vector field on Ω, depending only on Ω.Smooth solutions to Euler initially close to

ω|t=0 = γδh0

stay close toω = γδh(t),

cf. Turkington, Marchioro-Pulvirenti.


Definition of uΩ

Let ψ(h, ·) be the solution to :

∆ψ(h, ·) = 0 in Ω, ψ(h, ·) = G (· − h) on ∂Ω,

whereG (r) := − 1

2πln |r |.

The Kirchhoff-Routh stream function ψΩ is then defined by :

ψΩ(x) :=12ψ(x , x),

and the Kirchhoff-Routh velocity uΩ by

uΩ := ∇⊥ψΩ.


Case (i) : Dynamics of a solid shrinking to a pointwisemassive particle

The solid occupying the domain

Sε(q) := R(ϑ)Sε0 with Sε0 := εS0,

is assumed to have a mass and a moment of inertia of the form

mε = m and J ε = ε2J 1,

where m > 0 and J 1 > 0 are fixed.


Case (i)

We get at the limit :

m(h(i))′′ = γ

((h(i))

′ − γuΩ(h(i)))⊥.

We have : lim inf T ε ≥ T(i), and the convergence holds for any T ∈ (0,T(i)).

The convergence holds in W 2,∞([0,T ];R2) weak-*, and we think that thisconvergence is optimal.


Case (ii)

We then get the equation of a point vortex of intensity γ :

(h(ii))′ = γuΩ(h(ii)).

The solution is global (no collision with the boundary), and T ε −→ +∞.

The convergence holds in W 1,∞([0,T ];R2) weak-*, and, once again, wethink that this convergence is optimal.


The normal form

Withp =

(εϑ′, h′ − γ(uΩ(h) + εuc(θ, h))

),

the ODE can be recast as :

εmin(2,α)M∂Ω(ϑ, ε)(p)′

+ ε〈Γ∂Ω(ϑ), p, p〉 =

γp × B∂Ω(ϑ) + εγ2G (q) + O(εmin(2,α)),

where G (q) is weakly gyroscopic so that an extra ε comes out in the energyestimate.The two last lines of p × B∂Ω(ϑ) = 0 reads

h′ = γuΩ(h).


Multi-scale expansions

S(t)F(t) Ω

Fluid state in Ω \ Sε(t) = Fluid state as if ∂Ω

+ Corrector as if Sε(t)

+ Corrector(Corrector) as if ∂Ω

+ ...

Method : Potential Theory / Fredholm for ...


We expand uε1 and uε2 in

γ2E ε(qε), γ p × Bε(qε) and the main part of 〈Γε(qε), pε, pε〉,

which are all three expressions of the form∫∂Sε

0

((uε1 or uε2) · (uε1 or uε2))(ξj · n) ds,

and finally repeatedly use Lamb’s lemma.


Part 6. A mean-field limit.

In this part we introduce an Euler-Vlasov system which seems a plausible relevantcandidate to describe a cloud of small massive particles in a two dimensionalincompressible perfect fluid. We will provide a piece of beginning of justification ina regularized setting.


A mean-field limit

Let N solid bodies in a two dimensional incompressible perfect fluid, shrunk tomassive pointwise particles, with mass mi > 0, circulation γi and position hi (t),

∂tω + div x(ωu) = 0,

u(t, x) = KR2 [ω](t, x) +N∑

j=1

γjHR2(x − hj(t)),

mih′′i (t) = γi

(h′i (t)− vi (t, hi (t))

)⊥,

vi (t, x) = KR2 [ω](t, x) +∑j 6=i

γjHR2(x − hj(t)),

ω|t=0 = ω0, hi (0) = hi,0, h′i (0) = hi,1.


Mean-field

We want to study the limit system obtained by the empirical measure

fN(t) :=1N

N∑i=1

δ(hi (t),h′i (t))

when N goes to infinity, with an appropriate scaling of the amplitudes.


We therefore consider now the solutions of

∂tω + div x(ωu) = 0,

u(t, x) = KR2 [ω](t, x) +1N

N∑j=1

HR2(x − hj(t)),

h′′i (t) =(h′i (t)− vi (t, hi (t))

)⊥,

vi (t, x) = KR2 [ω](t, x) +1N

∑j 6=i

HR2(x − hj(t)),

ω|t=0 = ω0, hi (0) = hi,0, h′i (0) = hi,1.


A plausible limit system

Our guess is that one obtains in the limit the following PDE system :

∂tω + div x(ωu) = 0,∂t f + div x(f ξ) + div ξ(f (ξ − u)⊥) = 0,

where

u := KR2 [ω + ρ] and ρ :=

∫R2

fdξ.


An alternative formulation based on the fluid velocity

An alternative way to describe the system is to use the following velocityformulation :

∂tu + div x(u ⊗ u) +∇p = (ρu − j)⊥,div x u = 0

∂t f + div x(f ξ) + div ξ(f (ξ − u)⊥) = 0,

where

ρ :=

∫R2

fdξ and j :=

∫R2

f ξdξ.


A regularized version of the system

u := K [ω + ρ] and ρ :=

∫R2

fdξ,

where K is defined by

K [g ](x) :=

∫R2

H(x − y)g(y)dy ,

with H is in W 1,∞(R2) and satisfies H(0) = 0. With this regularization, a properproof of the mean field limit is possible thanks to optimal transportation.


Measures

We will denote byM(Rd) the set of signed measures,M+(Rd) the set of finite measures on Rd ,P(Rd) the set of the probability measures that is the subset ofM+(Rd)verifying ν(Rd) = 1,M1(Rd) the subspace of signed measures having a finite first moment

M1(Rd) :=

ν ∈M(Rd) :

∫Rd|x |1d |ν| <∞

,

and similarlyM+1 (Rd) and P1(Rd).


Pushforward

Givena measure ν inM(Rd1)

and a map τ : Rd1 → Rd2

we define the pushforward measure τ#ν ∈M(Rd2) of ν by τ by

τ#ν[B] := ν[τ−1(B)],

for any Borel subset B of Rd2 .


Measures

Given a measure γ over the product space

Rd × Rd = R2d

and the projections π1 and π2 on it factors, we define the first and secondmarginals of γ as the measures π1

#γ and π2#γ.

This means that for any Borel set B of Rd ,

π1#γ[B] = γ[B × Rd ] and π2

#γ[B] = γ[Rd × B].


Wasserstein distances for measures

Definition

Let ν1 and ν2 be inM+1 (Rd) two finite measures on Rd such as ν1(Rd) = ν2(Rd).

We define the Wasserstein distance W1(ν1, ν2) of order p between this twomeasures by

W1(ν1, ν2) := infγ

∫Rd×Rd

γ(x , y)|x − y |dxdy

where the infimum is taken over all γ ∈M+(Rd × Rd) with marginals ν1 and ν2.


Wasserstein distances for signed measures

Definition

We will say that two signed measures ν1 and ν2 on Rd are compatible wheneverν±1 (Rd) = ν±2 (Rd).

Definition

Given two compatible elements ν1, ν2 ofM1(Rd), we define the Wassersteindistance

W1(ν1, ν2) := W1(ν+1 + ν−2 , ν

−1 + ν+

2 ).


Kantorovitch duality

Proposition

Let ν1 and ν2 be two compatible elements ofM1(Rd). Then

W1(ν1, ν2) = supφ

∫Rdφ d(ν1 − ν2),

where the supremum is taken over the unit ball of Lip(Rd).


Translation invariance

Lemma

If ν1, ν2 and ν3, ν4 are two pairs of compatible elements ofM1(Rd), then the twomeasures ν1 + ν3 and ν2 + ν4 are compatible and we have

W1(ν1 + ν3, ν2 + ν4) 6 W1(ν1, ν2) + W1(ν3, ν4),

with equality in the particular case where ν3 = ν4.


Lemma

Consider two compatible elements ν1, ν2 ofM1(Rd1). Let τ : Rd1 → Rd2 be aLipschitz map. Then

W1(τ#ν1, τ#ν2) 6 ‖τ‖LipW1(ν1, ν2).


Lemma

Consider two compatible elements ν1, ν2 ofM1(Rd × Rd). We have, for i = 1, 2,

W1(πi#ν1, π

i#ν2) 6 W1(ν1, ν2).


Wasserstein distances for vector measures

Definition

Let d1 and d2 be two positive integers. Consider two pairs ν1, ν2 and σ1, σ2 ofcompatible measures (respectively inM1(Rd1) andM1(Rd2)).We introduce the couples

µi := (νi , σi ) ∈M1(Rd1)×M1(Rd2),

i = 1, 2 and define the associated Wasserstein distance W1(µ1, µ2) between µ1and µ2 by

W1(µ1, µ2) := W1(ν1, ν2) + W1(σ1, σ2).


A well-posedness result for the regularized system

Theorem (Moussa-S. 2012)

Existence and uniqueness. Assume that (ω0, f0) is inM1(R2)× P1(R2 × R2). Then there exists only one corresponding solution

(ω, f ) ∈ C 0([0,∞);M1(R2)× P1(R2 × R2)).

Stability. Consider two solutions

µ1 := (ω1, f1) and µ2 := (ω2, f2)

associated with two initial data

µ10 := (ω1

0 , f10 ) and µ2

0 := (ω20 , f

20 )

inM1(R2)× P1(R2 × R2).Then, for any t ≥ 0,

W1(µ1(t), µ2(t)) ≤ e2Ct W1(µ10, µ

20),

where C > 0 depends only on ‖H‖Lip and |ω0|(R2).Franck Sueur Rigid bodies in 2d perfect flows Prague, August 2014 141 / 145

Mean-field limit for regularized kernels

Assume that(ω0, f0) ∈M1(R2)× P1(R2 × R2)

and(h0

i , h1i )i∈N∗ ∈ (R2 × R2)N

∗

are such that

f N0 :=

1N

N∑i=1

δ(h0i ,h1i ) ∈ P1(R2 × R2),

satisfiesW1(f N

0 , f0)→ 0 when N → +∞.



Let us denote byµN := (ωN , f N)N∈N∗ and µ := (ω, f )

the solutions respectively associated with the initial data

(ω0, f N0 )N∈N∗ and (ω0, f0)

given by the previous theorem.



Then, for any t > 0, for any N ≥ 1,

f N(t) =1N

N∑i=1

δ(hi,N(t),h′i,N(t)),

where

h′′i,N(t) =(h′i,N(t)− uN(t, hi,N(t))

)⊥,

uN = K [ωN ] +1N

N∑j=1

H(· − hj,N(t)),

(hi,N(0), h′i,N(0)) = (h0i , h

1i ).



Moreover for any T > 0,W1(µN , µ)→ 0

when N → +∞.


prague sum · awell-posednessresultfortheregularizedsystem theorem(moussa-s.2012) existence and...

Documents