Mean Field Limits for Coulomb Type Dynamics Sylvia Serfaty Courant - - PowerPoint PPT Presentation

Mean Field Limits for Coulomb Type Dynamics

Sylvia Serfaty

Courant Institute, NYU

In celebration of Alessio Figalli, Fields Institute, October 20, 2020

The discrete coupled ODE system

Consider HN(x1, . . . , xN) = 1 2

• 1≤i=j≤N

w(xi − xj), xi ∈ Rd w(x) = − log |x| d = 1, 2 log case w(x) = 1 |x|s max(d − 2, 0) ≤ s < d Riesz case Evolution equation ˙ xi = − 1 N  ∇iHN(x1, . . . , xN) +

• j=i

v(xi − xj)   gradient flow ˙ xi = − 1 N J∇iHN(x1, . . . , xN) conservative flow (JT = −J) ¨ xi = − 1 N ∇iHN(x1, . . . , xN) Newton’s law possibly with added noise √ θdW t

i , N independent Brownian motions,

θ=temperature, v smooth force.

The discrete coupled ODE system

Consider HN(x1, . . . , xN) = 1 2

• 1≤i=j≤N

w(xi − xj), xi ∈ Rd w(x) = − log |x| d = 1, 2 log case w(x) = 1 |x|s max(d − 2, 0) ≤ s < d Riesz case Evolution equation ˙ xi = − 1 N  ∇iHN(x1, . . . , xN) +

• j=i

v(xi − xj)   gradient flow ˙ xi = − 1 N J∇iHN(x1, . . . , xN) conservative flow (JT = −J) ¨ xi = − 1 N ∇iHN(x1, . . . , xN) Newton’s law possibly with added noise √ θdW t

i , N independent Brownian motions,

θ=temperature, v smooth force.

Questions

For a general system ˙ xi = 1 N

• j=i

K(xi − xj) + √ θdW t

i ◮ What is the limit of the empirical measure? Is there µt such that for

each t 1 N

N

• i=1

δxt

i ⇀ µt

(1)

◮ if f 0 N(x1, . . . , xN) is the probability density of position of the system

at time 0, what is the limit behavior of f t

N? ◮ propagation of chaos (Boltzmann, Kac, Dobrushin): if

f 0

N(x1, . . . , xN) ≃ µ0(x1) . . . µ0(xN) is it true that

f t

N(x1, . . . , xN) ≃ µt(x1) . . . µt(xN)?

in the sense of convergence of the k-point marginal fN,k.

◮ In fact (1) is equivalent to propagation of chaos. Check : ϕ ∈ C ∞ c

• (fN,1−µt)ϕ =
• fN(x1, . . . , xN)
• 1

N

N

• i=1

ϕ(xi) −

• µϕ
• dx1 . . . dxN

Questions

For a general system ˙ xi = 1 N

• j=i

K(xi − xj) + √ θdW t

i ◮ What is the limit of the empirical measure? Is there µt such that for

each t 1 N

N

• i=1

δxt

i ⇀ µt

(1)

◮ if f 0 N(x1, . . . , xN) is the probability density of position of the system

at time 0, what is the limit behavior of f t

N? ◮ propagation of chaos (Boltzmann, Kac, Dobrushin): if

f 0

N(x1, . . . , xN) ≃ µ0(x1) . . . µ0(xN) is it true that

f t

N(x1, . . . , xN) ≃ µt(x1) . . . µt(xN)?

in the sense of convergence of the k-point marginal fN,k.

◮ In fact (1) is equivalent to propagation of chaos. Check : ϕ ∈ C ∞ c

• (fN,1−µt)ϕ =
• fN(x1, . . . , xN)
• 1

N

N

• i=1

ϕ(xi) −

• µϕ
• dx1 . . . dxN

Questions

For a general system ˙ xi = 1 N

• j=i

K(xi − xj) + √ θdW t

i ◮ What is the limit of the empirical measure? Is there µt such that for

each t 1 N

N

• i=1

δxt

i ⇀ µt

(1)

◮ if f 0 N(x1, . . . , xN) is the probability density of position of the system

at time 0, what is the limit behavior of f t

N? ◮ propagation of chaos (Boltzmann, Kac, Dobrushin): if

f 0

N(x1, . . . , xN) ≃ µ0(x1) . . . µ0(xN) is it true that

f t

N(x1, . . . , xN) ≃ µt(x1) . . . µt(xN)?

in the sense of convergence of the k-point marginal fN,k.

◮ In fact (1) is equivalent to propagation of chaos. Check : ϕ ∈ C ∞ c

• (fN,1−µt)ϕ =
• fN(x1, . . . , xN)
• 1

N

N

• i=1

ϕ(xi) −

• µϕ
• dx1 . . . dxN

Questions

For a general system ˙ xi = 1 N

• j=i

K(xi − xj) + √ θdW t

i ◮ What is the limit of the empirical measure? Is there µt such that for

each t 1 N

N

• i=1

δxt

i ⇀ µt

(1)

◮ if f 0 N(x1, . . . , xN) is the probability density of position of the system

at time 0, what is the limit behavior of f t

N? ◮ propagation of chaos (Boltzmann, Kac, Dobrushin): if

f 0

N(x1, . . . , xN) ≃ µ0(x1) . . . µ0(xN) is it true that

f t

N(x1, . . . , xN) ≃ µt(x1) . . . µt(xN)?

in the sense of convergence of the k-point marginal fN,k.

◮ In fact (1) is equivalent to propagation of chaos. Check : ϕ ∈ C ∞ c

• (fN,1−µt)ϕ =
• fN(x1, . . . , xN)
• 1

N

N

• i=1

ϕ(xi) −

• µϕ
• dx1 . . . dxN

Questions

For a general system ˙ xi = 1 N

• j=i

K(xi − xj) + √ θdW t

i ◮ What is the limit of the empirical measure? Is there µt such that for

each t 1 N

N

• i=1

δxt

i ⇀ µt

(1)

◮ if f 0 N(x1, . . . , xN) is the probability density of position of the system

at time 0, what is the limit behavior of f t

N? ◮ propagation of chaos (Boltzmann, Kac, Dobrushin): if

f 0

N(x1, . . . , xN) ≃ µ0(x1) . . . µ0(xN) is it true that

f t

N(x1, . . . , xN) ≃ µt(x1) . . . µt(xN)?

in the sense of convergence of the k-point marginal fN,k.

◮ In fact (1) is equivalent to propagation of chaos. Check : ϕ ∈ C ∞ c

• (fN,1−µt)ϕ =
• fN(x1, . . . , xN)
• 1

N

N

• i=1

ϕ(xi) −

• µϕ
• dx1 . . . dxN

Formal limit

Use ∂tδx(t) + div ( ˙ xδx(t)) = 0

• r Liouville equation + BBGKY hierarchy

∂tfN +

N

• i=1

∇xi

• fN

1 N

N

• i=1

K(xi − xj)

• = 0

We formally expect µt

N := 1 N

N

i=1 δxt

i ⇀ µt where µt solves the

mean-field equation ∂tµ = div ((K ∗ µ)µ) +1 2θ∆µ

• r in the second order case the Vlasov equation

∂tρ + v · ∇xρ + (K ∗ µ) · ∇vρ +1 2θ∆ρ = 0 µ =

• Rd ρ(x, v)dv

How to prove convergence?

◮ Classical method ([Sznitman]...): compare true trajectories of points

to trajectories following characteristics for the limit equation, show they remain close. OK for K Lipschitz. If K not regular, try to control the minimal distance between points in order to control K(xi − xj) [Hauray-Jabin]

◮ Find a good metric, typically Wasserstein W1 such that

∂tW1(µ1(t), µ2(t)) ≤ CW1(µ1(t), µ2(t)) for two solutions of the mean-field evolution. Apply to µt

N and µt.

[Braun-Hepp, Dobrushin, Neunzert-Wick]

◮ Use a relative entropy method: show a Gronwall relation for

0 ≤ HN(fN|ρ⊗N) := 1 N

• fN log fN

ρ⊗N dx1 . . . dxN. [Jabin-Wang ’16] for θ > 0, K not too irregular.

How to prove convergence?

◮ Classical method ([Sznitman]...): compare true trajectories of points

to trajectories following characteristics for the limit equation, show they remain close. OK for K Lipschitz. If K not regular, try to control the minimal distance between points in order to control K(xi − xj) [Hauray-Jabin]

◮ Find a good metric, typically Wasserstein W1 such that

∂tW1(µ1(t), µ2(t)) ≤ CW1(µ1(t), µ2(t)) for two solutions of the mean-field evolution. Apply to µt

N and µt.

[Braun-Hepp, Dobrushin, Neunzert-Wick]

◮ Use a relative entropy method: show a Gronwall relation for

0 ≤ HN(fN|ρ⊗N) := 1 N

• fN log fN

ρ⊗N dx1 . . . dxN. [Jabin-Wang ’16] for θ > 0, K not too irregular.

How to prove convergence?

◮ Classical method ([Sznitman]...): compare true trajectories of points

to trajectories following characteristics for the limit equation, show they remain close. OK for K Lipschitz. If K not regular, try to control the minimal distance between points in order to control K(xi − xj) [Hauray-Jabin]

◮ Find a good metric, typically Wasserstein W1 such that

∂tW1(µ1(t), µ2(t)) ≤ CW1(µ1(t), µ2(t)) for two solutions of the mean-field evolution. Apply to µt

N and µt.

[Braun-Hepp, Dobrushin, Neunzert-Wick]

◮ Use a relative entropy method: show a Gronwall relation for

0 ≤ HN(fN|ρ⊗N) := 1 N

• fN log fN

ρ⊗N dx1 . . . dxN. [Jabin-Wang ’16] for θ > 0, K not too irregular.

Specialization to the Coulomb or Riesz interaction

Limiting equations ∂tµ = div (∇(w ∗ µ)µ) (MFD) ∂tµ = div (J∇(w ∗ µ)µ) (MFC)

• r Vlasov-Poisson

∂tρ + v · ∇xρ + (∇w ∗ µ) · ∇vρ = 0 µ =

• Rd ρ(x, v)dv

Previous results

◮ d = 2 log, point vortex system → 2D incompressible Euler in

vorticity form [Goodman-Hou-Lowengrub ’90, Schochet ’96] with noise → 2D Navier-Stokes [Osada ’87, Fournier-Hauray-Mischler ’14]

◮ [Hauray’ 09, Carrillo-Choi-Hauray ’14] (s < d − 2) stability in

Wasserstein W∞

◮ [Carrillo-Ferreira-Precioso ’12, Berman-Onnheim ’15] (d = 1)

Wasserstein gradient flow, use convexity of the interaction in 1D

◮ [Duerinckx ’15] (d ≤ 2, s < 1) modulated energy method ◮ for convergence to Vlasov-Poisson [Hauray-Jabin ’15,

Jabin-Wang ’17] s < d − 2, relative entropy method. Coulomb interaction (or more singular) remains open.

◮ [Boers-Pickl ’16, Lazarovici ’16, Lazarovici-Pickl ’17] with

N-dependent cut-off of the interaction kernel

The modulated energy method

Idea: use Coulomb (or Riesz) based metric: µ − ν2 =

• Rd×Rd w(x − y)d(µ − ν)(x)d(µ − ν)(y).

Observe weak-strong uniqueness property of the solutions to (MFD)-(MFC) for · : µt

1 − µt 22 ≤ eCtµ0 1 − µ0 22

C = C(∇2(w ∗ µ2)L∞) In the discrete case, let XN denote (x1, . . . , xN) and take for modulated energy, FN(X t

N, µt) =

• Rd×Rd\△

w(x−y)d 1 N

N

• i=1

δxt

i −µt

(x)d 1 N

N

• i=1

δxt

i −µt

(y) where △ denotes the diagonal in Rd × Rd, and µt solves (MFD) or (MFC). Analogy with “relative entropy" and “modulated entropy" methods [Dafermos ’79] [DiPerna ’79] [Yau ’91] [Brenier ’00]....

The modulated energy method

Idea: use Coulomb (or Riesz) based metric: µ − ν2 =

• Rd×Rd w(x − y)d(µ − ν)(x)d(µ − ν)(y).

Observe weak-strong uniqueness property of the solutions to (MFD)-(MFC) for · : µt

1 − µt 22 ≤ eCtµ0 1 − µ0 22

C = C(∇2(w ∗ µ2)L∞) In the discrete case, let XN denote (x1, . . . , xN) and take for modulated energy, FN(X t

N, µt) =

• Rd×Rd\△

w(x−y)d 1 N

N

• i=1

δxt

i −µt

(x)d 1 N

N

• i=1

δxt

i −µt

(y) where △ denotes the diagonal in Rd × Rd, and µt solves (MFD) or (MFC). Analogy with “relative entropy" and “modulated entropy" methods [Dafermos ’79] [DiPerna ’79] [Yau ’91] [Brenier ’00]....

The modulated energy method

Idea: use Coulomb (or Riesz) based metric: µ − ν2 =

• Rd×Rd w(x − y)d(µ − ν)(x)d(µ − ν)(y).

Observe weak-strong uniqueness property of the solutions to (MFD)-(MFC) for · : µt

1 − µt 22 ≤ eCtµ0 1 − µ0 22

C = C(∇2(w ∗ µ2)L∞) In the discrete case, let XN denote (x1, . . . , xN) and take for modulated energy, FN(X t

N, µt) =

• Rd×Rd\△

w(x−y)d 1 N

N

• i=1

δxt

i −µt

(x)d 1 N

N

• i=1

δxt

i −µt

(y) where △ denotes the diagonal in Rd × Rd, and µt solves (MFD) or (MFC). Analogy with “relative entropy" and “modulated entropy" methods [Dafermos ’79] [DiPerna ’79] [Yau ’91] [Brenier ’00]....

Theorem (S. ’18)

Assume that v ∈ ˙ H

d−s 2 (Rd)∩C 0,α(Rd) for some α > 0 and ∇v ∈ Lq(Rd)

for some 1 ≤ q ≤ ∞. Assume (MFD) resp. (MFC) admits a solution

• µt ∈ L∞([0, T], L∞(Rd)),

if s < d − 1 µt ∈ L∞([0, T], C σ(Rd)) with σ > s − d + 1, if s ≥ d − 1. with ∇2w ∗ µt ∈ L∞([0, T], L∞(Rd)). There exist constants C1, C2 depending on the norms of µt and β > 0 depending on d, s, σ, s.t. ∀t ∈ [0, T] FN(X t

N, µt) ≤

• FN(X 0

N, µ0) + C1N−β

eC2t. In particular, if µ0

N ⇀ µ0 and is such that

(∗) lim

N→∞ FN(X 0 N, µ0) = 0,

then the same is true for every t ∈ [0, T] and µt

N ⇀ µt.

Theorem (S. ’18)

Assume that v ∈ ˙ H

d−s 2 (Rd)∩C 0,α(Rd) for some α > 0 and ∇v ∈ Lq(Rd)

for some 1 ≤ q ≤ ∞. Assume (MFD) resp. (MFC) admits a solution

• µt ∈ L∞([0, T], L∞(Rd)),

if s < d − 1 µt ∈ L∞([0, T], C σ(Rd)) with σ > s − d + 1, if s ≥ d − 1. with ∇2w ∗ µt ∈ L∞([0, T], L∞(Rd)). There exist constants C1, C2 depending on the norms of µt and β > 0 depending on d, s, σ, s.t. ∀t ∈ [0, T] FN(X t

N, µt) ≤

• FN(X 0

N, µ0) + C1N−β

eC2t. In particular, if µ0

N ⇀ µ0 and is such that

(∗) lim

N→∞ FN(X 0 N, µ0) = 0,

then the same is true for every t ∈ [0, T] and µt

N ⇀ µt.

Comments

◮ well-prepared assumption (∗) implied by

lim 1 N2 HN(X 0

N) =

• w(x − y)dµ0(x)dµ0(y).

◮ regularity assumption on µt allow for “patches" i.e. measures which

are only L∞, as in vortex patch solutions to Euler’s eq [Chemin, Serfati]

◮ Self-similar solutions of patch type are attractors in the Coulomb

case ([S-Vazquez]). For general s, self-similar Barenblatt solutions of the form t−

d 2+s (a − bx2t− 2 2+s ) s−d+2 2

+ ◮ [Rosenzweig ’20] improved the result in 2D log case: for regularity of

the initial data, µ0 ∈ L∞ suffices.

◮ limiting equation called fractional porous medium equation ◮ required propagation of regularity ok for s < d − 1 ([Lin-Zhang,

Xiao-Zhou, Caffarelli-Vazquez, Caffarelli-Soria-Vazquez, Ambrosio-S, S-Vazquez]

• pen for s > d − 1

Comments

◮ well-prepared assumption (∗) implied by

lim 1 N2 HN(X 0

N) =

• w(x − y)dµ0(x)dµ0(y).

◮ regularity assumption on µt allow for “patches" i.e. measures which

are only L∞, as in vortex patch solutions to Euler’s eq [Chemin, Serfati]

◮ Self-similar solutions of patch type are attractors in the Coulomb

case ([S-Vazquez]). For general s, self-similar Barenblatt solutions of the form t−

d 2+s (a − bx2t− 2 2+s ) s−d+2 2

+ ◮ [Rosenzweig ’20] improved the result in 2D log case: for regularity of

the initial data, µ0 ∈ L∞ suffices.

◮ limiting equation called fractional porous medium equation ◮ required propagation of regularity ok for s < d − 1 ([Lin-Zhang,

Xiao-Zhou, Caffarelli-Vazquez, Caffarelli-Soria-Vazquez, Ambrosio-S, S-Vazquez]

• pen for s > d − 1

Comments

◮ well-prepared assumption (∗) implied by

lim 1 N2 HN(X 0

N) =

• w(x − y)dµ0(x)dµ0(y).

◮ regularity assumption on µt allow for “patches" i.e. measures which

are only L∞, as in vortex patch solutions to Euler’s eq [Chemin, Serfati]

◮ Self-similar solutions of patch type are attractors in the Coulomb

case ([S-Vazquez]). For general s, self-similar Barenblatt solutions of the form t−

d 2+s (a − bx2t− 2 2+s ) s−d+2 2

+ ◮ [Rosenzweig ’20] improved the result in 2D log case: for regularity of

the initial data, µ0 ∈ L∞ suffices.

◮ limiting equation called fractional porous medium equation ◮ required propagation of regularity ok for s < d − 1 ([Lin-Zhang,

Xiao-Zhou, Caffarelli-Vazquez, Caffarelli-Soria-Vazquez, Ambrosio-S, S-Vazquez]

• pen for s > d − 1

Comments

◮ well-prepared assumption (∗) implied by

lim 1 N2 HN(X 0

N) =

• w(x − y)dµ0(x)dµ0(y).

◮ regularity assumption on µt allow for “patches" i.e. measures which

are only L∞, as in vortex patch solutions to Euler’s eq [Chemin, Serfati]

◮ Self-similar solutions of patch type are attractors in the Coulomb

case ([S-Vazquez]). For general s, self-similar Barenblatt solutions of the form t−

d 2+s (a − bx2t− 2 2+s ) s−d+2 2

+ ◮ [Rosenzweig ’20] improved the result in 2D log case: for regularity of

the initial data, µ0 ∈ L∞ suffices.

◮ limiting equation called fractional porous medium equation ◮ required propagation of regularity ok for s < d − 1 ([Lin-Zhang,

Xiao-Zhou, Caffarelli-Vazquez, Caffarelli-Soria-Vazquez, Ambrosio-S, S-Vazquez]

• pen for s > d − 1

Comments

◮ well-prepared assumption (∗) implied by

lim 1 N2 HN(X 0

N) =

• w(x − y)dµ0(x)dµ0(y).

◮ regularity assumption on µt allow for “patches" i.e. measures which

are only L∞, as in vortex patch solutions to Euler’s eq [Chemin, Serfati]

◮ Self-similar solutions of patch type are attractors in the Coulomb

case ([S-Vazquez]). For general s, self-similar Barenblatt solutions of the form t−

d 2+s (a − bx2t− 2 2+s ) s−d+2 2

+ ◮ [Rosenzweig ’20] improved the result in 2D log case: for regularity of

the initial data, µ0 ∈ L∞ suffices.

◮ limiting equation called fractional porous medium equation ◮ required propagation of regularity ok for s < d − 1 ([Lin-Zhang,

Xiao-Zhou, Caffarelli-Vazquez, Caffarelli-Soria-Vazquez, Ambrosio-S, S-Vazquez]

• pen for s > d − 1

The case with noise

[Bresch-Jabin-Wang ’19] incorporate the modulated energy into their relative entropy method: use a modulated free energy Fθ

N(fN, ρ) = θHN(fN|ρ⊗N) +

• fN(XN)FN(XN, ρ)dx1 . . . dxN

Theorem

Assume f 0

N is an initial density and f t N solves

∂tf t

N = N

• i=1

divi   1 N

• i=j

∇iHN(xi − xj)f t

N(XN)

  + 1 2θ

N

• i=1

∆if t

N.

Then for µt a solution to ∂tµ = div (∇(w ∗ µ)µ) + 1 2θ∆µ with µ0 regular enough (Lipschitz), we have Fθ

N(f t N, µt) ≤

• Fθ

N(f 0 N, µ0) + C1N−β)eC2t

and for all t, f t

N ⇀ (µt)⊗N.

Also allows them to extend the method to the case with moderate attraction (e.g. Patlak-Keller-Segel). Conservative case with noise open.

Theorem

Assume f 0

N is an initial density and f t N solves

∂tf t

N = N

• i=1

divi   1 N

• i=j

∇iHN(xi − xj)f t

N(XN)

  + 1 2θ

N

• i=1

∆if t

N.

Then for µt a solution to ∂tµ = div (∇(w ∗ µ)µ) + 1 2θ∆µ with µ0 regular enough (Lipschitz), we have Fθ

N(f t N, µt) ≤

• Fθ

N(f 0 N, µ0) + C1N−β)eC2t

and for all t, f t

N ⇀ (µt)⊗N.

Also allows them to extend the method to the case with moderate attraction (e.g. Patlak-Keller-Segel). Conservative case with noise open.

Convergence to Vlasov-Poisson in the monokinetic case

Let ZN = ((x1, v1), . . . , (xN, vN)) where vi = ˙ xi. Monokinetic version of (VP) (pressureless Euler-Poisson): ρt(x, v) = µt(x)δv=ut(x) ∂tµ + div (µu) = 0 ∂tu + u · ∇u = −∇w ∗ µ (PEP) Use modulated energy EN(ZN, (µ, u)) := 1 N

N

• i=1

|u(xi) − vi|2 + FN(XN, µ)

Theorem (Duerinckx-S ’18)

Assume Z t

N solves Newton’s law with initial data Z 0

• N. Assume (µ, u) is a

sufficiently regular solution to (PEP) on [0, T]. Then EN(Z t

N, (µt, ut)) ≤

• EN(Z 0

N, (µ0, u0)) + N−β

eC2t In particular if limN→∞ EN(Z 0

N, (µ0, u0)) = 0 then

µt

N := 1 N

N

i=1 δxt

i ⇀ µt for all t ∈ [0, T].

Convergence to Vlasov-Poisson in the monokinetic case

Let ZN = ((x1, v1), . . . , (xN, vN)) where vi = ˙ xi. Monokinetic version of (VP) (pressureless Euler-Poisson): ρt(x, v) = µt(x)δv=ut(x) ∂tµ + div (µu) = 0 ∂tu + u · ∇u = −∇w ∗ µ (PEP) Use modulated energy EN(ZN, (µ, u)) := 1 N

N

• i=1

|u(xi) − vi|2 + FN(XN, µ)

Theorem (Duerinckx-S ’18)

Assume Z t

N solves Newton’s law with initial data Z 0

• N. Assume (µ, u) is a

sufficiently regular solution to (PEP) on [0, T]. Then EN(Z t

N, (µt, ut)) ≤

• EN(Z 0

N, (µ0, u0)) + N−β

eC2t In particular if limN→∞ EN(Z 0

N, (µ0, u0)) = 0 then

µt

N := 1 N

N

i=1 δxt

i ⇀ µt for all t ∈ [0, T].

Convergence to Vlasov-Poisson in the monokinetic case

Let ZN = ((x1, v1), . . . , (xN, vN)) where vi = ˙ xi. Monokinetic version of (VP) (pressureless Euler-Poisson): ρt(x, v) = µt(x)δv=ut(x) ∂tµ + div (µu) = 0 ∂tu + u · ∇u = −∇w ∗ µ (PEP) Use modulated energy EN(ZN, (µ, u)) := 1 N

N

• i=1

|u(xi) − vi|2 + FN(XN, µ)

Theorem (Duerinckx-S ’18)

Assume Z t

N solves Newton’s law with initial data Z 0

• N. Assume (µ, u) is a

sufficiently regular solution to (PEP) on [0, T]. Then EN(Z t

N, (µt, ut)) ≤

• EN(Z 0

N, (µ0, u0)) + N−β

eC2t In particular if limN→∞ EN(Z 0

N, (µ0, u0)) = 0 then

µt

N := 1 N

N

i=1 δxt

i ⇀ µt for all t ∈ [0, T].

Proof of the weak-strong uniqueness principle

Set hµ = w ∗ µ. In the Coulomb case −∆hµ = cdµ We have by IBP

• Rd×Rd w(x−y)dµ(x)dµ(y) =
• Rd hµdµ = − 1

cd

• Rd hµ∆hµ = 1

cd

• Rd |∇hµ|2.

Stress-energy tensor [∇hµ]ij = 2∂ihµ∂jhµ − |∇hµ|2δij. For regular µ, div [∇hµ] = 2∆hµ∇hµ = − 2 cd µ∇hµ.

Proof of the weak-strong uniqueness principle

Set hµ = w ∗ µ. In the Coulomb case −∆hµ = cdµ We have by IBP

• Rd×Rd w(x−y)dµ(x)dµ(y) =
• Rd hµdµ = − 1

cd

• Rd hµ∆hµ = 1

cd

• Rd |∇hµ|2.

Stress-energy tensor [∇hµ]ij = 2∂ihµ∂jhµ − |∇hµ|2δij. For regular µ, div [∇hµ] = 2∆hµ∇hµ = − 2 cd µ∇hµ.

Let µ1 and µ2 be two solutions to (MFD) and hi = w ∗ µi. ∂t

• Rd |∇(h1 − h2)|2 = 2cd
• Rd(h1 − h2)∂t(µ1 − µ2)

= 2cd

• Rd(h1 − h2)div (µ1∇h1 − µ2∇h2)

= −2cd

• Rd(∇h1 − ∇h2) · (µ1∇h1 − µ2∇h2)

= −2cd

• Rd |∇(h1 − h2)|2µ1 − 2cd
• Rd ∇h2 · ∇(h1 − h2)(µ1 − µ2)

≤ −2cd

• Rd ∇h2 · div [∇(h1 − h2)]

so if ∇2h2 is bounded, we may IBP and bound by ∇2h2L∞

• Rd |[∇(h1 − h2)]| ≤ 2∇2h2L∞
• Rd |∇(h1 − h2)|2,

result by Gronwall’s lemma.

In discrete case, show instead crucial proposition

• Rd×Rd\△

(ψ(x)−ψ(y))·∇w(x−y)d( 1 N

N

• i=1

δxi−µ)(x)d( 1 N

N

• i=1

δxi−µ)(y) ≤ CDψL∞FN(XN, µ), applied with ψ = ∇hµt. Pbl: loss of positivity of FN, difficulty due to removal of △. Use suitable truncations of the potentials w ∗ ( 1

N

• i δxi − µ) at radius ηi

depending on the point, smaller than minimal distance. Use monotonicity property with respect to truncation parameter. This proposition allows to treat the quantum Coulomb case [Golse-Paul ’19]

In discrete case, show instead crucial proposition

• Rd×Rd\△

(ψ(x)−ψ(y))·∇w(x−y)d( 1 N

N

• i=1

δxi−µ)(x)d( 1 N

N

• i=1

δxi−µ)(y) ≤ CDψL∞FN(XN, µ), applied with ψ = ∇hµt. Pbl: loss of positivity of FN, difficulty due to removal of △. Use suitable truncations of the potentials w ∗ ( 1

N

• i δxi − µ) at radius ηi

depending on the point, smaller than minimal distance. Use monotonicity property with respect to truncation parameter. This proposition allows to treat the quantum Coulomb case [Golse-Paul ’19]

CONGRATULAZIONI, ALESSIO! THANK YOU FOR YOUR ATTENTION!

The Ginzburg-Landau equations

u : Ω ⊂ R2 → C −∆u = u ε2 (1 − |u|2) Ginzburg-Landau equation (GL) ∂tu = ∆u + u ε2 (1 − |u|2) parabolic GL equation (PGL) i∂tu = ∆u + u ε2 (1 − |u|2) Gross-Pitaevskii equation (GP) Associated energy Eε(u) = 1 2

• Ω

|∇u|2 + (1 − |u|2)2 2ε2 Models: superconductivity, superfluidity, Bose-Einstein condensates, nonlinear optics

Vortices

◮ in general |u| ≤ 1, |u| ≃ 1 = superconducting/superfluid phase,

|u| ≃ 0 = normal phase

◮ u has zeroes with nonzero degrees = vortices ◮ u = ρeiϕ, characteristic length scale of {ρ < 1} is ε = vortex core

size

◮ degree of the vortex at x0:

1 2π

• ∂B(x0,r)

∂ϕ ∂τ = d ∈ Z

◮ In the limit ε → 0 vortices become points, (or curves in dimension

3).

Vorticity

◮ In the case Nε → ∞, describe the vortices via the vorticity :

supercurrent jε := iuε, ∇uε a, b := 1 2(a¯ b + ¯ ab) vorticity µε := curl jε

◮ ≃ vorticity in fluids, but quantized: µε ≃ 2π i diδaε

i

◮ µε 2πNε → µ signed measure, or probability measure,

Dynamics in the case Nε ≫ 1

Nε | log ε|∂tu = ∆u + u ε2 (1 − |u|2) in R2 (PGL) iNε∂tu = ∆u + u ε2 (1 − |u|2) in R2 (GP)

◮ For (GP), by Madelung transform, the limit dynamics is expected to

be the 2D incompressible Euler equation. Vorticity form ∂tµ − div (µ∇⊥h) = 0 h = −∆−1µ (EV)

◮ For (PGL), formal model proposed by

[Chapman-Rubinstein-Schatzman ’96], [E ’95]: if µ ≥ 0 ∂tµ − div (µ∇h) = 0 h = −∆−1µ (CRSE) Studied by [Lin-Zhang ’00, Du-Zhang ’03, Masmoudi-Zhang ’05, Ambrosio-S ’08, S-Vazquez ’13]

Dynamics in the case Nε ≫ 1

Nε | log ε|∂tu = ∆u + u ε2 (1 − |u|2) in R2 (PGL) iNε∂tu = ∆u + u ε2 (1 − |u|2) in R2 (GP)

◮ For (GP), by Madelung transform, the limit dynamics is expected to

be the 2D incompressible Euler equation. Vorticity form ∂tµ − div (µ∇⊥h) = 0 h = −∆−1µ (EV)

◮ For (PGL), formal model proposed by

[Chapman-Rubinstein-Schatzman ’96], [E ’95]: if µ ≥ 0 ∂tµ − div (µ∇h) = 0 h = −∆−1µ (CRSE) Studied by [Lin-Zhang ’00, Du-Zhang ’03, Masmoudi-Zhang ’05, Ambrosio-S ’08, S-Vazquez ’13]

Previous rigorous convergence results

◮ (PGL) case : [Kurzke-Spirn ’14] convergence of µε/(2πNε) to µ

solving (CRSE) under assumption Nε ≤ (log log | log ε|)1/4 + well-preparedness

◮ (GP) case: [Jerrard-Spirn ’15] convergence to µ solving (EV) under

assumption Nε ≤ (log | log ε|)1/2 + well-preparedness

◮ both proofs “push" the fixed N proof (taking limits in the evolution

• f the energy density) by making it more quantitative

◮ difficult to go beyond these dilute regimes without controlling

distance between vortices, possible collisions, etc

Previous rigorous convergence results

◮ (PGL) case : [Kurzke-Spirn ’14] convergence of µε/(2πNε) to µ

solving (CRSE) under assumption Nε ≤ (log log | log ε|)1/4 + well-preparedness

◮ (GP) case: [Jerrard-Spirn ’15] convergence to µ solving (EV) under

assumption Nε ≤ (log | log ε|)1/2 + well-preparedness

◮ both proofs “push" the fixed N proof (taking limits in the evolution

• f the energy density) by making it more quantitative

◮ difficult to go beyond these dilute regimes without controlling

distance between vortices, possible collisions, etc

Modulated energy method

◮ Exploits the regularity and stability of the solution to the limit

equation

◮ Works for dissipative as well as conservative equations ◮ Works for gauged model as well

Let v(t) be the expected limiting velocity field (such that

1 Nε ∇uε, iuε ⇀ v and curl v = 2πµ). Define the modulated energy

Eε(u, t) = 1 2

• R2 |∇u − iuNεv(t)|2 + (1 − |u|2)2)

2ε2 , modelled on the Ginzburg-Landau energy.

Modulated energy method

◮ Exploits the regularity and stability of the solution to the limit

equation

◮ Works for dissipative as well as conservative equations ◮ Works for gauged model as well

Let v(t) be the expected limiting velocity field (such that

1 Nε ∇uε, iuε ⇀ v and curl v = 2πµ). Define the modulated energy

Eε(u, t) = 1 2

• R2 |∇u − iuNεv(t)|2 + (1 − |u|2)2)

2ε2 , modelled on the Ginzburg-Landau energy.

Main result: Gross-Pitaevskii case

Theorem (S. ’16)

Assume uε solves (GP) and let Nε be such that | log ε| ≪ Nε ≪ 1

ε. Let v

be a L∞(R+, C 0,1) solution to the incompressible Euler equation

• ∂tv = 2v⊥curl v + ∇p

in R2 div v = 0 in R2, (IE) with curl v ∈ L∞(L1). Let {uε}ε>0 be solutions associated to initial conditions u0

ε, with

Eε(u0

ε, 0) ≤ o(N2 ε). Then, for every t ≥ 0, we have

1 Nε ∇uε, iuε → v in L1

loc(R2).

Implies of course the convergence of the vorticity µε/Nε → curl v Works in 3D

Main result: parabolic case

Theorem (S. ’16)

Assume uε solves (PGL) and let Nε be such that 1 ≪ Nε ≤ O(| log ε|). Let v be a L∞([0, T], C 1,γ) solution to

• if Nε ≪ | log ε|
• ∂tv = −2vcurl v + ∇p

in R2 div v = 0 in R2, (L1)

• if Nε ∼ λ| log ε|

∂tv = −2vcurl v + 1 λ∇div v in R2. (L2) Assume Eε(u0

ε, 0) ≤ πNε| log ε| + o(N2 ε) and curl v(0) ≥ 0. Then ∀t ≥ 0

we have 1 Nε ∇uε, iuε → v in L1

loc(R2).

Taking the curl of the equation yields back the (CRSE) equation if Nε ≪ | log ε|, but not if Nε ∝ | log ε|! Long time existence proven by [Duerinckx ’16].

Proof method

◮ Go around the question of minimal vortex distances by using instead

the modulated energy and showing a Gronwall inequality on E.

◮ the proof relies on algebraic simplifications in computing d dt Eε(uε(t))

which reveal only quadratic terms

◮ Uses the regularity of v to bound corresponding terms ◮ An insight is to think of v as a spatial gauge vector and div v (resp.

p) as a temporal gauge

THANK YOU FOR YOUR ATTENTION!