Blowup Mechanism of 2D

Smoluchowski-Poisson Equation

2015. 05. 05

Takashi Suzuki

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infinite time quantization and finite

time simplicity

The model

– statistical mechanics, thermodynamics

Sire-Chavanis 02

motion of the mean field of many self-gravitating

Brownian particles

kinetic equation + maximum entropy production

1. Smoluchowski Part

2. Poisson Part

a) Debye system (DD model)

b) Childress-Percus-Jager-Luckhaus

model (chemotaxis)

other Poisson parts

Chavanis 08

relaxation to the equilibrium in the point vortices

BBGKY hierarchy + factorization

global-in-time existence with compact orbit

Biler-Hebisch-Nadzieja 94

blowup threshold

a. Biler 98, Gajewski-Zacharias 98,

Nagai-Senba-Yoshida 97

b. Nagai 01, Senba-S. 01b

transport

potential

SP equation

1. total mass conservation

2. free energy decreasing

3. weak form

density

flux

potential

attractive (chemotaxis, gravitation)

action at a distance (long range potential)

symmetry (action-reaction)

Green’s function

conservation law

Physical Model ~ Scaling Invariance

self-similar transformation

critical dimension

(Dual )Trudinger-Moser inequality

blowup threshold

stationary quantization (Nagasaki-S. 90) → dynamical quantization

Point Vortex Mean Field Equation

(Boltzmann-Poisson)

mean field limit

Stationary States

L. Onsager 49 point vortices

ordered structure in negative temperature

quantized blowup mechanism – spectral level (Boltzmann-Poisson equation)

u v duality

Hamiltonian

critical mass

Euler

Hamiltonian

Non-equilibrium statistical mechanics

Staniscia-Chavanis-Ninno-Fanelli 2009

relaxation

quasi-equilibrium equilibrium

Recursive hierarchy

Nagasaki-Suzuki 1990

Justification of the mean filed limit

Kinetic theory Static theory

recursive hierarchy

Theorem A1 [formation of collapse]

Theorem A2 [simplicity]

Theorem A3 [quantization of blowup in infinite time]

blowup in finite time

global-in-time behavior

Corollary

a.

b.

n=2

Results

blowup set (finite)

collapse mass

quantization

absolutely continuous part

quantized blowup

1. stationary

2. finite time

3. infinite time

1. Senba-S. 01 weak formulation

monotonicity formula formation of collapse

4. S. 05

backward self-similar transformation

scaling limit

parabolic envelope (1)

scaling invariance of the scaling limit

a local second moment

6. S. 08 scaling back

2. Senba-S. 02a weak solution

5. Senba 07

Naito-S. 08 parabolic envelope (2)

7. Senba-S. 11 translation limit

weak solution generation

instant blowup for over mass

concentrated initial data

collapse mass quantization

type II blowup rate

formation of sub-collapse

8. Espejo-Stevens-S. 12 simultaneous blowup

mass separation for systems

quantization without blowup threshold

3. Kurokiba-Ogawa 03 scaling invariance non-existence of over mass

entire solution without

concentration

limit equation simplification

concentration-cancelation

simplification

9. S. 13a limit equation classification boundary blowup exclusion

10. S. 13b improved regularity

concentration compactness sub-collapse quantization

cloud formation

11. S. 14

12. S. 15

residual vanishing tightness

simplicity, quantization of BUIT Kantorovich-Rubinstein metric

Contents (10)

1. weak form (2)

2. weak solution (1)

3. scaling limit (2)

4. parabolic envelope (1)

5. translation limit (2)

6. residual vanishing (1)

7. simplicity (1)

1. weak form

1) total mass conservation

2) symmetrization

1/10

Poisson - action at a distance

Smoluchowski

action –reaction law

monotonicity

formula

weak form

symmetry

2/10

3) weak continuation

4) epsilon regularity

Gagliard-Nirenberg inequality

5) formation of collapse

nice cut-off function

fundamental solution

Green’s function – potential of long range interaction due to the action at a distance

1. interior regularity

2. boundary regularity

discontinuity at the diagonal

2. weak solution

3/10

total mass of Radon

measure on locally

compact space

parabolic

envelope

sub-collapses

4/10 (weak) scaling limit

-1

0

1. local second moment

(weak) Liouville property

2. scaling

3. scaling limit

5/10 2) weak form to the scaled system

3) diagonal argument

detects loc. unif. behavior of u(x,t) in the backward

parabolic region around

1. pre-scaled collapse mass = total

mass of the weak scaling limit

(1st envelope)

4. parabolic envelope 2) second moment:

2. bounded total second moment

of the limit measure (2nd envelope)

3) estimate below

6/10 4) exclusion of boundary blowup

5. translation limit

1) scaling back

weak solution (half orbit) with uniformly

bounded multiplicate operator

2) concentration compactness principle

subsequence

1. compact

2. vanishing

3. dichotomy

s=0 a. dichotomy-compact

full orbit → (weak) Liouville property 7/10

3) improved ε regularity

1. Brezis-Strauss type estimate

no boundary blowup, scaling

gradient estimate for the principal part

2. parabolic regularity for Lp norm

3. localized Moser’s iteration scheme

scaling back

scaling limit

backward self-similar transformation

b. dichotomy-vanishing

improved ε regularity

(scaling limit, scaling back)

8/10

c

sub-collapse sub-collapse

parabolic

envelope

cloud

collision of sub-collapses

sequence → subsequence → continuation in t

6. residual (cloud) vanishing

crystal

crystal

attractive potential toward infinity valid to the cloud

smooth

smooth

smooth

tight

(microscopic) residual vanishing

collapse mass quantization

(Theorem A1)

2nd envelope

9/10

concentration compactness

1st envelope

7. simplicity Improved (dual) Trudinger-Moser inequality

Kantorovich-Rubinstein metric

by collapse mass quantization

parabolic envelope

(macroscopic residual vanishing)

10/10

simplicity (Theorem A2)

translation limit, Liouville

References

1. S. Mean Field Theories and Dual Variation, Atlantis Press, Amsterdam-Paris, second

edition, 2015 (September?)

2. S. and H. Ohtsuka, Elliptic Equations and Near from Equilibrium Dynamical Systems (in

Japanese), Asakura Shoten, 2015 (June?)

3. T. Senba and S., Applied Analysis – Mathematical Methods in Natural Science, second

edition, Imperial College Press, London, 2011

4. S. Free Energy and Self-Interacting Particles, Birkhauser, Boston, 2005

Proof of the quantization in blowup in infinite time (Theorem A3)

1. formation of collapse in infinite time (translation)

2. collapse mass quantization (scaling + weak Liouville)

3. improved TM inequality for unbounded free energy

4. macroscopic residual vanishing (Kantorovich-Rubinstein metric), a contradiction

5. singular stationary limits control bounded free energy

6. i.e. stationary quantization implies total mass quantization, a contradiction