Mean Field Theories and

Dual Variation

Takashi Suzuki(Osaka University)

KOSEF-JSPS Joint Project(The Japan-Korea Basic Scientific Cooperation Program-Joint Research)

2002. July - 2004. March

Mathematical Science and Mathematical Analysis for Self-Interacting Particles

principle

phenomenon physics mathematics

equation

Mathematical analysis for differential equations- Paradigm of Poincare-Hadamard

1. Fundamental theorem…well-posedness; existence, uniqueness, continuous dependence of the solution

2. Qualitative study

List of standard linear pde’s

Schrodinger(dispersive)

Elliptic (static)

Hyperbolic(conservative)

Parabolic (dissipative)

Linear – Semi-linear – Quasi-linear – Fully Nonlinear

Single – System

1. Method of Perturbation

2. Method of Energy / Variation

3. Method of Dynamical System

4. Method of Exact Solution

5. Method of Comparison

List of weak solutions valid to nonlinear problems

1. Schwartz

2. Lax-Kato

3. Minty-Browder

4. Nonlinear Semigroup

5. Viscosity

6. Renormalization

7. Varifold

Breaking down of the generation of weak solutions

1. flight

2. Concentration 3. Oscillation

Method of weak convergence in the theory of nonlinear partial differential equations controls those phenomena and gets the solution by excluding weak converging approximate solutions which are strongly non-compact.

General blowup mechanism for

1. Blowup…

2. Rigidness-Threshold…

3. Bubble – Quantization

2

max

0 max 0 max

( , , , , )

|| || ,|| ||

tu L D u Du u x t

T

u C T u C T∗∗

=

< ∞

< ⇒ = ∞ ∃ > ⇒ < ∞

Mean field equation of self-gravitating particles …movements of nebula, cellular slime molds,…

)( vuuut ∇−∇⋅∇=),0( T×Ω

0 v av u= ∆ − +

0// =∂∂=∂∂ νν vu ),0( T×Ω∂

bounded smooth2RΩ⊂ Ω∂

constant0a >

Quantized blowup mechanism

1. Formation of collapses

( )0max

0

max

0

1

( , ) ( ) ( ) ( )

0 ( ) \

xt Tx S

Tu x t dx m x dx f x dx

f L C S

δ→

∈

< +∞⇒

+

≤ ∈ Ω ∩ Ω

∑・ ・ ・ ・

0 * 0( ) ( )m x m x=2. mass quantization

84ππ

0

0

xx∈Ω∈∂Ω0( )m x∗ =

0 0 max | , , ( , ) k k k kS x x x t T u x t= ∈Ω ∃ → ∃ ↑ → +∞

…the blowup set.

1 0 1

0 1

|| ( ) || || ||2 #( ) #( ) || || /(4 )

u t uS S u

λπ

= =⇒ Ω ∩ + ∂Ω ∩ ≤

The number of collapses is estimated from above by the total mass.

Global existence by the free energy Blowup by the second moment

Space localization

Formation of collapses Blowup of the weak solution

Concentration lemma

Generation of theweak solution

Mass quantization and emergence of type (I)

blowup point

Parabolic envelopeForward self-similartransformation

Backward self-simlartransformation

Hyper-parabolicity of type (II) blowup point

Reverse second moment

Time discretization

vanishing of residual term

separation and convergence to the origin of sub-collapses

ODE

Kinetic

Conservation Euler of Mass

Equilibrium

Clustered Particles

Physics

Mathematics

Newton

Jeans-Vlasov

Semilinear elliptic eigenvalue problem(non-linearity unknown) (exponential non-linearity)Hamiltonian flow

Conservation laws

Chaotic motion

Langevin

Fokker-Planck

Keller-Segel

Semilinear ellipticeigenvalue problem

Gradient flow

Free energy

Quantized blow-up mechanism

Blowup mechanism in nonlinear e.v.p

Condensate.super-conductivity.Gauge theory

Non-equilibrium Statistical mechanics

Nonlinear quantum mechanics

Fencel-Moreau

Toland

Kuhn-Tucker

Phase separation Euler-Poisson Plasma Thomas-Fermi

Skew-gradient

Gierer-Meinhardt FitzHugh-Nagumo

Mean Field Theories and Dual Variation

Self-dual Gauge theory…super-conductivity,

condensate

Physical principle…mass conservation

free energyStatistical mechanics…turbulence,

transportation, self-gravitating particles

Mathematical principle…nonlinear quantum mechanics

quantized blowup mechanismdual variation

Mean field theories…phase separation,

phase transition,pattern formation,self-organization

System biology…backward causality

non-equilibriumemergencehyper-circle

Mathematical Biology…chemotaxis,

morphogenesisangiogenesis, pulse interaction