mean field theories and dual variation - osaka u
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Mean Field Theories and
Dual Variation
Takashi Suzuki(Osaka University)
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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
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principle
phenomenon physics mathematics
equation
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Mathematical analysis for differential equations- Paradigm of Poincare-Hadamard
1. Fundamental theorem…well-posedness; existence, uniqueness, continuous dependence of the solution
2. Qualitative study
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List of standard linear pde’s
Schrodinger(dispersive)
Elliptic (static)
Hyperbolic(conservative)
Parabolic (dissipative)
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Linear – Semi-linear – Quasi-linear – Fully Nonlinear
Single – System
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1. Method of Perturbation
2. Method of Energy / Variation
3. Method of Dynamical System
4. Method of Exact Solution
5. Method of Comparison
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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
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Breaking down of the generation of weak solutions
1. flight
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2. Concentration 3. Oscillation
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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.
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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∗∗
=
< ∞
< ⇒ = ∞ ∃ > ⇒ < ∞
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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 >
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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∗ =
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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.
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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
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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
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Blowup mechanism in nonlinear e.v.p
Condensate.super-conductivity.Gauge theory
Non-equilibrium Statistical mechanics
Nonlinear quantum mechanics
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Fencel-Moreau
Toland
Kuhn-Tucker
Phase separation Euler-Poisson Plasma Thomas-Fermi
Skew-gradient
Gierer-Meinhardt FitzHugh-Nagumo
Mean Field Theories and Dual Variation
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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