### January 14 ~ 17, 2011

### Purpose:

### The purpose of this workshop is to provide a platform for exchanging ideas, experiences,

### and current results, as well as on-going problems among researchers in the field of

### differential equations, dynamical systems, and their applications. Based on consideration

### of the geographical environment and according to a number of collaborations with the Japanese,

### Korean, German, and US researchers in past years, such as Professor Rainer Kress from

### Goettingen University, Professor Yoshio Tsutsumi from Kyoto University, and Professor Gen

### Nakamura from Hokkaido University who support the idea of hosting a workshop as an opportunity

### of communication. We hope this will enhance the original collaboration and create new

### relationships between the attendees can be expected in the near future.

### Venue:

### International Conference Hall, National Cheng Kung University

### Organizer:

### Min-Hung Chen, National Cheng Kung University

### Yung-Fu Fang, National Cheng Kung University

### Jong-Sheng Guo, Tamkang University

### Suchung Hou, National Cheng Kung University

### Kuo-Ming Lee, National Cheng Kung University

### Wen-Ching Lien, National Cheng Kung University

### Ching-Lung Lin, National Cheng Kung University

### Yoshio Tsutsumi, Kyoto University

### Chern-Shuh Wang, National Cheng Kung University

### Sponsors:

### NSC Math Research Promotion Center,

### NSC Department of International Cooperation,

### National Center for Theoretical Science (South), and

### National Cheng Kung University.

### Plenary Speakers:

### Rainer Kress, University Goettingen

### Tai-ping Liu, Academia Sinica

### Gen Nakamura, Hokkaido Univerity

### Yoshio Tsutsumi, Kyoto University

### Invited Speakers:

### Differential Equations:

### Chao-Nien Chen, National Changhua University of Education

### Yi-Chiuan Chen, Academia Sinica

### Jann-Long Chern, National Central University

### John M. Hong, National Central University

### Jin-Cheng Jiang, Academia Sinica

### Nobu Kishimoto, Kyoto University

### Hideo Kozono, Tohoku University

### Sanghyuk Lee, Seoul National University

### Ming-Chia Li, National Chiao Tung University

### Tai-chia Lin, National Taiwan University

### Kenji Nakanishi, Kyoto University

### Takayoshi Ogawa, Tohoku University

### Tohru Ozawa, Waseda University

### Hideo Takaoka, Hokkaido University

### Kotaro Tsugawa, Nagoya University

### Jenn-Nan Wang, National Taiwan University

### Applications:

### Jeng-Tzong Chen, National Taiwan Ocean University

### Jong-Shenq Guo, Tamkang University

### Sze-Bi Hsu, National Tsing Hua University

### Chien-Sen Huang, National Sun Yat-sen University

### Chi-Chuan Hwang, National Cheng Kung University

### Hideo Ikeda, Toyama University

### Yuusuke Iso, Kyoto University

### Li-Ren Lin, National Taiwan University

### Cheng-Chien Liu, National Cheng Kung University

### Chin-Yueh Liu, National University of Kaohsiung

### Masayasu Mimura, Meiji University

### Hirokazu Ninomiya, Meiji University

### Tsorng-Whay Pan, University of Houston

### Chun-Hao Teng, National Chiao Tung University

### Chin-Tien Wu, National Chiao Tung University

### Jonathan Wylie, City University of Hong Kong

### Shoji Yotsutani, Ryukoku University

### Time

### Chairman

### Speaker

### Title

### 15:00~16:00

### Registration

### 16:10~16:30

### Open Ceremony(Vice President: Hwung-Hweng Hwung)

### 16:40~17:30 G. Nakamura Tai-ping Liu

### Invariant Manifolds for Stationary Boltzmann Equation

### 18:00~

### Reception

### Jan. 15

### Time

### Chairman

### Speaker

### Title

### 08:55~09:45

### Hwaichiuan

### Wang

### Gen Nakamura

### Inverse boundary value problem for anisotropic heat operators

### (Dynamical probe method for anisotropic heat conductors)

### Hwaichiuan

### Wang

### Hideo Takaoka

### A priori estimates and weak solutions for the derivative nonlinear

### Schr\"odinger equations on torus

### 09:50~10:30

### Sze-Bi Hsu

### Masayasu

### Mimura

### Segregation property in a tumor growth PDE model with contact

### inhibition

### 10:30~11:00

### Coffee Break

### Hideo

### Takaoka

### Jenn-Nan

### Wang

### Quantitative uniqueness for Maxwell's equations with Lipschitz

### anisotropic media and asymptotic behaviors of nontrivial solutions

### 11:00~11:40

### Shin-H. Wang

### Sze-Bi Hsu

### A Lotka-Volterra Competition Model with Seasonal Succession

### H. Takaoka

### T. Ogawa

### Singular limit problem for a quantum drft-diffusion system

### 11:40~12:20

### Shin-H. Wang H. Ninomiya

### Non-planar traveling waves of reaction-diffusion equations

### 12:20~14:00

### Lunch

### Jen-H. Chang

### Tai-chia Lin

### Ground state of two-component Gross-Pitaevskii functionals

### 14:00~14:40

### Masayasu

### Mimura

### Hideo Ikeda

### Dynamics of traveling fronts in some heterogeneous diffusive

### media

### Jen-H. Chang Sanghyuk Lee On pointwise convergence of the Schr\"odinger equations

### 14:40~15:20

### M. Mimura Jong-S. Guo

### Motion by curvature of planar curves with two free end points

### 15:20~16:00

### Coffee Break

### Chiun-Chuan

### Chen

### Chao-Nien

### Chen

### Turing patterns and standing waves of FitzHugh-Nagumo type

### systems

### 16:00~16:40

### Chiun-Chuan

### Chen

### Kotaro

### Tsugawa

### Local well-posedness of the KdV equations with almost periodic

### initial data

### 16:40~17:20

### Ming-Chih Lai Jonathan Wylie Drawing of viscous threads with temperature-dependent viscosity

### 18:00~

### Banquet

### Jan. 16

### Time

### Chairman

### Speaker

### Title

### 08:55~09:45 Tai-ping Liu R. Kress

### Huygens' principle and iterative methods in inverse obstacle scattering

### M.C. Li

### T. Ozawa

### Life span of positive solutions to semilinear heat equations

### 09:50~10:30 Cheng-Chien

### Liu

### Jeng-Tzong

### Chen

### Focusing of seismic wave and harbor resonance by using null-field

### integral equations

### 10:30~11:00

### Coffee Break

### H. Kozono Ming-C. Li

### Chaos for multidimensional perturbations of dynamical systems

### 11:00~11:40

_{Chung }

### Kwong Law

### Shoji

### Yotsutani

### Multiplicity of solutions to a limiting system in the Lotka-Volterra

### competition with cross-diffusion

### Hideo

### Kozono

### Kenji

### Nakanishi

### Global dynamics beyond the ground energy for the focusing nonlinear

### Klein-Gordon equation

### 11:40~12:20

### C. K. Law

### C.-C. Liu Solving the radiative transfer equation for remote sensing of ocean color

### 12:20~14:00

### Lunch

### Jyh-Hao Lee

### Jann-Long

### Chern

### Uniqueness of Topological Solutions and the Structure of Solutions for

### the Chern-Simons System with Two Higgs Particles

### 14:00~14:40

### Chi-Tien Lin

### Chien-Sen

### Huang

### A LOCALLY CONSERVATIVE EULERIAN-LAGRANGIAN FINITE

### DIFFERENCE WENO METHOD FOR ADVECTION EQUATION

### Jyh-Hao Lee Yi-C. Chen

### Family of Julia Sets as Orbits of Differential Equations

### 14:40~15:20

### Chi-Tien Lin C.-H. Teng

### Pseudospectral penalty method for optical waveguide mode analysis

### 15:20~16:00

### Coffee Break

### 16:00~16:40

### Chin-Tien

### Wu

### A numerical study on Monge-Ampere equation arising from goemetric

### optical design

### 16:40~17:20

### Chien-Sen

### Huang

### Chin-Y. Liu

### Toward A Second Order Description of Neuronal Networks

### 16:00~16:30

### Kishimoto

### Local well-posedness for the Zakharov system on torus

### 16:30~17:00

### Tsung-fang

### Wu

### 17:00~17:30

### Li-Ren Lin

### Bose-Einstein condensates

### Jan. 17

### Time

### Chairman

### Speaker

### Title

### Y. Tsutsumi

### H. Kozono

### Leray's inequality in general multi-connected domains in R^n

### 09:00~9:50

### Chin-Tien Wu Yuusuke Iso

### Analysis of the transport equation as a mathematical model of the

### optical tomography

### 09:50~10:40

### Yoshio

### Tsutsumi

### John M.

### Hong

### Generalized Glimm method and geometric singular perturbations to

### nonlinear balance laws

### 10:40~11:10

### Coffee Break

### 11:10~12:00 Rainer Kress Y. Tsutsumi

### Stability of stationary solution for the Lugiato-Lefever equation

### 12:00~

### City Tour

**The 19th Workshop on Differential Equations and Its Applications **

### Hwung-Hweng Hwung

*Senior Executive Vice President *

*National Cheng Kung University *

### Good Afternoon and welcome to the National Cheng Kung University. I was asked by my

### colleague Professor Yung-Fu Fang, the chairman of the organizing committee, to make

### opening remarks at this very exciting Workshop.

### First of all, let me mention that this Workshop is co-sponsored by the National Science

### Council, the National Cheng Kung University, and the National Center for Theoretical

### Sciences, and co-organized by Professor Jong-Shenq Guo at the Department of Mathematics,

### Tamkang University, Professor Yoshio Tsutsumi at the Department of Mathematics, Kyoto

### University and our colleagues at the Department of Mathematics, and the chairman of the

### department Ruey-Lin Sheu. Let us thank their hard work.

### I am a Professor of Hydraulics and Ocean Engineering at the National Cheng Kung

### University. Obviously I am not a mathematician, but you may count me as an applied

### mathematician since I have used a lot of mathematics in my research, for example, wave

### modulation and wave dynamics. Real world problems contain comprehensive issues that

### require thorough understanding of the underlining principles of physics, mathematical

### modeling, theoretical analysis and numerical simulations, which is exactly what you are

### doing and is made abundantly clear in the range of topics covered in this Workshop.

### I believe that the Workshops on differential equations and its applications in the past 18

### years did provide an excellent platform for exchanging ideas, experiences and results among

### researchers. I hope that this year’s Workshop will not only play the same role, but also

### enhance existing collaborations and create new relationships and collaborations among the

### attendees in the near future.

### I also strongly believe that National Cheng Kung University can achieve prominence as a

### comprehensive university, and have an intellectually robust mathematics and applied

### mathematics program matters. For this reason, I am very pleased to see this Workshop is

### held here on campus.

### To our distinguished participants from abroad and domestic, I would like to welcome all

### of you to sunny Tainan. Besides our domestic participants, there are attendees and speakers

### from Germany, Japan, Korea and United States. We thank you for your participation.

### Especially we would like to thank Professor Rainer Kress from Goettingen University,

### Professor Yoshio Tsutsumi from Kyoto University, Professor Gen Nakamura from

### Hokkaido University and Professor Tai-ping Liu from the Institute of Mathematics,

### Academia Sinica. Thank you for your support.

### I hope you reach your goal in this Workshop. For the foreign guests, I

### hope you have an enjoyable time in Tainan.

### Equation

Tai-Ping Liu Shih-Hsien Yu

Academia Sinica, Taiwan, R.O.C. National University of Singapore

Jan. 14- Jan. 17, 2011, Tainan

### Invariant Manifolds for Steady Boltzmann Equation

Kinetic Theory

f (**x, t, ξ), density distribution function**
**x ∈ R**3space, t time,
ξ ∈ R3micorsocopic velocity.
Macroscopic variables
(
ρ(**x, t) ≡**R
R3f (**x, t, ξ)d ξ, density,**
ρ**v (x, t) ≡**R
R3ξf (**x, t, ξ)d ξ, momentum,**
ρE (**x, t) ≡**R
R3
|ξ|2
2 f (**x, t, ξ)d ξ, total energy.**
ρe(**x, t) ≡**R
R3
**|ξ−v |**2
2 f (**x, t, ξ)d ξ, internal energy,**
ρE = ρe +1_{2}ρ|**v |**2.
(
pij(**x, t) ≡**
R
R3(ξi− vi)(ξj− vj)f (**x, t, ξ)d ξ, 1 ≤ i, j ≤ 3, stress tensor**
qi(**x, t) ≡**
R
R3(ξi− vi)
**|v −ξ|**2
2 f (**x, t, ξ)d ξ, heat flux.**

Boltzmann equation
∂tf + ξ · ∂**x**f =
1
kQ(f , f ).
Transport: ∂tf + ξ · ∂**x**f
Collision operator:
Q(f , f )(ξ) ≡R
R3
R
S2
+[f (ξ
0_{)f (ξ}0
∗) −f (ξ)f (ξ∗)]B(|ξ − ξ∗|, θ)dΩdξ∗.
(
ξ0 = ξ − [(ξ − ξ_{∗}) · Ω]Ω,
ξ0∗ = ξ∗+ [(ξ − ξ∗) · Ω]Ω.

Hard sphere models B = |(ξ − ξ∗) · Ω| = |ξ − ξ∗| cos θ.

### Invariant Manifolds for Steady Boltzmann Equation

Boltzmann equation ∂tf + ξ · ∂**x**f = 1 kQ(f , f ). Conservation Laws Z R3 1 ξ 1 2|ξ|2 Q(f , f )d ξ = 0, mass momentum enegry . H-Theorem Z R3 log fQ(f , f )d ξ = 1 4k Z R3 Z S2 + log ff∗ f0

_{f}0 ∗ [f0f

_{∗}0 − ff∗]Bd Ωd ξ∗d ξ ≤ 0.

H-Theorem:
∂tH + ∂**x**· ~H ≤ 0, H ≡
Z
R3
f log fd ξ, ~H ≡
Z
R3
ξf log f ξ,
=0 if and only if
f (**x, t, ξ) =** ρ(**x, t)**
(2πRθ(**x, t))**3/2e
−**|ξ−v (x,t)|2**_{2Rθ(}** _{x,t)}**
≡ M

_{(ρ,}

**v ,θ)**,

Maxwellian, thermo-equilibrium states, Q(M, M) = 0.

### Invariant Manifolds for Steady Boltzmann Equation

Conservation laws

∂tρ + ∂x**· (ρv ) = 0, mass,**

∂t(ρ**v ) + ∂**x**· (ρv × v + P) = 0, momentum,**

∂t(ρE ) + ∂x**· (ρv E + Pv + q) = 0, energy.**

**1** _{More (14) unknowns than (5) equations. Need to know}_{the}

dependence on microscopic velocityto compute the stress

tensor**P and heat flux q.**

**2** _{Boltzmann equation is equivalent to a system of}_{infinite}

PDEs.

Boltzmann equation

∂tf + ξ · ∂**x**f =

1

kQ(f , f ).

At thermo-equilibrium, f = M, Q(M, M) = 0, the stress tensor

**P = pI is the pressure p = (3/2)e and heat flux become zero,**
**q = 0, and the conservation laws become the**Euler equations

in gas dynamics:
∂tρ + ∂x **· (ρv ) = 0,**
∂t(ρ**v ) + ∂**x**· (ρv × v + pI) = 0,**
∂t(ρE ) + ∂x**· (ρv E + pv ) = 0.**

### Invariant Manifolds for Steady Boltzmann Equation

Plane waves

f (ξ,**x, t) = f (ξ, x , t), x = (x , y , z), ξ = (ξ**1, ξ2, ξ3).
f (ξ, x , t) even in ξ2and ξ3; **v = (u, 0, 0).**

Boltzmann equation, plane waves

∂tf + ξ1∂xf =

1

kQ(f , f ).

Steady Boltzmann equation, plane waves

ξ1∂xf =

1

kQ(f , f ).

Steady Boltzmann equation, plane waves
ξ1∂xf =
1
kQ(f , f ). (1)
Function space
L∞_{ξ,3}≡nf ∈ L∞(R3)| kfkL∞
ξ,α ≡ sup
ξ∈R3
(1 + |ξ|)3|f(ξ)| < ∞o.
Invariant manifolds

A manifold**N in L**∞_{ξ,3}isinvariantfor (1) if for any g0**∈ N, there**

exists a solution (flow) g(x ) of (1) with initial value g(0) = g0

and satisfying g(x ) ∈**N, |x | < δ, for some positive δ.**Stable

manifold consists of states g0yielding flows g(x ), which

converges exponentially to an equilibrium state as x → ∞. Analogously, a state on theunstablemanifold gives rise to a converging flow as x → −∞.

### Invariant Manifolds for Steady Boltzmann Equation

Invariant manifold theory for differential equations

u0(x ) = f(u), u ∈ Rn. Use spectral consideration.

Invariant manifolds for steady Boltzmann equation

ξ1∂xf =

1

kQ(f , f ),

ininfinite dimensionalspace L∞_{ξ,3}.Spectral consideration not
sufficient. UseGreen’s function approach.

Steady Boltzmann equation, plane waves

ξ1∂xf =

1 kQ(f , f )

Critical statesQ(f , f ) = 0, f = M(ρ,**v ,θ)**=M(ρ,u,θ) the

Maxwellian, equilibriumstates,3−dimensional manifold.

Linearization
f = M0+
√
M0g, gt + ξ1∂xg = 1_{κ}(L(g) + Γ(g, g)),
L(g) = √2
M0
Q(M0,
√

M0g), linearized collision operator,

Γ(g) = √1

M0

Q(√M0g,

√

M0g), nonlinear term.

Linearized Boltzmann equation

gt + ξ1∂xg =

1 κL(g)

ξ1∂xg =

1

κL(g), linearized steady equation.

### Invariant Manifolds for Steady Boltzmann Equation

Linearized Boltzmann equation

gt+ ξ1∂xg =

1 κL(g).

Collision invariants, kernel of L

√ M, ξ1 √ M, |ξ|2 √ M. Macro-Micro decomposition

macro projection P0is the projection onto the kernel of L.

The micro projection is P1≡ I − P0.

Linearized Euler equations

(P0h)t + (P0ξ1P0h)x =0.

Euler characteristics: speeds λi, directions Ei, i = 1, 2, 3 :

P0ξ1P0Ei = λiEi,
{λ_{1}, λ2, λ3**} = {−c + u, u, c + u} ,**
**c =**
q
5θ

3, (sound speed at rest),

E1≡ −
r
1
2
(ξ1− u)
√
θ +
1
√
30
|ξ − ~u|2
θ
! r
M_{[1,u,θ]}
θ3 ,
E2≡ −
r
5
2+
1
√
10
|ξ − ~u|2
θ
!
q
M_{[1,u,θ]},
E3≡
r
1
2
(ξ1− u)
√
θ +
1
√
30
|ξ − u|2
θ
! r
M[1,u,θ]
θ3 .
M[1,u,θ]≡ e
−(ξ1−u)2+|ξ2|2+|ξ3|2
√
(2πθ)3 .

### Invariant Manifolds for Steady Boltzmann Equation

Euler projections Bi, 1 = 1, 2, 3, B_{i}k ≡ (E

_{i},k)E

_{i}, P0≡ 3 X j=1 Bj,macro projection.

Euler Flux Projections ˜Bi, i = 1, 2, 3,

˜ P0≡ 3 X k=1 ˜ Bk, ˜ Bkg ≡ (Ek, ξ1g)Ek λk , ˜ B±≡ X ±λk>0 ˜

Bk,upwind, downwind projections.

The Euler flux projections become singular when one of the Euler characteristic speeds approaches zero.

Green’s function G(x, y, t, τ, ξ, ξ∗) = G(x − y, t − τ, ξ, ξ∗)
(
(−∂τ − ξ∗∂y− L)G(x − y, t − τ, ξ, ξ∗) =0,
G(x − y , 0, ξ, ξ∗) = δ1(y − x )δ3(ξ∗− ξ).
G(x , t , ξ; ξ∗) =e−ν(ξ∗)tδ(x − ξ1t)δ3(ξ − ξ∗)
+
3
X
k =1
e−
(_{x −λk t)}2
4Ak (t+1)
p4Akπ(t + 1)
Ek(ξ)Ek(ξ∗) + · · · ; (2)
kGtP1gin(x )kL2
ξ =O(1) |||gin|||
3
X
i=1
e−
|x−λi t|2
C0(t+1)
(t + 1) +e
−(|x|+t)/C1
,
kP1GtP1gin(x )kL2
ξ =O(1) |||gin|||
3
X
i=1
e−|x−λi t|
2
C0(t+1)
(t + 1)3/2 +e
−(|x|+t)/C1
.
(3)

### Invariant Manifolds for Steady Boltzmann Equation

Green’s identitySuppose that there are solutions f+(x ) of the

linearized Boltzmann equation in x > 0 with the full boundary values b+∈ L∞ξ,3: ξ1∂xf+=Lf+for x > 0, lim x →∞f+(x ) = 0, Then f+(x ) = Z t 0 G(x , t −τ )[ξ1b+]d τ + Z ∞ 0 G(x −y , t )f+(y )dy , x , t > 0.

Linear Stable-Center-Unstable Decomposition Theorem

L∞_{ξ,3}=** _{S ⊕ C ⊕ U, S ≡ S}**0+(L∞ξ,3),

**C ≡ ˜**P0(L∞ξ,3),

**U ≡ U**0−(L∞ξ,3) Stable steady linear Boltzmann flows

( Sxh ≡

R∞

0 G(x , s)ξ1(1 − ˜B+)hds,

Sxh = O(1)e−α|x|, x → ∞

Unstable steady linear Boltzmann flows

(

Uxh ≡ −R_{0}∞G(x , s)ξ1(1 − ˜B−)hds,

Uxh = O(1)e−α|x|, x → −∞

Here α is some positive constant which tends to zero as one of the Euler characteristics tends to zero.

### Invariant Manifolds for Steady Boltzmann Equation

Nonlinear invariant manifolds

The nonlinear center-stable manifold M+, center-unstable

manifold M−, stable manifold Ms, and unstable manifold Muare

defined as graphs:

F+:Range(S0+) ⊕Range(C0) 7−→Range(U0−),

F−:Range(C0) ⊕Range(U0−) 7−→Range(S0+),

Fs :Range(S0+) 7−→Range(C0) ⊕Range(U0−),

Fu:Range(U0−) 7−→Range(S0+) ⊕Range(C0),

with F+(0) = ∇Fs(0) = 0, etc., using the spectral gap:

Sx =O(1)e−αx, x > 0, Ux =O(1)e−α|x|, x < 0, and that the

nonlinear source Γ(g) = √1

M0

Q(√M0g,

√

M0g) is microscopic.

Resonance cases

Euler characteristics λi near zero, for some i ∈ {1, 2, 3}; e.g.

λ3= , transonic condensation. Slowly decaying solution

(

ψ(x ) = φ_{e}η()x_{, η() =}_{O(1),}
1

ξ1Lφ = ηφ.

Uniformly bounded operator for subsonic condensation, > 0

B],_{3} f0≡
(E
3, ξ1f0)
(E
3, ξ1`3)
`_{3}, `_{3}≡ φ
_{− E}
3
,
S],x f0≡
Z ∞
0
G(x , τ )[ξ1(1 − ˜B1− ˜B2− B
],
3 )f0]d τ, x > 0,

S],x =O(1)e−αx, for some α > 0 independent of .

### Invariant Manifolds for Steady Boltzmann Equation

Uniformly bounded operator for subsonic condensation, < 0

U],x f0≡ − Z ∞ 0 G(x , τ )[ξ1(1 − B],3 )f0]d τ, x < 0,

U],x =O(1)e−α|x|, for some α > 0 independent of .

Uniformly bounded operator for subsonic condensation, > 0

S],x f0≡ Z ∞ 0 G(x , τ )[ξ1(1 − ˜B1− ˜B2− B ], 3 )f0]d τ, x > 0,

S],x =O(1)e−αx, for some α > 0 independent of .

Conjugate operator ¯ S],x h ≡ s M M−S ], x s M− M h ,x > 0 Knudsen operator S[,−x , − <0 :

There exist unique bounded operators S[,−x , x > 0, and Λ−on

Range(S],−0+ )satisfying, for any b ∈ Range(S
],−
0+ ),
b = S[,−_{0+} b + Λ−(b)φ−, Λ−(b) ∈ R,
kS[,−x bkL∞
ξ,3 ≤ O(1)kbkL∞ξ,3e
−αx _{for x > 0,}
kS[,−_{0+} b − ¯S],_{0+}bkL∞
ξ,3 ≤ O(1)| log |kbkL
∞
ξ,3,
(ξ1∂x − L−)S[,−x b = 0, x > 0.

### Invariant Manifolds for Steady Boltzmann Equation

Discontinuity of linear operators

˚_{S}_{0+}_{b ≡ lim}
→0+S
],
0+b, ˚Sxb ≡ lim_{→0+}S
],
x b, x > 0; Stable flow,
˚
U0−b ≡ lim
→0−U
],
0−b, ˚Uxb ≡ lim
→0−U
],
x b, x < 0; Unstable flow,
˚
`3≡ lim
→0`
3: Degenerated eigenvector,
˚
C0b =
2
X
j=1
˜
B0_{j}b + B0_{3}b + (ξ
1_{E}0
3,P01b)
(ξ1_{E}0
3, ˚`3)
˚
`3− B03(˚S0++ ˚U0−)P01b, Center component.
lim
→0−Range(S
],
0+) =Range(˚S0+) ⊕span(E
0
3),
lim
→0+Range(S
],
0+) =Range(˚S0+),
lim
→0+Range(U
],
0−) =Range(˚U0−) ⊕span(E
0
3),
lim
→0−Range(U
],
0−) =Range(˚U0−).

Nonlinear invariant manifoldsM_{u}, M_{+}, ( < 0); M_{s},M−, ( > 0):

the nonlinear unstable manifold, the center-stable manifold, (supersonic); the nonlinear stable manifold, the nonlinear center-unstable, (subsonic) defined as graphs

F_{u} :Range(U],0−) 7−→Range
2
X
j=1
˜
B_{j} +B],_{3} + S],0+
, <0
F_{s} :Range(S],0+) 7−→Range
U
],
0−+
2
X
j=1
˜
B_{j} +B],_{3}
, >0,
F−:Range
U
],
0−+
2
X
j=1
˜
B_{j} +B],_{3}
7−→ Range
S],0+
, >0,
F_{+}:Range
2
X
j=1
˜
B_{j} +B],_{3} _{+ S}],_{0+}
7−→ Range
U],0−
, <0.

### Invariant Manifolds for Steady Boltzmann Equation

Center manifold=intersection of center-stable with

center-unstable manifolds. With local Maxwellian coordinates, the flows on center manifold are governed byBurgers type equations.

Bifurcation manifold= nonlinear manifold based on subsonic condensation stable operator S],x , x > 0, > 0,

Sone manifold= nonlinear manifold based on supersonic Knudsen operator S[,x , x > 0, < 0.

Two-scale flows: Fast, Knudsen type flows on Bifurcation and

Sone manifolds; slow fluid-like flows on center manifold; two scale flows in general.

Monotonicity of Boltzmann shock profilesdue to Burgers type dynamics on the center manifold.

*q*
+
r

### M

*qr*

### M

Sone Manifold Bifurcation Manifold 1*C*

*C*2

*C*3 4

*C*5

*C*1

*A*2

*A*3

*A*4

*A*

*A*5 0 0 M , M M M M+

*q*

*knus*+ + +

*q*−

*q*r r r

### M

0 0 M M M M M−*q*

*s*−+ −

*q*−

*q*r r r

### M

### Invariant Manifolds for Steady Boltzmann Equation

Bifurcation phenomena.

The flux is conserved for steady flows

(Φi, ξ1f )x = (Φ,

1

κQ(f , f )) = 0, Phii, i = 1, 2, 3, collision invariant. The invariant manifolds depend smoothly on the flux. For the Euler equations,

~

Ut + ~F (~U) = 0

a perturbation in the characteristic direction ~

U2= ~U1+ ~ri(~U1) +O(1)2,

with theresonancecase λ = O(1), the flux changes little: ~

F (~U2) − ~F (~U1) = λi(~U1) +O(1)2=O(1)2.

Thus a small change O(1)2_{of the flux can induce a relatively}

large change of the states. This implies the large changes of Sone and Bifurcation manifolds in the transonic condensation case, for instance. And the bifurcation phenomena occur.

Sone States

Bifurcation Manifold

**Inverse Boundary Value Problem for**

**Anisotropic Heat Operators**

**(Dynamical probe method for anisotropic**

**heat conductors)**

**Gen Nakamura and Kim Kyoungsun**

gnaka@math.sci.hokudai.ac.jp

Department of Mathematics, Hokkaido University, Japan Department of Mathematics, Ewha University, Korea

19th Workshop on Diﬀerential Equations and Its Applications, National Cheng-Kung Univ., Tainan, January 15, 2011

.

. .**1** **Important Preliminary Estimates**

Gradient estimate of solutions for parabolic equations Gradient estimate of fundamental solution

Remarks About the proof

.

. .**2** **Dynamical Probe Method**

Active thermography Forward problem

Dynamical probe method Seperated inclusions case result Outline of the proof

**Important Preliminary Estimates**

**Gradient estimate of solutions for parabolic equations**

**Domain and operators**

Ω*⊂ Rn _{: b’dd domain (heat conductor), ∂Ω : C}*2

_{.}

*γ(x) = (γjk(x)) : deﬁned*a.e. in Ω,symm, pos. def. (conductivity)
*λ|ξ|*2*≤*∑*γjk(x)ξjξk≤ Λ|ξ|*2*.*

**Domain and operators (continued)**

Let Ω =
( _{∪}

_{L}*m=1*

*Dm*)

*\ ∂Ω.*

*γ(m)*

_{∈ C}µ_{(D}*m) (0 < µ < 1), γ(x) = γ(m)(x) (x∈ Dm*).

Each*separated Dm* is of*C1,α* smooth*with 0 < α≤ 1 and*

non-separated one is the limit of the separated one.

*D*1 _{D}

**Important Preliminary Estimates**

**Gradient estimate of solutions for parabolic equations**

**Gradient estimate**

.

**Theorem 1 (Fan, Kim, Nagayasu and N)**

. .

.

*Let*Ω*′* *b Ω, 0 < t*0*< T. Any sol u to(P): ∂tu− ∇ · (γ∇u) = 0 in*

Ω*× (0, T )has the following interior regularity est:*
*sup _{t}*

0*<t<T∥u(·, t)∥C1,α′*(Ω*′∩Dm*)*≤ C∥u∥L*2(Ω*×(0,T )),*

*where 0 < α′≤ min(µ,* _{2(α+1)}α*) andC is indep of the dist between*
*inclusions.*

**Gradient estimate of fundamental solution**

By applying our main theorem and ascaling argument, we obtain

pointwise grad. est. *for 0 < t− s < T ,*

*|∇xE(x, t; y, s)| ≤*
*CT*
*(t− s)n+1*2
exp
(
*−c|x − y|*2
*t− s*
)

**Important Preliminary Estimates**

**Remarks**

**Remarks**

(i) We can obtain a similar estimate fornon-homog parabolic eq:

*∂tu− ∇ · (γ∇u) = g + ∇ · f.*

(ii) Thetime dependent inclusionscase is an open problem.

(iii) Theelliptic casewas proved by Li-Vogelius for scalar equations and Li-Nirenberg ([LN]) for systems, which answered to theBabuˇska’s conjecture. Babuˇska et al (1999) numerically observed that the gradient est of sol is indep of the distances between inclusions.

**Idea of Proof**

**Idea of proof:**

*• Some interior estimates (Lemma).*
*· (ref. Ladyzenskaja-Rivkind-Uralceva)*
*• Apply [LN] to (P).*

**Important Preliminary Estimates**

**About the proof**

**Proof**

.
**Lemma 2**. . .

*Let* *eΩ b Ω, 0 < t*0*< T. Any sol u to*

*∂tu− ∇ · (A∇u) = 0 in*Ω*× (0, T ) =: Q*
*has the following estimates:*

*sup*
*t*0*<t<T*

*∥u(·, t)∥L2(e*Ω)*≤ C∥u∥L2(Q)* *(standard),*
*∥u∥L∞(e*Ω*×(t*0*,T ))≤ C∥u∥L2(Q)* *(Di Giorgi’s arg.),*
*∥ut∥ _{L2(e}*

_{Ω}

*0*

_{×(t}*,T ))≤ C∥u∥L2(Q)*

*([LRU]).*.

**Remark 3**. . .

*(i)This lemma holds forA∈ L∞.*

**Proof**

Let eΩ3b eΩ2b eΩ1b eΩ0*:= Ω, 0 < δ*1*< δ*2*< T*. Then

(*∗) supδ*2*<t<T∥u(·, t)∥L*2_{(e}_{Ω}_{2}_{)}*≤ C∥u∥L*2*(Q),*

(*∗∗) ∥ut∥ _{L}*2

_{(e}

_{Ω}

1*×(δ*1*,T ))≤ C∥u∥L*2*(Q).*

*Since ∂tut− ∇ · (A∇ut*) = 0, we have

(*∗ ∗ ∗) ∥ut∥ _{L∞(e}*

_{Ω}

2*×(δ*2*,T ))≤ C∥ut∥L2(e*Ω1*×(δ*1*,T ))≤ C*
*′ _{∥u∥}*

*L*2_{(Q)}.

**Important Preliminary Estimates**

**About the proof**

**Proof**

Then by [LN], we have
*∥u(·, t)∥C1,α′*

_{(D}*m∩e*Ω3)

*≤ C*(

*∥u(·, t)∥L*2

_{(e}

_{Ω}2)+

*∥ut*(

*·, t)∥L∞*(eΩ2) ) . Taking sup

*2*

_{δ}*<t<T*, we have by (*), (**), (***), sup

*2*

_{δ}*<t<T∥u(·, t)∥C1,α′*

_{(D}_{m}_{∩e}_{Ω}

_{3}

_{)}

*≤ C*(sup

*2*

_{δ}*<t<T∥u(·, t)∥L*2(eΩ2)+

*∥ut∥L∞*(eΩ2

*×(δ*2

*,T ))*)

*≤ C∥u∥L*2

_{(Q)}.**Dynamical Probe Method**
**Active thermography**

**Active thermography**

### D

### ∂

ν### u|

∂Ω### = f

### u(f )|

_{∂Ω}

### Ω

**Principle of active thermography**

infrared
camera
heater /
flash lam p
inclusion
**Dynamical Probe Method**

**Forward problem**

**Mixed problem (set up)**

Ω*⊂ Rn* _{(1}_{≤ n ≤ 3) : bounded domain,}

*∂Ω : C*2*(n = 2, 3),* *∂Ω = ΓD _{∪ Γ}N_{,}*

where Γ*D*_{, Γ}*N* _{are open subsets of ∂Ω such that Γ}D_{∩ Γ}N_{=}_{∅ and}*∂ΓD _{, ∂Γ}N*

*2*

_{are C}_{if they are nonempty.}

*D⊂ Ω : open set (***separated inclusion(s)***), D⊂ Ω,*

*∂D : C1,α* _{(0 < α}_{≤ 1), Ω \ D : connected.}

Heat conductivity:

*γ(x) = A(x) + ( ˜A(x)− A(x))χD* *: positive deﬁnite for each x∈ Ω,*

where*A, ˜A∈ C*1(Ω)are positive deﬁnite and ˜*A− A is always positive*

*Hp _{(∂Ω), H}p,q*

_{(Ω}

_{× (0, T )):}_{usual Sobolev spaces}

*(p, q∈ Z*+:=*N ∪ {0} or p =*

1 2)

ex. *For p, q∈ Z*+*, g∈ Hp,q*(Ω*× (0, T )) iﬀ*

*||g||Hp,q*_{(Ω}* _{×(0,T ))}*:=
∑

*|α|+2k≤p*

*k≤q*∫ Ω

_{×(0,T )}*∂xα∂tkg*2

*dtdx*

*1/2*

*<∞*

*L*2

*((0, T ); Hp(∂Ω)) :={f ;*∫

_{0}

*T||f(·, t)||*2

_{H}p_{(∂Ω)}dt <∞}**Dynamical Probe Method**

**Forward problem**

**Mixed problem (forward problem)**

*Given f∈ L*2* _{((0, T ); H}*12

_{(Γ}

*D*2

_{)), g}_{∈ L}*1 2(Γ*

_{((0, T ); ˙}_{H}−*N*)), (?)

*∃! weak solution*

*u = u(f, g)∈ W (ΩT*) :=

*{u ∈ H1,0*(Ω

*T), ∂tu∈ L*2

*((0, T ); H*1(Ω)

*∗*)

*}*:

*PDu(x, t) := ∂tu(x, t)− divx(γ(x)∇xu(x, t)) = 0 in ΩT*

*u(x, t) = f (x, t) on ΓD*

*T,* *∂Au(x, t):= ν· A∇u(x, t) = g(x, t) on ΓNT*
*u(x, 0) = 0 for x∈ Ω,*

*where ν is the outer unit normal of ∂Ω,*

*H*
1

2_{(Γ}*D*_{), ˙}* _{H}_{−}*1

2(Γ*N*) are H¨ormander’s notations of Sobolev sp,

Ω*T* = Ω*(0,T )*:= Ω*× (0, T ), ∂ΩT* *= ∂Ω(0,T ):= ∂Ω× (0, T ).*

(cylindrical sets)

**Measured data**

Neumann-to-Dirichlet mapΛ*D*:

For*ﬁxed f∈ L*2* _{((0, T ); H}*12

_{(Γ}

*D*

_{))}

_{, deﬁne}Λ

*D: L*2

*((0, T ); ˙H−*1 2(Γ

*N*))

*→ L*2

*((0, T ); H*1 2

_{(Γ}

*N*

_{))}

*g7→ u(f, g)|*Γ

*N*

*T.*

**Inverse boundary value problem**

**Dynamical Probe Method**

**Forward problem**

**Known results I**

*∗*H. Bellout(1992): Local uniqueness and stability.

*∗*A. Elayyan and V. Isakov (1997): Global uniqueness using the
*localized Neumann-to-Dirichlet map.*

*∗*M. Di Cristo and S. Vessella(2010): Optimal stability estimate (i.e.
log type stability estimate) even for time dependent inclusions.

*∗*Y. Daido, H. Kang and G. Nakamura (2007) (Inverse Problems) :
Introduced the dynamical probing method for 1-D case.

*∗*Y. Daido, Y. Lei, J. Liu and G. Nakamura(2009) (Applied
Mathematics and Computation) Numerical implementations of 1-D
dynamical probe method for non-stationary heat equation.

**Known results II**

*∗*Y. Lei, K. Kim and G. Nakamura(2009) (Journal of Computational
Mathematics) Theoretical and numerical studies for 2-D dynamical probe
method.

*∗*M. Ikehata and M. Kawashita(2009) (Inverse Problems) Extracted
some geometric information of an unknown cavity using CGO solution
and asymptotic analysis.

*∗*V. Isakov, K. Kim and G. Nakamura(2010) (Ann. Scola Superior di
Pisa) Gave the theoretical basis of dynamical probe method.

**Dynamical Probe Method**

**Dynamical probe method**

**Dynamical probe method (fundamental solutions)**

*For (y, s), (y′, s′*)*∈ Rn _{× R, (x, t) ∈ Ω}*

*T*,

*Γ(x, t; y, s) : fundamental solution ofP _{∅}:= ∂t− ∇ · (A(x)∇)*

Γ*∗(x, t; y′, s′*) : fundamental solution of*P _{∅}∗*:=

*−∂t− ∇ · (A(x)∇)*

*G(x, t; y, s)*,*G∗(x, t; y, s′*):
*P∅G(x, t; y, s) = δ(x− y)δ(t − s) in ΩT,*
*G(·, ·; y, s) = 0 on ΓD*
*T,*
*G(x, t; y, s) = 0 for x∈ Ω, t ≤ s*
*P∗*
*∅G∗(x, t; y, s′) = δ(x− y)δ(t − s′*) in Ω*T,*
*G∗*(*·, ·; y, s′*) = 0 on Γ*D*
*T,*
*G∗(x, t; y, s′) = 0 for x∈ Ω, t ≥ s′*
*G(x, t; y, s)− Γ(x, t; y, s), G∗(x, t; y, s′*)*− Γ∗(x, t; y, s′*)*∈ C∞*(Ω*T*)*.*

**Dynamical probe method (Runge’s approximation)**

*∃{v0j*

*(y,s)}*,

*{φ*

*0j*

*(y′,s′*)

*}∈ H*

*2,1*

_{(Ω}(

*) for*

_{−ε,T +ε)}*∀ε > 0 s.t.*

*P∅v0j(y,s)*= 0 in Ω(

*−ε,T +ε),*

*v*= 0 on Γ

_{(y,s)}0j*D*

_{× (−ε, T + ε),}*v*

_{(y,s)}0j*(x, t) = 0*if

*− ε < t ≤ 0,*

*v*

_{(y,s)}0j*→ Γ(·, ·; y, s)*

*in H2,1*

_{(}

_{U}_{× (−ε}′_{, T + ε}′_{)}

* *

_{) as j}_{→ ∞,}*P∗*

*∅φ0j(y′,s′*)= 0 in Ω(

*−ε,T +ε),*

*ψ*

_{(y,s}0j*′*)= 0 on Γ

*D*

_{× (−ε, T + ε),}*φ0j*)

_{(y}′,s′*(x, t) = 0*

*if T*

*≤ t < T + ε,*

*φ0j*)

_{(y}′,s′*→ Γ∗*(

*·, ·; y′, s′*)

*in H*

*2,1*

_{(U}_{× (−ε}_{′}_{, T + ε}_{′}_{)) as j}_{→ ∞}*for 0 <∀ε′< ε,* *∀U ⊂ Ω : open*s.t.

*U* *⊂ Ω, Ω \ U : connected, ∂U : Lipschitz, U ̸∋ y, y′*,
and*−ε < s, s′< T + ε.*

**Dynamical Probe Method**

**Dynamical probe method**

**Dynamical probe method (Runge approx funcs)**

Let*v, ψ*satisfy
*P∅v = 0 in ΩT,*
*v = f on ΓD*
*T,*
*∂νv = 0 on ΓNT,*
*v(x, 0) = 0 for x∈ Ω,*
*P∗*
*∅ψ = 0 in ΩT,*
*ψ = 0 on ΓD*
*T,*
*∂νψ = g on ΓNT,*
*ψ(x, T ) = 0 for x∈ Ω.*
*For j = 1, 2,· · · , we deﬁne*
{

*v _{(y,s)}j*

*:= v + v*

_{(y,s)}0j*→V(y,s):= v + G(·, ·; y, s)*

*ψj _{(y,s}′*)

*:= ψ + ψ*

*0j*

*(y,s′*)

*→*Ψ

*(y,s′*)

*:= ψ + G∗*(

*·, ·; y, s′).*

*in H2,1*

_{(U}*T) as j→ ∞.*

*{vj*

*(y,s)}, {φ*

*j*

**Pre-indicator function**
.
**Definition 4**
.
.
.
*(y, s), (y′, s′*)*∈ ΩT*
*{vj*
*(y,s)}, {φ*
*j*

*(y′,s′*)*} ⊂ W (ΩT) : Runge’s approximation functions*

*Pre-indicator function:*
*I(y′, s′; y, s)*
= lim
*j→∞*
∫
Γ*N*
*T*
[
*∂νv*
*j*
*(y,s)|*Γ*N*
*T* *φ*
*j*
*(y′,s′*)*|*Γ*N*
*T* *−*Λ*D(∂νv*
*j*
*(y,s)*)*|*Γ*N*
*T* *∂νφ*
*j*
*(y′s′*)*|*Γ*N*
*T*
]

**Dynamical Probe Method**

**Dynamical probe method**

**Reflected solution**

.
**Lemma 5**. . .

*y̸∈ D, 0 < s < T , {vj _{(y,s)}} ⊂ W (ΩT) : Runge’s approximation functions,*

*uj*Γ

_{(y,s)}:= u(f, ∂Avj_{(y,s)}|*N*

*T),w*
*j*
*(y,s):= u*
*j*
*(y,s)− v*
*j*
*(y,s)*

*Then,wj _{(y,s)}has a limitw(y,s)∈ W (ΩT) satisfying*

*PDw(y,s)*= div*x*(( ˜*A− A)χD∇xV(y,s)*) in Ω*T,*
*w(y,s)*= 0 on Γ*DT, ∂Aw(y,s)*= 0 on Γ*NT*
*w(y,s)(x, 0) = 0 for x∈ Ω.*

**Representation formula**

.
**Theorem 6**. . .

*Fory, y′̸∈ D, 0 < s, s′< T* *such that(y, s)̸= (y′, s′*)*, the*

*pre-indicator function I(y′, s′; y, s) has the representation formula in*

*terms of the reflected solution w(y,s)* *:*

*I(y′, s′; y, s) =−w(y,s)(y′, s′*)*−*

∫

*∂ΩT*

**Dynamical Probe Method**

**Seperated inclusions case result**

**Main result (indicator function)**

.
**Definition 7**
.
.
.
*C :={c(λ) ; 0 ≤ λ ≤ 1} : non-selfintersecting C*1_{curve in Ω,}

*c(0), c(1)∈ ∂Ω (We call this C aneedle.)*

*Then, for each c(λ)∈ Ω and each fixed s ∈ (0, T ),*

*indicator function* *(mathematical testing machine)*

*J (c(λ), s) := lim*

*ϵ↓0*lim sup_{δ}_{↓0}*|I(c(λ − δ), s + ϵ*

2_{; c(λ}_{− δ), s)|}

*D*
Ω
*C*
*c(0)*
*c(λ− δ)*
*c(1)*
*c(λ)*

**Dynamical Probe Method**

**Seperated inclusions case result**

**Seperated inclusions case result (theorem)**

.

**Theorem 8**

. .

.

*Let D consist ofseparated inclusions, and C, c(λ) be as in the definition*
*above.*

*Fix s∈ (0, T ).*

**(i)** *C⊂ Ω \ D except c(0) and c(1)*

=*⇒ J(c(λ), s) < ∞ for all λ, 0 ≤ λ ≤ 1*

**(ii)** *C∩ D ̸= ∅*

*λs(0 < λs< 1) s.t. c(λs*)*∈ ∂D, c(λ) ∈ Ω \ D (0 < λ < λs*)

=*⇒*

**Remark :**

(i) Anumerical realizationof this reconstruction scheme has been done for isotropic conductivities.

(ii) IfΓ*D _{̸= ∅ and f(·, t) = 0 = g(·, t) (t > T}′_{) with 0 < T}′_{< T}*

_{,}

*then u(f, g) has the decaying property. That is u(f, g) decays*

*exponentially after t = T′*. Hence, in this case, we can guarantee the

**Dynamical Probe Method**

**Outline of the proof**

**Proof of Theorem 6:**

*Consider only the case n = 3 in the rest of the arguments.*
First, we recall the previous two facts.

*(i) w(y,s)∈ W (ΩT*) : solution to

*PDw(y,s)*= div*x*(( ˜*A− A)χD∇xV(y,s)*) in Ω*T,*
*w(y,s)*= 0 on Γ*DT, ∂Aw(y,s)*= 0 on Γ*NT*
*w(y,s)(x, 0) = 0 for x∈ Ω.*

(ii)

*I(y, s′; y, s) =−w(y,s)(y, s′*)*−*

∫

*∂ΩT*

*w(y,s)∂ν*Γ*∗*(*·, ·; y, s′)dσdt*

*If y = c(λ)̸∈ ∂D, it is easy to see the indicator function is ﬁnite at y.*

**Setup**

Note that

*PDw(y,s)*= div*x*(( ˜*A− A)χD∇xV(y,s)*) in Ω*T*

Hence,

*E(x, t; y, s):= w(y,s)(x, t) + V(y,s)(x, t)*

(*⇒*fundamental solution for*PD.)*
Let*P = c(λ*0)*∈ ∂D* *for some λ*0

*x = y = c(λ*0*− δ) ∈ C \ D* *for δ > 0.*

Φ :R3* _{→ R}*3

_{with Φ(P ) = O (C}1,α

_{diﬀeomorphism, 0 < α}_{≤ 1),}*Φ(D)⊂ R*3

*−* =*{ξ = (ξ*1*, ξ*2*, ξ*3)*∈ R*3*; ξ*3*< 0},*

**Dynamical Probe Method**

**Outline of the proof**

Let

*E : ∂t− ∇ · ((A(x) + ( ˜A(x)− A(x))χD*)*∇)*

Γ*P* *: ∂t− ∇ · ((A(x) + ( ˜A(P )− A(x))χD*)*∇)*

Γ_{−}*: ∂t− ∇ · ((A(Φ−1(ξ)) + ( ˜A(P )− A(Φ−1(ξ)))χ−*)*∇)*

Γ0_{−}*: ∂t− ∇ · ((A(P ) + ( ˜A(P )− A(P ))χ−*)*∇)*

Γ0 *: ∂t− ∇ · (A(P )∇)*

*Γ : ∂t− ∇ · (A(x)∇).*

be thefund. sol. and corresponding operators, where*χ _{−}* is
the characteristic function of the spaceR3

**Main part of the proof**

*Decompose w(y,s)*as follows:

*w(y,s)(x, t) = E(x, t; y, s)− Γ(x, t; y, s)*
=*{E(x, t; y, s) − ΓP(x, t; y, s)} + {ΓP(x, t; y, s)− Γ−(Φ(x), t; Φ(y), s)}*
+*{Γ _{−}(Φ(x), t; Φ(y), s)− Γ*0

*+*

_{−}(Φ(x), t; Φ(y), s)}*{Γ*0

*0*

_{−}(Φ(x), t; Φ(y), s)− Γ*(Φ(x), t; Φ(y), s)}*+

*{Γ*0

*(Φ(x), t; Φ(y), s)− Γ*0

*(x, t; y, s)}*+

*{Γ*0

*(x, t; y, s)− Γ(x, t; y, s)} + +{Γ(x, t; y, s) − V(y,s)(x, t)},*

To show : *|w(y,s)(y, s′*)*| → ∞ as s′* *→ s, y → ∂D*

*Let ξ = η = Φ(x) = Φ(y)→ O (δ ↓ 0) and consider the case, for*
*example n = 3.*

**Dynamical Probe Method**

**Outline of the proof**

**Behavior of each term**

1.
lim sup
*δ→0*
*|E(x, s + ε*2_{; y, s)}_{− Γ}*P(x, s + ε*2*; y, s)| = O(εµ−3),*
*as ε→ 0.*
2.
lim sup
*δ↓0*
*|(˜Γp− Γ−)(ξ, s + ε*2*; η, s)| = O(εα−3*) as *ε→ 0.*
3.
lim sup
*δ↓0*
*|Γ−(ξ, t + ε*2*; η, s)− Γ*0*−(ξ, t + ε*2*; η, s)| = O(εµ−3*) as *ε→ 0.*

(In 1,2,3, we used a pointwise space gradient estimate for a fundamental solution of parabolic equation with disconti. coeﬀ..)

4. Put

*W (ξ, t; η, s) := Γ*0* _{−}(ξ, t; η, s)− Γ*0

*(ξ, t; η, s)*(dominant)

*Denote W (ξ, t; η, s) for±ξn* *> 0 by W±(ξ, t; η, s).*

*Then, there exist a constant C > 0 such that*
lim
*δ _{↓0}|W*
+

*2*

_{(η, s + ε}

_{; η, s)}_{| ≥ Cε}_{−3}_{as}

*5. lim sup*

_{ε}_{→ 0.}*δ*

_{↓0}*|Γ*0

*0*

_{(Φ(x), t; Φ(y), s)}_{− Γ}

_{(x, t; y, s)}_{| = 0.}*6. Let G(x, t; y, s) = Γ*0

*lim sup*

_{(x, t; y, s)}_{− Γ(x, t; y, s). Then,}*δ↓0*

*|G(y, s + ε*2

_{; y, s)}_{| = O(ε}−2_{)}

_{as}

_{ε}_{→ 0.}**Dynamical Probe Method**

**Outline of the proof**

*7. It follows from the deﬁnitions of Γ and v that*
*Γ(x, t; y, s)− V(y,s)(x, t)*

**Remark for non-separated inclusions (open question)**

The previous proof for the separated inclusions case works well except
the*estimate for W (ξ, t; η, s)*.

**Dynamical Probe Method**

**Remark for non-separated inclusions Case**

**A priori estimates and weak solutions for the**

**derivative nonlinear Schr ¨odinger equations on torus**

Hideo Takaoka

Hokkaido University

. . . .

We consider the initial value problem for a nonlinear Schr ¨odinger equation. The most basic question is whether the initial value problem is well-posed in a certain classes of data. How about if we start from data in a certain regularity class, say in Sobolev spaces

Equation

Consider the 1-d derivative nonlinear Schr ¨odinger equation:

* i*∂

**t****+**∂

**2**

_{x}**u****=**

*∂*

**i**

**x****(**|

*|*

**u****2**

**u****)**

and initial data:

**u****(0**,**x****) =ϕ(****x****)**∈ **H****s****(T)**

whereTis the periodic b.c. **T = R/2**πZand

**u****:R × T ∋ (*** t*,

**x****)**7−→

**u****(**

*,*

**t**

**x****)**∈ C

Physical model

Longwavelength dynamics of dispersive Alfv ´en waves along an ambient magnetic field (D. J. Kaup, A. C. Newell, 1978).

Our interest

How about if we start from low-regularity data? How about long-time solutions?

. . . .

Classical existence

I * Local well-posed for s* >

**3**(M.Tsutsumi, I.Fukuda, 1980).

_{2}I * For s* ≥

**1**(S. Herr, 2006).

_{2}I * Global well-posed for s* >

**1**(Yin Yin Su Win, 2010).

_{2}Techniques

Energy method, Gauge transformation, Fourier restriction norm method, I-method.

Remark

I * The follow map: Hs* ∋ ϕ 7→

*∈*

**u**

**C****([0**,

**T****]**,

**H****s****)**does not

**C****3*** -uniformly for s* <

**1**.

_{2}I Gibbs measure was constructed by L.Thomann, N.Tzvetkov
(2010). This measure is well-defined for Gaussian data of the
form∑**g****n****(**_{n}**ω)*** einx* ∈

**H****12**−ε

Related to the real line setting

Tsutsumi-Fukuda (1980), N.Hayashi-T.Ozawa (1992), Ozawa-Y.Tsustumi (1998), H.T (1999),

J.Colliander-M.Keel-G.Staffilani-H.T-T.Tao (2002). Theorem (R-case)

**The initial value problem for DNLS is locally well-posed in H****s****(R)**

* for s*≥

**1**. Furthermore, the solutions become global in time when

_{2}* s*>

**1**.

_{2}Techniques: Gauge transformation (Wadati-Sogo, 1983,

. . . .

**In the periodic b.c. case**
Herr’s gauge transformation
Let
**G(****u****)(*** t*,

**x****) =**

*−*

**e***I*

**i**

**u****(**

*,*

**t**

**x****)**

**u****(**

*,*

**t**

**x****)**where I

**f****(**

**x****) =**∫ −

*θ ∫*

**d***θ ( |*

**x**

**f****(**

**y****)**|

**2**−

**1**

**2**π∥

*∥*

**f****2**

**L****2**)

*.*

**dy****I :**

**L****2(T) ∋**

*7→ I*

**f***∈*

**f***∞*

**L****(T)**

**Write w****=**

*−*

**e***I*

**i**

**u****u, compute***∂*

**i**

**t****w****+**∂

**2**

_{x}**w****=**

**iw****2**∂

*−*

**x****w****1**

**2**|

*|*

**w****4**

_{w}

_{+}**1**

**2µ(**

**w****)**|

*|*

**w****2**

_{w}

_{− ψ(}

_{w}

_{)}*where*

_{w}**µ(**

**w****) =**∫ −|

*|*

**w****2**

**dx****=**

**µ(ϕ)**

**ψ(**

**w****) =**∫ −

**2Im(**

*∂*

**w**

_{x}**w****)**

*− ∫ −*

**dx****1**

**2**|

*|*

**w****4**

_{dx}

_{+}_{µ(}

_{w}

_{)}2Remark
Non-linear term**=****iw****2**∂* xw* −

*∫ −*

**w****2Im(**

*∂*

**w**

**x****w****)**

*| {z } Leading term*

**dx****+**(Non derivative term)

* Work in the Fourier restriction norm space Xs*,

*where*

**b**∥* f*∥

*,*

_{X}**s**

**b****=**∥⟨ξ⟩

*⟨τ − ξ*

**s****2**⟩

*b*

**b**

**f****(τ, ξ)∥**

_{l}**2**ξ

**L****2**
τ

Claim (Herr): The estimate

∫_{0}**t****e****i****(*** t*−

*′*

**t****)**∂

**2**

**x****(**Leading term

**)(**

*′*

**t****)**

*′*

**dt*** Xs*,

**b**. ∥* w*∥

**3**

* Xs*,

**b**. . . .

Theorem (S. Herr, 2006)

**The Cauchy problem of DNLS is locally well-posed in H****s**_{for s}_{≥} **1**
**2**.

Motivation

I * Scaling indicates that s* ≥

**0 is necessary.**

I **Gibbs measure for DNLS is constructed in H****s*** for s* <

**1**. (This

_{2}*<*

**measure along with LWP with s****1**gives a global existence result.)

_{2}Theorem (H.T., 2010)

* Let s*>

**4**

_{9}*>*

**. For any r**

**0, there exist T****=**

**T****(**

**r****)**>

**0 and metric**

**space Z****s**

* T* s.t. for dataϕ ∈

**H****s**_{with}_{∥ϕ∥}

* Hs* <

**r there exists a solution***∈*

**u**

**Z****s*** T* ,→

**C****([0**,

**T****]**,

**H****s**_{)}_{of DNLS. Moreover the map}_{ϕ 7→}_{u is}

Uniqueness

**Let w****1****and w****2** be solutions for DNLS with same data. Either

ξ| b**w****j****(****t****, ξ)|2** ∈ * L_{T}*∞ℓ

_{ξ}∞,

**j****=1 or 2**

or

|c**w****1(****t****, ξ)| = |c****w****2(****t****, ξ)|for 0**≤* t* ≤

**T**. . . .

Sketch of proof

I A priori estimate for∥* w*∥

*∞*

**L****T****H*** s* depending on∥

*∥*

**w***,*

**X****a****1**/

**2**

*for some*

**T***<*

**low regularity a**

**s.**I A priori estimate for∥* w*∥

*,*

_{Y}**a****1**/

**2**

* T*
depending on∥

*∥*

**w***∞*

**L**

**T****H***. Here*

**s***,*

**Y****a****1**/

**2**

*⊂*

**T**

**X***,*

**a****1**/

**2**

*is complementary spaces.*

**T**I **Compactness theorem gives the existence of solution w of*** DNLS in L*∞

**T****H**

**s**_{∩}*_{Y}a*,

**1**/

**2**

* T* .

I **Non-concentration estimate and Y****a**

* T* ,→

**C****([0**,

**T****] :**

**H****a**_{)}_{gives}

**Estimate for**∥* w*∥

*∞*

**L****T****H****s****x**

We observe, using the fundamental theorem of calculus and the
integration by parts:
∥**w****(****t****)**∥**2**
**˙**
* Hs* − ∥

**w****(0)**∥

**2**

**˙**

**H****s****=**∫

**t****0**

**d***∥*

**dt**

**w****(**

**t****)**∥

**2**

**˙**

**H****s****dt****=**

**2Re**∫

**t****0**∑ ξ

**1+**ξ

**2=0**|ξ

**1**|

*|ξ*

**s****2**|

*b*

**s****w**

**t****(**

*, ξ*

**t****1)b**

**w****(**

*, ξ*

**t****2)**

*Remark*

**dt**Fourier transform of leading non-linear term

**iw****2**∂*_{x}w* −

*−∫*

**w****2Im(**

*∂*

**w**

_{x}**w****)**

**dx is written as****1**

**2**π ∑

**ξ=ξ1+**ξ

**2+**ξ

**3**

**(**ξ

**1−ξ)(ξ2−ξ),0**ξ

**3**

*b*

**w****(**ξ

**1)**

*b*

**w****(**ξ

**2)b**

**w****(**ξ

**3) +**

**1**

**2**πξ|b

**w****(ξ)|**

**2**

_{w}_{b}

_{(}_{ξ)}. . . .

and by using the equation (eliminating no derivative nonlinearity)

**=** **2**
∫ **t****0**
∑
|ξ**1**|**2s****Re*** i*ξ

_{1}2*b*

**w****(**ξ

**1)b**

**w****(**ξ

**2)**| {z } canceled out

**+1**π

**Re**∫

**t****0**∑ ξ

**14**ξ

**24**,

**0**|ξ

**4**|

*ξ*

**2s****3**

*b*

**w****(**ξ

**1)**

*b*

**w****(**ξ

**2)b**

**w****(**ξ

**3)b**

**w****(**ξ

**4)**

**+1**π ∫

**t****0**∑

**Re**

*ξ*

**i****|ξ**

_{1}**|**

_{1}*|b*

**2s**

**w****(**ξ

**1)**|

**2**

*b*

**w****(**ξ

**1)b**

**w****(**ξ

**2)**| {z } canceled out Here∑

**=**∑

_{ξ}

_{1}_{+}_{ξ}

_{2}_{+}_{...+ξ}

_{n}**andξ**

_{=}_{0}

_{jk}**=**ξ

**j****+**ξ

*.*

**k**Then by symmetrization under the interchange of

**=** **1**
**4**π* i*
∫

**t****0**∑ ξ

**14**ξ

**24**,

**0**( ξ

**1**|ξ

**2**|

**2s****+**ξ

**2**|ξ

**1**|

**2s****+**ξ

**3**|ξ

**4**|

**2s****+**ξ

**4**|ξ

**3**|

*) ×b*

**2s**

**w****(**ξ

**1)**

*b*

**w****(**ξ

**2)b**

**w****(**ξ

**3)b**

**w****(**ξ

**4)**

We write* w*b

**(**

**t****, ξ) =**

*−*

**e***ξ*

**it****2**(

*ξ*

**e****it****2**

*b*

**w****(**

**t****, ξ)**)to achieve a smoothing effect.

With this, we write

b
**w****(**ξ**1)*** w*b

**(**ξ

**2)b**

**w****(**ξ

**3)b**

**w****(**ξ

**4) =**

*−*

**e**

**it****(**ξ

**2**

**1+**ξ

**2**

**2**−ξ

**2**

**3**−ξ

**2**

**4)**(

*ξ*

**e****it****12**

*b*

**w****(**

*, ξ*

**t****) . . .**

_{1})Expanding: underξ** _{1}+**ξ

**2+**ξ

**3+**ξ

**4**

**=0, we have**

ξ**2**
**1** **+**ξ
**2**
**2**− ξ
**2**
**3** − ξ
**2**
**4** **=2**ξ**14**ξ**24**

. . . .
we rewrite
**=** * c*∑

**M****(**ξ

**1**, ξ

**2**, ξ

**3**, ξ

**4)**[ b

**w****(**

*, ξ*

**s****1)**b

**w****(**

*, ξ*

**s****2)b**

**w****(**

*, ξ*

**s****3)b**

**w****(**

*, ξ*

**s****4)**]

**s****=**

**t**

**s****=0**

**+**

*∫*

**c**

**t****0**∑

**M****(**ξ

**1+**ξ

**2**

**+**ξ

**3**, ξ

**4**, ξ

**5**, ξ

**6)**× b

**w**_{| {z }}

**(**ξ

**1)**

*b*

**w****(**ξ

**2)**ξ

**3**

*b*

**w****(**ξ

**3)**non-resonance b

**w****(**ξ

**4)b**

**w****(**ξ

**5)b**

**w****(**ξ

**6)**

*∫*

**c**

**t****0**∑

**M****(**ξ

**1**, ξ

**2**, ξ

**3**, ξ

**4)**

_{| {z }}ξ

**1**|b

**w****(**ξ

**1)**|

**2**b

**w****(**ξ

**1)**resonance b

**w****(**ξ

**2)b**

**w****(**ξ

**3)b**

**w****(**ξ

**4)**

**+(**symmetrizing terms

**)**where

**M****(**ξ

**1**, ξ

**2**, ξ

**3**, ξ

**4) =**ξ

**1**|ξ

**2**|

**2s****+**ξ

**2**|ξ

**1**|

**2s****+**ξ

**3**|ξ

**4**|

**2s****+**ξ

**4**|ξ

**3**|

*ξ*

**2s****14**ξ

**24**

Lemma 1

Forξ** _{1}+**ξ

**2+**ξ

**3+**ξ

**4**

**=**

**0 and**ξ

**14**ξ

**24**,

**0, we have**

|**M****(**ξ**1**, ξ**2**, ξ**3**, ξ**4)**| . |ξ∗|* 2s*−

**1**where|ξ

_{∗}

**| =max**{|ξ

*|}.*

**j***−*

**Remark: 2s****1=4**(

**s****−**

_{2}**1**)

_{4}

**and H****14+**,→

**L****4**Lemma 2

*>*

**For s****4**/

**9, we have**

**ww**_{| {z }}∂

*non-resonance*

**x****w***/*

_{X}**s****2+**,−

**1**/

**2**. ∥

*∥*

**w****3**

*∞*

**L**

**t**

**H**

**s**

**x****+**∥

*∥*

**w****3**

*/2*

**X****s****+**,1/2

**We apply Lemma 1 to bound M****(. . .)**, and apply Lemma 2 to
estimate the non-resonance term via duality relationship between

. . . .
∥* w*∥

*∞*

_{L}

**T****H**

**s***. ∥*

**x**

**w****(0)**∥

**H****s****+**δ∥

*∥*

**w****4**

*∞*

**L**

**T****H***/*

**s****2+**

**x****+**

**1**δ∥

*∥*

**w****4**

*∞*

**L**

**T****L****2**

**x****+**∥

**w****(0)**∥

**4**

*/*

**H****s**_{x}**2+**

**+**

*δ∥*

**T***∥*

**w****3**

*∞*

**L**

**T****H**

**s***∥*

**x***∥*

**w****3**

*/*

**X****s****2+**,

**1**/

**2+**∥

*∥*

**w****6**

*/*

**X****s****2+**,

**1**/

**2**

**+**

*δ∥*

**T***∥*

**w****4**

*∞*

**L**

**T****H***/*

**s****2+**

*∥*

**x***∥*

**w****2**

*∞*

**L**

**T****L****2**

**x****for some 0**< δ ≪

**1.**

The estimate for∥* w*∥

*/*

_{Y}**s****2+**

**T**

.is similar, thus we have no difficulty in proving the desired apriori estimate

∥* w*∥

*∞*

_{L}

**T****H**

**s**

**x****+**∥

*∥*

**w***/*

**Y****s****2+**

*.*

**T**

**C**_{∥}

_{w}

_{(}_{0}_{)}_{∥}

**Hs**### WITH SEASONAL SUCCESSION.

Sze-Bi Hsu

National Tsing-Hua University, Hsinchu, Taiwan

Joint Work with Xiaoqiang Zhao , Memorial University of Newfoundland,Canada

19th DE workshop,Jan. 15,2011

### Classical Lotka-Volterra Two-Species Competition Model

dx1 dt = r1x1(1 − x1 K1) − αx1x2 dx2 dt = r2x2(1 − x2 K2) − βx1x2 x1(0) > 0, x2(0) > 0,Four competition outcomes

L1 : r1(1 −_{K}x1_{1}) − αx2= 0 , L2 : r2(1 −_{K}x2_{2}) − βx1 = 0

(I) Species 1 win: r1

α > K2, r2

β < K1

(III) Stable coexistence:K1 < r_{β}2, K2 < r_{α}1

(IV) Bistability:K1 > r_{β}2, K2 > r_{α}1

Let 0 < φ < 1 , be the proportion of good season,ω be the period. dx1 dt = −λ1x1 bad season dx2 dt = −λ2x2 mω ≤ t ≤ (1 − φ)ω + mω dx1 dt = r1x1(1 − x1 K1) − αx1x2 good season (1) dx2 dt = r2x2(1 − x2 K2) − βx1x2 mω + (1 − φ)ω ≤ t ≤ (m + 1)ω (x1(0), x2(0)) = (x10, x20), x10, x20 > 0

Q1:Does the periodic system have same four outcomes as the classical Lotka-Volterra model?

Q2:How do the parameters φ, λ1, λ2, change the competition

outcomes?

### Single Species Growth With Seasonal Succession:

(2) dx dt = −λx mω ≤ t ≤ (1 − φ)ω + mω dx dt = rx (1 − x K) mω + (1 − φ)ω ≤ t ≤ (m + 1)ω x (0) = x0 > 0let y (t, y0) be the unique solution of dy dt = ry (1 − y K) y (0, y0) = y0> 0 then x (t, x0) = e−λtx (mω, x0), t ∈ [mω, mω + (1 − φ)ω] x (t, x0) = y (t −[mω+(1−φ)ω], x (mω+(1−φ)ω, x0)) ∀t[mω + (1 − φ)ω, (m + 1)ω]

Let x (t, x0) be the unique solution of (2).Then

(i) If r φ − λ(1 − φ) ≤ 0 then lim

t→∞x (t, x0) = 0 for all x0 > 0.

(ii) If r φ − λ(1 − φ) > 0 then (2) admits a unique positive ω -periodic solution x∗(t) and lim

t→∞(x (t, x0) − x

∗_{(t)) = 0}

pf:

Consider period ω − map associated with (2)
M(x0) = x (ω, x0) = y (φω, e−λ(1−φ)ωx0)
M0(0) = er φω· e−λ(1−φ)ω= e(r φ−λ(1−φ))ω,
|M0(0)| < 1 ⇐⇒ r φ − λ(1 − φ) < 0
y (t, y0) = _{(K −y}_{0}Ky_{)e}0−rt_{+y}
0
M(x0) = Kx0e
−λ(1−φ)ω
(K −x0e−λ(1−φ)ω)e−r φω+x0e−λ(1−φ)ω