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 ∈ R3space, t time, ξ ∈ R3micorsocopic velocity. Macroscopic variables ( ρ(x, t) ≡R R3f (x, t, ξ)d ξ, density, ρv (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 +12ρ|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 · ∂xf = 1 kQ(f , f ). Transport: ∂tf + ξ · ∂xf Collision operator: Q(f , f )(ξ) ≡R R3 R S2 +[f (ξ 0)f (ξ0 ∗) −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 · ∂xf = 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∗ f0f0 ∗ [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)|22Rθ(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(ρv ) + ∂x· (ρv × v + P) = 0, momentum,
∂t(ρE ) + ∂x· (ρv E + Pv + q) = 0, energy.
1 More (14) unknowns than (5) equations. Need to knowthe
dependence on microscopic velocityto compute the stress
tensorP and heat flux q.
2 Boltzmann equation is equivalent to a system ofinfinite
PDEs.
Boltzmann equation
∂tf + ξ · ∂xf =
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 theEuler equations
in gas dynamics: ∂tρ + ∂x · (ρv ) = 0, ∂t(ρv ) + ∂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 = (ξ1, 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 manifoldN in L∞ξ,3isinvariantfor (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 = 1κ(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, {λ1, 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, Bik ≡ (Ei,k)Ei, 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 ≡ S0+(L∞ξ,3), C ≡ ˜P0(L∞ξ,3), U ≡ U0−(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 ≡ −R0∞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 ) = φeη()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
˚S0+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 ˜ B0jb + B03b + (ξ 1E0 3,P01b) (ξ1E0 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 manifoldsMu, M+, ( < 0); Ms,M−, ( > 0):
the nonlinear unstable manifold, the center-stable manifold, (supersonic); the nonlinear stable manifold, the nonlinear center-unstable, (subsonic) defined as graphs
Fu :Range(U],0−) 7−→Range 2 X j=1 ˜ Bj +B],3 + S],0+ , <0 Fs :Range(S],0+) 7−→Range U ], 0−+ 2 X j=1 ˜ Bj +B],3 , >0, F−:Range U ], 0−+ 2 X j=1 ˜ Bj +B],3 7−→ Range S],0+ , >0, F+:Range 2 X j=1 ˜ Bj +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
qrM
Sone Manifold Bifurcation Manifold 1 C C2 C3 4 C 5 C 1 A 2 A 3 A 4 A A5 0 0 M , M M M M+q knus + + +q− q r r rM
0 0 M M M M M−q s −+ −q− q r r rM
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)2of 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 : C2.
γ(x) = (γjk(x)) : defineda.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)∈ Cµ(D m) (0 < µ < 1), γ(x) = γ(m)(x) (x∈ Dm).Eachseparated Dm is ofC1,α smoothwith 0 < α≤ 1 and
non-separated one is the limit of the separated one.
D1 D
Important Preliminary Estimates
Gradient estimate of solutions for parabolic equations
Gradient estimate
.
Theorem 1 (Fan, Kim, Nagayasu and N)
. .
.
LetΩ′ b Ω, 0 < t0< T. Any sol u to(P): ∂tu− ∇ · (γ∇u) = 0 in
Ω× (0, T )has the following interior regularity est: supt
0<t<T∥u(·, t)∥C1,α′(Ω′∩Dm)≤ C∥u∥L2(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+12 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 < t0< T. Any sol u to
∂tu− ∇ · (A∇u) = 0 inΩ× (0, T ) =: Q has the following estimates:
sup t0<t<T
∥u(·, t)∥L2(eΩ)≤ C∥u∥L2(Q) (standard), ∥u∥L∞(eΩ×(t0,T ))≤ C∥u∥L2(Q) (Di Giorgi’s arg.), ∥ut∥L2(eΩ×(t 0,T ))≤ C∥u∥L2(Q) ([LRU]). . Remark 3 . . .
(i)This lemma holds forA∈ L∞.
Proof
Let eΩ3b eΩ2b eΩ1b eΩ0:= Ω, 0 < δ1< δ2< T. Then
(∗) supδ2<t<T∥u(·, t)∥L2(eΩ2)≤ C∥u∥L2(Q),
(∗∗) ∥ut∥L2(eΩ
1×(δ1,T ))≤ C∥u∥L2(Q).
Since ∂tut− ∇ · (A∇ut) = 0, we have
(∗ ∗ ∗) ∥ut∥L∞(eΩ
2×(δ2,T ))≤ C∥ut∥L2(eΩ1×(δ1,T ))≤ C ′∥u∥
L2(Q).
Important Preliminary Estimates
About the proof
Proof
Then by [LN], we have ∥u(·, t)∥C1,α′(D m∩eΩ3) ≤ C(∥u(·, t)∥L2(eΩ 2)+∥ut(·, t)∥L∞(eΩ2) ) . Taking supδ 2<t<T, we have by (*), (**), (***), supδ 2<t<T∥u(·, t)∥C1,α′(Dm∩eΩ3) ≤ C(supδ 2<t<T∥u(·, t)∥L2(eΩ2)+∥ut∥L∞(eΩ2×(δ2,T )) ) ≤ C∥u∥L2(Q).Dynamical Probe Method Active thermography
Active thermography
D
∂
νu|
∂Ω= f
u(f )|
∂ΩΩ
Principle of active thermography
infrared camera heater / flash lam p inclusionDynamical Probe Method
Forward problem
Mixed problem (set up)
Ω⊂ Rn (1≤ n ≤ 3) : bounded domain,
∂Ω : C2(n = 2, 3), ∂Ω = ΓD∪ ΓN,
where ΓD, ΓN are open subsets of ∂Ω such that ΓD∩ ΓN =∅ and ∂ΓD, ∂ΓN are C2 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,
whereA, ˜A∈ C1(Ω)are positive definite and ˜A− A is always positive
Hp(∂Ω), Hp,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 <∞ L2((0, T ); Hp(∂Ω)) :={f ; ∫0T||f(·, t)||2Hp(∂Ω)dt <∞}
Dynamical Probe Method
Forward problem
Mixed problem (forward problem)
Given f∈ L2((0, T ); H12(ΓD)), g∈ L2((0, T ); ˙H−1 2(ΓN)), (?)∃! weak solution u = u(f, g)∈ W (ΩT) :={u ∈ H1,0(ΩT), ∂tu∈ L2((0, T ); H1(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(ΓD), ˙H−1
2(ΓN) are H¨ormander’s notations of Sobolev sp,
ΩT = Ω(0,T ):= Ω× (0, T ), ∂ΩT = ∂Ω(0,T ):= ∂Ω× (0, T ).
(cylindrical sets)
Measured data
Neumann-to-Dirichlet mapΛD:
Forfixed f∈ L2((0, T ); H12(ΓD)), define ΛD: L2((0, T ); ˙H− 1 2(ΓN))→ L2((0, T ); H 1 2(ΓN)) g7→ u(f, g)|ΓN 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 ofP∅∗:=−∂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 ΩT, G∗(·, ·; y, s′) = 0 on ΓD 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∞(ΩT).
Dynamical probe method (Runge’s approximation)
∃{v0j (y,s)},{φ 0j (y′,s′)}∈ H 2,1(Ω (−ε,T +ε)) for∀ε > 0 s.t. P∅v0j(y,s)= 0 in Ω(−ε,T +ε), v(y,s)0j = 0 on ΓD× (−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,s0j ′)= 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 : opens.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)
Letv, ψsatisfy 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→∞ ∫ ΓN T [ ∂νv j (y,s)|ΓN T φ j (y′,s′)|ΓN T −ΛD(∂νv j (y,s))|ΓN T ∂νφ j (y′s′)|ΓN 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(y,s):= u(f, ∂Avj(y,s)|ΓN
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)= divx(( ˜A− A)χD∇xV(y,s)) in ΩT, w(y,s)= 0 on ΓDT, ∂Aw(y,s)= 0 on ΓNT 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 C1curve 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↓0lim supδ↓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) IfΓD̸= ∅ 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)= divx(( ˜A− A)χD∇xV(y,s)) in ΩT, w(y,s)= 0 on ΓDT, ∂Aw(y,s)= 0 on ΓNT w(y,s)(x, 0) = 0 for x∈ Ω.
(ii)
I(y, s′; y, s) =−w(y,s)(y, s′)−
∫
∂ΩT
w(y,s)∂νΓ∗(·, ·; 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)= divx(( ˜A− A)χD∇xV(y,s)) in ΩT
Hence,
E(x, t; y, s):= w(y,s)(x, t) + V(y,s)(x, t)
(⇒fundamental solution forPD.) LetP = c(λ0)∈ ∂D for some λ0
x = y = c(λ0− δ) ∈ C \ D for δ > 0.
Φ :R3→ R3 with Φ(P ) = O (C1,α diffeomorphism, 0 < α≤ 1),
Φ(D)⊂ R3
− ={ξ = (ξ1, ξ2, ξ3)∈ R3; ξ3< 0},
Dynamical Probe Method
Outline of the proof
Let
E : ∂t− ∇ · ((A(x) + ( ˜A(x)− A(x))χD)∇)
ΓP : ∂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, whereχ− 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)− Γ0−(Φ(x), t; Φ(y), s)} +{Γ0−(Φ(x), t; Φ(y), s)− Γ0(Φ(x), t; Φ(y), s)} +{Γ0(Φ(x), t; Φ(y), s)− Γ0(x, t; y, s)} +{Γ0(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 + ε2; y, s)− Γ P(x, s + ε2; y, s)| = O(εµ−3), as ε→ 0. 2. lim sup δ↓0 |(˜Γp− Γ−)(ξ, s + ε2; η, s)| = O(εα−3) as ε→ 0. 3. lim sup δ↓0 |Γ−(ξ, t + ε2; η, s)− Γ0−(ξ, t + ε2; 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) := Γ0−(ξ, t; η, s)− Γ0(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 +(η, s + ε2; η, s)| ≥ Cε−3 as ε→ 0. 5. lim sup δ↓0 |Γ 0(Φ(x), t; Φ(y), s)− Γ0(x, t; y, s)| = 0. 6. Let G(x, t; y, s) = Γ0(x, t; y, s)− Γ(x, t; y, s). Then, lim sup δ↓0 |G(y, s + ε2; 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 theestimate 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 +∂2xu =i∂x(|u|2u)
and initial data:
u(0,x) =ϕ(x)∈ Hs(T)
whereTis the periodic b.c. T = R/2πZand
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 > 32 (M.Tsutsumi, I.Fukuda, 1980).
I For s ≥ 12 (S. Herr, 2006).
I Global well-posed for s > 12 (Yin Yin Su Win, 2010).
Techniques
Energy method, Gauge transformation, Fourier restriction norm method, I-method.
Remark
I The follow map: Hs ∋ ϕ 7→u ∈ C([0,T],Hs)does not
C3-uniformly for s < 12.
I Gibbs measure was constructed by L.Thomann, N.Tzvetkov (2010). This measure is well-defined for Gaussian data of the form∑gn(nω)einx ∈ H12−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 Hs(R)
for s≥ 12. Furthermore, the solutions become global in time when
s> 12.
Techniques: Gauge transformation (Wadati-Sogo, 1983,
. . . .
In the periodic b.c. case Herr’s gauge transformation Let G(u)(t,x) =e−iIu(t,x)u(t,x) where If(x) = ∫ −dθ ∫ x θ ( |f(y)|2− 1 2π∥f∥ 2 L2 ) dy. I : L2(T) ∋f 7→ If ∈L∞(T) Write w =e−iIuu, compute i∂tw+∂2xw =iw2∂xw − 1 2|w| 4w + 1 2µ(w)|w| 2w − ψ(w)w where µ(w) = ∫ −|w|2dx = µ(ϕ) ψ(w) = ∫ −2Im(w∂xw)dx − ∫ −1 2|w| 4dx +µ(w)2
Remark Non-linear term=iw2∂xw −w ∫ −2Im(w∂xw)dx | {z } Leading term
+(Non derivative term)
Work in the Fourier restriction norm space Xs,b where
∥f∥Xs,b =∥⟨ξ⟩s⟨τ − ξ2⟩bbf(τ, ξ)∥l2 ξL
2 τ
Claim (Herr): The estimate
∫0tei(t−t′)∂2x(Leading term)(t′)dt′
Xs,b
. ∥w∥3
Xs,b
. . . .
Theorem (S. Herr, 2006)
The Cauchy problem of DNLS is locally well-posed in Hs for s≥ 1 2.
Motivation
I Scaling indicates that s ≥0 is necessary.
I Gibbs measure for DNLS is constructed in Hs for s < 12. (This measure along with LWP with s < 12 gives a global existence result.)
Theorem (H.T., 2010)
Let s> 49. For any r >0, there exist T = T(r)>0 and metric
space Zs
T s.t. for dataϕ ∈ H
s with∥ϕ∥
Hs < r there exists a solution u ∈Zs
T ,→C([0,T], H
s)of DNLS. Moreover the mapϕ 7→u is
Uniqueness
Let w1and w2 be solutions for DNLS with same data. Either
ξ| bwj(t, ξ)|2 ∈ LT∞ℓξ∞, j =1 or 2
or
|cw1(t, ξ)| = |cw2(t, ξ)|for 0≤t ≤T
. . . .
Sketch of proof
I A priori estimate for∥w∥L∞
TH s depending on∥w∥ Xa,1/2 T for some low regularity a <s.
I A priori estimate for∥w∥Ya,1/2
T depending on∥w∥L∞ TH s. Here Ya,1/2 T ⊂X a,1/2 T is complementary spaces.
I Compactness theorem gives the existence of solution w of DNLS in L∞
T H
s ∩Ya,1/2
T .
I Non-concentration estimate and Ya
T ,→C([0,T] :H
a)gives
Estimate for∥w∥L∞
TH s x
We observe, using the fundamental theorem of calculus and the integration by parts: ∥w(t)∥2 ˙ Hs − ∥w(0)∥ 2 ˙ Hs = ∫ t 0 d dt∥w(t)∥ 2 ˙ Hsdt = 2Re ∫ t 0 ∑ ξ1+ξ2=0 |ξ1|s|ξ2|swbt(t, ξ1)bw(t, ξ2)dt Remark
Fourier transform of leading non-linear term
iw2∂xw −w−∫2Im(w∂xw)dx is written as 1 2π ∑ ξ=ξ1+ξ2+ξ3 (ξ1−ξ)(ξ2−ξ),0 ξ3wb(ξ1)wb(ξ2)bw(ξ3) + 1 2πξ|bw(ξ)| 2wb(ξ)
. . . .
and by using the equation (eliminating no derivative nonlinearity)
= 2 ∫ t 0 ∑ |ξ1|2sReiξ12wb(ξ1)bw(ξ2) | {z } canceled out +1 πRe ∫ t 0 ∑ ξ14ξ24,0 |ξ4|2sξ3wb(ξ1)wb(ξ2)bw(ξ3)bw(ξ4) +1 π ∫ t 0 ∑ Reiξ1|ξ1|2s|bw(ξ1)|2wb(ξ1)bw(ξ2) | {z } canceled out Here∑=∑ξ1+ξ2+...+ξn=0andξjk =ξj +ξk.
Then by symmetrization under the interchange of
= 1 4πi ∫ t 0 ∑ ξ14ξ24,0 ( ξ1|ξ2|2s +ξ2|ξ1|2s +ξ3|ξ4|2s +ξ4|ξ3|2s ) ×bw(ξ1)wb(ξ2)bw(ξ3)bw(ξ4)
We writewb(t, ξ) =e−itξ2(eitξ2wb(t, ξ))to achieve a smoothing effect.
With this, we write
b w(ξ1)wb(ξ2)bw(ξ3)bw(ξ4) =e−it(ξ 2 1+ξ 2 2−ξ 2 3−ξ 2 4) ( eitξ12wb(t, ξ1) ) . . .
Expanding: underξ1+ξ2+ξ3+ξ4 =0, we have
ξ2 1 +ξ 2 2− ξ 2 3 − ξ 2 4 =2ξ14ξ24
. . . . we rewrite = c∑M(ξ1, ξ2, ξ3, ξ4) [ b w(s, ξ1)bw(s, ξ2)bw(s, ξ3)bw(s, ξ4) ]s=t s=0 +c ∫ t 0 ∑ M(ξ1+ξ2 +ξ3, ξ4, ξ5, ξ6) × bw| {z }(ξ1)wb(ξ2)ξ3wb(ξ3) non-resonance b w(ξ4)bw(ξ5)bw(ξ6) c ∫ t 0 ∑ M(ξ1, ξ2, ξ3, ξ4)| {z }ξ1|bw(ξ1)|2bw(ξ1) resonance b w(ξ2)bw(ξ3)bw(ξ4) +(symmetrizing terms) where M(ξ1, ξ2, ξ3, ξ4) = ξ1|ξ2|2s+ξ2|ξ1|2s+ξ3|ξ4|2s+ξ4|ξ3|2s ξ14ξ24
Lemma 1
Forξ1+ξ2+ξ3+ξ4 = 0 andξ14ξ24 ,0, we have
|M(ξ1, ξ2, ξ3, ξ4)| . |ξ∗|2s−1 where|ξ∗| =max{|ξj|}. Remark: 2s−1=4(s2 − 14)and H14+ ,→L4 Lemma 2 For s> 4/9, we have ww| {z }∂xw non-resonance Xs/2+,−1/2 . ∥w∥3 L∞ t H s x +∥w∥3 Xs/2+,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∞ TH s x . ∥w(0)∥Hs +δ∥w∥ 4 L∞ TH s/2+ x + 1 δ∥w∥ 4 L∞ TL 2 x +∥w(0)∥4 Hsx/2+ +Tδ∥w∥3 L∞ TH s x ∥w∥3 Xs/2+,1/2+∥w∥ 6 Xs/2+,1/2 +Tδ∥w∥4 L∞ TH s/2+ x ∥w∥2 L∞ TL 2 x for some 0< δ ≪1.
The estimate for∥w∥Ys/2+
T
.is similar, thus we have no difficulty in proving the desired apriori estimate
∥w∥L∞ TH s x +∥w∥Ys/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 −Kx11) − αx2= 0 , L2 : r2(1 −Kx22) − βx1 = 0
(I) Species 1 win: r1
α > K2, r2
β < K1
(III) Stable coexistence:K1 < rβ2, K2 < rα1
(IV) Bistability:K1 > rβ2, K2 > rα1
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 −y0Ky)e0−rt+y 0 M(x0) = Kx0e −λ(1−φ)ω (K −x0e−λ(1−φ)ω)e−r φω+x0e−λ(1−φ)ω
