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Some Recent Investigations on Interfacial Propagation in Inhomogeneous Medium

Aaron N. K. Yip Department of Mathematics Purdue University

Some Recent Investigations on Interfacial Propagation in Inhomogeneous Medium – p.1/56

Inhomogeneous Motion by Mean Curvature IMMC: Vn = κ + f(p) + F

V = + f(p) + F κ

N

Γ( ) t ν V( , f, F) ν

Vn = Normal Velocity, κ = Mean Curvature f(p) = Background Heterogeneity, F = External Forcing V (ν, f, F) = Effective Normal Velocity

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Some Physical Applications

κ: reduction of surface energy f(p): impurities or defects in the background (f(·) depends on the actual location of the interface.) F: external control parameter

• Grain Boundary Motion, Surface Growth
• Dislocation Lines
• Fluid Contact Lines
• Vortex Filaments in Super-Conductivity
• Biological Growth
• D. S. Fisher: Physics Reports, Vol 301(1999), 113-150
• M. Kardar: Physics Reports, Vol 301(1999), 85-112

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Two Main Types of Results of Interests

Effective Property and Homogenization

V = + f(p) + F κ

N

Γ( ) t ν V( , f, F) ν

Existence, uniqueness, and stability properties of V (ν, f, F) as a function of the background inhomo- geneity (f) and external control parameter (F).

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Two Main Types of Result - Cont’ Critical Phenomena – Transition between pinned and propagation states

F V Pinned Propagation F

*

V ∼ C(F − F∗)α, F > F∗. Properties and characterization of the threshold forcing F∗ and the critical exponent α.

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Dynamics on Heterogeneous Landscape

* u

E (u)

ε

E (u)

*

E ( . ) E ( . )

ε

duǫ dt = −∇Eǫ(uǫ)(?) =

⇒ (?)du

dt = −∇E∗(u)

• ǫ = length scale of inhomogeneity
• The presence of local minima and metastable states.

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Outline of Talk

• One Dimensional Example:

Pinning and De-Pinning Transition

• Extension to Semilinear PDE Case (Dirr-Y.)
• Effective Interface and Its Propagation (IMMC)

(Dirr-Karali-Y.)

• Pinning Threshold
• Contact Line Dynamics
• Interface between Patterns
• Random Walk in Random Medium

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An Illustrative One Dimensional Example dx dt = − cos(x) + F

• For 0 ≤ F ≤ 1, the solution get pinned.
• For F > 1, there exists continued propagation.

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Behavior Near F ∼ 1+

dx dt = 1 − cos(x) + F − 1 ≈ x2 2 + o(x2) + δ (near x ≈ 0) where δ = F − 1. Let 0 < η ≪ 1 be fixed, independent of δ. T = 2π dx 1 − cos(x) + δ = η dx

x2 2 + δ +

2π−η

η

dx 1 − cos(x) + δ + 2π

2π−η

dx

(x−2π)2 2

+ δ ≈ O(δ− 1

2)

V = 1 T ≈ δ

1 2 = (F − 1) 1 2

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Discrete Allen-Cahn Equation

dui dt = ui+1 − 2ui + ui−1 − A (W ′(ui) − g) where W is of bistable type: W(u) = (1 − u2)2. The above equation and behavior also appears in the modeling of charge density waves, dissipative Frenkel-Kantorova and many

• thers:

dui dt = ui+1 − 2ui + ui−1 + A sin(ui) + B Compared with the continuum case: ut = uxx − W ′(u) + F which always a travelling wave solution: u(x, t) = U(x − ct), for A ≫ 1, the discrete system exhibits the presence of propagation failure, i.e. the existence of stationary solutions (Keener; Bonilla-Carpio).

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Extension to PDE Setting: Bifurcation Theory

dX dt = N(X) + F

• For 0 < F < F∗, there exists stable stationary solution:

N(XF) + F = 0

• At F = F∗, the stationary solution XF∗ becomes unstable.

Perform center-manifold analysis of the dynamical equation: X(t) = s0(t)Φ0 +

• n≥1

sn(t)Φn so that: ds0(t) dt ≈ D2N(XF∗)Φ0, Φ0 Φ0, Φ0 s2

0(t) + F − F∗, Φ0

Φ0, Φ0

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Pinning and De-Pinning Transition. I Semi-Linear Equation for Linearized IMMC:

ut = △u + f(x, u) + F

u(x,t) x u(x,t)

• f(·, ·) is 1-periodic in x and u.
• At F = 0, the above eqn has a stable stationary solution.
• At F = F∗ – the threshold value, the above eqn. is

non-degenerate.

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Semi-Linear Equation - Cont’

(Dirr-Y.) There exists an F∗ > 0 such that

• For 0 ≤ F ≤ F∗, there exists pinned states:

△u + f(x, u) + F = 0

• For F > F∗, there exists unique pulsating wave –

space-time periodic solution U(x, t): U(·, · + TF) = U(·, ·) + 1

• Scaling of the velocity:

V = 1 TF ∼ C(F − F∗)

1 2,

for 0 ≤ F − F∗ ≪ 1 (Results expected to be true also for IMMC for rational direction.)

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Pinning and De-Pinning Transition. II

Reaction-Diffusion Equation for Front Propagation: ϕt = ϕxx − W ′(ϕ)

2

+ δ (g(x) + F) W(ϕ) = (1 − ϕ2)2, g(·) is 1-periodic and δ ≪ 1.

ϕ

location of front x (x) ~ 1 (x) ~ −1

ϕ

(The above Allen-Cahn or Ginzburg-Landau type equation is commonly used in the modeling of phase boundary motion.)

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Reaction-Diffusion Equation- Cont’

(Dirr-Y.) There exists an F∗ > 0 such that

• For 0 ≤ F ≤ F∗, there exists pinned states:

ϕxx − W ′(ϕ) 2 + δ(g(x) + F) = 0

• For F > F∗, there exists pulsating wave – space-time

periodic solution Φ(x, t): Φ(x, t + TF) = Φ(x − 1, t)

• Scaling of the velocity:

V = 1 TF ∼ C(F − F∗)

1 2,

for δ ≪ F − F∗ ≪ 1 (The F∗ and C can be computed asymptotically.)

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Fully Nonlinear IMMC:

ν u(x,t) x Γ( ) t

where Γ(t) = {(x, u(x, t)) : x ∈ Rn, t ∈ R+} and u satisfies: ∂u ∂t =

• 1 + |∇u|2
• div
• ∇u
• 1 + |∇u|2
• + δf(x, u)
• a degenerate quasilinear parabolic equation

– graph might not stay as a graph.

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Statement of Result for IMMC

Effective Description of Propagation (Dirr-Karali-Y.) Vn = κ + δf Let f(·) be periodic in the spatial variable, and δ small enough, i.e. weak heterogeneity, then for any effective normal direction ν, we have

• uniform space-time oscillation and gradient bounds for the

evolving interface in an appropriate moving frame

• existence and uniqueness of effective speed of propagation
• Lipschitz continuity of speed w.r.t. the normal direction
• existence, uniqueness, and stability of the pulsating waves

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Uniform Space Time Oscillation and Gradient Bound

• sc(u(t)) =

sup

x∈Rn u(x, t) − inf x∈Rn u(x, t)

• sup

t∈R+

• sc(u(t)) ≤ C(n, δ, f) < ∞

∇uL∞(Rn×R+) ≤ C(n, δ, f) < ∞

• The oscillation bound implies that the evolving surface lies

within finite distance from a moving hyper-plane. The notion of effective front is well-defined.

• The gradient bound implies that the graph representation is

valid for all times and the equation is uniformly parabolic.

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Statement of Result for IMMC - Cont’

Pulsating Wave (cν = 0): Γ(t) satisfies: Γ(t + τ) = Γ(t) + z, for all z ∈ Zn+1, τ = ν · z cν

z t+ t z t Γ( τ) = Γ( ) + Γ( ) ν

Existence, Uniqueness, Stability Properties

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Proof of Uniform Oscillation and Gradient Bounds

Birkhoff Property – reduction to finite domain consideration: Let Γ(0) be a hyper-plane with normal ν. If a unit cube Q is above(below) Γ(t), so is any “tangential” translation of Q.

t ν Γ( )

The above property implies:

• scRn(Γ(t)) ≤ A(n)osc[0,5]n(Γ(t)) + B(n)

where oscΩ(Γ)) = supp,q∈Γ∩Ω {(p − q) · ν}.

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Proof of Uniform Oscillation and Gradient Bounds - Cont’

• I. Bernstein’s Technique – Bound on Gradient in terms of

Oscillation: sup

t∈[0,T]

∇u(t)∞ ≤ ∇u(0)∞ + sup

t∈[0,T]

λ(δ, f)osc(u(t)) where λ ∼ δ

1 2.

• II. Birkhoff Property – Bound on Oscillation in terms of

Gradient:

• sc(u(t)) ≤ Aosc[0,5]n(u(t)) + B ≤ A ∇u(t)∞ + B

The above imply for δ small enough that: sup

t∈[0,∞)

∇u(t)∞ ≤ C(n, f) and sup

t∈[0,∞)

• sc(u(t)) ≤ C(n, f)

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Related Results in the Stationary Case Planelike Minimizers for Minimal Surfaces in Periodic Media Caffarelli-De la Llave: the existence of an energy minimizing surface in a periodic inhomogeneous medium which satisfies the Birkhoff Property and the Uniform Oscillation Bound. E(S) =

• S

F(p, ν) dσ(p) (F represents the background metric.)

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Related Results in the Stationary Case – Cont’ Planelike Minimizers for Reaction-Diffusion Equations in Periodic Media (E. Valdinoci) E(u) =

• aij(x)∂iu∂ju + Q(x)χ(−1,1)(u)
• r

E(u) =

• aij(x)∂iu∂ju + F(x, u)

Birkhoff Property, Lower Density Estimate and Flatness of Interface.

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Recent Results on Homogenization

∂u ∂t = |∇u|

• ǫdiv

∇u |∇u|

• + f

x ǫ

• Caffarelli-Souganidis-Wang: fully nonlinear

uniformly elliptic/parabolic equations, in periodic and stationary random medium

• Bhattacharya-Craciun: 1st-order Hamilton-Jacobi

and 2nd order curvature flow under the assumption of the existence of solution of cell problem with sufficient regularity property

• Lions-Souganidis: MMC in periodic and

stationary random medium, with positive forcing

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Necessity of Weak Forcing: δ ≪ O(1)

Pinch-Off Phenomena

t Γ( )

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Necessity of Weak Forcing: δ ≪ O(1) - Cont’ Fingering Phenomena

t

• sc(u(t))

Γ( )

Loss of finite oscillation bound. Definition of a front?

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Necessity of Weak Forcing: δ ≪ O(1) - Cont’ Example in Laminate Environments Using Translation Invariant Solutions – grim-reapers:

Effective Front?

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A Recent Work of Cardaliague-Lions-Souganidis    uǫ

t = F

• ǫD2u, Duǫ, x

ǫ

• ,

in RN × [0, ∞) uǫ = u0

• n RN × {0}

Three scenerios are classified:

• Homogenization takes place if uǫ −

→ ¯ u in C(RN × (0, ∞)) with ¯ u solves ¯ ut = ¯ F(¯ u).

• Trapping(pinning) occurs if uǫ −

→ u0 in C(RN × (0, ∞)).

• Homogenization does not take place if for (y, s) → (x, t),

ǫ → 0, lim sup uǫ(y, s) = lim inf uǫ(y, s). There are essential the two dichotomy for conditions: Smallness or Positivity of the Inhomgeneous Forcing

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Pinning Threshold 2D Discrete Allen-Cahn Equation:

(Cahn–Mallet-Paret–Van Vleck) dui,j dt = α  

• (x,y)∈N.N.(i,j)

ux,y − 4ui,j   − f(ui,j) where f(u) is given by: (and a = 1

2 gives the bias or

the tilt of the double well po- tential)

1 f(u) a u

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2D Discrete Allen-Cahn Equation-Cont’ Look for travelling wave solution of the form ui,j(t) = ϕ(iκ + jσ − ct), κ = cos θ, σ = sin θ Then ϕ(·) satisfies: −cϕ′(ξ) = α

• ϕ(ξ+κ)+ϕ(ξ−κ)+ϕ(ξ+σ)+ϕ(ξ−σ)

− 4ϕ(ξ)

• − f(ϕ(ξ))

where ϕ(ξ) − → 0 as ξ − → −∞ and ϕ(ξ) − → 1 as ξ − → ∞.

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Solution Formula — Fourier Transform

ˆ ϕ(s) = ∞

−∞

e−isξϕ(ξ), dξ, for −1 ≪ Ims < 0. The solution is given by: ϕ(ξ) = 1 2 + 1 π ∞ A(s) sin(sξ) A2(s) + c2s2ds + c π ∞ A(s) cos(sξ) A2(s) + c2s2ds where A(s) = 1 + 2α [2 − cos κs − cos σs] and ϕ(ξ) < (>) a for ξ < (>) 0. which gives the relationship between the speed c and the bias a

• f the potential:

a − 1 2 = c π ∞ ds A(s)2 + c2s2 := Γθ(c)

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Solution Formula — Pinning Threshold

θ

γ(θ) c Γ ( ) c lim

c→∞ Γθ(c) = 1

2 lim

c→0+ Γθ(c) = γ(θ) (= a∗ = Pinning Threshold)

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Directional Dependence of the Pinning Threshold The function γ(θ) is

• continuous and equal to a constant at

irrational direction: γ(θ) = β0,0 2 ;

• discontinous at rational direction:

γ(θ) = 1 2

• (m,n):mκ+nσ=0

βm,n 2 .

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Asymptotic Computation of Pinning Threshold (Dirr-Y.) Consider the linearized version of IMMC: ut = uxx + δ sin u sin x + f. where u is the graph of the interface. Its rotated version is: ut = uxx + δ sin(au + bx) sin(bu − ax) + f where a2 + b2 = 1. Consider the regime: 0 < δ ≪ 1 u(x) = c + δg(x) + δ2h(x) + δ3k(x) + · · ·

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Asymptotic Computation — Cont’

The expression N(u) = uxx + δ sin(au + bx) sin(bu − ax) + f equals: δgxx + δ2hxx + δ3kxx +δ

• sin(ac + bx) + cos(ac + bx)[aδg + aδ2h + · · · ]

− sin(ac+bx)

2

[aδg + aδ2h + · · · ]2 × ×

• sin(bc − ax) + cos(bc − ax)[aδg + aδ2h + · · · ]

− sin(bc−ax)

2

[aδg + aδ2h + · · · ]2 +f Try to find a constant c and bounded functions g, h, k that N(u) = 0, i.e. pinning states.

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Asymptotic Computation — O(δ)

gxx + 1 2

• cos
• (a−b)c+(a+b)x
• −cos
• (a+b)c+(b−a)x
• +

f δ

• ≈ 0
• If |a| = |b|, i.e. ±45◦, then for example

gxx = −1 2 cos

• 2bx
• + 1

2 cos

• 2bc
• + f

δ so that the pinning threshold f∗ ∼ δ 2.

• If |a| = |b|, then the pinning threshold f∗ ≪ δ and

g = cos

• (a − b)c + (a + b)x
• 2(a + b)2

− cos

• (a + b)c + (b − a)x
• 2(a − b)2

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Asymptotic Computation — O(δ2)

hxx+

• b sin(ac+bx) cos(bc−ax)+a sin(bc−ax) cos(ac+bx)
• g

+ f δ2 ≈ 0 so that hxx + 1 8

• 1

a + b − 1 b − a

• sin(2ac + 2bx)

+ 1 8

• 1

a + b + 1 b − a

• sin(2bc − 2ax)

+ b − a 8(a + b)2 sin

• 2(a − b)c + 2(a + b)x
• −

a + b 8(a − b)2 sin

• 2(a + b)c + 2(b − a)x
• + f

δ2 ≈ 0

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Asymptotic Computation — O(δ2) and O(δ3) Hence, we have:

• If a = 0 or b = 0, then f∗ = Cδ2;
• If a, b = 0, then f∗ ≪ δ2.

Similar computation at the O(δ3)-level: leads to

• If 3|a| = |b| or 3|b| = |a|, then f∗ = Cδ3;
• otherwise, f∗ ≪ δ3.

......

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Asymptotic Computation – Full Version F(x, u) = δ

• m,n

Am,nei(mx+nu) + f where A−m,−n = Am,n. Consider again the rotated version: F(au + bx, bu − ax) = δ

• m,n

Am,nei

• m(au+bx)+n(bu−ax)
• + f

and u = δg + δ2h + δ3k + · · ·

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Asymptotic Computation – Full Version O(δ): gxx +

• m,n

Am,nei(mb−na)xei(ma+nb)c + f δ = 0? For mb − na = 0, i.e. b/a irrational, then g(x) =

• m,n

Am,n (mb − na)2ei(mb−na)xei(ma+nb)c and f∗ ≪ O(δ).

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Asymptotic Computation – Full Version

O(δ2): hxx + i

• m,m′,n,n′

i(ma + nb) (m′b − n′a)2Am,nAm′,n′× × ei(ma+nb)cei(m′a+n′b)cei(mb−na)xei(m′b−n′a)x + f δ2 = 0? For (m + m′)b − (n + n′)a = 0, i.e. b/a irrational, then h(x) = i

• m+m′,n+n′=0

i(ma + nb) (m′b − n′a)2Am,nAm′,n′× × ei

• (m+m′)a+(n+n′)b
• cei
• (m+m′)b−(n+n′)a
• x
• (m + m′)b − (n + n′)a

2 and f∗ ≪ O(δ2).

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Contact Line Dynamics

△u =

• n {u > 0};

ut = |∇u|

• |∇u| − g

x ǫ

• n ∂ {u > 0}

with g(·) 1-periodic.

• (Kim) Homogenization of free boundary velocity:

as ǫ − → 0: Vn = F(∇u).

• (Grunewald-Kim) Use of a gradient flow structure:
• E(χ) =
• χ |∇uχ|2 + Ln(χ) + σ |∂χ|.
• Dist2(χ1, χ2) =
• χ1△χ2 dist(x, ∂χ1) dx
• Time discretization: given χn, χn+1 minimizes

F(χ) = E(χ) + 1 2△tDist2(χn, χ)

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Contact Line Dynamics – Cont’

Pinning and DePinning Transition (Kim-Y., in progress) Use the far-field slope as the de-pinning parameter: u(x) − → −mx+

1

as x1 − → −∞

• there is a unique maximum m∗ such that there is a

stationary solution with m = m∗;

• for m = m∗ + δ, there exists a pulsating wave:

u(x, t + Tδ) = u(x − e1, t)

• The speed Vδ =

1 Tδ of the pulsating wave satisfies:

Vδ ≫ δ, i.e. super-linear growth in δ.

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Boundary between Patterns

t Γ( ) t Γ( )

Effective Front?

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Boundary between Patterns Mixture of Phases

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Boundary between Patterns

Some recent related works:

• (Dirr-Lucia-Novaga) Γ-convergence of Allen-Cahn

functional with oscillatory forcing: Eǫ(u) =

• ǫ|∇u|2 + 1

ǫW(u) + 1 ǫαg x ǫα

• u dx

(Dirr-Orlandi) and with random forcing.

• (Alicandro-Braides-Cicalese) Phase and anti-phase

boundaries in binary discrete systems. Nearest and Next-Nearest Neighbor Interactions

• (Oono-Puri)

Cell-Dynamical Systems

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Lattice Model (Alicandro-Braides-Cicalese) (Related to some earlier works of Cahn, et. al.) Nearest Neighbour Interaction

• Ferromagnetic:

E(u) = −

• i,j∈Z2,|i−j|=1

uiuj, ui = ±1 leads to pure phases.

• Anti-ferromagnetic:

E(u) =

• i,j∈Z2,|i−j|=1

uiuj, ui = ±1 leads to checker-board phases.

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Lattice Model (Alicandro-Braides-Cicalese) Next Nearest Neighbour Interaction E(u) = c1

• i,j∈Z2,|i−j|=1

uiuj + c2

• i,j∈Z2,|i−j|=

√ 2

uiuj ui = ±1. Depending on the values of c1 and c2, it can leads to checker-board phases or stripe patterns.

(Braides-Cicalese-Y., in progress) on the dynamics of fronts between the patterns using discrete space-discrete time variational approach (based on Braides-Gelli-Novaga) leading to Crystalline Motion by Mean Curvature.

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Homogenization on Longer Time Scales

The dynamics V = κ + f(p) is the cell problem of the following motion law: V = ǫκ + f p ǫ

• under the space-time scaling: ˜

p = p ǫ and ˜ t = p ǫ. The previous result can lead to the following homogenized equation (in the {x : u(x, t) = 0} = Γ(t) = limǫ Γǫ(t)): ut = c(∇u) |∇u| — 1st Order Hamilton-Jacobi-Equation

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Longer (Diffusive) Time Scales.

Consider: ǫV = ǫκ + f p ǫ

• r V = κ + 1

ǫf p ǫ

• .

It is plausible to have: ut = F(∇2u, ∇u) + H(∇u) where F represents some 2nd order curvature information.

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Random Walk in Random Medium

Nearest Neighbour Jumps (Solomon) {Xn}n≥1 P (Xn+1 = j + 1|Xn = j) = αj P (Xn+1 = j − 1|Xn = j) = βj (= 1 − αj) where the αj’s are iid random variables. Let σj = βj

αj . Then

E ln σ < 0 = ⇒ Xn − → +∞ a.s. E ln σ > 0 = ⇒ Xn − → −∞ a.s. E ln σ = 0 = ⇒ −∞ = lim inf

n

Xn < lim sup

n

Xn = +∞ a.s.

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Random Walk in Random Medium – Cont’

Nearest Neighbour Jumps (Solomon) Furthermore, Eσ < 1 = ⇒ lim

n

Xn n = 1 − Eσ 1 + Eσ, a.s. E(σ−1) < 1 = ⇒ lim

n

Xn n = −1 − Eσ 1 + Eσ, a.s. (Eσ)−1 ≤ 1 ≤ E(σ−1) = ⇒ lim

n

Xn n = 0 a.s.

• (Sinai) When E ln σ = 0, then Xn ∼ (log n)2.
• (Kesten-Kozlov-Spitzer) When E ln σ < 0 and Eσ ≥ 1,

then Xn ∼ nκ for some κ.

• (Key) Extends some of the above results to random walks

with longer jumps.

Some Recent Investigations on Interfacial Propagation in Inhomogeneous Medium – p.52/56

Random Walk in Random Medium

Another Version of Nearest Neighbour Jumps (Derrida) Consider the following continuous-time-discrete-space random walk: dPn dt = Wn,n+1Pn+1 + Wn,n−1Pn−1 − (Wn+1,n + Wn−1,n)Pn Similar results as before: If E ln Wn,n+1 Wn+1,n < 0 and E Wn,n+1 Wn+1,n < 1 then V =

• E

1 Wn+1,n −1 1 − E Wn,n+1 Wn+1,n

• Some Recent Investigations on Interfacial Propagation in Inhomogeneous Medium – p.53/56

Random Walk in Random Medium

Explicit Computations for Next Nearest Neighbour Jumps (Wang-Y.) dQn dt = Wn,n+1Qn+1 + Wn,n−1Qn−1 − (Wn+1,n + Wn−1,n)Qn +ǫ ˜ Wn,n+2Qn+2 + ǫ ˜ Wn,n−2Qn−2 −ǫ( ˜ Wn+2,n + ˜ Wn−2,n)Qn with all the W’s and ˜ W’s independent. Set Qn = Pn + ǫ ˜ Pn. Then V = V0 + ǫV1 where V1 = 1 (ER)2

• E((Un + Un+1)Rn)E ˜

Wn+2,n− E((Un + Un+1)Rn+2)E ˜ Wn,n+2

• Some Recent Investigations on Interfacial Propagation in Inhomogeneous Medium – p.54/56

Random Walk in Random Medium

Some Simulations for Next Nearest Neighbour Jumps (Wang-Y.) For example, Pr(Wn,n+1 = 1, Wn+1,n = W) = α and Pr(Wn,n+1 = W, Wn+1,n = 1) = 1 − α.

Some Recent Investigations on Interfacial Propagation in Inhomogeneous Medium – p.55/56

THANK YOU

Some Recent Investigations on Interfacial Propagation in Inhomogeneous Medium – p.56/56