Front propagation in spatially ergodic media Hiroshi Matano (Univ. - - PowerPoint PPT Presentation

Anogia, 2009

Front propagation in spatially ergodic media

Hiroshi Matano

(Univ. of Tokyo)

“Mathematical Challenges Motivated by Multi-Phase Materials” Anogia, June 21-26, 2009

Anogia, 2009

Outline of the talk

• 1. Introduction
• 2. Basic concepts

ergodicity and its properties

• 3. Applications

(1) recovery of disturbed planar fronts (2) speed of traveling waves in spatially ergogic media (3) asymptotic shape of spreading fronts

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• 1. Introduction

motivation and topics to be discussed

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• 1. Introduction motivation and basic concepts

Types of heterogeneity: periodic quasi-periodic (QP) almost periodic (AP) ergodic Spatial heterogeneity in the equation Spatial heterogeneity in the initial data How are they reflected in the solution?

evolution equation

• r

Topics to be discussed:

• 1. Perturbation of planar fronts.
• 2. Speed of TW in spatially ergodic media
• 3. Spreding fronts in ergodic media.

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Topic 1: Disturbed planar fronts

Can the disturbed front recover its planar shape uniformly? Allen-Cahn equation

f : bistable nonlinearity planar front

disturbance

But yes, if the perturbation is uniquely ergodic.

1 u f(u)

In general, NO.

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Topic 2: Speed of TW in heterogeneous media

periodic non-periodic

Does the (generalized) TW has the average speed? propagation in an undulating cylindar Yes, if the medium is ergodic.

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What is the asymptotic shape

• f the front ?

Topic 3: Spreading fronts in stratified environments

Periodic case: Non-periodic case:

KPP (Liang-Lin-M.) Epidemic model (Ducrot-M) Interface equation (motion by curvature)

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• 2. Basic concepts

ergodicity and its properties

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Hull of a function

closure of the set of all translations of g(x) in the local uniform topology L∞

loc

bounded continuous function on R Multi-dimensional case Hull in the direction :

Y

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periodic quasi-periodic almost periodic recurrent

Moreover, every orbit is dense.

Various classes of heterogeneity

uniquely ergodic

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Ergodicity

multi-D

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Transmission of ergodicity

continuous deformation

Ergodic function Ergodic function

x x x

vertical deformation horizontal deformation E E, AP AP E E, AP AP X

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Example of ergodic functions

A B A B B A B B B B B B A A A A B B

: irrational

A B

x

uniquely ergodic + recurrent = strictly ergodic

non-periodic

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Penrose tiling

Any finite pattern is distributed uniformly.

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Penrose tiling

Any finite pattern is distributed uniformly.

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Penrose tiling

Any finite pattern is distributed uniformly.

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Pinwheel tiling

(Radin 1994)

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• 2a. Preservation of ergodicity

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Assumptions

• 1. Well-posed.
• 2. Equivariance w.r.t. spatial translations

Proposition: If u0 (x) is ergodic in the direction Y, then for any t > 0, the same holds for u(x, t ; u0 )

evolution equation on

Ergodic function Ergodic function continuous deformation

Proof

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• 3. Applications

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Topic 1.

Recovery of disturbed planar fronts

Joint work with M. Nara

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Topic 1: Disturbed planar fronts

Can the disturbed front recover its planar shape uniformly? Allen-Cahn equation

planar front

disturbance

1 u f(u)

In general, NO.

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Theorem (recovery of planar front).

If is uniquely ergodic in x, and if then uniformly in RN for some α. x y

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x y

Outline of the proof

Approximation of γt by the mean curvature flow (for all large t ). u0 (x,y) unif ergodic in x implies γt unif ergodic. Solutions of MCF with unif ergodic data converges to drifting hyperplane uniformly.

Let be the front position.

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Topic 2.

Speed of traveling waves in spatially heterogeneous media

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A simple model problem Key concept: Hull of b(x)

x

Conventional notion of “traveling wave”.

(A) constant speed (B) constant profile

Q

What if the medium is spatially inhomogeneous?

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Definition of TW

Current landscape Current profile

The current profile depends on the current landscape continuously.

T W

def x

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Definition of TW

The current profile depends on the current landscape continuously.

TW

def

Law of motion

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More about the law of motion

Law of motion The front speed depends mainly on local environments. Hence is continuous in the topology of Key observations

x

Consequently b : ergodic (resp. AP, QP, P) g : ergodic (resp. AP, QP, P)

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Law of motion

Existence of average speed

Q

What condition on guarantees that has the above property?

• Theorem. TW has the average speed if the medium is

uniquely ergodic in the direction of propagation.

has mean on

The front speed depends mainly on local environments. Hence is continuous in the topology of

Proof

Therefore g (hence 1 / g ) is uniquely ergodic.

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Effect of geometry on front propagation

Propagation: Which direction is faster?

Q

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Motion by curvature in a sawtoothed cylinder

recurrent functions normal velocity curvature constant

Joint work with B. Lou and K.-I. Nakamura

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maximal opening angles maximal closing angles Notation

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GOAL

To study the speed of propagation (TW) as What determines the speed of the limit TW ?

large small

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Theorem 1 (existence and stability).

If , then a periodic TW exists for any small . This periodic TW is unique up to time shift and is asymptotically stable.

Theorem 2 (non-existence).

If , then there exists no periodic TW. Moreover any time-global solution converges to a stationary solution as .

Hereafter, we assume for simplicity.

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Theorem 3 (homogenization). Assume

and let be the recurrent TW which is normalized to satisfy . Then (i) converges to a function of the form as whose contact angle is ; (ii) the limit speed is determined by

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• Corollary. The limit speed satisfies

The larger the

• pening angle ,

the slower the speed

.

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Concluding remarks:

• TW has an average speed in uniformly ergodic environments.
• Planar waves are asymptotically stable w.r.t. UE perturbations.

Open problems:

• Does there exist an asymptotic shape of spreading fronts in

the Penrose tiling or other non-periodic media?

• Propagation in (stochastically) random media?