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 Anogia, 2009
1. Introduction motivation and topics to be discussed Anogia, 2009
1. Introduction motivation and basic concepts evolution equation Spatial heterogeneity in the equation How are they reflected or in the solution? Spatial heterogeneity in the initial data periodic quasi-periodic (QP) Types of heterogeneity: almost periodic (AP) ergodic Topics to be discussed: 1. Perturbation of planar fronts. 2. Speed of TW in spatially ergodic media 3. Spreding fronts in ergodic media. Anogia, 2009
f(u) Topic 1: Disturbed planar fronts u 0 1 Allen-Cahn equation f : bistable nonlinearity disturbance planar front Can the disturbed front recover its planar shape uniformly? In general, NO. But yes, if the perturbation is uniquely ergodic. Anogia, 2009
Topic 2: Speed of TW in heterogeneous media periodic non-periodic propagation in an undulating cylindar Does the (generalized) TW Yes, if the medium is ergodic. has the average speed? Anogia, 2009
Topic 3: Spreading fronts in stratified environments What is the asymptotic shape of the front ? Periodic case: KPP (Liang-Lin-M.) Epidemic model (Ducrot-M) Non-periodic case: Interface equation (motion by curvature) Anogia, 2009
2. Basic concepts ergodicity and its properties Anogia, 2009
Hull of a function bounded continuous function on R closure of the set of all translations of g(x) in the local uniform topology L ∞ loc Multi-dimensional case Y Hull in the direction : Anogia, 2009
Various classes of heterogeneity periodic quasi-periodic almost periodic uniquely ergodic recurrent Moreover, every orbit is dense. Anogia, 2009
Anogia, 2009 multi-D Ergodicity
Transmission of ergodicity Ergodic function Ergodic function continuous deformation x vertical deformation E E, AP AP x horizontal deformation E E, AP AP X x Anogia, 2009
Example of ergodic functions : irrational A B non-periodic A A A A A A A B B B B B B B B B B B x uniquely ergodic + recurrent = strictly ergodic Anogia, 2009
Any finite pattern is Penrose tiling distributed uniformly. Anogia, 2009
Any finite pattern is Penrose tiling distributed uniformly. Anogia, 2009
Any finite pattern is Penrose tiling distributed uniformly. Anogia, 2009
Anogia, 2009 (Radin 1994) Pinwheel tiling
Anogia, 2009 2a. Preservation of ergodicity
evolution equation on 1. Well-posed. Assumptions 2. Equivariance w.r.t. spatial translations Proposition: If u 0 ( x ) is ergodic in the direction Y , then for any t > 0, the same holds for u ( x , t ; u 0 ) Proof Ergodic function Ergodic function continuous deformation Anogia, 2009
Anogia, 2009 3. Applications
Topic 1. Recovery of disturbed planar fronts Joint work with M. Nara Anogia, 2009
f(u) Topic 1: Disturbed planar fronts u 0 1 Allen-Cahn equation disturbance planar front Can the disturbed front recover its planar shape uniformly? In general, NO. Anogia, 2009
Theorem (recovery of planar front). If is uniquely ergodic in x , and if then uniformly in R N for some α . x y Anogia, 2009
Outline of the proof Let be the front position. x y Approximation of γ t by the mean curvature flow (for all large t ). (x,y) unif ergodic in x implies γ t unif ergodic. u 0 Solutions of MCF with unif ergodic data converges to drifting hyperplane uniformly. Anogia, 2009
Topic 2. Speed of traveling waves in spatially heterogeneous media Anogia, 2009
Conventional notion of “traveling wave”. (A) constant speed (B) constant profile Q What if the medium is spatially inhomogeneous? A simple model problem x Key concept: Hull of b(x) Anogia, 2009
Definition of TW x Current landscape Current profile The current profile depends on the T W current landscape continuously. def Anogia, 2009
Definition of TW The current profile depends on the TW current landscape continuously. def Law of motion Anogia, 2009
More about the law of motion Law of motion x Key observations The front speed depends mainly on local environments. Hence is continuous in the topology of Consequently b : ergodic (resp. AP, QP, P) g : ergodic (resp. AP, QP, P) Anogia, 2009
Law of motion Existence of has mean on average speed What condition on guarantees that Q has the above property? Theorem . TW has the average speed if the medium is uniquely ergodic in the direction of propagation. Proof The front speed depends mainly on local environments. Hence is continuous in the topology of Therefore g (hence 1 / g ) is uniquely ergodic. Anogia, 2009
Effect of geometry on front propagation Q Propagation: Which direction is faster? Anogia, 2009
Motion by curvature in a sawtoothed cylinder Joint work with B. Lou and K.-I. Nakamura normal velocity curvature constant recurrent functions Anogia, 2009
Notation maximal opening angles maximal closing angles Anogia, 2009
GOAL To study the speed of propagation (TW) as large small What determines the speed of the limit TW ? Anogia, 2009
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. Anogia, 2009
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 Anogia, 2009
Corollary. The limit speed satisfies The larger the opening angle , the slower the speed . Anogia, 2009
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? Anogia, 2009
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