Stochastic Homogenization for Reaction-Diffusion Equations

Stochastic Homogenization for Reaction-Diffusion Equations Jessica Lin McGill University Joint Work with Andrej Zlatoš June 18, 2018

Motivation: Forest Fires

ç ç ç ç ç ç ç ç ç ç Motivation: Forest Fires ç ç ç ç ç ç ç ç ç ç ç

1. A PDE Let u(t, x) denote the temperature at (t, x) (0, ) R d. { u t u = f (x, u) in (0, ) R d, u(0, x) χ Θ0 on R d, for Θ 0 R d open and bounded. For each x R d, f (x, ) : Ignition KPP f(x,u) f(x,u) 0 u 1 0 u 1

1. A PDE to Model Combustion Let u(t, x) denote the temperature at (t, x) (0, ) R d. { u t u = f (x, u, ω) in (0, ) R d, u(0, x) = χ Θ0 on R d, for Θ 0 R d open and bounded. For each x R d, f (x, ) : Ignition KPP f(x,u) f(x,u) 0 u 1 0 u 1

(Ω, F, P) 2. Conveying a Random Environment

2. Conveying a Random Environment (Ω, F, P) For each ω Ω, f (x, u, ω) satisfies f (x, u, ω) is an ignition reaction OR KPP reaction, f 0 (u) f (x, u, ω) f 1 (u), where f 0, f 1 : [0, 1] R are some fixed deterministic, homogeneous reactions of the same type as f (x, u, ω). Stationarity and Ergodicity (SE): f (, u, ) is stationary, i.e. there exists a measure-preserving group of transformations {T y } y R d : Ω Ω so that for all u R, f (x + y, u, ω) = f (x, u, T y ω).

u(,, ω) solves 3. Describing the Asymptotics u t u = f (x, u, ω).

u(,, ω) solves Let 3. Describing the Asymptotics u t u = f (x, u, ω). ( t u ε (t, x, ω) := u ε, x ) ε, ω

u(,, ω) solves Let 3. Describing the Asymptotics u t u = f (x, u, ω). ( t u ε (t, x, ω) := u ε, x ) ε, ω ut ε ε u ε = 1 ( x ) ε f ε, uε, ω

u(,, ω) solves Let 3. Describing the Asymptotics What about the initial condition? u t u = f (x, u, ω). ( t u ε (t, x, ω) := u ε, x ) ε, ω ut ε ε u ε = 1 ( x ) ε f ε, uε, ω u ε (0, x, ω) χ Θ0 (x)

u(,, ω) solves Let 3. Describing the Asymptotics What about the initial condition? This implies u t u = f (x, u, ω). ( t u ε (t, x, ω) := u ε, x ) ε, ω ut ε ε u ε = 1 ( x ) ε f ε, uε, ω u ε (0, x, ω) χ Θ0 (x) { u t u = f (x, u, ω) in (0, ) R d, u(0, x, ω) χ 1 ε on Rd. So initial fire is large compared to the size of the heterogeneities

Goal of Homogenization Identify deterministic open sets {Θ t } t>0 such that almost surely and locally uniformly away from the boundary Γ t := Θ t, { 1 if x Θ t lim ε 0 uε (t, x, ω) = 0 if x R d \ Θ t. {Θ t } t>0 represents the effective front propagation taking place on average in the random, heterogeneous environment. Q: What governs {Θ t } t>0? The normal velocities for the tangent planes. Q: Can we identify the sets {Θ t } t>0 according to a PDE?

Equivalent Goal: Identify a deterministic function c : S d 1 (0, ) such that almost surely and locally uniformly in space-time (away from certain boundaries), lim ε 0 uε (t, x, ω) = u(t, x), where u is the unique viscosity solution of { ( ) ut = c Du Du Du in (0, ) R d, u(0, x) = χ Θ0 (x) on R d. c (e)=the normal velocity in direction e S d 1 governing the front propagation

Results Theorem (Lions, Souganidis, 05) Assume (Ω, F, P) is (SE), f (,, ω) is KPP. Then for P-a.e. ω, homogenization holds.

Results Theorem (Lions, Souganidis, 05) Assume (Ω, F, P) is (SE), f (,, ω) is KPP. Then for P-a.e. ω, homogenization holds. Approach: KPP Reaction-Diffusion Equations can be compared to solutions of v t v = f u (x, 0)v. Hopf-Cole transformation: Converts this PDE into a viscous Hamilton-Jacobi equation with a convex Hamiltonian.

Theorem (L., Zlatoš, 17) Assume (Ω, F, P) is (SE), f (,, ω) is ignition, d 3, and certain additional assumptions. Then for P-a.e. ω, homogenization holds.

Theorem (L., Zlatoš, 17) Assume (Ω, F, P) is (SE), f (,, ω) is ignition, d 3, and certain additional assumptions. Then for P-a.e. ω, homogenization holds. Theorem (Lions, Souganidis, 05; L., Zlatoš, in prep) Assume (Ω, F, P) is (SE), f (,, ω) is KPP. Then for P-a.e. ω, homogenization holds.

Why d 3? How can we expect to see a sharp interface (u(t, x) = χ Θt (x)) in the limit as ε 0?

Why d 3? How can we expect to see a sharp interface (u(t, x) = χ Θt (x)) in the limit as ε 0? We need to control the width of the transition zone.

Why d 3? How can we expect to see a sharp interface (u(t, x) = χ Θt (x)) in the limit as ε 0? We need to control the width of the transition zone. 1D: { u t u xx = f (x, u), u(0, x, ω) = χ Θ0. u=1 u=0 For η ( 0, 1 2), let L u,η (t) := dist H ({x : u(t, x) 1 η}, {x : u(t, x) η})

Theorem (Zlatoš, 14) Let u solve { u t u = f (x, u) in (0, ) R d, u t 0 in (0, ) R d. If d 3, then lim sup L u,η (t) <. t

Theorem (Zlatoš, 14) Let u solve { u t u = f (x, u) in (0, ) R d, u t 0 in (0, ) R d. If d 3, then lim sup L u,η (t) <. t In fact, for d 3, there exists C > 0 such that for P-a.e. ω, lim sup L u,η,ω (t) < C. t

Theorem (Zlatoš, 14) Let u solve { u t u = f (x, u) in (0, ) R d, u t 0 in (0, ) R d. If d 3, then lim sup L u,η (t) <. t In fact, for d 3, there exists C > 0 such that for P-a.e. ω, lim sup L u,η,ω (t) < C. t For d > 3, this is not in general true! There exist reactions f (,, ω) with ω Ω such that L u,η,ω (t) Ct

Main Steps 1. Identify a deterministic candidate c : S d 1 (0, ). 2. Given c, show that u ε u in the appropriate sense.

Main Steps 1. Identify a deterministic candidate c : S d 1 (0, ). How will we relate c to the reaction-diffusion PDE? 2. Given c, show that u ε u in the appropriate sense.

Main Steps 1. Identify a deterministic candidate c : S d 1 (0, ). How will we relate c to the reaction-diffusion PDE? 2. Given c, show that u ε u in the appropriate sense. This is a completely deterministic PDE argument relying upon the theory of viscosity solutions and generalized front propagation.

Definition: Front Speeds Fix e S d 1, and let u(,, ω) solve { u t u = f (x, u, ω) in (0, ) R d, u(0, x, ω) = χ {x e 0} (x) on R d. The front speed c (e) > 0 is the deterministic constant such that for P-a.e. ω, for any K R d compact, for any δ > 0, lim inf t K {x e c (e) δ} lim sup t K {x e c (e)+δ} Roughly speaking, this says that for P-a.e. ω, u(t, x, ω) t u(t, xt, ω) = 1 u(t, xt, ω) = 0. χ {x e<c (e)t}(x)

Key Difficulties Heterogeneous Setting. If the right hand side is f (u), a traveling front with speed c satisfies solves the PDE and u(t, x) = U(x e ct) lim U(s) = 1 s lim U(s) = 0. s If (U, c) is a traveling front pair, then c satisfies our definition of front speeds. There is an analogous type of solution (pulsating front) for right hand side f (x, u) when f (, u) is periodic.

Front-Like Initial Data and Higher Dimensions: K Front speeds in random media in one dimension: Nolen and Ryzhik, Zlatoš

Definition: Spreading Speeds Fix e S d 1, and let u(,, ω) solve { u t u = f (x, u, ω) in (0, ) R d, u(0, x) = θ 0 χ BR on R d, for R sufficiently large. Then we say w(e) is the spreading speed in direction e if for P-a.e. ω, for any δ > 0, lim u(t, (w(e) δ)te, ω) = 1, t lim u(t, (w(e) + δ)te, ω) = 0. t

First Passage Times for Reaction-Diffusion Equations Define τ(0, y, ω) := inf { t : u(t, x, ω) θ 0 χ BR (y)}. By the subadditive ergodic theorem, there exists a deterministic τ(e) such that for P-a.e. ω, Then τ(0, ne, ω) lim = τ(e). n n w(e) := 1 τ(e) satisfies the definition of spreading speed.

All Directions at Once: The Wulff Shape Proposition Let u(,, ω) solve { u t u = f (x, u, ω) in (0, ) R d, u(0, x) = Θ 0 χ BR on R d, for R sufficiently large. Define S := {se : 0 s w(e)}, a convex set. For P-a.e. ω, for every δ > 0, for t sufficiently large, { (1 δ)ts x : u(t, x, ω) = 1 } (1 + δ)ts. 2

Recovery of Front Speeds In the periodic setting, Freidlin-Gärtner formula says: w(e) = c (e ) inf e S d 1, e e e e>0

Recovery of Front Speeds In the periodic setting, Freidlin-Gärtner formula says: w(e) = c (e ) inf e S d 1, e e e e>0 For us, we do not have front speeds, but we DO have spreading speeds!

Recovery of Front Speeds In the periodic setting, Freidlin-Gärtner formula says: w(e) = c (e ) inf e S d 1, e e e e>0 For us, we do not have front speeds, but we DO have spreading speeds! Let c (e) := sup w(e )e e e S d 1, e e>0 The additional assumptions guarantee that the Wulff Shape S has no corners, so it has tangents in all directions. This is enough to show that c (e) defined in this way is the front speed.

Example where Homogenization Holds: Isotropic Environment (I) The random environment is isotropic. This guarantees that P is invariant with respect to rotations in physical space.

Example where Homogenization Holds: Isotropic Environment (I) The random environment is isotropic. This guarantees that P is invariant with respect to rotations in physical space. Canonical Example: Poisson Point Process Let P(ω) := {x n (ω)} n N R d denote a collection of points distributed by a Poisson point process with intensity 1. Then we have f (x, u, ω) f 1 (u)χ B1(P(ω)) + f 0 (u)(1 χ B1(P(ω)))

Common Theme: Convexity Let ( ) p H(p) := c p. p For all solvable cases of stochastic homogenization for reaction-diffusion equations (solvable ignition and all KPP), H(p) is convex.

Common Theme: Convexity Let ( ) p H(p) := c p. p For all solvable cases of stochastic homogenization for reaction-diffusion equations (solvable ignition and all KPP), H(p) is convex. In such cases, we recover a Huygen s Principle guiding the front propagation, and we can explicitly describe {Θ t } t>0. For the stochastic homogenization of Hamilton-Jacobi equations, there are counterexamples to homogenization when the random Hamiltonians are nonconvex (Ziliotto [ 16], Feldman-Souganidis [ 16]).

Future Directions: Can we impose stronger assumptions on the random environment (finite range of dependence) to eliminate some of the restrictions (dimension, no corners on Wulff shape, etc.)?

Future Directions: Can we impose stronger assumptions on the random environment (finite range of dependence) to eliminate some of the restrictions (dimension, no corners on Wulff shape, etc.)? Can we extend to time-dependent reactions? More general coefficients? Can we quantify the convergence in these statements? In particular, can we quantify the fluctuations to the front-like interface?

Thank you very much for your attention!