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 V n = κ + f ( p ) + F IMMC : ν V( , f, F) κ V = + f(p) + F N ν Γ( ) t V n = Normal Velocity , κ = Mean Curvature f ( p ) = Background Heterogeneity , F = External Forcing V ( ν, f, F ) = Effective Normal Velocity Some Recent Investigations on Interfacial Propagation in Inhomogeneous Medium – p.2/56
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 Some Recent Investigations on Interfacial Propagation in Inhomogeneous Medium – p.3/56
Two Main Types of Results of Interests Effective Property and Homogenization ν V( , f, F) κ V = + f(p) + F N ν Γ( ) t Existence, uniqueness, and stability properties of V ( ν, f, F ) as a function of the background inhomo- geneity ( f ) and external control parameter ( F ). Some Recent Investigations on Interfacial Propagation in Inhomogeneous Medium – p.4/56
Two Main Types of Result - Cont’ Critical Phenomena – Transition between pinned and propagation states V Propagation Pinned F F * V ∼ C ( F − F ∗ ) α , F > F ∗ . Properties and characterization of the threshold forcing F ∗ and the critical exponent α . Some Recent Investigations on Interfacial Propagation in Inhomogeneous Medium – p.5/56
Dynamics on Heterogeneous Landscape E (u) ε E ( . ) E ( . ) ε * E (u) * u du ǫ ⇒ (?) du dt = −∇ E ǫ ( u ǫ )(?) = dt = −∇ E ∗ ( u ) � � ǫ = length scale of inhomogeneity The presence of local minima and metastable states. Some Recent Investigations on Interfacial Propagation in Inhomogeneous Medium – p.6/56
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 Some Recent Investigations on Interfacial Propagation in Inhomogeneous Medium – p.7/56
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. Some Recent Investigations on Interfacial Propagation in Inhomogeneous Medium – p.8/56
Behavior Near F ∼ 1 + dt = 1 − cos( x ) + F − 1 ≈ x 2 dx 2 + o ( x 2 ) + δ (near x ≈ 0 ) where δ = F − 1 . Let 0 < η ≪ 1 be fixed, independent of δ . � 2 π dx T = 1 − cos( x ) + δ 0 � 2 π − η � 2 π � η dx dx dx = 2 + δ + 1 − cos( x ) + δ + x 2 ( x − 2 π ) 2 + δ 0 2 π − η η 2 O ( δ − 1 ≈ 2 ) 1 1 1 2 = ( F − 1) V = ≈ δ 2 T Some Recent Investigations on Interfacial Propagation in Inhomogeneous Medium – p.9/56
Discrete Allen-Cahn Equation du i dt = u i +1 − 2 u i + u i − 1 − A ( W ′ ( u i ) − g ) where W is of bistable type: W ( u ) = (1 − u 2 ) 2 . The above equation and behavior also appears in the modeling of charge density waves, dissipative Frenkel-Kantorova and many others: du i dt = u i +1 − 2 u i + u i − 1 + A sin( u i ) + B Compared with the continuum case: u t = u xx − 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). Some Recent Investigations on Interfacial Propagation in Inhomogeneous Medium – p.10/56
Extension to PDE Setting: Bifurcation Theory dX dt = N ( X ) + F • For 0 < F < F ∗ , there exists stable stationary solution: N ( X F ) + F = 0 • At F = F ∗ , the stationary solution X F ∗ becomes unstable. Perform center-manifold analysis of the dynamical equation: � X ( t ) = s 0 ( t )Φ 0 + s n ( t )Φ n n ≥ 1 ≈ � D 2 N ( X F ∗ )Φ 0 , Φ 0 � so that: ds 0 ( t ) 0 ( t ) + � F − F ∗ , Φ 0 � s 2 dt � Φ 0 , Φ 0 � � Φ 0 , Φ 0 � Some Recent Investigations on Interfacial Propagation in Inhomogeneous Medium – p.11/56
Pinning and De-Pinning Transition. I Semi-Linear Equation for Linearized IMMC : u t = △ u + f ( x, u ) + F u(x,t) u(x,t) x • 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. Some Recent Investigations on Interfacial Propagation in Inhomogeneous Medium – p.12/56
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 ( · , · + T F ) = U ( · , · ) + 1 • Scaling of the velocity: V = 1 1 ∼ C ( F − F ∗ ) 2 , for 0 ≤ F − F ∗ ≪ 1 T F (Results expected to be true also for IMMC for rational direction.) Some Recent Investigations on Interfacial Propagation in Inhomogeneous Medium – p.13/56
Pinning and De-Pinning Transition. II Reaction-Diffusion Equation for Front Propagation : ϕ t = ϕ xx − W ′ ( ϕ ) + δ ( g ( x ) + F ) 2 W ( ϕ ) = (1 − ϕ 2 ) 2 , g ( · ) is 1 -periodic and δ ≪ 1 . ϕ (x) ~ 1 location of front x ϕ (x) ~ −1 (The above Allen-Cahn or Ginzburg-Landau type equation is commonly used in the modeling of phase boundary motion.) Some Recent Investigations on Interfacial Propagation in Inhomogeneous Medium – p.14/56
Reaction-Diffusion Equation- Cont’ (Dirr-Y.) There exists an F ∗ > 0 such that • For 0 ≤ F ≤ F ∗ , there exists pinned states: ϕ xx − W ′ ( ϕ ) + δ ( g ( x ) + F ) = 0 2 • For F > F ∗ , there exists pulsating wave – space-time periodic solution Φ( x, t ) : Φ( x, t + T F ) = Φ( x − 1 , t ) • Scaling of the velocity: V = 1 1 ∼ C ( F − F ∗ ) 2 , for δ ≪ F − F ∗ ≪ 1 T F (The F ∗ and C can be computed asymptotically.) Some Recent Investigations on Interfacial Propagation in Inhomogeneous Medium – p.15/56
Fully Nonlinear IMMC: x u(x,t) ν Γ( ) t where Γ( t ) = { ( x, u ( x, t )) : x ∈ R n , t ∈ R + } and u satisfies: � � � � ∂u ∇ u � 1 + |∇ u | 2 ∂t = + δf ( x, u ) div � 1 + |∇ u | 2 a degenerate quasilinear parabolic equation – graph might not stay as a graph. Some Recent Investigations on Interfacial Propagation in Inhomogeneous Medium – p.16/56
Statement of Result for IMMC Effective Description of Propagation (Dirr-Karali-Y.) V n = 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 Some Recent Investigations on Interfacial Propagation in Inhomogeneous Medium – p.17/56
Uniform Space Time Oscillation and Gradient Bound � � osc ( u ( t )) = x ∈ R n u ( x, t ) − inf sup x ∈ R n u ( x, t ) sup osc ( u ( t )) ≤ C ( n, δ, f ) < ∞ t ∈ R + �∇ u � L ∞ ( R n × 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. Some Recent Investigations on Interfacial Propagation in Inhomogeneous Medium – p.18/56
Statement of Result for IMMC - Cont’ Pulsating Wave ( c ν � = 0 ) : Γ( t ) satisfies: Γ( t + τ ) = Γ( t ) + z, for all z ∈ Z n +1 , τ = ν · z c ν Γ( τ) = Γ( ) + z t+ t Γ( ) t z ν Existence, Uniqueness, Stability Properties Some Recent Investigations on Interfacial Propagation in Inhomogeneous Medium – p.19/56
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 . Γ( ) t ν The above property implies: osc R n (Γ( t )) ≤ A ( n ) osc [0 , 5] n (Γ( t )) + B ( n ) where osc Ω (Γ)) = sup p,q ∈ Γ ∩ Ω { ( p − q ) · ν } . Some Recent Investigations on Interfacial Propagation in Inhomogeneous Medium – p.20/56
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