Nonlinear Stability of Sources

Transcription

1 Nonlinear Stability of Sources Björn Sandstede Arnd Scheel Margaret Beck Toan Nguyen Kevin Zumbrun Spiral waves [Li, Ouyang, Petrov, Swinney] [Nettesheim, von Oertzen, Rotermund, Ertl] Dynamics of core / spiral tip Modulations of wave trains in far field [Li, Ouyang, Petrov, Swinney]

2 One-dimensional defects space time Chloride-iodide-malonic acid reaction (CIMA) [Perraud, De Wit, Dulos, De Kepper, Dewel, Borckmans] Standing time-periodic structures space One-dimensional defects Surface waves [Pastur et al.] space Light-sensitive BZ-reaction [Yoneyama, Fujii, Maeda]

3 One-dimensional defects wave train defect wave train time asymptotically periodic in space time-periodic in co-moving frame space Overview: wave trains & group velocities sources existence & bifurcations spectral & nonlinear stability Dynamics of wave trains c wave train k wavenumber ω=ω(k) temporal frequency local wavenumber slowly varying modulations of wavenumber Spectrum of wave trains λ(iγ) Im λ Re λ λ(iγ) = -ic g γ - dγ + Ο( γ 3 )

4 Dynamics of wave trains c wave train k wavenumber ω=ω(k) temporal frequency local wavenumber slowly varying modulations of wavenumber c g q(x,t) q t = c g v x group velocity: direction of transport Spectrum of wave trains λ(iγ) Im λ Re λ λ(iγ) = -ic g γ - dγ + Ο( γ 3 ) Dynamics of wave trains q(x,t) local wavenumber c g slowly varying modulations of wavenumber near k 0 on scale X=ε(x-c gt) and T=ε t/ for 0<ε<<1 Viscous = (k 0 ) q X Formal derivation: [Howard & Kopell], [Kuramoto] Validity over natural time scale 1/ε : [Doelman, S., Scheel, Schneider] Stability of wave trains: [S., Scheel, Schneider, Uecker], [Johnson, Zumbrun] Anticipated dynamics: Zero-mean perturbations converge to zero Lax shocks and rarefaction waves

5 Sources Sources: outgoing transport group velocities point away from core c g transport c g Existence: how do sources arise? Spectral and linear stability: linearized equation is time-periodic Nonlinear stability: previous methods do not apply Essential Hopf instabilities of pulses standing pulse + Im λ Re λ Hopf instability of rest state wave trains Theorem [S., Scheel] k flip-flop target source μ target flip-flop

6 Spatial dynamics time u x = v v x = D 1 [u t c d v f(u)] space u H 1 (S 1 ) L (S 1 ) v wave train = periodic orbit defect = heteroclinic orbit wave train = periodic orbit Spectra of sources Reaction-diffusion system: Standing sources are time-periodic: Floquet spectrum determines spectral stability time space Spectrum of wave trains Evans-function analysis (eigenfunctions u x and u t) L space t>>1 c g<0 exponential weight c g>0 t>>1 exponentially weighted L space defect core x

7 Expected dynamics Burgers equation with advection in far field exponential adjustment of position and phase x=-c g t t position/phase adjustment x=c g t defect core Gaussians error terms x Nonlinear stability Theorem [Beck, Nguyen, S., Zumbrun]: Assume u * (x,t) is a spectrally stable source and let u(x,0)=u * (x,0)+v 0 (x) where v 0 (x)exp(x /M) <ε is sufficiently small. Then there are constants p, φ <ε such that u(x,t)-u * (x-p,t-φ ) < εc exp(-ηt) for (x,t) in Ω 1 and u(x,t)-u * (x,t) < εc exp(-ηt) for (x,t) in Ω. x=-c g t t x=c g t Ω 1 Ω Ω defect core x

8 Nonlinear stability proofs Define appropriate offset from source: Derive equation for offset: Variation-of-constants formula: Fixed-point argument in appropriate function space: No decay in L spaces Decay in weighted L spaces, but nonlinearity not well defined Caveats Long-time dynamics for small localized initial data Heat equation Reaction-diffusion equation Burgers equation Heat equation Reaction diffusion Burgers equation Reason: Gaussian * Gaussian Gaussian Differentiated Gaussian * Gaussian Gaussian

9 Nonlinear stability proofs Define appropriate offset from source: Derive equation for offset: Variation-of-constants formula: Fixed-point argument in appropriate function space: No decay in L spaces Decay in weighted L spaces, but nonlinearity not well defined Define offsets Let u(x,t) be a solution of u t = Du xx + cu x + f(u) near a given defect u * (x,t) Define p(x,t) and φ(x,t) so that p(x,t) and φ(x,t): space-time shift v(x,t): profile changes Substitution gives the following system for the functions p, φ, and v:

10 Solve linearized system: Green s function Linearization about defect: Solve via Green s function: Expansion of Green s function: x=-c gt projection x=c gt t error function plateau y Gaussians error terms x a j(x,t;y,s): G R(x,t;y,s): scalar functions composed of error function plateaus in (x,t-s) times a localized projection function in y plus Gaussian errors sum of moving differentiated Gaussians Variation-of-constants formula Equation for offsets: Variation-of-constants formula: Expansion of Green s function: Collect terms:

11 Nonlinear iterations Equation for offsets: differentiated Gaussian * Gaussian Gaussian Expansion of Green s function: x=-c gt w j(y) x=c gt t Gaussians err(x,t-s) y a j(x,t;y,s) = err(x,t-s) w j(y) + Gaussians G R(x,t;y,s) differentiated Gaussians x Nonlinear iterations Equation for offsets: differentiated Gaussian * Gaussian Gaussian Expansion of Green s function: a j (x,t;y,s) = err(x,t-s) w j (y) + Gaussians G R(x,t;y,s) differentiated Gaussians Nonlinear iteration using templates: v and (p,φ) x sum of moving Gaussians Completes nonlinear stability result

12 Expansion of Green s function Green s function G(x,t;y,s) satisfies Laplace transform: Resolvent kernel (x,t;y,s,λ) is π-periodic in (t,s) and satisfies Makes connection with spatial-dynamics formulation of defects: u x = v u v x = D 1 H 1 (S 1 ) L (S 1 ) [u t c d v f(u)] v Linearize about defect and expand in λ gives bounds needed for (x,t;y,s,λ) Summary: Summary + Outlook Proved that spectrally stable sources are nonlinearly stable in an appropriate sense Outlook: Obtain expansions of dynamics in wedge-shape interface regions: Stability analysis of contact defects (c g =0): contact defect Interaction of sink-source pairs: