Nonlinear stability of time-periodic viscous shocks. Margaret Beck Brown University

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1 Nonlinear stability of time-periodic viscous shocks Margaret Beck Brown University

2 Motivation Time-periodic patterns in reaction-diffusion systems: t x Experiment: chemical reaction chlorite-iodite-malonic-acid (CIMA) Numerical simulation: reaction-diffusion equation u t = Du xx + f (u) [Perraud et. al., Phys. Rev. Lett. 1993]

3 Motivation Diagram of a source: Defect Perturbations Perturbations u(x,t) t x x Defect Importance in applications: Widely observed in experiments and numerics Defect created spontaneously; not caused by inhomogeneity Organize dynamics in rest of spatial domain Mathematical interest: Linear stability: embedded zero eigenvalue; time-periodic linear operator Nonlinear stability: weighted spaces don t work, no current methods apply

4 Linear stability overview Stability Ansatz: u t = F(u) u(t) = ū + v(t) v t = f u(ū)v + N(v) ODE: finite dimensions C Stable Unstable Neutral PDE: infinite dimensions C Stable Unstable Neutral Neutral

5 Stability of sources Linear stability: two key issues Continuous spectrum up to imaginary axis; embedded zero eigenvalue Time-periodic operator; no Floquet theory C u t = Du xx + f(u) v t = Dv xx + f u (ū(x, t))v +N(v) }{{} L(t)v Nonlinear stability: Source: perturbations moving outward Current methods can t be applied Focus on linear issues: time-periodic shocks Same linear issues Nonlinearity can be dealt with

6 Time-periodic shocks Parabolic system of conservation laws: u t = u xx F (u) x, u(t) X, u(x, t) R N Time-periodic shock: ū(x, t), lim x ± ū(x, t) = u ±, ū(x, t) = ū(x, t + 2π) ū(x, t) u + N p p t N +1 p p 1 u - x outgoing incoming shock incoming outgoing x Lax shock: Eigenvalues a ± 1 <... < a± N There is a number p {1,..., N} so that of Fu(u±) are real, distinct, nonzero. a N p < 0 < a N p+1, a + N p+1 < 0 < a + N p+2

7 Time-periodic shocks u t = u xx F (u) x Analyze stability: u(x, t) = ū(x, t) + v(x, t) v t = L(t)v + N(v) x, L(t)v = v xx (F u(ū(x, t))v) x Assume spectrum of L(t): Double zero eigenvalue; eigenfunctions ū x and ū t Motivation: Hopf bifurcation [Texier, umbrun 08], [Sandstede, Scheel 08] i -i C Stability: solution converges to space and time translate of wave lim u(x, t) = ū(x t q, t τ )

8 Time-Periodic Shocks: statement of result Theorem [B., Sandstede, umbrun 08] Under appropriate spectral stability assumptions, the profile ū is nonlinearly asymptotically stable with respect to perturbations v 0 with (1 + ) 3/2 v 0( ) H 3 ɛ. Also, q(t), τ(t), q, τ so that u(, t) ū( q q(t), t τ Cɛ τ(t)) L p (R) (1 + t) p for 1 p, where u is the solution to the conservation law with initial data u 0(x) = ū(x, 0) + v 0(x), and Proof: (q, τ ) + (1 + t) 1/2 (q, τ)(t) Cɛ. i) Develop a contour integral representation of linear evolution and derive pointwise bounds ii) Extract neutral behavior due to zero eigenvalues iii) Use fixed-point iteration scheme to prove nonlinear stability

9 Sketch of proof: i) contour integral and pointwise bounds Recall contour integral for time-independent operators: v t = Lv, v(0) = v 0 Laplace transform: ˆv(x, λ) = R 0 e tλ v(x, t)dt Solve with resolvent Invert Laplace transform λˆv v 0 = Lˆv ˆv = (λ L) 1 v 0 v(t) = e Lt v 0 = 1 2πi Γ e tλ (λ L) 1 v 0dλ C Γ

10 Sketch of proof: i) contour integral and pointwise bounds Pointwise estimates via resolvent kernel, G(x, y, λ): λg δ( y) = LG so that ((λ L) 1 v 0)(x) = G(x, y, λ)v 0(y)dy R This leads to the Green s function, G(x, t, y): G(x, t, y) = 1 e tλ G(x, y, λ)dλ 2πi Γ v(x, t) = G(x, t, y)v 0(y)dy R Key points: Resolvent kernel still defined pointwise inside continuous spectrum Can deform contour Γ into spectrum to obtain sharp pointwise bounds Stationary shocks: [umbrun, Howard 98]

11 Sketch of proof: i) Floquet spectrum for contour integral Floquet spectrum: Φ 2π is time 2π map for linear flow Σ = {σ C : e 2πσ spectrum of Φ 2π} Eigenvalue equation: v t = L(t)v for t (0, 2π), v(2π) = e 2πσ v(0) Equivalently, v(t) = e σt u(t), u is spatially localized and solves σu + u t = u xx (F u(ū(x, t))u) x, u(x, t) = u(x, t + 2π) Two eigenfunctions for σ = 0: ū x(x, t) and ū t(x, t) i C -i

12 Sketch of proof: i) definition of resolvent kernel Recall: Green s function solved inhomogeneous eigenvalue equation: (λ L)ˆv = v 0, ˆv(x) = G(x, y, λ)v 0(y)dy Floquet eigenvalue equation: σv + v t = v xx (F u(ū)v) x, v(x, t) = v(x, t + 2π) R Write as first-order spatial dynamical system: ««0 1 v V x = V, V = t + σ + F uu(ū)[ū x, ] F u(ū) Work in Y = H 1 (S 1 ) H 1/2 (S 1 ). v x Inhomogeneous equation: «0 1 V x = V + F t + σ + F uu(ū)[ū x, ] F u(ū)

13 Sketch of proof: i) exponential dichotomies Ill-posed equation: U x = «0 1 U has eigenvalues ± ik, k t 0 C Define exponential dichotomies on Y [Sandstede and Scheel 01]: xφ s,u = A(x, t, σ)φ s,u, Φ s,u (x, y, σ) L(Y, Y ) Φ s (x, y, σ)v Y Ke η(x y) V Y, for x y Φ u (x, y, σ)v Y Ke η(y x) V Y, for y x Solution operators in forwards and backwards time

14 Sketch of proof: i) definition of resolvent kernel Resolvent kernel defines solutions, for arbitrary F, to «0 1 V x = V + F t + σ + F uu(ū)[ū x, ] F u(ū) Solve using exponential dichotomies [Sandstede, Scheel 01]: V (x, σ) = x Φ s (x, y, σ)f(y)dy + x + Φ u (x, y, σ)f(y)dy Work in Y = H 1 (S 1 ) H 1/2 (S 1 ), so well defined Fourier series: V (x, σ, t) = X n e int ˆV n (x, σ) This will give convergence of the contour integral.

15 Sketch of proof: i) contour integral Recall time-independent case: λg δ( y) = LG, G(x, t, y) = 1 2πi Γ e tλ G(x, y, λ)dλ Need to solve: for «0 1 xg = G + t + σ + F uu(ū)[ū x, ] F u(ū) «0 (x, t) = δ(t)δ(x y) ( ) Green s function: G(x, t, y) = 1 2πi = 1 2πi µ+i µ i µ+ i 2 µ i 2 e λt G(x, y, λ)dλ e X σt e int Ĝ n (x, y, σ) dσ n {z } G(x,y,σ,t) solves ( ) i i/2 -i/2 -i C

16 Sketch of proof: i) pointwise bounds Green s function: G(x, t, y) = 1 µ+ i 2 e σt G(x, y, σ, t)dσ 2πi µ 2 i i/2 Meromorphically extend resolvent kernel near σ 0: G(x, y, σ, t) A + B A B -i/2 A: ero eigenvalues Similar to a spectral projection B: Continuous spectrum Related to weak spatial decay of G

17 Sketch of proof: i) B, continuous spectrum Recall G solves: «0 1 xg = G + t + σ + F uu(ū)[ū x, ] F u(ū) Asymptotic limits of operator: «0 1 t + σ F u(u ±) «0 1 σ F u(u ±) Spatial eigenvalues for σ 0: C C N-p-1 p+1 N-p p ν a ± j ν σ a ± j p+1 N-p-1 p N-p + Weak decay becomes weak growth as σ crosses into essential spectrum Leads to leading order decay of Green s function

18 Sketch of proof: i) A, eigenvalues i/2 Meromorphically extend resolvent kernel near σ 0: G(x, y, σ, t) A + B A B -i/2 A: eigenvalues, similar to a spectral projection A 1 σ ūx(x, t) ψ1(y), + 1 ūt(x, t) ψ2(y), σ 1 σ ūx(x, t) X a in c 1(y)e (σ/a in )y + 1 σ ūt(x, t) X a in c 2(y)e (σ/a in )y

19 Sketch of proof: i) pointwise bounds Invert Laplace transform: G(x, t, y) = 1 µ+ i 2 e σt G(x, y, σ, t)dσ 2πi µ 2 i = 1 2πi µ+ i 2 µ i 2 e σt (A + B)dσ = [E 1(x, t, y) + E 2(x, t, y)] + G(x, t, y) {z } {z } A B

20 Sketch of proof: i) pointwise bounds Invert Laplace transform: G(x, t, y) = 1 µ+ i 2 e σt G(x, y, σ, t)dσ 2πi µ 2 i = 1 2πi µ+ i 2 µ i 2 e σt (A + B)dσ = [E 1(x, t, y) + E 2(x, t, y)] + G(x, t, y) Eigenfunction terms: E i (x, t, y) = ū i (x, t)π i (y, t) π i (y, t) = X a in c i "errfn y + a in t p 4(t + 1)! errfn!# y a in t p 4(t + 1)

21 Sketch of proof: i) pointwise bounds Invert Laplace transform: G(x, t, y) = 1 µ+ i 2 e σt G(x, y, σ, t)dσ 2πi µ 2 i = 1 2πi µ+ i 2 µ i 2 e σt (A + B)dσ = [E 1(x, t, y) + E 2(x, t, y)] + G(x, t, y) G(x, t, y) X c 1 e (x y a 4t 4πt out t)2 + X c 2 4πt e (x a out (t y/a in ))2 4t a out + X a in,a+ out c 3 4πt e (x a a out,a in + out (t y/a in ))2 4t

22 Sketch of proof: ii) neutral behavior, iii) nonlinear stability Remove neutral part: define time-dependent phase shift for space and time u(x, t) = ū(x q(t), t τ(t)) + v(x, t) v t = L(t)v + N(v) x + q(ū x + v x) + τū t + (O(τ)v) x Using variation of constants, solution must satisfy: G = ū xπ 1 + ū tπ 2 + G t t q(t) = π 1v 0dy + (π 1) y (N + qv)dyds + (π 1) y (O(τ)v)dyds τ(t) = π 2v 0dy + v(x, t) = R R Gv 0dy t 0 t 0 R R (π 2) y (N + qv)dyds + G y (N + qv)dyds t R 0 R 0 R 0 t R 0 R (π 2) y (O(τ)v)dyds G y (O(τ)v)dyds Set up fixed point iteration scheme to prove existence and convergence result.

23 Summary and Extensions Goal: prove nonlinear stability of time-periodic shocks u(x, t) = ū(x, t) + v(x, t), lim u(x, t) = ū(x q, t τ ) t Mathematical issues overcome Continuous spectrum up to imaginary axis; embedded zero eigenvalue Time-periodic operator; no Floquet theory C v t = L(t)v + N(v) x Extensions: Non-Lax shocks Real viscosity, u xx (B(u)u x) x Sources...

24 Outlook: sources in reaction-diffusion systems Source: u t = Du xx + f (u) Defect Perturbations Perturbations u(x,t) t x x Defect Mathematical Issues: Similar spectral and linear stability issues Spatial end states are spatially-periodic: spatial Floquet spectrum No conservation-law structure, F (u) x; used in nonlinear estimates