Spectral stability of small-amplitude traveling waves via geometric - - PowerPoint PPT Presentation

Spectral stability of small-amplitude traveling waves via geometric singular perturbation theory

Johannes W¨ achtler

Konstanz University, Germany

Padova, 28 June 2012

Overview

• Spectral stability, Evans function techniques
• Spectral stability of small shock waves associated with a

degenerate mode

joint work with H. Freist¨ uhler and P. Szmolyan

Spectral stability of shock waves

System of viscous conservation laws ut + f (u)x = uxx, u ∈ Rn Shock wave u(x, t) = φ(x − st), φ(±∞) = u±, ξ = x − st Eigenvalue problem for φ: Non-autonomous linear system on C2n W ′ = Df (φ) − sI I κI

• A(φ,κ)

W ,

′ = d

dξ (EVP) κ ∈ C eigenvalue :⇔ ∃ non-trivial sol. W (ξ, κ), W (±∞, κ) = 0. Due to shift invariance of φ: κ = 0 is an eigenvalue.

φ spectrally stable :⇔ (i) There are no eigenvalues in H := {κ ∈ C : Re κ ≥ 0} except at κ = 0. (ii) The trivial eigenvalue κ = 0 is simple. Spectral stability ⇒ nonlinear stability [Zumbrun, Howard 1998] Tool to find unstable eigenvalues: Evans function E(κ), analytic function with κ eigenvalue ⇐ ⇒ E(κ) = 0.

Evans function techniques

Assume

• φ(ξ) → u± at exponential rate.
• ∀ κ ∈ H \ {0}: A(u±, κ) is hyperbolic with n-dimensional

unstable space U±(κ) and n-dimensional stable space S±(κ) (consistent splitting). Need to track the evolutions of U−(κ) and S+(κ). G2n

d (C) Grassmann manifold of d-dimensional subspaces of C2n,

d = 1, . . . , 2n. (EVP) induces a flow on G2n

d (C):

X ′ = Ad(φ, κ)(X). For fixed φ = u0: d-dimensional invariant spaces of A(u0, κ) fixed points of the induced system.

Autonomous end-systems W ′ = A(u±, κ)W

• n C2n

U±(κ), S±(κ) invariant spaces

Autonomous end-systems W ′ = A(u±, κ)W

• n C2n

U±(κ), S±(κ) invariant spaces

Autonomous end-systems X ′ = An(u±, κ)(X)

• n G2n

n (C)

U±(κ), S±(κ) invariant spaces U±(κ) hyperbolic attractor, S±(κ) hyperbolic repeller

Augmented eigenvalue problem: profile equation + (EVP) u′ = f (u) − su − c, X ′ = An(u, κ)(X)

• n Rn × G2n

n (C).

H± : H → G2n

n (C) unstable, stable Evans bundles for φ (analytic):

W (±∞, κ) = 0 ⇐ ⇒ W (0, κ) ∈ H±(κ) for all κ ∈ H \ {0} and any sol. W (ξ, κ) of (EVP). κ is an eigenvalue. ⇐ ⇒ H−(κ) ∩ H+(κ) = {0}

Evans function for φ: E(κ) := det

• η−

1 (κ), . . . , η− n (κ), η+ 1 (κ), . . . , η+ n (κ)

• with {η±

1 (κ), . . . , η± n (κ)} analytic bases of H±(κ).

E analytic function with κ eigenvalue ⇐ ⇒ E(κ) = 0. φ spectrally stable ⇐ ⇒ (i) E(κ) = 0 for all κ ∈ H \ {0} (ii) E′(0) = 0 [Alexander, Gardner, Jones 1990; Gardner, Zumbrun 1998, . . . ]

Small shock waves

Consider a strictly hyperbolic system ut + f (u)x = uxx, u ∈ Rn. λ1(u) < · · · < λn(u) eigenvalues of Df (u); right eigenvectors rj(u), j = 1, . . . , n.

• Small shock waves: |u± − u∗| ≪ 1
• Genuinely nonlinear mode

∇λk(u∗) · rk(u∗) = 0 : Limit equation ut + (u2)x = uxx, u ∈ R. [Freist¨ uhler, Szmolyan 2002; Plaza, Zumbrun 2004]

Small shock waves associated with a degenerate mode

Let Σ ⊂ Rn be a smooth hypersurface with ∀u ∈ Σ : ∇λk(u) · rk(u) = 0, (rk(u) · ∇)2 λk(u) = 0. (Σ transversal to the vector field rk, outside Σ: ∇λk · rk = 0) Family of small shock waves close to Σ, φε(x − sεt), φε(±∞) = u±

ε ,

0 < ε ≤ ε0, with λk(u−

ε ) > sε > λk(u+ ε ) (non-characteristic) and end states

u−

ε = u∗ − εrk(u∗),

u+

ε = u∗ + εαrk(u∗) + ε2w(u∗, ε, α),

u∗ ∈ Σ, α ∈ (−1, 1

2) fixed, and a vector w(u∗, ε, α) ⊥ rk(u∗).

Theorem

Let H±

ε : H → G2n n (C) be the Evans bundles of φε and let

H±

0 : H → G2 1(C) be the Evans bundles of the shock wave

φ0, φ0(−∞) = −1, φ0(+∞) = α,

• f the scalar viscous conservation law

ut + (u3)x = uxx, u ∈ R. Define H±

ε : H → G2n n (C) by H± ε (κ) = H± ε (ε4κ). It holds:

(i) H±

ε converge for ε → 0 as analytic functions,

limε→0 H±

ε = H± 0 ; in suitable coordinates of C2n:

H−

0 (κ) = H− 0 (κ) ⊕ (C × {0})n−1,

H+

0 (κ) = H+ 0 (κ) ⊕ ({0} × C)n−1.

(ii) There exist R > 0, ε0 > 0 s. t. ∀ ε ∈ [0, ε0], |κ| ≥ R : H−

ε (κ) ∩ H+ ε (κ) = {0}.

• Scaled Evans functions

Eε(κ) = Eε(ε4κ)

• f φε converge to Evans function E0 of φ0 (as analytic

functions).

• As φ0 is stable: Theorem implies spectral stability of φε.
• Nonlinear stability of φε via energy methods: [Fries 2000]
• Systems with a degenerate mode appear in applications

(MHD, elasticity, . . . ).

Sketch of the proof

General idea

• Problem has a slow-fast structure.
• Use geometric singular perturbation theory [Fenichel 1979] to

construct the Evans bundles directly.

• Find scalings to separate the distinct time scales.
• Bundles split into slow and fast components. Use GSPT to

study the limits of the Evans bundles H±

ε for ε → 0.

Assume without loss of generality u∗ = 0, f (0) = 0, Df (0) = diag(λ0

1, . . . , λ0 n) with λ0 j = λj(0),

λ0

k = 0.

Reduction of the profile equation

φε governed by u′ = f (u) − su − c. Scaling u = ε¯ u, s = ε2¯ s, c = ε3¯ c yields ¯ u′ = ε−1f (ε¯ u) − ε2¯ s¯ u − ε2¯ c. Slow-fast system in standard form: fast ¯ u′

j = λ0 j ¯

uj + O(ε), j = k, slow ¯ u′

k = O(ε)

As λ0

1 < · · · < λ0 k−1 < 0 < λ0 k+1 < · · · < λ0 n:

For ε = 0: M0 = {¯ uj = 0, j = k} normally hyperbolic critical manifold

GSPT (Fenichel): M0 perturbs smoothly to one-dimensional invariant slow manifolds Mε = {¯ u ∈ Rn : ¯ uj = εhj(¯ uk, ε), j = k}, ε ∈ [0, ε1], with hj(¯ uk, ε) = − 1

2λ0

j

∂2fj ∂u2

k (0)¯

u2

k + O(ε),

j = k. Flow on Mε: ¯ u′

k = εa¯

u2

k + ε2

b¯ u3

k − ¯

s0¯ uk − ¯ c0

k + O(ε)

• with

a = 1 2 ∂2fk ∂u2

k

(0) = 1 2∇λk(0) · rk(0)= 0 b = 1 6 ∂3fk ∂u3

k

(0) − 1 2

• j=k

1 λ0

j

∂2fk ∂uk∂uj (0)∂2fj ∂u2

k

(0) = 1 6(rk · ∇)2λk|u=0= 0.

Assume from now on b = 1 and set τ = ¯ uk. Flow on Mε governed by τ ′ = ε2 τ 3 − ¯ s0τ − ¯ c0

k + O(ε)

• ,

τ ∈ [−1, α]. Fixed points: τ − = −1 (repelling), τ + = α (attracting). Parametrization of the profile by τ (center-manifold coordinate) Dividing out a factor of ε2: Regular perturbation of the profile equation for φ0

Analysis of the eigenvalue problem

Couple the eigenvalue problem with the reduced profile equation to

• btain an autonomous system on [−1, α] × C2n:

τ ′ = ε2 τ 3 − ¯ s0τ − ¯ c0

k + O(ε)

• ,

W ′ = Aε,κ[τ]W , (P) with Aε,κ[τ] = Df (ε¯ u) − ε2¯ sI I κI

• ∈ C2n×2n.

For ε ≥ 0, κ ∈ H \ {0}: Aε,κ[τ] is hyperbolic. GSPT: ∀R > 0 : ∃¯ ε > 0 : No eigenvalues in {|κ| > R} for ε < ¯ ε. Argument breaks down in κ = 0: A0,0[τ] is not hyperbolic, n + 1 eigenvalues vanish. Point (ε, κ) = (0, 0) not accessible

Blow-up of (ε, κ) = (0, 0)

Scaling regimes I. ε = r1ε1, κ = r2

1 eiϕ

II. κ = ε2ζ2 III. ε = δ3r3, κ = δ4

3r2 3 eiϕ

IV. κ = ε4ζ4 In each regime: (P) has three separated time scales. GSPT is applicable (slow manifolds, . . . ) Bundles split into fast, fast-slow and slow subbundles.

There are no eigenvalues for φε if ε ≪ 1 and

• I. ε2R1 ≤ |κ| ≤ R0 with certain 0 < R1 < R0
• II. ε2R2 ≤ |κ| ≤ ε2R1 with arbitrary 0 < R2 < R1
• III. ε4R3 ≤ |κ| ≤ ε2R2 with certain 0 < R3 < R2

⇒ ∃¯ R > 0: No eigenvalues in {κ : |κ| ≥ ε4 ¯ R} if ε ≪ 1. In regime IV: Scaled Evans bundles H±

ε (κ) = H± ε (ε4κ) converge for ε → 0 to

H−

0 (κ) = Uf 0 ⊕ ˆ

H−

0 (κ) ⊕ Uss 0 ,

H+

0 (κ) = Sf 0 ⊕ ˆ

H+

0 (κ) ⊕ Sss 0 ,

uniformly on compact subsets of H.