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 u ∈ R n u t + f ( u ) x = u xx , Shock wave φ ( ±∞ ) = u ± , u ( x , t ) = φ ( x − st ) , ξ = x − st Non-autonomous linear system on C 2 n Eigenvalue problem for φ : � Df ( φ ) − sI � I ′ = d W ′ = W , (EVP) 0 κ I d ξ � �� � A ( φ,κ ) 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 ). G 2 n d ( C ) Grassmann manifold of d -dimensional subspaces of C 2 n , d = 1 , . . . , 2 n . (EVP) induces a flow on G 2 n d ( C ): X ′ = A d ( φ, κ )( X ) . For fixed φ = u 0 : d -dimensional invariant spaces of A ( u 0 , κ ) fixed points of the induced system.

Autonomous end-systems W ′ = A ( u ± , κ ) W on C 2 n U ± ( κ ), S ± ( κ ) invariant spaces

Autonomous end-systems W ′ = A ( u ± , κ ) W on C 2 n U ± ( κ ), S ± ( κ ) invariant spaces

Autonomous end-systems X ′ = A n ( u ± , κ )( X ) on G 2 n 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 ′ = A n ( u , κ )( X ) on R n × G 2 n n ( C ). H ± : H → G 2 n n ( C ) unstable, stable Evans bundles for φ (analytic): ⇒ W (0 , κ ) ∈ H ± ( κ ) W ( ±∞ , κ ) = 0 ⇐ for all κ ∈ H \ { 0 } and any sol. W ( ξ, κ ) of (EVP). ⇒ H − ( κ ) ∩ H + ( κ ) � = { 0 } κ is an eigenvalue. ⇐

Evans function for φ : � � η − 1 ( κ ) , . . . , η − n ( κ ) , η + 1 ( κ ) , . . . , η + E ( κ ) := det 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 u ∈ R n . u t + f ( u ) x = u xx , λ 1 ( u ) < · · · < λ n ( u ) eigenvalues of Df ( u ); right eigenvectors r j ( u ), j = 1 , . . . , n . • Small shock waves: | u ± − u ∗ | ≪ 1 • Genuinely nonlinear mode ∇ λ k ( u ∗ ) · r k ( u ∗ ) � = 0 : Limit equation u t + ( u 2 ) x = u xx , u ∈ R . [Freist¨ uhler, Szmolyan 2002; Plaza, Zumbrun 2004]

Small shock waves associated with a degenerate mode Let Σ ⊂ R n be a smooth hypersurface with ( r k ( u ) · ∇ ) 2 λ k ( u ) � = 0 . ∀ u ∈ Σ : ∇ λ k ( u ) · r k ( u ) = 0 , (Σ transversal to the vector field r k , outside Σ: ∇ λ k · r k � = 0) Family of small shock waves close to Σ, φ ε ( ±∞ ) = u ± φ ε ( x − s ε t ) , ε , 0 < ε ≤ ε 0 , with λ k ( u − ε ) > s ε > λ k ( u + ε ) (non-characteristic) and end states u − ε = u ∗ − ε r k ( u ∗ ) , u + ε = u ∗ + εα r k ( u ∗ ) + ε 2 w ( u ∗ , ε, α ) , u ∗ ∈ Σ, α ∈ ( − 1 , 1 2 ) fixed, and a vector w ( u ∗ , ε, α ) ⊥ r k ( u ∗ ).

Theorem Let H ± ε : H → G 2 n n ( C ) be the Evans bundles of φ ε and let H ± 0 : H → G 2 1 ( C ) be the Evans bundles of the shock wave φ 0 , φ 0 ( −∞ ) = − 1 , φ 0 (+ ∞ ) = α, of the scalar viscous conservation law u t + ( u 3 ) x = u xx , u ∈ R . Define H ± ε : H → G 2 n n ( C ) by H ± ε ( κ ) = H ± ε ( ε 4 κ ) . It holds: (i) H ± ε converge for ε → 0 as analytic functions, ε = H ± lim ε → 0 H ± 0 ; in suitable coordinates of C 2 n : 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 , ε 0 > 0 s. t. H − ε ( κ ) ∩ H + ∀ ε ∈ [0 , ε 0 ] , | κ | ≥ R : ε ( κ ) = { 0 } .

• Scaled Evans functions E ε ( κ ) = E ε ( ε 4 κ ) of φ ε converge to Evans function E 0 of φ 0 (as analytic functions). • As φ 0 is stable: Theorem implies spectral stability of φ ε . • Nonlinear stability of φ ε 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( λ 0 1 , . . . , λ 0 n ) with λ 0 λ 0 j = λ j (0) , k = 0 .

Reduction of the profile equation φ ε governed by u ′ = f ( u ) − su − c . s = ε 2 ¯ c = ε 3 ¯ Scaling u = ε ¯ yields u , s , c u ′ = ε − 1 f ( ε ¯ u ) − ε 2 ¯ u − ε 2 ¯ ¯ s ¯ c . Slow-fast system in standard form : u ′ j = λ 0 fast ¯ j ¯ u j + O ( ε ) , j � = k , u ′ slow ¯ k = O ( ε ) As λ 0 1 < · · · < λ 0 k − 1 < 0 < λ 0 k +1 < · · · < λ 0 n : For ε = 0: M 0 = { ¯ u j = 0 , j � = k } normally hyperbolic critical manifold

GSPT (Fenichel): M 0 perturbs smoothly to one-dimensional invariant slow manifolds u ∈ R n : ¯ M ε = { ¯ u j = ε h j (¯ u k , ε ) , j � = k } , ε ∈ [0 , ε 1 ] , ∂ 2 f j u k , ε ) = − 1 u 2 with h j (¯ k (0)¯ k + O ( ε ) , j � = k . 2 λ 0 ∂ u 2 j Flow on M ε : k + ε 2 � � u ′ u 2 u 3 s 0 ¯ c 0 ¯ k = ε a ¯ b ¯ k − ¯ u k − ¯ k + O ( ε ) with ∂ 2 f k a = 1 (0) = 1 2 ∇ λ k (0) · r k (0)= 0 ∂ u 2 2 k ∂ 3 f k ∂ 2 f k (0) ∂ 2 f j b = 1 (0) − 1 1 (0) = 1 � 6( r k · ∇ ) 2 λ k | u =0 � = 0 . ∂ u 3 λ 0 ∂ u 2 6 2 ∂ u k ∂ u j k j k j � = k

Assume from now on b = 1 and set τ = ¯ u k . Flow on M ε governed by τ ′ = ε 2 � τ 3 − ¯ � s 0 τ − ¯ c 0 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 ε 2 : Regular perturbation of the profile equation for φ 0

Analysis of the eigenvalue problem Couple the eigenvalue problem with the reduced profile equation to obtain an autonomous system on [ − 1 , α ] × C 2 n : τ ′ = ε 2 � τ 3 − ¯ � s 0 τ − ¯ c 0 k + O ( ε ) , (P) W ′ = A ε,κ [ τ ] W , with � Df ( ε ¯ � u ) − ε 2 ¯ sI I ∈ C 2 n × 2 n . A ε,κ [ τ ] = κ I 0 For ε ≥ 0, κ ∈ H \ { 0 } : A ε,κ [ τ ] is hyperbolic. GSPT: ∀ R > 0 : ∃ ¯ ε > 0 : No eigenvalues in {| κ | > R } for ε < ¯ ε . Argument breaks down in κ = 0: A 0 , 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 ε = r 1 ε 1 , κ = r 2 1 e i ϕ I. κ = ε 2 ζ 2 II. ε = δ 3 r 3 , κ = δ 4 3 r 2 3 e i ϕ III. κ = ε 4 ζ 4 IV. 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 φ ε if ε ≪ 1 and I. ε 2 R 1 ≤ | κ | ≤ R 0 with certain 0 < R 1 < R 0 II. ε 2 R 2 ≤ | κ | ≤ ε 2 R 1 with arbitrary 0 < R 2 < R 1 III. ε 4 R 3 ≤ | κ | ≤ ε 2 R 2 with certain 0 < R 3 < R 2 R > 0: No eigenvalues in { κ : | κ | ≥ ε 4 ¯ ⇒ ∃ ¯ R } if ε ≪ 1. In regime IV: Scaled Evans bundles H ± ε ( κ ) = H ± ε ( ε 4 κ ) converge for ε → 0 to H − 0 ⊕ ˆ H − H + 0 ⊕ ˆ H + 0 ( κ ) = U f 0 ( κ ) ⊕ U ss 0 ( κ ) = S f 0 ( κ ) ⊕ S ss 0 , 0 , uniformly on compact subsets of H . �