Stability of Small Shock Waves Associated with M´ etivier-Convex Modes uhler 1 Heinrich Freist¨ Joint work with Peter Szmolyan 2 1 University of Konstanz, Germany 2 Vienna University of Technology, Austria Padova HYP, June 2012
Overview Spectral stability of extreme shocks The case of non-extreme shocks
I. Spectral stability of extreme shocks
Hyperbolic systems of conservation laws d ∂ ∂ � ∂tu + ( f j ( u )) = 0 ∂x j j =1 system: u ∈ R n , multidimensional: x ∈ R d d � u ∈ U ⊂ R n , ζ ∈ R d . ζ j Df j ( u ) symmetric, j =1 eigenvalues κ l ( u, ζ ) , eigenvectors r l ( u, ζ ) , l = 1 , . . . , n Parabolic extension d d ∂ 2 ∂ ∂ � � ∂tu + ( f j ( u )) = u ∂x 2 ∂x j j j =1 j =1
Inviscid shock wave planar shock wave � u − , x · n − st < 0 u ( x, t ) = u + , x · n − st > 0 direction n ∈ R d , speed s ∈ R left state u − , right state u + jump condition + � d � � � − su + n j f j ( u ) = 0 � � j =1 � −
Viscous shock wave Travelling wave solution ξ →±∞ φ ( ξ ) = u ± u ( x, t ) = φ ( ξ ) , ξ := x · n − st, lim w.l.o.g.: n = (1 , 0 , . . . , 0) φ ′ = f 1 ( φ ) − sφ − c Profile φ is a heteroclinic orbit connecting equilibria u − and u + :
Stability of multidimensional viscous shock waves Theorem (Zumbrun 2001): If a multi-dimensional planar viscous shock wave satisfies the spectral stability conditions D ( λ, ω ) � = 0 , Reλ ≥ 0 , ω ∈ R d − 1 , ( λ, ω ) � = (0 , 0) (E) λ | 2 + | ¯ ω | 2 = 1 , ∆(¯ ω ) � = 0 , Re ¯ ω ∈ R d − 1 , | ¯ (L) λ, ¯ λ ≥ 0 , ¯ then the shock wave is nonlinearly stable. Here, D is the Evans function and ∆ is the Majda determinant (cf. below), with λ spectral parameter, ω transverse wave number. Briefly: spectral stability ⇒ linear stability ⇒ nonlinear stability Remark: According to Gues, M´ etivier, Williams, Zumbrun (2002, ..., 2009, ...), (E) and (L) imply also the validity of the vanishing viscosity limit, even for curved shock fronts.
Spectral stability of small-amplitude shocks Theorem 1 (F. and Szmolyan 2010): For a symmetric system of viscous conservation laws d d ∂ 2 ∂ ∂ � � ∂tu + ( f j ( u )) = u ∂x 2 ∂x j j j =1 j =1 assume near some state u ∗ and ζ near n : i) The smallest (or largest) eigenvalue κ ( u, ζ ) of � d j =1 ζ j Df j ( u ) is simple with eigenvector r ( u, ζ ) ii) κ is genuinely nonlinear: r ( u ∗ , n ) D u κ ( u ∗ , n ) > 0 etivier convex with resp. to ζ : D 2 iii) κ is M´ ζ κ ( u ∗ , n ) | n ⊥ > 0 ⇒ family of small-amplitude viscous κ -shocks φ ε ( x · n − st ) with u ± ε = φ ε ( ±∞ ) = u ∗ − εr ( u ∗ , n ) + O ( ε 2 ) satisfy spectral stability conditions (E) and (L).
The Majda determinant The (Kreiss-)Majda (Lopatinski) determinant ∆(¯ λ, ¯ ω ) ≡ 1 (¯ p − 1 (¯ ω ) , ¯ ω ( u )] , R + p +1 (¯ n (¯ det( R − ω ) , .., R − λ [ u ] + i [ f ¯ ω ) , .., R + λ, ¯ λ, ¯ λ, ¯ λ, ¯ ω )) of the given shock is defined on ω ) ∈ C × R d − 1 : Re ¯ λ | 2 + ¯ ω 2 = 1 } , H ≡ { (¯ λ ≥ 0 , | ¯ λ, ¯ ω ≡ � d with f ¯ j =2 ¯ ω j f j and { R − 1 (¯ ω ) , . . . , R − p − 1 (¯ p +1 (¯ n (¯ { R + ω ) , . . . , R + λ, ¯ λ, ¯ ω ) } , λ, ¯ λ, ¯ ω ) } continuous bases for the (extensions to H of) the stable/unstable spaces E − (¯ ω ) , E + (¯ λ, ¯ λ, ¯ ω ) of A ∓ (¯ ω ) ≡ (¯ λI + iDf ¯ ω ( u ∓ ))( Df 1 ( u ∓ )) − 1 . λ, ¯
Eigenvalue problem linearization along profile in co-moving frame d ∂p ∂t = Lp := ∆ p − ∂ f j ( φ ) ∂p � ∂ξ [( d f 1 ( φ ) − sI ) p ] − d ∂x j j =2 eigenvalue problem: Lp = λp , lim ξ →±∞ p = 0 translation invariance: eigenvalue λ = 0 , p = φ ′ spectral stability condition (E): no spectrum in {ℜ λ ≥ 0 , λ � = 0 }
Eigenvalue problem as a first order ODE system Fourier-transform in transversal directions ( x 2 , . . . , x d ) → ( ω 2 , . . . , ω d ) =: ω ∈ R d − 1 and introduce q := p ′ − ( Df 1 ( φ ) − sI ) p to obtain p ′ = ( Df 1 ( φ ) − sI ) p + q ( λI + iB ω ( φ ) + | ω | 2 I ) p q ′ = B ω ( φ ) := � d ( p, q ) ∈ C 2 n , λ ∈ C , j =2 ω j Df j ( φ ) briefly: z ′ = A ( λ, ω, ξ ) z, z := ( p, q ) asymptotically constant coefficients: lim ξ →±∞ A ( λ, ω, ξ ) = A ± ( λ, ω ) , exponential rate! ω = 0 ⇔ stability problem in one space dimension ( d = 1 )
Unstable and stable bundles, Evans function Theorem: for ℜ λ ≥ 0 , ( λ, ω ) � = (0 , 0) : ∃ n -dimensional unstable and stable spaces U ( λ, ω ) , S ( λ, ω ) analytic in λ , i.e. spaces of initial values (at ξ = 0 ) of solutions to z ′ = A ( λ, ω, ξ ) z decaying for ξ → −∞ , ξ → ∞ , respectively. λ eigenvalue of L ⇔ ∃ ω : U ( λ, ω ) ∩ S ( λ, ω ) � = 0 Evans function D ( λ, ω ) := det [ U ( λ, ω ) , S ( λ, ω )] i) analytic in λ , ii) λ eigenvalue of L ⇔ D ( λ, ω ) = 0 , iii) algebraic multiplicity of eigenvalue equals order of zero Evans, Jones, Gardner, Alexander, Sandstede, Kapitula, Zumbrun, . . .
Evans function, unstable and stable bundles intersection only along ( p, q ) = (0 , 0) ⇒ λ no eigenvalue
First difficulty: singular perturbation w. r. t. ε → 0 ! Assume f (0) = 0 and Df (0) = diag ( κ 0 1 , . . . , κ 0 n ) with κ 0 k = 0 . Via scaling φ = εu etc. , spectral problem (incl. profile equation) reads u ′ ε − 1 f 1 ( εu ) − εsu − εc = p ′ = ( Df 1 ( εu )) − εsI ) p + εq ε ( λI + iB ω ( u ) + | ω | 2 I ) p q ′ =
Different scales for ε → 0 : Fast scale u ′ ε − 1 f 1 ( εu ) − εsu − εc = p ′ = ( Df 1 ( εu )) − εsI ) p + εq q ′ ε ( λI + iB ω ( u ) + | ω | 2 I ) p = Slow scale ε − 1 f 1 ( εu ) − εsu − εc ε ˙ u = ε ˙ p = ( Df 1 ( εu )) − εsI ) p + εq ( λI + iB ω ( u ) + | ω | 2 I ) p ˙ = q
Tool: Geometric Singular Perturbation Theory slow problem fast problem x ′ x ˙ = f ( x, y ) = εf ( x, y ) y ′ ε ˙ = g ( x, y ) = g ( x, y ) y reduced problem layer problem x ′ ˙ = f ( x, y ) = 0 x y ′ 0 = g ( x, y ) = g ( x, y ) Assume on M 0 := g − 1 (0) : det ∂g ∂y � = 0 . Then g ( x, y ) = 0 ⇔ y = h 0 ( x ) and reduced flow on M 0 = graph ( h 0 ) : x = ˆ ˙ f 0 ( x ) := f ( x, h 0 ( x )) Question: Letting 0 < ε < < 1 , what happens to the reduced flow?
Answer: Theorem (Fenichel, 1971): If ∂g ∂y normally hyperbolic, i. e., σ ( ∂g ∂y ) ∩ i R = ∅ , then M 0 perturbs regularly to an invariant manifold M ε = graph ( h ε ) with slow flow x = ˆ ˙ f ε ( x ) = f ( x, h ε ( x )) .
Second difficulty: D ( λ, ω ) close to (0 , 0) polar coordinates λ | 2 + | ¯ ω | 2 = 1 , λ = ρ ¯ | ¯ ℜ ¯ λ, ω = ρ ¯ ω, λ ≥ 0 , ρ ≥ 0 spectral problem coupled to profile equation u ′ = f 1 ( u ) − su − c p ′ = ( Df 1 ( u )) − sI ) p + q ρ (¯ q ′ λI + iB ¯ ω ( u ) + ρ | ¯ ω | 2 I ) p = ρ → 0 long wave limit ρ small : another singular perturbation!
Summary of the proof of Theorem 1 ǫ → 0 for ρ fixed: singular perturbation problem, rescaled profile governed by Burgers equation ( ǫ, ρ ) → (0 , 0) , multi-scale singular perturbation problem inner regime: 0 < ρ ≤ ε 2 r 0 middle regime: ε 2 r 0 ≤ ρ ≤ r 1 outer regime: r 1 ≤ ρ slow-fast decomposition of unstable and stable spaces, viewed as points or manifolds in suitable Grassmann manifolds various rescalings to regain hyperbolicity and transversality at certain points in (¯ λ, ¯ ω ) space Melnikov type arguments to show transversal breaking of ρ = 0 intersections of unstable and stable spaces.
II. The case of non-extreme shocks
Theorem 2 (F. and Szmolyan 2011): The assumption that the shock be extreme (i. e., correspond to smallest or largest char- acteristic speed), cannot be removed without losing something. There are systems for which arbitrarily small shocks ( U − ǫ , U + ǫ ) associated with a non-extreme M´ etivier convex mode have ∆( λ ǫ , ω ǫ ) = 0 for certain (¯ ω ǫ ) with Re ¯ λ ǫ = 0 , Im ¯ λ ǫ , ¯ λ ǫ � = 0 .
Theorem 2 (F. and Szmolyan 2011): The assumption that the shock be extreme (i. e., correspond to smallest or largest char- acteristic speed), cannot be removed without losing something. There are systems for which arbitrarily small shocks ( U − ǫ , U + ǫ ) associated with a non-extreme M´ etivier convex mode have ∆( λ ǫ , ω ǫ ) = 0 for certain (¯ ω ǫ ) with Re ¯ λ ǫ = 0 , Im ¯ λ ǫ , ¯ λ ǫ � = 0 . Remark: One might have thought that a 1989 result of G. M´ etivier would imply that this cannot happen ... .
Theorem 2 (F. and Szmolyan 2011): The assumption that the shock be extreme (i. e., correspond to smallest or largest char- acteristic speed), cannot be removed without losing something. There are systems for which arbitrarily small shocks ( U − ǫ , U + ǫ ) associated with a non-extreme M´ etivier convex mode have ∆( λ ǫ , ω ǫ ) = 0 for certain (¯ ω ǫ ) with Re ¯ λ ǫ = 0 , Im ¯ λ ǫ , ¯ λ ǫ � = 0 . Proof: By example:
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