Stability of Small Shock Waves Associated with M etivier-Convex - - PowerPoint PPT Presentation

Stability of Small Shock Waves Associated with M´ etivier-Convex Modes

Heinrich Freist¨ uhler1 Joint work with Peter Szmolyan2

1University of Konstanz, Germany 2Vienna 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

∂ ∂tu +

d

• j=1

∂ ∂xj (fj(u)) = 0 system: u ∈ Rn, multidimensional: x ∈ Rd

d

• j=1

ζjDfj(u) symmetric, u ∈ U ⊂ Rn, ζ ∈ Rd. eigenvalues κl(u, ζ), eigenvectors rl(u, ζ), l = 1, . . . , n

Parabolic extension

∂ ∂tu +

d

• j=1

∂ ∂xj (fj(u)) =

d

• j=1

∂2 ∂x2

j

u

Inviscid shock wave

planar shock wave u(x, t) = u−, x · n − st < 0 u+, x · n − st > 0 direction n ∈ Rd, speed s ∈ R left state u−, right state u+ jump condition  −su +

d

• j=1

njfj(u)  

• +

−

= 0

Viscous shock wave

Travelling wave solution u(x, t) = φ(ξ), ξ := x · n − st, lim

ξ→±∞ φ(ξ) = u±

w.l.o.g.: n = (1, 0, . . . , 0) φ′ = f1(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 (E) D(λ, ω) = 0, Reλ ≥ 0, ω ∈ Rd−1, (λ, ω) = (0, 0) (L) ∆(¯ λ, ¯ ω) = 0, Re¯ λ ≥ 0, ¯ ω ∈ Rd−1, |¯ λ|2 + |¯ ω|2 = 1, 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 ∂ ∂tu +

d

• j=1

∂ ∂xj (fj(u)) =

d

• j=1

∂2 ∂x2

j

u assume near some state u∗ and ζ near n : i) The smallest (or largest) eigenvalue κ(u, ζ) of d

j=1 ζjDfj(u)

is simple with eigenvector r(u, ζ) ii) κ is genuinely nonlinear: r(u∗, n)Duκ(u∗, n) > 0 iii) κ is M´ etivier convex with resp. to ζ: D2

ζκ(u∗, n)|n⊥ > 0

⇒ family of small-amplitude viscous κ-shocks φε(x · n − st) with u±

ε = φε(±∞) = u∗ − εr(u∗, n) + O(ε2)

satisfy spectral stability conditions (E) and (L).

The Majda determinant

The (Kreiss-)Majda (Lopatinski) determinant ∆(¯ λ, ¯ ω) ≡ det(R−

1 (¯

λ, ¯ ω), .., R−

p−1(¯

λ, ¯ ω), ¯ λ[u] + i[f ¯

ω(u)], R+ p+1(¯

λ, ¯ ω), .., R+

n (¯

λ, ¯ ω))

• f the given shock is defined on

H ≡ {(¯ λ, ¯ ω) ∈ C × Rd−1 : Re ¯ λ ≥ 0, |¯ λ|2 + ¯ ω2 = 1}, with f ¯

ω ≡ d j=2 ¯

ωjfj and {R−

1 (¯

λ, ¯ ω), . . . , R−

p−1(¯

λ, ¯ ω)}, {R+

p+1(¯

λ, ¯ ω), . . . , R+

n (¯

λ, ¯ ω)} continuous bases for the (extensions to H of) the stable/unstable spaces E−(¯ λ, ¯ ω), E+(¯ λ, ¯ ω) of A∓(¯ λ, ¯ ω) ≡ (¯ λI + iDf ¯

ω(u∓))(Df1(u∓))−1.

Eigenvalue problem

linearization along profile in co-moving frame ∂p ∂t = Lp := ∆p − ∂ ∂ξ [(d f1(φ) − sI)p] −

d

• j=2

d fj(φ) ∂p ∂xj eigenvalue problem: Lp = λp, limξ→±∞ p = 0 translation invariance: eigenvalue λ = 0, p = φ′ spectral stability condition (E): no spectrum in {ℜλ ≥ 0, λ = 0}

Eigenvalue problem as a first order ODE system

Fourier-transform in transversal directions (x2, . . . , xd) → (ω2, . . . , ωd) =: ω ∈ Rd−1 and introduce q := p′ − (Df1(φ) − sI)p to obtain p′ = (Df1(φ) − sI)p + q q′ = (λI + iBω(φ) + |ω|2I)p λ ∈ C, (p, q) ∈ C2n, Bω(φ) := d

j=2 ωjDfj(φ)

briefly: z′ = A(λ, ω, ξ)z, z := (p, q) asymptotically constant coefficients: limξ→±∞ 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 ℜλ ≥ 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(κ0

1, . . . , κ0 n) with κ0 k = 0.

Via scaling φ = εu etc., spectral problem (incl. profile equation) reads u′ = ε−1f1(εu) − εsu − εc p′ = (Df1(εu)) − εsI)p + εq q′ = ε(λI + iBω(u) + |ω|2I)p

Different scales for ε → 0 :

Fast scale u′ = ε−1f1(εu) − εsu − εc p′ = (Df1(εu)) − εsI)p + εq q′ = ε(λI + iBω(u) + |ω|2I)p Slow scale ε ˙ u = ε−1f1(εu) − εsu − εc ε ˙ p = (Df1(εu)) − εsI)p + εq ˙ q = (λI + iBω(u) + |ω|2I)p

Tool: Geometric Singular Perturbation Theory

slow problem fast problem ˙ x = f(x, y) x′ = εf(x, y) ε ˙ y = g(x, y) y′ = g(x, y) reduced problem layer problem ˙ x = f(x, y) x′ = = g(x, y) y′ = g(x, y) Assume on M0 := g−1(0) : det ∂g

∂y = 0. Then

g(x, y) = 0 ⇔ y = h0(x) and reduced flow on M0 = graph(h0): ˙ x = ˆ f0(x) := f(x, h0(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) ∩ iR = ∅, then M0 perturbs regularly to an invariant manifold Mε = graph(hε) with slow flow ˙ x = ˆ fε(x) = f(x, hε(x)).

Second difficulty: D(λ, ω) close to (0, 0)

polar coordinates λ = ρ¯ λ, ω = ρ¯ ω, |¯ λ|2 + |¯ ω|2 = 1, ℜ¯ λ ≥ 0, ρ ≥ 0 spectral problem coupled to profile equation u′ = f1(u) − su − c p′ = (Df1(u)) − sI)p + q q′ = ρ(¯ λI + iB ¯

ω(u) + ρ|¯

ω|2I)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 ≤ ε2r0 middle regime: ε2r0 ≤ ρ ≤ r1

• uter regime: r1 ≤ 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 ∆(λǫ, ωǫ) = 0 for certain (¯ λǫ, ¯ ωǫ) with Re ¯ λǫ = 0, Im ¯ λǫ = 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 ∆(λǫ, ωǫ) = 0 for certain (¯ λǫ, ¯ ωǫ) with Re ¯ λǫ = 0, Im ¯ λǫ = 0. Remark: One might have thought that a 1989 result

• f 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 ∆(λǫ, ωǫ) = 0 for certain (¯ λǫ, ¯ ωǫ) with Re ¯ λǫ = 0, Im ¯ λǫ = 0. Proof: By example:

The counterexample

Consider a system ∂tu + ∂x1f1(u) + ∂x2f2(u) = 0, (1) ∂tv + ∂x1g1(v) + ∂x2g2(v) = 0, (2) where (1) by itself is a symmetrizable hyperbolic system of conservation laws with modes of constant multiplicity, among which at least one, λp, is M´ etivier convex. Assume that for some point u∗ in the state space of (1) and for propagation direction N∗ = (1, 0)⊤, this mode has zero speed, λp(u∗, N∗, 0) = 0.

• E. g., (1) could be the Euler equations for compressible fluid flow

and λp the acoustic mode.

Tune (2) by choosing g1, g2 : R2 → R2 as g1(v1, v2) = s −s v1 v2

• ,

g2(v1, v2) = s s v1 v2

• .

with some s > max{|λ| : λ eigenvalue of Df1(u∗)}. (3) Fix a family of small-amplitude p-shock waves (u−

ǫ , u+ ǫ ) of (1) with

speed 0 and propagation direction N∗, perturbing from u±

0 = u∗.

Augmenting u to U = (u, 0), this family trivially induces a family

• f small-amplitude shock waves (U −

ǫ , U+ ǫ ) of (1). Since the

characteristic speeds of (2) are −s and +s and by virtue of (3), the augmentation increases both the number of outgoing and that

• f incoming modes by 1, on either side of each shock wave.

As (1) and (2) are completely independent from each other ...

..., the Lopatinski determinant ∆ǫ of the shock wave (U −

ǫ , U+ ǫ )

contains a factor δ = det(r−(τ, ξ), r+(τ, ξ)) with r−(τ, ξ), r+(τ, ξ) spanning the stable/unstable spaces of b(τ, ξ) =

• τI + iξ

s s s −s −1 . Writing τ = iσ, and µ = iβ for the possible eigenvalue µ of b(τ, ξ), we see that as det

• σI + ξ

s s

• + β

s −s

• = −s2β2 + (σ2 − s2ξ2),

these spaces coincide at the branchpoints given by σ ∈ {±s(1 + s2)−1/2} and ξ ∈ {±(1 + s2)−1/2}. At these (four) points, δ and thus ∆ǫ vanish!

Work in progress ...

Trying to show that

there are M´ etivier convex modes with arbitrarily small shocks that violate even the weak Kreiss-Lopatinski condition, or all small shocks for M´ etivier convex modes are weakly stable at the inviscid level, and perhaps even all small shocks for M´ etivier convex modes are stable at the viscous level

THANK YOU

PS to I. The Zumbrun-Serre Lemma

Evans function and Majda determinant

Lemma (Zumbrun & Serre): Consider a fixed viscous profile. Then: D(ρ¯ λ, ρ¯ ω) = ρ Γ ∆(¯ λ, ¯ ω) + O(ρ2) as ρ → 0. ∆(¯ λ, ¯ ω) is the Lopatinski-Kreiss-Majda determinant governing inviscid stability of the planar shock wave (u−, u+) as a solution of the hyperbolic conservation law ∂u ∂t +

d

• j=1

∂ ∂xj (fj(u)) = 0. The constant Γ is nonzero iff the profil φ is realized as transverse intersection of the unstable and stable manifolds of the end-states u− resp. u+ of the profile equation u′ = f1(u) − su − c.

Geometric proof of the Zumbrun-Serre lemma

u′ = f1(u) − su − c p′ = (Df1(u)) − sI)p + q q′ = ρ(¯ λI + iB ¯

ω(u) + ρ|¯

ω|2I)p is for ρ → 0 a singularly perturbed system on fast time scale ξ layer problem (fast dynamics): u′ = f1(u) − su − c p′ = (Df1(u) − sI)p + q q′ = normally hyperbolic manifold M of equilibria u = u±, p = −(Df1(u±) − sI)−1q, q ∈ Cn reduced dynamics on M: ˙ q = − ¯ λI + iB ¯

ω(u±)

• (Df1(u±) − sI)−1 q

reduced problem: slow dynamics on M u = u±, p = −(Df1(u±) − sI)−1q, q ∈ Cn ˙ q = − ¯ λI + iB ¯

ω(u±)

Df1(u±) − sI −1 q define slow unstable space U s

0 at u−, slow stable space Ss 0 at u+.

slow - fast decomposition of stable and unstable spaces: D(ρ¯ λ, ρ¯ ω) = det(Uρ, Sρ) = ρ det(U f

0 , Sf 0 ) det(U s 0, ¯

λ[u] + i[f ¯

ω], Ss 0) + O(ρ2)

= ρ Γ ∆(¯ λ, ¯ ω) + O(ρ2)