Stability of viscous shock waves and beyond

Kevin Zumbrun

Department of MathematicsIndiana University

Sponsored by NSF Grants no. DMS-0300487 and DMS-0801745

Short course, IHP: Lecture 15a

Zumbrun Stability of viscous shock waves

Reprise

In this lecture, RETURN TO ROOTS (= shock waves):

• Original motivations for study: what progress have we made?

• Thermodynamical assumptions: Bethe-Weyl conditions and therole of entropy in viscous and inviscid shock stability.

• Numerical proof: toward viscous Majda’s Theorem for γ-law gas.

Zumbrun Stability of viscous shock waves

I. A 25-year summer research project: inviscid instability,shock splitting, and the viscous selection principle

POSED BY BURTON WENDROFF (Los Alamos internship,summer 1990):

Interesting examples of Smith (more later) of “real gas” satisfyingusual Bethe-Weyl conditions but exhibiting nonuniqueness ofRiemann solutions. Numerics indicate physical selection principlevia stability of component shocks. But, not inviscid stability (infact, both inviscid-stable); rather, numerical viscosity...

QUESTION: Can we validate/confirm this observation by rigorousby rigorous stability analysis?

Interest redoubled in postdoc investigations: undercompressiveshocks in multi-phase flow (Glimm, Marchesin, Plohr).

Zumbrun Stability of viscous shock waves

Revisiting an observation of C. Gardner: the signedLopatinski condition and uniqueness of Riemann solutions

Post-world war II years: D’yakov, Kondrachov, Erpenbeck derivedinviscid stability (Lopatinski) condition. Relation of 1D conditionto uniqueness was pointed out by C. Garner for gas dynamics,giving simple sufficient condition for stability (monotonicity ofHugoniot curves).

For a general system of conservation laws

ut + f (u)x = 0,

Rankine-Hugoniot (shock) conditions:

σ[u] = [f ], (RH)

yielding free-boundary/transmission problem, n constraints,unknowns consisting of speed σ and outgoing modes.

Zumbrun Stability of viscous shock waves

The Lopatinski condition

Counting variables yields Lax condition, number of outgoing modes= n − 1. Checking rank yields Lopatinski (determinant) condition

δ := det(r−1 , . . . , r−p−1, [u], r+

p+1, . . . , r+n ) 6= 0, (1)

where r−j , j = 1, . . . , p − 1 are the “outgoing” modes of

A− := fu(U−) associated with eigenvalues a−j − σ < 0 and r+j ,

j = p + 1, . . . , n are the “outgoing” modes of A+ := fu(U+),associated with eigenvalues a+

j − σ > 0.

(Multi-D analog by Laplace-Fourier analysis, following approachdiscussed for hyperbolic IVBP in lectures of Metivier.)

Zumbrun Stability of viscous shock waves

Relation to Riemann solutions

Riemann (“shock tube”) problem: self-similar solution u(x/t) fordiscontinuous data u0 = u− for x < 0 and u+ for x ≥ 0.Constructed from Riemann (=rarefaction + shock) curves incharacteristic families j = 1, . . . , n, as

u+ = Ψθnn ·Ψ

θ11 (u−),

where θj parametrize motion along Riemann curves. Fixing u−,uniqueness amounts to invertibility of the Rn → Rn map

R : (θ1, . . . , θj)→ u+.

At a viscous p-shock, θ = (0, . . . , 0, θp, 0, . . . 0), Jacobian of R ∼Lopatinski determinant δ = det(r−1 , . . . , r

−p−1, [u], r+

p+1, . . . , r+n )!

Zumbrun Stability of viscous shock waves

Shock splitting: relation to nonuniqueness

Now, fix u− and move along the pth Hugoniot curve

θ = (0, · · · , 0, θp, 0, . . . , 0).

Proposition ([Barker-Freistuhler-Z14)

Change in sign of δ as θp crosses θ∗p implies nonuniquess ofRiemann solutions in the vicinity of the shock solution (u−, u

∗+)

associated with θ∗p.

Proof.

Uniqueness would imply constancy of Brouwer degree of R in avicinity of θ∗ = (0, · · · , 0, θ∗p, 0, . . . , 0) while at the same timeequality of the degree to the sign of the Jacobian of R.

Zumbrun Stability of viscous shock waves

Result: double-valued solution with alternating-sign shocks

Generically, corresponds to fold singularity, double (resp. zero)valued solution on one side of image of the singularity curve.

Moreover, preimages (since Brouwer degree zero is invariant) haveJacobians of opposite sign (summing to zero).

(Simple example is 2× 2 case [Z01].)

Zumbrun Stability of viscous shock waves

Now recall Evans function theory

For small θ, associated shock profiles exist/are transversalconnections of u± in general [Majda-Pego, Freistuhler]. Assumingthat these properties continue to large θ, as holds for example for aWeyl-Bethe gas [Gilbarg51, Menikoff-Plohr89], we may considerthe related question of viscous stability.

In particular, associated to the linearized problem about a viscousprofile, the is an Evans function D(λ), analytic on <λ ≥ 0, andsatisfying the low-frequency relation

D ′(0) = γδ,

where γ is a (nonvanishing by assumption) Wronskian associatedwith the linearized traveling-wave ODE. Moreover, D may be takenreal-valued for γ real.

Zumbrun Stability of viscous shock waves

Conclusion

Evidently, the Stability index

sgnD ′(0)D(+∞real)

counts parity of the number of nonzero nonstable roots <λ ≥ 0.

Hence, change of sign in δ is generically associated with passage ofa root of the Evans function from the unstable (<λ < 0) to thestable (<λ > 0) complex half plane.

Corollary: One of the two Riemann solutions features a viscousunstable shock (the one with wrong sign of δ). Note that neither isinviscid unstable since δ 6= 0. In the case that this passage of aroot through λ = 0 is the first crossing for the viscous problem(typically the case), then the other shock is stable and theassociated Riemann solution is indeed selected by the viscousselection principle.

Zumbrun Stability of viscous shock waves

Comments

• Rather strong conclusions drawn from topological considerations,essentially no computation. No counterpart in inviscid theory.

• Stability index is related to pioneering work of Pego-Weinstein indispersive scalar case. In some cases, directly computable, yieldinginstability/stability results [Gardner-Z, homoclinic uc case; Oh-Z,periodic case]. Surprisingly, noncharacteristic viscous boundarylayers are always unstable for γ-law gase, in the large-amplitudestanding-shock limit [Serre-Z, Z].

• The corresponding low-frequency relation for multi-D yields theperhaps surprising result that viscosity can only destabilize and notstabilize an inviscid shock wave [Z-Serre99].

Zumbrun Stability of viscous shock waves

II. Uniqueness, entropy, and viscous/inviscid shock stability

(with Barker, Freistuhler, Texier)

Quasilinear hyperbolic–parabolic conservation laws:

ut + f (u)x =

{0

ν(b(u)ux)x, u ∈ Rn, ν > 0,

govern compressible (gas, plasma, solid) mechanics. Interestingsolutions [Riemann1856] [Rayleigh1919,Gilbarg1951] are shockwaves

u(x , t) = u(x − σt

ν

), lim

x→±∞u(x) = u±,

propagating large energies coherently over great distance,Stationary solutions of ut − σux + f (u)x = ν(b(u)ux)x .

Zumbrun Stability of viscous shock waves

Stability Theory

Studied since 1960’s (1980’s) [Erpenbeck, Landau, Dy’akov,...][Oleinik, Matsumura, Nishihara, Kawashima, Goodman, Liu, ...] Inprinciple, well understood.

Stability criteria: reduce to spectra of linearized equations,computable as zeros of a Lopatinski (Evans) determinant[Majda1983, Metivier1992,...][Gardner,Howard, Mascia, Serre,Zumbrun, 1998-2006].

All-parameters stability analyses: Combining asympototicanalysis/singular perturbations and numerical Evans analysis,determine stability across full parameter range [Barker,Humpherys, Lafitte, Lewicka, Lyng, Zumbrun, 2009-2012].

Zumbrun Stability of viscous shock waves

Stability Practice

Essentially all waves studied so far (gas, MHD, viscoelasticity, withpolytropic gas laws) are stable!

Could there be a simple mechanism/structure for this?

In particular, could thermodynamic stability, or convexity of theequation of state e = e(τ,S), imply shock stability?

More generally, existence of convex entropy? Forut + f (u)x = ν(bux)x ,

η(u), dηdf = dq, d2η(u)b ≥ 0, ∂t

∫η(u) ≤ 0.

(Associated with symmetrizability, stability of constant solutions.)

Zumbrun Stability of viscous shock waves

Further questions

It is known [Gardner-Z1998,Z-Serre1999] that viscous stabilityimplies inviscid stability. Is this strict? Or do they coincide?

“Viscous destabilization” must occur through passage of nonzeroimaginary roots (same low-frequency analysis), so amounts to Hopfbifurcation, or “galloping,” familiar in detonation.

Could convex entropy prevent existence of nonzero imaginaryeigenvalues for the viscous problem? Or, give a “principleeigenvalue property” preventing imaginary leading mode?

(In either case, viscous and inviscid stability would coincide.)

Zumbrun Stability of viscous shock waves

III. Results: Inviscid gas dynamics

Our inviscid analysis is related to/partly based on important workof R. Smith [Smith1979] on uniqueness of Riemann solutions.

Equations. In Lagrangian coordinates, the Euler equations are

τt − vx = 0,

vt + px = 0,

(e + v2/2)t + (vp)x = 0,

(2)

where τ denotes specific volume, v velocity, e specific internalenergy, and p pressure. Here, p = p(τ, e) is obtained by invertingS = S(τ, e), using T = eS > 0 and p = −eτ .

Zumbrun Stability of viscous shock waves

Assumptions

Ideal gas assumptions [Weyl, Bethe]:

p = p(τ, e) > 0. (Positivity) (J1)

(∂τ − p∂e)p < 0. (Hyperbolicity) (J2)

(∂τ − p∂e)2p > 0. (Genuine nonlinearity) (J3)

pe > 0. (Weyl condition) (J4)

Zumbrun Stability of viscous shock waves

Conditions

Stability/uniqueness conditions [Smith, BFZ]:

− eτseS eττ

< − 1

eτ, or pτ < 0, (Strong)

− eτSeS eττ

< −e2τ

2eeττ+ 1

eτ, or pτ <

p2

2e, (MediumU)

− eτSeS eττ

< −− eτ√

2eeττ+ 1

eτ, or pτ < cp/

√2e, (MediumS)

− eτseS eττ

< − 2

eτ, or pτ <

ppe2. (Weak)

Zumbrun Stability of viscous shock waves

Main inviscid results

Theorem

Assuming (J1)–(J4), stability (for all shocks) is equivalent to(MediumS) while [Smith] uniqueness of Riemann solutions (for anydata) is equivalent to (MediumU). The four conditions are relatedby

(Strong) ⇒ (MediumU) ⇒ (MediumS) ⇒ (Weak).

In particular, condition (Strong) by itself is sufficient to implystability of all shocks, while violation of (Weak) implies existenceof unstable ones.

Corollary

The equation of state e(τ,S) = eS

τ + C 2eS/C2−τ/C , C >> 1, is

convex, satisfies (J1)–(J4), but admits (inviscid) unstable shocks.

Zumbrun Stability of viscous shock waves

Viscous results (i) (numerical)

Numerical Observation A (on gas dynamics). For the aboveequation of state, (a) viscous [in]stability is equivalent to inviscid[in]stability,(b) the viscous-stability problem has no non-zero imaginaryeigenvalues; in particular, passing through the origin, and(c) in situations of instability, the eigenvalue with largest real partis real and simple.

Zumbrun Stability of viscous shock waves

Viscous results (ii) (numerical)

Numerical Observation B (on general systems). There exist3× 3 viscous systems of conservation laws with convex entropy,that(a) admit shocks that are inviscidly stable, but viscously unstable,(b) the viscous-stability problem sometimes does have non-zeroimaginary eigenvalues, while(c) in all situations of instability we investigated numerically, theeigenvalue with largest real part is real, and transitions fromstability to instability occur exclusively by real eigenvalues passingthrough the origin,(d) in some cases, there are an even number of unstable (and allreal) eigenvalues, and(e) in some cases the eigenvalue with largest real part is notsimple.

Zumbrun Stability of viscous shock waves

Discussion

• [C. Gardner1963] as a footnote states that instability is possiblefor convex entropy, but with incorrect examplee(τ,S) = eS/τ + f (S), f ′ >> 1 (satisfies strong, so stable).

• Relative entropy, [Diperna1979,Leger-Vasseur2011]; see also[Golse-Saint Raymond2004]. A question of current interest!

Zumbrun Stability of viscous shock waves

IV. Inviscid analysis: Hugoniot, Lax, and Lopatinski

Integrating TW-ODE −σu′ + f (u)′ = 0 gives jump conditions(Hugoniot relations)

σ[u] = [f ], (RH)

yielding free-boundary/transmission problem, n constraints,unknowns consisting of speed σ and outgoing modes.

Counting variables yields Lax condition, number of outgoing modes= n − 1. Checking rank yields Lopatinski (determinant) condition

δ := det(r−1 , . . . , r−p−1, [u], r+

p+1, . . . , r+n ) 6= 0, (3)

where r−j , j = 1, . . . , p − 1 are the “outgoing” modes of

A− := fu(U−) associated with eigenvalues a−j − σ < 0 and r+j ,

j = p + 1, . . . , n are the “outgoing” modes of A+ := fu(U+),associated with eigenvalues a+

j − σ > 0.

Zumbrun Stability of viscous shock waves

Specialization to gas dynamics

1-Hugoniot curve: Combining the three equations (RH), we obtain[e] + (1/2)(p + p−)[τ ] = 0, with p = p(τ, e) determining a curvein τ , e.

Lax condition: Characteristics 0, ±c , 0, ⇒ Lax 1-shock condition|σ| < c+, where c =

√eττ = sound speed. (Here, σ < 0.)

(Signed) Lopatinski condition: Straightforward (*) computation

gives −(

eτSeS eττ

)+<

|σ|c+

+1

[p] .

Monotonicity condition: IFT gives −(

eτSeS eττ

)+<

σ2

c2+

+1

[p] .

Zumbrun Stability of viscous shock waves

Simplified global conditions

Define now

−( eτSeS eττ

)+<

1

[p]. (Strong’)

−( eτSeS eττ

)+<

2

[p]. (Weak’)

Using 0 < |σ|/c+ < 1, we have the string of implications

(Strong’)⇒monotone⇒Lopatinski ⇒(Weak’). (4)

Zumbrun Stability of viscous shock waves

Proof of Main Theorem

Proof.

Under (J1)–(J4) we have [Weyl, Bethe], denoting as the backward1-Hugoniot through U+, H ′1(U+), the set of all left states U−connected to U+ by a Lax 1-shock,

[τ ] < 0 on H ′1(U+), (P1)

p → 0 as U progresses along H ′1(U+), (P2)

e → 0 as U progresses along H ′1(U+), (P3)

τ is increasing and p decreasing along H ′1(U+). (P4)

(Recall, [e] + (1/2)(p + p−)[τ ] = 0.) Thus, worst case is(e−, p−) = (0, 0), τ− = 2e+/p+ + τ+, giving result.

Zumbrun Stability of viscous shock waves

Proof of Corollary (example of instability)

Proof.

T = eS = eS/τ + eS/C2−τ/C > 0,

p = −eτ = eS/τ2 + CeS/C2−τ/C > 0,

pS = −eSτ = eS/τ2 + C−1eS/C2−τ/C < 0,

pτ = −eττ = −2eS/τ3 − eS/C2−τ/C < 0,

pττ = −eτττ = 6eS/τ4 + C−1eS/C2−τ/C > 0,

At τ+,S+ = 1, 0, by C >> 1, we have failure of (Weak):

−eSτeττ

− 2eS−eτ

=1 + O(C−1)

3 + O(C−1)− 2 + O(C−1)

C + O(1)∼ 1

3> 0.

Zumbrun Stability of viscous shock waves

Numerics: Hugoniot and transition point

(a) −6 −5 −4 −3 −2 −1 0−5

0

5

10

15

S

τ

T

σ

v

(b) −6 −5 −4 −3 −2 −1 0

0

2

4

6

8

10

12

S

τ

(c) −6 −5 −4 −3 −2 −1 00.8

1

1.2

1.4

1.6

1.8

2

S

T

(d) −6 −5 −4 −3 −2 −1 0

85

90

95

100

105

S

p

(e) −6 −5 −4 −3 −2 −1 0−1.8

−1.6

−1.4

−1.2

−1

S

σ

(f ) −6 −5 −4 −3 −2 −1 0

0

2

4

6

8

10

S

v

Figure: The backward 1-Hugoniot curve through (τ+, S+) = (1, 0) for global model

e(τ, S) = eS/τ + eS/C2−τ/C of points (τ, S) connecting to (τ+, S+) by a Lax 1-shock, displayed as a graph

(τ, p, v, e, σ) over S plotted with respective colors (black, green, blue, red, cyan). We zoom in to see (b) the

Hugoniot curve, (c) T over S , (d) p over S , (e) σ over S , and (f) v over S . The value of (τ−, S−) along the

backward Hugoniot curve at which transition to instability occurs is marked by a black X.

Zumbrun Stability of viscous shock waves

Canonical local model

(a)−6 −5 −4 −3 −2 −1 00

5

10

15

S

τ

(b)−6 −5 −4 −3 −2 −1 00

5

10

15

S

τ

Figure: We mark transition to instability with a thick X. (a) Plot of theHugoniot curve for the local model, e(S , τ) = eS/τ + S + τ 2/2,(τ0,S0) = (1, 0). (b) Plot of the the Hugoniot curve for the local modelwith a thick red line and that of the global model for C = 40, 100, 250.Note that as C →∞, the Hugoniot curve of the local model matcheswell that of the global model.

Zumbrun Stability of viscous shock waves

Pause for second lecture

BREAK

Zumbrun Stability of viscous shock waves