Stability of isentropic viscous shock profiles in the high-Mach number limit
aa r X i v : . [ m a t h . A P ] M a y Stability of isentropic viscous shock profiles inthe high-Mach number limit
Jeffrey Humpherys , Olivier Laffite , Kevin Zumbrun Department of Mathematics, Brigham Young University, Provo, UT 84602 LAGA, Institut Galilee, Universite Paris 13, 93 430 Villetaneuse and CEASaclay, DM2S/DIR, 91 191 Gif sur Yvette Cedex Department of Mathematics, Indiana University, Bloomington, IN 47402Received: date / Revised version: date
Abstract.
By a combination of asymptotic ODE estimates and nu-merical Evans function calculations, we establish stability of vis-cous shock solutions of the isentropic compressible Navier–Stokesequations with γ -law pressure (i) in the limit as Mach number M goes to infinity, for any γ ≥ M ≥ , γ ∈ [1 , .
5] (demonstrated numerically). This builds onand completes earlier studies by Matsumura–Nishihara and Barker–Humpherys–Rudd–Zumbrun establishing stability for low and inter-mediate Mach numbers, respectively, indicating unconditional stabil-ity, independent of shock amplitude, of viscous shock waves for γ -lawgas dynamics in the range γ ∈ [1 , . γ -values may be treatedsimilarly, but have not been checked numerically. The main idea is toestablish convergence of the Evans function in the high-Mach numberlimit to that of a pressureless, or “infinitely compressible”, gas withadditional upstream boundary condition determined by a boundary-layer analysis. Recall that low-Mach number behavior is incompress-ible.
1. Introduction
The isentropic compressible Navier-Stokes equations in one spatialdimension expressed in Lagrangian coordinates take the form v t − u x = 0 ,u t + p ( v ) x = (cid:16) u x v (cid:17) x , (1) Jeffrey Humpherys et al. where v is specific volume, u is velocity, and p pressure. We assumean adiabatic pressure law p ( v ) = a v − γ (2)corresponding to a γ -law gas, for some constants a > γ ≥ γ > γ = ( n + 2) /n , where n is thenumber of internal degrees of freedom of an individual particle [4]: n = 3 ( γ = 1 . ... ) for monatomic, n = 5 ( γ = 1 .
4) for diatomicgas. For dense fluids, γ is typically determined phenomenologically[19]. In general, γ is usually taken within 1 ≤ γ ≤ viscous shock waves ,or asymptotically-constant traveling-wave solutions( v, u )( x, t ) = (ˆ v, ˆ u )( x − st ) , lim z →±∞ (ˆ v, ˆ u )( z ) = ( v, u ) ± , (3)in agreement with physically-observed phenomena. In nature, suchwaves are seen to be quite stable, even for large variations in pressurebetween v ± . However, it is a long-standing mathematical question towhat extent this is reflected in the continuum-mechanical model (1),that is, for which choice of parameters ( v ± , u ± , γ ) are solutions of (3)time-evolutionarily stable in the sense of PDE; see, for example, thediscussions in [23, 3].The first result on this problem was obtained by Matsumura andNishihara in 1985 [34] using clever energy estimates on the integralsof perturbations in v and u , by which they established stability withrespect to the restricted class of perturbations with “zero mass”, i.e.,perturbations whose integral is zero, for shocks with sufficiently smallamplitude | p ( v + ) − p ( v − ) | ≤ C ( v − , γ ) , with C → ∞ as γ →
1, but
C << ∞ for γ = 1. There followed anumber of works by Liu, Goodman, Szepessy-Xin, and others [28, 15,43, 29] toward the treatment of general, nonzero mass perturbations;see [47, 45, 46] and references therein. A complete result of stabilitywith respect to general L ∩ H perturbations of small-amplitudeshocks of system (1) was finally obtained in 2004 by Mascia andZumbrun [33] using pointwise semigroup techniques introduced byZumbrun and Howard [47, 44] in the strictly parabolic case.The result of [33], together with the small-amplitude spectral sta-bility result of Humpherys and Zumbrun [22] generalizing that of The result of [21] is obtained by energy estimates combining the techniques of[34] with those of [14,15]; a similar approach has been used in [30] to obtain small-amplitude zero-mass stability of Boltzmann shocks. See [37,12] for an alternativeapproach based on asymptotic ODE methods.tability of isentropic viscous shock profiles 3 [34], in fact yields stability of small-amplitude shocks of general sym-metric hyperbolic-parabolic systems, largely settling the problem ofsmall-amplitude shock stability for continuum mechanical systems.However, there remains the interesting question of large-amplitudestability . The main result in this direction, following a general strat-egy proposed in [47], is a “refined Lyapunov theorem” establishedby Mascia and Zumbrun [32, 46] for general symmetric hyperbolic–parabolic systems, stating that linearized and nonlinear L ∩ H → L ∩ H orbital stability (the standard notions of stability) are equiv-alent to spectral stability , or nonexistence of nonstable (nonnegativereal part) eigenvalues of the linearized operator L about the wave,other than the single zero eigenvalue arising through translationalinvariance of the underlying equations.This reduces the problem of large-amplitude stability to the studyof the associated eigenvalue equation ( L − λ ) u = 0, a standard analyt-ically and numerically well-posed (boundary value) problem in ODE,which can be attacked by the large body of techniques developed forasymptotic, exact, and numerical study of ODE. In particular, thereexist well-developed and efficient numerical algorithms to determinethe number of unstable roots for any specific linearized operator L ,independent of its origins in the PDE setting; see, e.g., [9, 10, 11, 8, 22]and references therein. In this sense, the problem of determining sta-bility of any single wave is satisfactorily resolved, or, for that matter,of any compact family of waves. To determine stability of a familyof waves across an unbounded parameter regime, however, is anothermatter. It is this issue that we confront in attempting to decide thestability of general isentropic Navier–Stokes shocks.As pointed out in [47, 21], zero-mass stability implies (and in apractical sense is roughly equivalent to) spectral stability. Thus, theoriginal results of Matsumura and Nishida [34] imply small-amplitudeshock stability for general γ and large-amplitude stability as γ → γ ∈ [1 , ≤ M ≤ , M is the Mach number associated with the shock. For dis-cussion of the Evans function, see Section 2.4. Recall that Machnumber is an alternative measure of shock strength, with 1 corre-sponding to | p ( v + ) − p ( v − ) | = 0 and M → ∞ corresponding to | p ( v + ) − p ( v − ) | → ∞ ; see Appendix A, [2], or Section 2.1 below.Mach 3 ,
000 is far beyond the hypersonic regime M ∼ encoun-tered in current aerodynamics. However, the mathematical questionof stability across arbitrary γ , M remains open up to now. “Refined” because the linearized operator L does not possess a spectral gap,hence e Lt decays time-algebraically and not exponentially; see [47,46] for furtherdiscussion. Jeffrey Humpherys et al. In this paper, we resolve this question, using a combination ofasymptotic ODE estimates and Evans function calculations to con-clude, first, stability of isentropic Navier–Stokes shocks in the largeMach number limit M → ∞ for any γ ≥
1, and, second, stability forall M ≥ ,
500 for γ ∈ [1 , .
5] (for γ ∈ [1 , M ≥ unconditional sta-bility for γ ∈ [1 , . , independent of shock amplitude . As in [2], ournumerical study is not a numerical proof, but contains the neces-sary ingredients for one; see discussion, Section 6. The restriction to γ ∈ [1 , .
5] is an arbitrary one coming from the choice of parameterson which the numerical study [2] was carried out; stability for other γ can be easily checked as well.Our method of analysis is straightforward, though somewhat del-icate to carry out. Working with the rescaled and conveniently rear-ranged versions of the equations introduced in [2], we observe that theassociated eigenvalue equations converge uniformly as Mach numbergoes to infinity on a “regular region” x ≤ L , for any fixed L >
0, toa limiting system that is well-behaved (hence treatable by the stan-dard methods of [31, 37, 2]) in the sense that its coefficient matrixconverges uniformly exponentially in x to limits at x = ±∞ , but isunderdetermined at x = + ∞ .On the complementary “singular region” x ≥ L , the convergenceis only pointwise due to a fast “inner structure” featuring rapid vari-ation of the converging coefficient matrices near x = + ∞ , but the be-havior at x = + ∞ is of course determinate. Performing a boundary-layer analysis on the singular region and matching across x = L , weare able to show convergence of the Evans function of the originalsystem as the Mach number goes to infinity to an Evans function ofthe limiting system with an appropriately imposed additional condi-tion at x = + ∞ , upstream of the shock. This reduces the question ofstability in the high-Mach number limit to existence or nonexistenceof zeroes of the limiting Evans function on ℜ eλ ≥
0, a question thatcan be resolved by routine numerical computation as in [2], or byenergy estimates as in Appendix B.The limiting system can be recognized as the eigenvalue equationassociated with a pressure-less ( γ = 0) gas, that is, the “infinitely-compressible” limit one might expect as the Mach number goes toinfinity. Recall that behavior in the low Mach number limit is incom-pressible [25, 27, 18]. However, the upstream boundary condition hasto our knowledge no such simple interpretation. Indeed, to carry outthe boundary-layer analysis by which we derive this condition is themain technical difficulty of the paper.Besides their independent interest, the results of this paper seemsignificant as prototypes for future analyses. Our calculations use tability of isentropic viscous shock profiles 5 some properties specific to the structure of (1). In particular, wemake use of surprisingly strong energy estimates carried out in [2]in confining unstable eigenvalues to a bounded set independent ofshock strength (or Mach number). Also, we use the extremely simplestructure of the eigenvalue equation to carry out the key analysis ofthe eigenvalue flow in the singular region near x = + ∞ essentiallyby hand. However, these appear to be conveniences rather than es-sential aspects of the analysis. It is our hope that the basic argumentstructure of this paper together with [2] can serve as a blueprint forthe study of large-amplitude stability in more general situations.In particular, we expect that the analysis will carry over to the full(nonisentropic) equations of gas dynamics with ideal gas equation ofstate, which, formally, decouple in the high-Mach number limit intothe equations of isentropic pressureless gas dynamics studied here,augmented with a separate temperature equation governed by simplediffusion/Fourier’s law.
2. Preliminaries
We begin by recalling a number of preliminary steps carried out in [2].Making the standard change of coordinates x → x − st , we considerinstead stationary solutions ( v, u )( x, t ) ≡ (ˆ v, ˆ u )( x ) of v t − sv x − u x = 0 ,u t − su x + ( a v − γ ) x = (cid:16) u x v (cid:17) x . (4)Under the rescaling( x, t, v, u ) → ( − εsx, εs t, v/ε, − u/ ( εs )) , (5)where ε is chosen so that 0 < v + < v − = 1, our system takes theconvenient form v t + v x − u x = 0 ,u t + u x + ( av − γ ) x = (cid:16) u x v (cid:17) x , (6)where a = a ε − γ − s − . Steady shock profiles of (6) satisfy v ′ − u ′ = 0 ,u ′ + ( av − γ ) ′ = (cid:18) u ′ v (cid:19) ′ , Jeffrey Humpherys et al. subject to boundary conditions ( v, u )( ±∞ ) = ( v ± , u ± ), or, simplify-ing, v ′ + ( av − γ ) ′ = (cid:18) v ′ v (cid:19) ′ . Integrating from −∞ to x , we get the profile equation v ′ = H ( v, v + ) := v ( v − a ( v − γ − , (7)where a is found by setting x = + ∞ , thus yielding the Rankine-Hugoniot condition a = − v + − v − γ + − v γ + − v + − v γ + . (8)Evidently, a → γ − in the weak shock limit v + →
1, while a ∼ v γ + inthe strong shock limit v + →
0. The associated Mach number M maybe computed as in [2], Appendix A, as M = ( γa ) − / (9)so that M ∼ γ − / v − γ/ → + ∞ as v + → M → v + → that is, the high-Mach number limit corresponds to the limit v + → .2.2. Eigenvalue equations Linearizing (6) about the profile (ˆ v, ˆ u ), we obtain the eigenvalue prob-lem λv + v ′ − u ′ = 0 ,λu + u ′ − (cid:18) h (ˆ v )ˆ v γ +1 v (cid:19) ′ = (cid:18) u ′ ˆ v (cid:19) ′ , (10)where h (ˆ v ) = − ˆ v γ +1 + a ( γ −
1) + ( a + 1)ˆ v γ . (11)We seek nonstable eigenvalues λ ∈ {ℜ e ( λ ) ≥ } \ { } , i.e., λ forwhich (10) possess a nontrivial solution ( v, u ) decaying at plus andminus spatial infinity. As pointed out in [47, 21], by divergence formof the equations, such solutions necessarily satisfy R + ∞−∞ v ( x ) dx = R + ∞−∞ u ( x ) dx = 0, from which we may deduce that˜ u ( x ) = Z x −∞ u ( z ) dz, ˜ v ( x ) = Z x −∞ v ( z ) dz tability of isentropic viscous shock profiles 7 and their derivatives decay exponentially as x → ∞ . Substituting andthen integrating, we find that (˜ u, ˜ v ) satisfies the integrated eigenvalueequations (suppressing the tilde) λv + v ′ − u ′ = 0 , (12a) λu + u ′ − h (ˆ v )ˆ v γ +1 v ′ = u ′′ ˆ v . (12b)This new eigenvalue problem differs spectrally from (10) only at λ =0, hence spectral stability of (10) is equivalent to spectral stability of(12). Moreover, since (12) has no eigenvalue at λ = 0, one can expectmore uniform stability estimates for the integrated equations in thevicinity of λ = 0 [34, 14, 47]. The following estimates established in [2] indicate the suitability ofthe rescaling chosen in Section 2.1. For completeness, we prove thesein Appendix A.
Proposition 1 ([2]).
For each γ ≥ , < v + ≤ , (7) has a unique(up to translation) monotone decreasing solution ˆ v decaying to itsendstates with a uniform exponential rate, independent of v + , γ . Inparticular, for < v + ≤ and ˆ v (0) := v + + , | ˆ v ( x ) − v + | ≤ (cid:16) (cid:17) e − x x ≥ , (13a) | ˆ v ( x ) − v − | ≤ (cid:16) (cid:17) e x +122 x ≤ . (13b) Proposition 2 ([2]).
Nonstable eigenvalues λ of (12) , i.e., eigen-values with nonnegative real part, are confined for any < v + ≤ tothe region ℜ e ( λ ) + |ℑ m ( λ ) | ≤ (cid:16) √ γ + 12 (cid:17) . (14) Following [2], we may express (12) concisely as a first-order system W ′ = A ( x, λ ) W, (15) A ( x, λ ) = λ
10 0 1 λ ˆ v λ ˆ v f (ˆ v ) − λ ! , W = uvv ′ ! , ′ = ddx , (16) Jeffrey Humpherys et al. where f (ˆ v ) = ˆ v − ˆ v − γ h (ˆ v ) = 2ˆ v − a ( γ − v − γ − ( a + 1) , (17)with h as in (11) and a as in (8), or, equivalently, f (ˆ v ) = 2ˆ v − ( γ − (cid:16) − v + − v γ + (cid:17)(cid:16) v + ˆ v (cid:17) γ − (cid:16) − v + − v γ + (cid:17) v γ + − . (18)Eigenvalues of (12) correspond to nontrivial solutions W for whichthe boundary conditions W ( ±∞ ) = 0 are satisfied. Because A ( x, λ )as a function of ˆ v is asymptotically constant in x , the behavior near x = ±∞ of solutions of (16) is governed by the limiting constant-coefficient systems W ′ = A ± ( λ ) W, A ± ( λ ) := A ( ±∞ , λ ) , (19)from which we readily find on the (nonstable) domain ℜ eλ ≥ λ = 0 of interest that there is a one-dimensional unstable manifold W − ( x ) of solutions decaying at x = −∞ and a two-dimensional stablemanifold W +2 ( x ) ∧ W +3 ( x ) of solutions decaying at x = + ∞ , each ofwhich may be chosen analytically in λ . With additional care, thesemay be extended analytically to the whole set ℜ eλ ≥
0, i.e., to λ = 0[13]. Defining the Evans function D associated with operator L asthe analytic function D ( λ ) := det( W − W +2 W +3 ) | x =0 , (20)we find that eigenvalues of L correspond to zeroes of D both in loca-tion and multiplicity; see, e.g., [1, 13, 31, 46] for further details.Equivalently, following [36, 2], we may express the Evans functionas D ( λ ) = (cid:0)f W +1 · W − (cid:1) | x =0 , (21)where f W +1 ( x ) spans the one-dimensional unstable manifold of solu-tions decaying at x = + ∞ (necessarily orthogonal to the span of W +2 ( x ) and W +3 ( x )) of the adjoint eigenvalue ODE f W ′ = − A ( x, λ ) ∗ f W . (22)The simpler representation (21) is the one that we shall use here.
3. Description of the main results
We can now state precisely our main results. tability of isentropic viscous shock profiles 9
Under the strategic rescaling (5), both profile and eigenvalues equa-tions converge pointwise as v + → v + = 0.The limiting profile equation (the limit of (7)) is evidently v ′ = v ( v − , (23)with explicit solution ˆ v ( x ) = 1 − tanh( x/ , (24)while the limiting eigenvalue system (the limit of (16)) is W ′ = A ( x, λ ) W, (25) A ( x, λ ) = λ
10 0 1 λ ˆ v λ ˆ v f (ˆ v ) − λ ! , (26)where f (ˆ v ) = 2ˆ v − − tanh( x/ . (27)Indeed, this convergence is uniform on any interval ˆ v ≥ ǫ >
0, or,equivalently, x ≤ L , for L any positive constant, where the sequence istherefore a regular perturbation of its limit. We will call x ∈ ( −∞ , L ]the “regular region” or “regular side”. For ˆ v → x → ∞ , the limit is less well-behaved, as may be seen by the factthat ∂f /∂ ˆ v ∼ ˆ v − as ˆ v → v + , a consequence of the appearance of (cid:0) v + ˆ v (cid:1) in the expression (18) for f . Similarly, in contrast to ˆ v , A ( x, λ )does not converge to A + ( λ ) as x → + ∞ with uniform exponentialrate independent of v + , γ , but rather as C ˆ v − e − x/ . We call x ∈ [ L, + ∞ ) therefore the “singular region ” or “singular side”. (This isnot a singular perturbation in the usual sense but a weaker type ofsingularity, at least as we have framed the problem here.) We should now like to define a limiting Evans function following theasymptotic Evans function framework introduced in [37] and establishconvergence to this function in the v + → v + → For, the limiting coefficient matrix A ( λ ) := A (+ ∞ , λ ) = λ
10 0 10 0 − − λ ! (28)is nonhyperbolic (in ODE sense) for all λ , having eigenvalues 0 , , − − λ ; in particular, the stable manifold drops to dimension one in thelimit v + →
0. Thus, the subspace in which W +2 and W +3 should beinitialized at x = + ∞ is not self-determined by (28), but must bededuced by a careful study of the double limit v + → x → + ∞ .But, the computationlim v + → A (+ ∞ , λ ) = λ
10 0 10 0 − γ − λ ! = A ( λ ) = lim x →∞ lim v + → A ( x, λ )(29)shows that these limits do not commute, except in the special case γ = 1 already treated in [34] by other methods.The rigorous treatment of this issue is the main work of the paper.However, the end result can be easily motivated on heuristic grounds.A study of lim v + → A (+ ∞ , λ ) on the set ℜ eλ ≥ ∗ , ∗ , T associated with eigenvalue − γ − λ of strictlynegative real part and a two-dimensional (center) subspace of neutralmodes ( r, T associated with Jordan block (cid:18) λ (cid:19) , of which there isonly a single genuine eigenvector (1 , , T . For v + small, therefore, A + ( λ ) has also a single fast, decaying, eigenmode with eigenvaluenear − γ − A + has dimension two for ℜ eλ ≥ λ = 0 and theunstable subspace dimension one).Focusing on the single slow decaying eigenvector of A + , and con-sidering its limiting behavior as v + →
0, we see immediately that itmust converge in direction to ± (1 , , T . For, the sequence of direc-tion vectors, since continuously varying and restricted to a compactset, has a nonempty, connected set of accumulation points, and thesemust be eigenvectors of lim v + → A + with eigenvalues near zero. Since ± (1 , , T are the unique candidates, we obtain the result. Indeed,both growing and decaying slow eigenmodes must converge to thiscommon direction, making the limiting analysis trivial.The same argument shows that ± (1 , , T is the limiting directionof the slow stable eigenmode of A ( x, λ ) as x → + ∞ , that is, in thealternate limit lim x →∞ lim v + → A ( x, λ ). That is, V +2 := (1 , , T isthe common limit of the slow decaying eigenmode in either of the tability of isentropic viscous shock profiles 11 two alternative limits lim v + → A + and lim v + → A + ; it thus seems areasonable choice to use this limiting slow direction to define an Evansfunction for the limiting system (26). On the other hand, the stableeigenmode V := ( a − ( λ/a + 1) , a − , T ,a = − − λ , of A is forced on us by the system itself, independentof the limiting process.Combining these two observations, we require that solutions W and W of the limiting eigenvalue system (26) lie asympotically indirections V and V , respectively, thus determining a limiting, or“reduced” Evans function D ( λ ) := det( W − W W ) | x =0 , (30)or alternatively D ( λ ) = (cid:0)f W · W − (cid:1) | x =0 , (31)with f W defined analogously as a solution of the adjoint limitingsystem lying asymptotically at x = + ∞ in direction e V := (0 , , ¯ a − ) T = (0 , , (1 + ¯ λ ) − ) T (32)orthogonal to the span of V and V , where “ ¯ ” denotes complexconjugate. (The prescription of W − in the regular region is straight-forward: it must lie on the one-dimensional unstable manifold of A − as in the v + > Alternatively, the limiting equations may be derived by taking a for-mal limit as v + → a ∼ v γ + , to obtain a limiting evolution equation v t + v x − u x = 0 ,u t + u x = (cid:16) u x v (cid:17) x (33)corresponding to a pressure-less gas , or γ = 0, then deriving profileand eigenvalue equations from (33) in the usual way. This gives someadditional insight on behavior, of which we make important mathe-matical use in Appendix B. Physically, it has the interpretation that,in the high-Mach number limit v + →
0, effects of pressure are con-centrated near x = + ∞ on the infinite-density side, as encoded inthe special upstream boundary condition ( u, u ′ , v, v ′ ) → c (1 , , , x → + ∞ for the integrated eigenvalue equation, which may be seento be equivalent to the conditions imposed on W + j in the previoussubsection. Defining D as in (30)–(31), we have the following main theorems. Theorem 1.
For λ in any compact subset of ℜ eλ ≥ , D ( λ ) con-verges uniformly to D ( λ ) as v + → . Corollary 1.
For any compact subset Λ of ℜ eλ ≥ , D is nonvan-ishing on Λ for v + sufficiently small if D is nonvanishing on Λ , andis nonvanishing on the interior of Λ only if D is nonvanishing there.Proof. Standard properties of uniform limit of analytic functions.
Corollary 2.
Isentropic Navier–Stokes shocks are stable in the high-Mach number limit v + → if D is nonvanishing on the wedge Λ : ℜ e ( λ ) + |ℑ m ( λ ) | ≤ (cid:16) √ γ + 12 (cid:17) , ℜ eλ ≥ and only if D is nonvanishing on the interior of Λ .Proof. Corollary 1 together with Proposition 2.
Remark 1.
Likewise, on any compact subset of ℜ eλ ≥ | D | is uni-formly bounded from zero for v + sufficiently small ( M sufficientlylarge) if and only if | D | is uniformly bounded from zero. Thus, isen-tropic Navier–Stokes shocks are “uniformly stable” for sufficientlysmall v + , in the sense that | D | is bounded from below independentof v + , if and only if D is nonvanishing on Λ as defined in (34).The following result completes our abstract stability analysis. Theproof, given in Appendix B, is by an energy estimate analogous tothat of [34]. Proposition 3.
The limiting Evans function D is nonzero on ℜ eλ ≥ . Corollary 3.
For any γ ≥ , isentropic Navier–Stokes shocks arestable for Mach number M sufficiently large (equivalently, v + suffi-ciently small). Stability for γ = 1, proved in [34], already implies nonvanishing of D outsidethe imaginary interval [ − i p / , + i p / Unfortunately, the energy estimate used to establish 3, though math-ematically elegant, yields only the stated, abstract result and notquantitative estimates. A simpler and more general approach, thatdoes yield quantitative information, is to compute the reduced Evansfunction numerically. We carry out this by-now routine numericalcomputation using the methods of [2]. Specifically, we map a semi-circle ∂ ( {ℜ eλ ≥ } ∩ {| λ | ≤ } ) enclosing Λ for γ ∈ [1 ,
3] by D and compute the winding number of its image about the origin todetermine the number of zeroes of D within the semicircle, and thuswithin Λ . For details of the numerical algorithm, see [2, 11, 21].The result is displayed in Figure 1, clearly indicating stability.More precisely, the minimum of | D | on the semicircle is found to be ≈ . M ≥
50 for γ ∈ [1 , M ≥ , γ ∈ [2 , . M ≥ ,
000 for γ ∈ [2 . , v + = 10 − . In Figure 2, we superimpose on the imageof the semicircle by D its (again numerically computed) image bythe full Evans function D , for a monatomic gas γ = 1 . ... at succes-sively higher Mach numbers v + = ,graphically demonstrating the convergence of D to D as v + ap-proaches zero.Indeed, we can see a great deal more from Figure 2. For, note thatthe displayed contours are, to the scale visible by eye, “monotone” in v + , or nested, one within the other (they do not appear so at smallerscales). Thus, lower-Mach number contours are essentially “trapped”within higher-Mach number contours, with the worst-case, outmostcontour corresponding to the limiting Evans function D . From thisobservation, we may conclude with confidence stability down to thesmallest value M ≈ . v + = 10 − . That is, a great deal of topological informa-tion is encoded in the analytic family of Evans functions indexed by v + , from which stability may be deduced almost by inspection. Be-havior for other γ ∈ [0 ,
3] is similar. See, for example, the case γ = 3displayed in Figure 4, which is virtually identical to Figure 2. Such topological information does not seem to be available fromother methods of investigating stability such as direct discretationof the linearized operator about the wave [26] or studies based onlinearized time-evolution or power methods [5, 6]. This represents inour view a significant advantage of the Evans function formulation. In particular, Figure 4 indicates stability down to v + = 10 − , or Mach number ∼
20, from which we may conclude unconditional stability on the whole range γ ∈ [1 ,
3] of [2].4 Jeffrey Humpherys et al. −2 −1 0 1 2 3 4 5−4−3−2−101234 Re I m Fig. 1.
The image of the semi-circular contour via the Evans function for thereduced system. Note that the winding number of this graph is zero. Hence, thereare no unstable eigenvalues in the semi-circle.
Remark 2.
Recall that the Evans function is not determined uniquely,but only up to a nonvanishing analytic factor [1, 13]. The simple con-tour structure in Figure 2 is thus partly due to a favorable choiceof D induced by the initialization at ±∞ by Kato’s ODE [24], asdescribed in [11, 22, 2]. A canonical algorithm for tracking bases ofevolving subspaces, this in some sense minimizes “action”; see [20]for further discussion. Remark 3.
Note that the limiting equations, and the limiting Evansfunction D are both independent of γ . To study high-Mach numberstability for γ >
3, therefore, requires only to examine D on suc-cessively larger semicircles. Thus, our methods in combination withthe those of [2], allow us to determine stability in principle over anybounded interval in γ , for γ > M ≥ Remark 4.
As Figure 2 suggests, an alternative method for determin-ing stability, without reference to D , is to compute the full Evansfunction for sufficiently high Mach number. That is, nonvanishing of D , and thus stability of sufficiently high Mach number shocks for γ ∈ [1 , tability of isentropic viscous shock profiles 15 −2 −1 0 1 2 3 4 5−4−3−2−101234 Re I m Fig. 2.
Convergence to the limiting Evans function as v + → γ = 1 . ... . The contours depicted, going from inner to outer, are images ofthe semicircle under D for v + = . The outermostcontour is the image under D , which is nearly indistinguishable from the imagefor v + = . study of [2] together with the fact that a limit D → D exists (seealso Remark 1). The analytical result of Corollary 3 guarantees stability for γ ≥ M sufficiently large. For γ ∈ [1 , . M ≥ ,
500 by a crude Rouche bound, and indeed muchlower if further structure is taken into account. Together with thesmall and intermediate Mach number studies of [34, 2] for M ≤ , γ ∈ [1 , .
5] and M ≥
1. There is no inherent restriction to γ ∈ [1 , . γ to determine stability (or instability)for all M ≥
1. Indeed, our method of analysis indicates that thelarge- γ limit is quite analogous to the high-Mach number limit v + → −2 −1 0 1 2 3 4 5−4−3−2−101234 Re I m Fig. 3.
Convergence to the limiting Evans function as v + → γ = 1. Thecontours depicted, going from inner to outer, are images of the semicircle under D for v + = . The outermost contour is the imageunder D , which is nearly indistinguishable from the image for v + = .
0, suggesting the possibility to establish still more general resultsencompassing all γ ≥ M ≥ v + is virtually independent of the value of γ .This also indicates that v + and not M is the more useful measure ofshock strength in this regime.
4. Boundary-layer analysis
We now carry out the main work of the paper, analyzing the flow of(16) in the singular region. Our starting point is the observation that A ( x, λ ) = λ
10 0 1 λ ˆ v λ ˆ v f (ˆ v ) − λ ! (35) tability of isentropic viscous shock profiles 17 −2 −1 0 1 2 3 4 5−4−3−2−101234 Re I m Fig. 4.
Convergence to the limiting Evans function as v + → γ = 3. Thecontours depicted, going from inner to outer, are images of the semicircle under D for v + = . The outermost contour is the imageunder D , which is nearly indistinguishable from the image for v + = . is approximately block upper-triangular for ˆ v sufficiently small, withdiagonal blocks (cid:18) λ (cid:19) and ( f (ˆ v ) − λ ) that are uniformly spectrallyseparated on ℜ eλ ≥
0, as follows by f (ˆ v ) ≤ v − ≤ − / . (36)We exploit this structure by a judicious coordinate change convert-ing (16) to a system in exact upper triangular form, for which thedecoupled “slow” upper lefthand 2 × regular per-turbation that can be analyzed by standard tools introduced in [37].Meanwhile, the fast, lower righthand 1 × ond, we introduce a more stable method of block-reduction taking ac-count of usually negligible derivative terms in the definition of block-triangularizing transformations, which, if ignored, would in this caselead to unacceptably large errors. We first block upper-triangularize by a static (constant) coordinatetransformation the limiting matrix A + = A (+ ∞ , λ ) = λ
10 0 1 λv + λv + f ( v + ) − λ ! (37)at x = + ∞ using special block lower-triangular transformations R + := (cid:18) I λv + θ + (cid:19) , L + := R − = (cid:18) I − λv + θ + (cid:19) , (38)where I denotes the 2 × θ + ∈ C × is a 1 × Lemma 1.
On any compact subset of ℜ eλ ≥ , for each v + > sufficiently small, there exists a unique θ + = θ + ( v + , λ ) such that ˆ A + := L + A + R + is upper block-triangular, ˆ A + = (cid:18) λ ( J + v + θ + ) 110 f (ˆ v ) − λ − λv + θ + (cid:19) , (39) where J = (cid:18) (cid:19) and
11 = (cid:18) (cid:19) , satisfying a uniform bound | θ + | ≤ C. (40) Proof.
Setting the 2 − A + to zero, we obtain the matrixequation θ + ( aI − λJ ) = − T + λv + θ + θ + , where a = f ( v + ) − λ , or, equivalently, the fixed-point equation θ + = ( aI − λJ ) − (cid:16) − T + λv + θ + θ + (cid:17) . (41)By det( aI − λJ ) = a = 0, ( aI − λJ ) − is uniformly bounded oncompact subsets of ℜ eλ ≥ ℜ eλ ≥ | λ | bounded and v + sufficiently small,there exists a unique solution by the Contraction Mapping Theorem,which, moreover, satisfies (40). tability of isentropic viscous shock profiles 19 Defining now Y := L + W andˆ A ( x, λ ) = L + A ( x, λ ) R + ( x, λ )= λ ( J + v + θ + ) 11 λ (ˆ v − v + )11 T − λv + ( f (ˆ v ) − f ( v + )) θ + f (ˆ v ) − λ − λv + θ + , (42)we have converted (16) to an asymptotically block upper-triangularsystem Y ′ = ˆ A ( x, λ ) Y, (43)with ˆ A + = ˆ A (+ ∞ , λ ) as in (39). Our next step is to choose a dynamic transformation of the same form˜ R := (cid:18) I Θ (cid:19) , ˜ L := ˜ R − = (cid:18) I − ˜ Θ (cid:19) , (44)converting (43) to an exactly block upper-triangular system, with˜ Θ uniformly exponentially decaying at x = + ∞ : that is, a regularperturbation of the identity. Lemma 2.
On any compact subset of ℜ eλ ≥ , for L sufficientlylarge and each v + > sufficiently small, there exists a unique Θ = Θ + ( x, λ, v + ) such that ˜ A := ˜ L ˆ A ( x, λ ) ˜ R + ˜ L ′ ˜ R is upper block-triangular, ˜ A = (cid:18) λ ( J + v + θ + ) + 11 ˜ Θ f (ˆ v ) − λ − λθ + − ˜ Θ (cid:19) , (45) and ˜ Θ ( L ) = 0 , satisfying a uniform bound | ˜ Θ ( x, λ, v + ) | ≤ Ce − ηx , η > , x ≥ L, (46) independent of the choice of L , v + .Proof. Setting the 2 − A to zero and computing˜ L ′ ˜ R = (cid:18) − ˜ Θ ′ (cid:19) (cid:18) I Θ I (cid:19) = (cid:18) − ˜ Θ ′ , (cid:19) we obtain the matrix equation˜ Θ ′ − ˜ Θ (cid:0) aI − λ ( J + v + θ + ) (cid:1) = ζ + ˜ Θ
11 ˜ Θ, (47)where the forcing term ζ := − λ (ˆ v − v + )11 T + λv + ( f (ˆ v ) − f ( v + )) θ + by derivative estimate df /d ˆ v ≤ C ˆ v − together with the Mean ValueTheorem is uniformly exponentially decaying: | ζ | ≤ C | ˆ v − v + | ≤ C e − ηx , η > . (48)Initializing ˜ Θ ( L ) = 0, we obtain by Duhamel’s Principle/Variationof Constants the representation (supressing the argument λ )˜ Θ ( x ) = Z xL S y → x ( ζ + ˜ Θ
11 ˜ Θ )( y ) dy, (49)where S y → x is the solution operator for the homogeneous equation˜ Θ ′ − ˜ Θ (cid:0) aI − λ ( J + v + θ + ) (cid:1) = 0 , or, explicitly, S y → x = e R xy a ( y ) dy e − λ ( J + v + θ + )( x − y ) . For | λ | bounded and v + sufficiently small, we have by matrix per-turbation theory that the eigenvalues of − λ ( J + v + θ + ) are smalland the entries are bounded, hence | e − λ ( J + v + θ + ) z | ≤ Ce ǫz for z ≥
0. Recalling the uniform spectral gap ℜ ea = f (ˆ v ) − ℜ eλ ≤− / ℜ eλ ≥
0, we thus have | S y → x | ≤ Ce η ( x − y ) (50)for some C , η >
0. Combining (48) and (50), we obtain (cid:12)(cid:12)(cid:12) Z xL S y → x ζ ( y ) dy (cid:12)(cid:12)(cid:12) ≤ Z xL C e − η ( x − y ) e − ( η/ y dy = C e − ( η/ x . (51)Defining ˜ Θ ( x ) =: ˜ θ ( x ) e − ( η/ x and recalling (49) we thus have˜ θ ( x ) = f + e ( η/ x Z xL S y → x e − ηy ˜ θ θ ( y ) dy, (52)where f := e ( η/ x R xL S y → x ζ ( y ) dy is uniformly bounded, | f | ≤ C ,and e ( η/ x R xL S y → x e − ηy ˜ θ θ ( y ) dy is contractive with arbitrarily small tability of isentropic viscous shock profiles 21 contraction constant ǫ > L ∞ [ L, + ∞ ) for | ˜ θ | ≤ C for L suffi-ciently large, by the calculation (cid:12)(cid:12)(cid:12) e ( η/ x Z xL S y → x e − ηy ˜ θ θ ( y ) − e ( η/ x Z xL S y → x e − ηy ˜ θ θ ( y ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) e ( η/ x Z xL Ce − η ( x − y ) e − ηy dy (cid:12)(cid:12)(cid:12) k ˜ θ − ˜ θ k ∞ max j k ˜ θ j k ∞ ≤ e − ( η/ L (cid:12)(cid:12)(cid:12) Z xL Ce − ( η/ x − y ) dy (cid:12)(cid:12)(cid:12) k ˜ θ − ˜ θ k ∞ max j k ˜ θ j k ∞ = C e − ( η/ L k ˜ θ − ˜ θ k ∞ max j k ˜ θ j k ∞ . It follows by the Contraction Mapping Principle that there exists aunique solution ˜ θ of fixed point equation (52) with | ˜ θ ( x ) | ≤ C for x ≥ L , or, equivalently (redefining the unspecified constant η ), (46). Remark 5.
The above calculation is the most delicate part of the anal-ysis, and the main technical point of the paper. The interested readermay verify that a “quasi-static” transformation treating term ˜ Θ ′ in(47) as an error, as is typically used in situations of slowly-varying co-efficients (see for example [31, 37]), would lead to unnaceptable errorsof magnitude O ( | f (ˆ v )) ′ || ˆ v − v + | ) = O ( | df /d ˆ v ) || ˆ v − v + | ) = O ( | ˆ v | − | ˆ v − v + | ) . One may think of the exact ODE solution (44) as “averaging” theeffects of rapidly-varying coefficients by integration of (47).
Making now the further change of coordinates Z = ˜ LY and computing( ˜ LY ) ′ = ˜ LY ′ + ˜ L ′ Y = ( ˜ LA + + ˜ L ′ ) Y, = ( ˜ LA + ˜ R + ˜ L ′ ˜ R ) Z, we find that we have converted (43) to a block-triangular system Z ′ = ˜ AZ = (cid:18) λ ( J + v + θ + ) + 11 ˜ Θ f (ˆ v ) − λ − λv + θ + − ˜ Θ (cid:19) Z, (53) related to the original eigenvalue system (16) by W = LZ, R := R + R = (cid:18) I Θ (cid:19) , L := R − = (cid:18) I − Θ (cid:19) , (54)where Θ = ˜ Θ + λv + θ + . (55)Since it is triangular, (53) may be solved completely if we can solvethe component systems associated with its diagonal blocks. The fastsystem z ′ = (cid:16) f (ˆ v ) − λ − λv + θ + − ˜ Θ (cid:17) z associated to the lower righthand block features rapidly-varying co-efficients. However, because it is scalar, it can be solved explicitly byexponentiation.The slow system z ′ = (cid:16) λ ( J + v + θ + ) + 11 ˜ Θ (cid:17) z (56)associated to the upper lefthand block, on the other hand, by (46),is an exponentially decaying perturbation of a constant-coefficientsystem z ′ = λ ( J + v + θ + ) z (57)that can be explicitly solved by exponentiation, and thus can be well-estimated by comparison with (57). A rigorous version of this state-ment is given by the conjugation lemma of [35]: Proposition 4 ([35]).
Let M ( x, λ ) = M + ( λ ) + Θ ( x, λ ) , with M + continuous in λ and | Θ ( x, λ ) | ≤ Ce − ηx , for λ in some compactset Λ . Then, there exists a globally invertible matrix P ( x, λ ) = I + Q ( x, λ ) such that the coordinate change z = P v converts the variable-coefficient ODE z ′ = M ( x, λ ) z to a constant-coefficient equation v ′ = M + ( λ ) v, satisfying for any L , < ˆ η < η a uniform bound | Q ( x, λ ) | ≤ C ( L, ˆ η, η, max | ( M + ) ij | , dim M + ) e − ˆ ηx for x ≥ L. (58) Proof.
See [35, 46], or Appendix C.By Proposition 4, the solution operator for (56) is given by P ( y, λ ) e λ ( J + v + θ + ( λ,v + ))( x − y ) P ( x, λ ) − , (59)where P is a uniformly small perturbation of the identity for x ≥ L and L > tability of isentropic viscous shock profiles 23
5. Proof of the main theorem
With these preparations, we turn now to the proof of the main the-orem.
We begin by establishing the following key estimates on f W +1 ( L ), thatis, the value of the dual mode f W +1 appearing in (21) at the boundary x = L between regular and singular regions. Lemma 3.
For λ on any compact subset of ℜ eλ ≥ , and L > sufficiently large, with f W +1 normalized as in [13, 37, 2], | f W +1 ( L ) − e V | ≤ Ce − ηL (60) as v + → , uniformly in λ , where C , η > are independent of L and e V := (0 , , (1 + ¯ λ ) − ) T is the limiting direction vector (32) appearing in the definition of D . Corollary 4.
Under the hypotheses of Lemma 3, | ˜ W ( L ) − e V | ≤ Ce − ηL (61) and | f W +1 ( L ) − f W ( L ) | ≤ Ce − ηL (62) as v + → , uniformly in λ , where C , η > are independent of L and f W is the solution of the limiting adjoint eigenvalue systemappearing in definition (31) of D .Proof (Proof of Lemma 3). Making the coordinate-change˜ Z := R ∗ ˜ W , (63) R as in (54), reduces the adjoint equation ˜ W ′ = − A ∗ ˜ W to blocklower-triangular form,˜ Z ′ = − ˜ A ∗ ˜ Z = (cid:18) − ¯ λJ T + ( λv + θ + −
11 ˜ Θ ) ∗ − T − f (ˆ v ) + ¯ λ + ¯ λv + ( θ + ∗ − ( ˜ Θ ∗ (cid:19) Z, (64)with “¯” denoting complex conjugate. Denoting by ˜ V +1 a suitably normalized element of the one-dimensional(slow) stable subspace of − ˜ A ∗ , we find, similarly as in the discussionof Section 3.2 that, without loss of generality,˜ V +1 → (0 , , ( γ + ¯ λ ) − ) T (65)as v + →
0, while the associated eigenvalue ˜ µ +1 → , uniformly for λ on an compact subset of ℜ eλ ≥
0. The dual mode ˜ Z +1 = R ∗ ˜ W +1 isuniquely determined by the property that it is asymptotic as x → + ∞ to the corresponding constant-coefficient solution e ˜ µ +1 ˜ V +1 (thestandard normalization of [13, 37, 2]).By lower block-triangular form (64), the equations for the slowvariable ˜ z T := ( ˜ Z , ˜ Z ) decouples as a slow system˜ z ′ = − (cid:16) λ ( J + v + θ + ) + 11 ˜ Θ (cid:17) ∗ ˜ z (66)dual to (56), with solution operator P ∗ ( x, λ ) − e − ¯ λ ( J + v + θ + ) ∗ )( x − y ) P ( y, λ ) ∗ (67)dual to (59), i.e. (fixing y = L , say), solutions of general form˜ z ( λ, x ) = P ∗ ( x, λ ) − e − ¯ λ ( J + v + θ + ) ∗ )( x − y ) ˜ v, (68)˜ v ∈ C arbitrary.Denoting by ˜ Z +1 ( L ) := R ∗ ˜ W +1 ( L ) , therefore, the unique (up to constant factor) decaying solution at+ ∞ , and ˜ v +1 := (( ˜ V +1 ) , ( ˜ V +1 ) ) T , we thus have evidently˜ z +1 ( x, λ ) = P ∗ ( x, λ ) − e − ¯ λ ( J + v + θ + ) ∗ ) x ˜ v +1 , which, as v + →
0, is uniformly bounded by | ˜ z +1 ( x, λ ) | ≤ Ce ǫx (69)for arbitrarily small ǫ > x less than orequal to any fixed X simply tolim v + → ˜ z +1 ( x, λ ) = P ∗ ( x, λ ) − (0 , T . (70)Defining by ˜ q := ( ˜ Z +1 ) the fast coordinate of ˜ Z +1 , we have, by(64), ˜ q ′ + (cid:16) f (ˆ v ) − ¯ λ − ( λv + θ +
11 + ˜ Θ ∗ (cid:17) ˜ q = 11 T ˜ z +1 , tability of isentropic viscous shock profiles 25 whence, by Duhamel’s principle, any decaying solution is given by˜ q ( x, λ ) = Z + ∞ x e R xy a ( z,λ,v + ) dz T z +1 ( y ) dy, where a ( y, λ, v + ) := − (cid:16) f (ˆ v ) − ¯ λ − ( λv + θ +
11 + ˜ Θ ∗ (cid:17) . Recalling, for ℜ eλ ≥
0, that ℜ ea ≥ /
2, combining (69) and (70),and noting that a converges uniformly on y ≤ Y as v + → Y > a ( y, λ ) := − f (ˆ v ) + ¯ λ + ( ˜ Θ ∗ = (1 + ¯ λ ) + O ( e − ηy )we obtain by the Lebesgue Dominated Convergence Theorem that˜ q ( L, λ ) → Z + ∞ L e R Ly a ( z,λ ) dz T (0 , T dy = Z + ∞ L e (1+¯ λ )( L − y )+ R Ly O ( e − ηz ) dz dy = (1 + ¯ λ ) − (1 + O ( e − ηL )) . Recalling, finally, (70), and the fact that | P − Id | ( L, λ ) , | R − Id | ( L, λ ) ≤ Ce − ηL for v + sufficiently small, we obtain (60) as claimed. Proof (Proof of Corollary 4).
Applying Proposition 4 to the limitingadjoint system˜ W ′ = − ( A ) ∗ ˜ W = − ¯ λ − − λ ˜ W + O ( e − ηx ) ˜ W , we find that, up to an Id + O ( e − ηx ) coordinate change, ˜ W ( x )is given by the exact solution ˜ W ≡ ˜ V of the limiting, constant-coefficient system˜ W ′ = − ( A ) ∗ ˜ W = − ¯ λ − − λ ˜ W .
This yields immediately (61), which, together with (60), yields (62).
Remark 6.
Noting that (61) is sharp, we see from (62) that the errorbetween ˜ W +1 ( L ) and ˜ W ( L ) is already within the error toleranceof the numerical scheme used to approximate D , in which f W isinitialized at x = L with approximate value ˜ V [11, 37, 2]. Thus, solong as the flow on the regular region x ≤ L well-approximates theexact limiting flow as v + →
0, we can expect convergence of D to D based on the known convergence of the numerical approximationscheme. D As hinted by Remark 6, the rest of our analysis is standard if notentirely routine.
Lemma 4. On x ≤ L for any fixed L > , there exists a coordinate-change W = T Z conjugating (16) to the limiting equations (26) , T = T ( x, λ, v + ) , satisfying a uniform bound | T − Id | ≤ C ( L ) v + (71) for all v + > sufficiently small.Proof. For x ∈ ( −∞ , ConvergenceLemma of [37], a variation on Proposition 4, together with uniformconvergence of the profile and eigenvalue equations. For x ∈ [0 , L ], itis essentially continuous dependence; more precisely, observing that | A − A | ≤ C ( L ) v + for x ∈ [0 , L ], setting S := T − Id , and writingthe homological equation expressing conjugacy of (16) and (26), weobtain S ′ − ( AS − SA ) = ( A − A ) , which, considered as an inhomogeneous linear matrix-valued equa-tion, yields an exponential growth bound S ( x ) ≤ e Cx ( S (0) + C − C ( L ) v + )for some C >
0, giving the result.
Proof (Proof of Theorem 1).
Lemma 4, together with convergence as v + → A − to the unstable subspace of A − at the same rate O ( v + ) (as follows by spectral separation of theunstable eigenvalue of A and standard matrix perturbation theory)yields | W − (0 , λ ) − W − (0 , λ ) | ≤ C ( L ) v + . (72)Likewise, Lemma 4 gives | ˜ W +1 (0 , λ ) − ˜ W (0 , λ ) | ≤ C ( L ) v + | ˜ W +1 (0 , λ ) | + | S L → || ˜ W +1 ( L, λ ) − ˜ W ( L, λ ) | , (73) tability of isentropic viscous shock profiles 27 where S y → x denotes the solution operator of the limiting adjointeigenvalue equation ˜ W ′ = − ( A ) ∗ ˜ W . Applying Proposition 4 to thelimiting system, we obtain | S L → | ≤ C e − A L ≤ C L | λ | by direct computation of e − A L , where C is independent of L > | ˜ W +1 (0 , λ ) − ˜ W (0 , λ ) | ≤ C ( L ) v + | ˜ W +1 (0 , λ ) | + L | λ | C Ce − ηL , hence, for | λ | bounded, | ˜ W +1 (0 , λ ) − ˜ W (0 , λ ) | ≤ C ( L ) v + | ˜ W (0 , λ ) | + LC e − ηL ≤ C ( L ) v + + LC e − ηL . (74)Taking first L → ∞ and then v + →
0, we obtain therefore conver-gence of W +1 (0 , λ ) and ˜ W +1 (0 , λ ) to W (0 , λ ) and ˜ W (0 , λ ), yield-ing the result by definitions (21) and (31).
6. Numerical convergence
Having established analytically convergence of D to D as M → ∞ ,we turn finally to numerics to obtain quantitative information yield-ing a concrete stability threshold. Specifically, for fixed γ , we com-pute the “Rouch´e bound” v + at which the maximum relative error | D − D | / | D | over the semicircular contour ∂ {ℜ eλ ≥ , | λ | ≤ } around which we perform our winding number calculations becomes1 /
2. Recall that Rouch´e’s Theorem guarantees for relative error < D is equal to the winding number of D ,which we have shown to be zero, hence we may conservatively con-clude stability for v + less than or equal to this bound, or M greaterthan or equal to the corresponding Mach number. Computations areperformed using the algorithm of [2]; results are displayed in Table1. More detailed results are displayed for the monatomic gas case, γ = 1 . ... , in Table 2. Results are similar for other γ ∈ [1 , | D − D | , by tracking constants carefullythrough the estimates of the previous sections. Indeed, one could domuch better than the rather crude bounds stated for the general caseby taking into account the eigenstructure of the actual matrices A ± , A ± appearing in our analysis. That is, there are contained in our γ v + Error Number3.0 1.27e-3 .5009 127652.5 1.36e-3 .5006 24232.0 1.49e-3 .5001 4741.5 1.75e-3 .4999 95.51.0 2.8e-3 .4995 18.9
Table 1.
Rouch´e bounds for various γ .Mach Relative Absolute v + Number Difference Difference1.0(-6) 7.71(4) 0.1221 0.06011.0(-5) 1.13(4) 0.1236 0.14451.0(-4) 1.64(3) 0.1487 0.47141.0(-3) 2.44(2) 0.4098 1.34641.0(-2) 36.1 0.9046 2.82531.0(-1) 5.50 1.2386 3.8688
Table 2.
Maximum relative and absolute differences between D and D , for γ = 1 . ... and λ on the semicircle of radius 10. analysis, as in the study of [2], all of the ingredients needed for a nu-merical proof. Given the fundamental nature of the problem studied,this would be a very interesting program to carry out.
7. Discussion and open problems
Besides long-time stability, our results have application also to ex-istence of shock layers in the small-viscosity limit, which likewisereduces to the question of stability of the Evans function [38, 17].Indeed, spectral stability has been a key missing piece in several di-rections [45, 46]. Our methods should have application also to spectralstability of large-amplitude noncharacteristic boundary layers, com-pleting the investigations of [41, 16, 35]. It may be hoped that theywill extend also to full gas dynamics and multi-dimensions, two im-portant directions for further investigation. As discussed in the text,the problems of numerical proof and of stability in the large- γ limitare two other interesting directions for further study.More speculatively, our results suggest the possibility of a large-variation version of the results obtained by quite different methodsin [7] on general viscous solutions (including not only noninteractingshocks, but shocks, rarefactions, and their interactions), and, throughthe physical insight provided into the high-Mach number limit, per- tability of isentropic viscous shock profiles 29 haps even a hint toward possible methods of analysis. This would bean extremely interesting direction for further investigation. A. Proofs of Preliminary Estimates
Proof (Proof of Proposition 1).
Existence and monotonicity followtrivially by the fact that (7) is a scalar first-order ODE with convexrighthand side. Exponential convergence as x → + ∞ follows, forexample, by the computation H ( v, v + ) = v (cid:16) ( v − − ( v + − v − γ − v − γ + − (cid:17) = v (cid:16) ( v − v + ) + (cid:16) − v + − v γ + (cid:17)(cid:16)(cid:0) v + v (cid:1) γ − (cid:17)(cid:17) = ( v − v + ) (cid:16) v − (cid:16) − v + − v γ + (cid:17)(cid:16) − (cid:0) v + v (cid:1) γ − (cid:0) v + v (cid:1) (cid:17)(cid:17) , yielding v − γ ≤ H ( v, v + ) v − v + ≤ v − (1 − v + )by the elementary estimate 1 ≤ − x γ − x ≤ γ for 0 ≤ x ≤
1. Convergenceas x → −∞ follows by a similar, but simpler computation; see [2]. Lemma 5.
The following identity holds for ℜ eλ ≥ : ( ℜ e ( λ ) + |ℑ m ( λ ) | ) Z R ˆ v | u | − Z R ˆ v x | u | + Z R | u ′ | ≤ √ Z R h (ˆ v )ˆ v γ | v ′ || u | + Z R ˆ v | u ′ || u | . (75) Proof.
We multiply (12b) by ˆ v ¯ u and integrate along x . This yields λ Z R ˆ v | u | + Z R ˆ vu ′ ¯ u + Z R | u ′ | = Z R h (ˆ v )ˆ v γ v ′ ¯ u. We get (75) by taking the real and imaginary parts and adding themtogether, and noting that |ℜ e ( z ) | + |ℑ m ( z ) | ≤ √ | z | . ⊓⊔ Lemma 6.
The following identity holds for ℜ eλ ≥ : Z R | u ′ | = 2 ℜ e ( λ ) Z R | v | + ℜ e ( λ ) Z R | v ′ | ˆ v + 12 Z R (cid:20) h (ˆ v )ˆ v γ +1 + aγ ˆ v γ +1 (cid:21) | v ′ | (76) Proof.
We multiply (12b) by ¯ v ′ and integrate along x . This yields λ Z R u ¯ v ′ + Z R u ′ ¯ v ′ − Z R h (ˆ v )ˆ v γ +1 | v ′ | = Z R v u ′′ ¯ v ′ = Z R v ( λv ′ + v ′′ )¯ v ′ . Using (12a) on the right-hand side, integrating by parts, and takingthe real part gives ℜ e (cid:20) λ Z R u ¯ v ′ + Z R u ′ ¯ v ′ (cid:21) = Z R (cid:20) h (ˆ v )ˆ v γ +1 + ˆ v x v (cid:21) | v ′ | + ℜ e ( λ ) Z R | v ′ | ˆ v . The right hand side can be rewritten as ℜ e (cid:20) λ Z R u ¯ v ′ + Z R u ′ ¯ v ′ (cid:21) = 12 Z R (cid:20) h (ˆ v )ˆ v γ +1 + aγ ˆ v γ +1 (cid:21) | v ′ | + ℜ e ( λ ) Z R | v ′ | ˆ v . (77)Now we manipulate the left-hand side. Note that λ Z R u ¯ v ′ + Z R u ′ ¯ v ′ = ( λ + ¯ λ ) Z R u ¯ v ′ − Z R u (¯ λ ¯ v ′ + ¯ v ′′ )= − ℜ e ( λ ) Z R u ′ ¯ v − Z R u ¯ u ′′ = − ℜ e ( λ ) Z R ( λv + v ′ )¯ v + Z R | u ′ | . Hence, by taking the real part we get ℜ e (cid:20) λ Z R u ¯ v ′ + Z R u ′ ¯ v ′ (cid:21) = Z R | u ′ | − ℜ e ( λ ) Z R | v | . This combines with (77) to give (76). ⊓⊔ Lemma 7.
For h (ˆ v ) as in (11) , we have sup ˆ v (cid:12)(cid:12)(cid:12)(cid:12) h (ˆ v )ˆ v γ (cid:12)(cid:12)(cid:12)(cid:12) = γ − v + − v γ + ≤ γ. (78) Proof.
Defining g (ˆ v ) := h (ˆ v )ˆ v − γ = − ˆ v + a ( γ − v − γ + ( a + 1) , (79)we have g ′ (ˆ v ) = − − aγ ( γ − v − γ − < < v + ≤ ˆ v ≤ v − = 1,hence the maximum of g on ˆ v ∈ [ v + , v − ] is achieved at ˆ v = v + .Substituting (8) into (79) and simplifying yields (78). tability of isentropic viscous shock profiles 31 Proof (Proof of Proposition 2).
Using Young’s inequality twice onright-hand side of (75) together with (78), we get( ℜ e ( λ ) + |ℑ m ( λ ) | ) Z R ˆ v | u | − Z R ˆ v x | u | + Z R | u ′ | ≤ √ Z R h (ˆ v )ˆ v γ | v ′ || u | + Z R ˆ v | u ′ || u |≤ θ Z R h (ˆ v )ˆ v γ +1 | v ′ | + ( √ θ Z R h (ˆ v )ˆ v γ ˆ v | u | + ǫ Z R ˆ v | u ′ | + 14 ǫ Z R ˆ v | u | < θ Z R h (ˆ v )ˆ v γ +1 | v ′ | + ǫ Z R | u ′ | + (cid:20) γ θ + 14 ǫ (cid:21) Z R ˆ v | u | . Assuming that 0 < ǫ < θ = (1 − ǫ ) /
2, this simplifies to( ℜ e ( λ ) + |ℑ m ( λ ) | ) Z R ˆ v | u | + (1 − ǫ ) Z R | u ′ | < − ǫ Z R h (ˆ v )ˆ v γ +1 | v ′ | + (cid:20) γ θ + 14 ǫ (cid:21) Z R ˆ v | u | . Applying (76) yields( ℜ e ( λ ) + |ℑ m ( λ ) | ) Z R ˆ v | u | < (cid:20) γ − ǫ + 14 ǫ (cid:21) Z R ˆ v | u | , or equivalently, ( ℜ e ( λ ) + |ℑ m ( λ ) | ) < (4 γ − ǫ − ǫ (1 − ǫ ) . Setting ǫ = 1 / (2 √ γ + 1) gives (14). ⊓⊔ B. Nonvanishing of D As pointed out in Section 3.3, the limiting eigenvalue system (25),(26), together with the limiting boundary conditions derived in Sec-tion 3.2 may be expressed equivalently as the integrated eigenvalueproblem λv + v ′ − u ′ = 0 , (80a) λu + u ′ − − ˆ v ˆ v v ′ = u ′′ ˆ v . (80b)corresponding to a pressureless gas, γ = 0, with special boundaryconditions( u, u ′ , v, v ′ )( −∞ ) = (0 , , , , ( u, u ′ , v, v ′ )(+ ∞ ) = ( c, , , . (81) Motivated by this observation, we establish stability of the limit-ing system by a Matsumura–Nishihara-type spectral energy estimateexactly analogous to that used to prove stability for γ = 1 in [34, 2]. Proof (Proof of Proposition 3).
Multiplying (80b) by ˆ v ¯ u/ (1 − ˆ v ) andintegrating on some subinterval [ a.b ] ⊂ R , we obtain λ Z ba ˆ v − ˆ v | u | dx + Z ba ˆ v − ˆ v u ′ ¯ udx − Z ba v ′ ¯ udx = Z ba u ′′ ¯ u − ˆ v dx. Integrating the third and fourth terms by parts yields λ Z ba ˆ v − ˆ v | u | dx + Z ba (cid:20) ˆ v − ˆ v + (cid:18) − ˆ v (cid:19) ′ (cid:21) u ′ ¯ udx + Z ba | u ′ | − ˆ v dx + Z ba v ( λv + v ′ ) dx = (cid:20) v ¯ u + u ′ ¯ u − ˆ v (cid:21) (cid:12)(cid:12)(cid:12) ba . Taking the real part, we have ℜ e ( λ ) Z ba (cid:18) ˆ v − ˆ v | u | + | v | (cid:19) dx + Z ba g (ˆ v ) | u | dx + Z ba | u ′ | − ˆ v dx = ℜ e (cid:20) v ¯ u + u ′ ¯ u − ˆ v − (cid:20) ˆ v − ˆ v + (cid:18) − ˆ v (cid:19) ′ (cid:21) | u | − | v | (cid:21) (cid:12)(cid:12)(cid:12) ba , (82)where g (ˆ v ) = − (cid:20)(cid:18) ˆ v − ˆ v (cid:19) ′ + (cid:18) − ˆ v (cid:19) ′′ (cid:21) . Note that ddx (cid:18) − ˆ v (cid:19) = − (1 − ˆ v ) ′ (1 − ˆ v )2 = ˆ v x (1 − ˆ v )2 = ˆ v (ˆ v − − ˆ v )2 = − ˆ v − ˆ v . Thus, g (ˆ v ) ≡ ℜ e ( λ ) Z ba (cid:18) ˆ v − ˆ v | u | + | v | (cid:19) dx + Z ba | u ′ | − ˆ v dx = (cid:20) ℜ e ( v ¯ u ) + ℜ e ( u ′ ¯ u )1 − ˆ v − | v | (cid:21) (cid:12)(cid:12)(cid:12) ba . We show next that the right-hand side goes to zero in the limitas a → −∞ and b → ∞ . By Proposition 4, the behavior of u , v near tability of isentropic viscous shock profiles 33 ±∞ is governed by the limiting constant–coefficient systems W ′ = A ± ( λ ) W , where W = ( u, v, v ′ ) T and A ± = A ( ±∞ , λ ). In particular,solutions W asymptotic to (1 , ,
0) at x = + ∞ decay exponentiallyin ( u ′ , v, v ′ ) and are bounded in coordinate u as x → + ∞ . Observingthat 1 − ˆ v → x → + ∞ , we thus see immediately that theboundary contribution at b vanishes as b → + ∞ .The situation at −∞ is more delicate, since the denominator 1 − ˆ v of the second term goes to zero at rate e x as x → −∞ , the rateof convergence of the limiting profile ˆ v . By inspection, the limitingcoefficient matrix A − = λ
10 0 1 λ λ − λ ! , (83)has eigenvalues µ = − λ, ± √ λ , hence for ℜ eλ ≥ x = + ∞ is theunstable eigenvector corresponding to µ = √ λ , with growthrate ℜ e (cid:18) √ λ (cid:19) = 12 + 12 ℜ e √ λ > / . Thus, | u | , | u ′ | , | v ′ | , | v | ≤ Ce (1+ ǫ ) x/ as x → −∞ , ǫ >
0, and in partic-ular (cid:12)(cid:12)(cid:12) ℜ e ( u ′ ¯ u )1 − ˆ v (cid:12)(cid:12)(cid:12) ≤ Ce (1+ ǫ ) x /e x ≤ Ce ǫx → x → −∞ . It follows that the boundary contribution at x = a vanishes also as a → −∞ , hence, in the limit as a → −∞ , b → + ∞ , ℜ e ( λ ) Z + ∞−∞ (cid:18) ˆ v − ˆ v | u | + | v | (cid:19) dx + Z + ∞−∞ | u ′ | − ˆ v dx = 0 . (84)But, for ℜ eλ ≥
0, this implies u ′ ≡
0, or u ≡ constant, which, by u ( −∞ ) = 0, implies u ≡
0. This reduces (80a) to v ′ = λv , yieldingthe explicit solution v = Ce λx . By v ( ±∞ ) = 0, therefore, v ≡ ℜ eλ ≥
0. It follows that there are no nontrivial solutions of (80), (81)for ℜ eλ ≥ Remark 7.
The above energy estimate is equivalent to multiplying thesystem by the special symmetrizer (cid:18) v/ (1 − ˆ v ) (cid:19) , then taking the L inner product with ( v, u ) T . The analog of the high-frequency es-timates of Appendix A would be obtained using the alternative sym-metrizer (cid:18) − ˆ v
00 ˆ v (cid:19) optimized for its effect on second-order derivativeterm u ′′ / ˆ v . This may clarify somewhat the strategy of the energy es-timates used in [34, 2]. C. Quantitative conjugation estimates
Consider a general first-order system W ′ = A ( x, λ ) W. (85) Proposition 5 (Quantitative Gap Lemma [13, 47]).
Let V + and µ + be an eigenvector and associated eigenvalue of A + ( λ ) and supposethat there exist complementary generalized eigenprojections (i.e., A -invariant projections) P and Q such that | P e ( A + − µ + ) x | ≤ C e − ˆ ηx x ≤ , | Qe ( A + − µ + ) x | ≤ C e − ˆ ηx x ≥ , | ( A − A + )( x ) | ≤ C e − ηx x ≥ , (86) with ≤ ˆ η < η . Then, there exists a solution W = e µ + x V ( x, λ ) of (85) with | V ( x, λ ) − V + ( λ ) || V + ( λ ) | ≤ C C e − ηx ( η − ˆ η )(1 − ǫ ) for x ≥ L (87) provided ( η − ˆ η ) − C C e − ηL ≤ ǫ .Proof. Writing V ′ = ( A + − µ + ) V + ( A − A + ) V and imposing thelimiting behavior V (+ ∞ , λ ) = V + , we seek a solution in the form V = T V , T V ( x ) := V + − Z + ∞ x P e ( A + − µ + )( x − y ) ( A − A + ) V ( y ) dy + Z xL Qe ( A + − µ + )( x − y ) ( A − A + ) V ( y ) dy, from which the result follows by a straightforward Contraction Map-ping argument, using (86) to compute that | T V − T V | ( x ) = (cid:12)(cid:12)(cid:12) − Z + ∞ x P e ( A + − µ + )( x − y ) ( A − A + )( V − V )( y ) dy + Z xL Qe ( A + − µ + )( x − y ) ( A − A + )( V − V )( y ) dy (cid:12)(cid:12)(cid:12) ≤ C C Z + ∞ L e − ˆ η ( x − y ) e − ηy dy k V − V k L ∞ [ L, + ∞ ) = C C e − ˆ ηx e − ( η − ˆ η ) L η − ˆ η k V − V k L ∞ [ L, + ∞ ) , and thus k T V − T V k L ∞ [ L, + ∞ ) ≤ C C e − ηL η − ˆ η k V − V k L ∞ [ L, + ∞ ) . tability of isentropic viscous shock profiles 35 Corollary 5.
Let V + and µ + be an eigenvector and associated eigen-value of A + ( λ ) , where A + is n × n with at most k stable eigenvaluesand max | ( A + − µ ) ij | ≤ C ; | ( A − A + )( x ) | ≤ C e − ηx x ≥ , (88)0 < ˆ η < η . Then, there exists a solution W = e µ + x V ( x, λ ) of (85) with | V ( x, λ ) − V + ( λ ) || V + ( λ ) | ≤ nn !( C ) n C e − ˆ ηx δ n ( η − ˆ η )(1 − ǫ ) for x ≥ L, (89) δ := η − ˆ η k +2 , provided nn !( C ) n C e − ˆ ηL δ n ( η − ˆ η ) ≤ ǫ .Proof. Without loss of generality, take µ ≡
0. Dividing [ − η, − ˆ η ] into k + 1 equal subintervals, we find by the pigeonhole principle that atleast one subinterval contains the real part of no eigenvalue of A + .Denoting the midpoint of this interval by − ˜ η > ˆ η , we havemin |ℜ eσ ( A + ) − ˜ η | ≥ δ := η − ˆ η k + 2 . (90)Defining P to be the total eigenprojection of A + associated witheigenvalues of real part greater than ˆ η and Q the total eigenprojectionassociated with eigenvalues of real part less than ˆ η , and estimating P e A + x , Qe A + x using the the inverse Laplace transform representation e A + x = 12 πi I Γ e zx ( z − A + ) − dz, with Γ chosen to be a rectangle of side 4 nC centered about thereal axis, with one vertical side passing through ℜ eλ ≡ − ˜ η and theother respectively lying respectively to the right and to the left, andestimating | ( λ − A + ) − | ≤ n ! C n − δ − n crudely by Kramer’s rule, we obtain (86) with C = 16 nn ! C n δ − n , whence the result follows by Proposition 5. Corollary 6 (Quantitative Conjugation Lemma).
Proposition4 holds with C ( L, ˆ η, η, max | ( M + ) ij | , dim M + ) = 16 nn !( C ) n C e − ˆ ηx δ n ( η − ˆ η )(1 − ǫ ) ,n := (dim M + ) , k := (dim M + ) − dim M , when nn !( C ) n C e − ˆ ηL δ n ( η − ˆ η ) ≤ ǫ . Proof.
Writing the homological equation expressing conjugacy of variable-and constant-coefficient systems following [35], we have P ′ = M + P − P M + + ΘM.
Considering this as an asymptotically constant-coefficient system onthe n -dimensional vector space of matrices P , noting that the linearoperator M + P := M + P − P M + , as a Sylvester matrix, has at least n zero eigenvalues and equal numbers of stable and unstable eigen-values, we see that the number of its stable eigenvalues is not morethan k := n − n , whence the result follows by Corollary 5. References
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