Stability of radiative shock profiles for hyperbolic–elliptic coupled systems

Extending previous work with Lattanzio and Mascia on the scalar in fluid-dynamical variables Hamer model for a radiative gas, we show nonlinear orbital asymptotic stability of small- amp

arXiv:0908.1566v1 [math.AP] 11 Aug 2009

HYPERBOLIC-ELLIPTIC COUPLED SYSTEMS

TOAN NGUYEN, RAM ´ ON G PLAZA, AND KEVIN ZUMBRUN

Abstract Extending previous work with Lattanzio and Mascia on the scalar (in fluid-dynamical variables) Hamer model for a radiative gas, we show nonlinear orbital asymptotic stability of small- amplitude shock profiles of general systems of coupled hyperbolic–eliptic equations of the type modeling a radiative gas, that is, systems of conservation laws coupled with an elliptic equation for the radiation flux, including in particular the standard Euler–Poisson model for a radiating gas The method is based on the derivation of pointwise Green function bounds and description

of the linearized solution operator, with the main difficulty being the construction of the resolvent kernel in the case of an eigenvalue system of equations of degenerate type Nonlinear stability then follows in standard fashion by linear estimates derived from these pointwise bounds, combined with nonlinear-damping type energy estimates.

1 Introduction

In the theory of non-equilibrium radiative hydrodynamics, it is often assumed that an inviscidcompressible fluid interacts with radiation through energy exchanges One widely accepted model[37] considers the one dimensional Euler system of equations coupled with an elliptic equation forthe radiative energy, or Euler–Poisson equation With this system in mind, this paper considersgeneral hyperbolic-elliptic coupled systems of the form,

ut+ f (u)x+ Lqx= 0,

with (x, t) ∈ R×[0, +∞) denoting space and time, respectively, and where the unknowns u ∈ U ⊆ Rn,

n ≥ 1, play the role of state variables, whereas q ∈ R represents a general heat flux In addition,

L ∈ Rn×1 is a constant vector, and f ∈ C2(U; Rn) and g ∈ C2(U; R) are nonlinear vector- andscalar-valued flux functions, respectively

The study of general systems like (1.1) has been the subject of active research in recent years[10, 11, 13, 17] There exist, however, more complete results regarding the simplified model of aradiating gas, also known as the Hamer model [6], consisting of a scalar velocity equation (usuallyendowed with a Burgers’ flux function which approximates the Euler system), coupled with a scalarelliptic equation for the heat flux Following the authors’ concurrent analysis with Lattanzio andMascia of the reduced scalar model [16], this work studies the asymptotic stability of general radiativeshock profiles, which are traveling wave solutions to system (1.1) of the form

Date : August 11, 2009.

The research of TN and KZ was supported in part by the National Science Foundation, award number

DMS-0300487 The research of RGP was partially supported by DGAPA-UNAM through the program PAPIIT, grant IN-109008 RGP is warmly grateful to the Department of Mathematics, Indiana University, for their hospitality and financial support during two short visits in May 2008 and April 2009, when this research was carried out TN, RGP, and KZ are warmly grateful to Corrado Lattanzio and Corrado Mascia for their interest in this work and for many helpful conversations, as well as their collaboration in concurrent work on the scalar case.

with asymptotic limits

U (x) → u±, Q(x) → 0, as x → ±∞,being u± ∈ U ⊆ Rn constant states and s ∈ R the shock speed The main assumption is that thetriple (u+, u−, s) constitutes a shock front [19] for the underlying “inviscid” system of conservationlaws

A(u) := Df (u) ∈ Rn×n, B(u) := Dg(u) ∈ R1×n, u ∈ U

Right and left eigenvectors of A will be denoted as r ∈ Rn×1 and l ∈ R1×n, and we suppose thatsystem (1.3) is hyperbolic, so that A has real eigenvalues a1≤ · · · ≤ an

It is assumed that system (1.1) represents some sort of regularization of the inviscid system (1.3)

in the following sense Formally, if we eliminate the q variable, then we end up with a system ofform

ut+ f (u)x= (LB(u)ux)x+ (ut+ f (u)x)xx,which requires a nondegeneracy hypothesis

for some 1 ≤ p ≤ n, in order to provide a good dissipation term along the p-th characteristic field

in its Chapman-Enskog expansion [34]

More precisely, we make the following structural assumptions:

For all u ∈ U there exists A0 symmetric, positive definite such that A0A

is symmetric, and A0LB is symmetric, positive semi-definite of rank one

(symmetric dissipativity ⇒ non-strict hyperbolicity) Moreover, we assume

that the principal eigenvalue ap of A is simple

(S1)

Remark 1.1 Assumption (S1) assures non-strict hyperbolicty of the system, with simple principalcharacteristic field Notice that (S1) also implies that (A0)1/2A(A0)−1/2 is symmetric, with realand semi-simple spectrum, and that, likewise, (A0)1/2B(A0)−1/2preserves symmetric positive semi-definiteness with rank one Assumption (S2) defines a general class of hyperbolic-elliptic equationsanalogous to the class defined by Kawashima and Shizuta [9, 14, 36] and compatible with (1.5).Moreover, there is an equivalent condition to (S2) given by the following

Lemma 1.2 (Shizuta–Kawashima [14, 36]) Under (S0) - (S1), assumption (S2) is equivalent tothe existence of a skew-symmetric matrix valued function K : U → Rn×n such that

for all u ∈ U

As usual, we can reduce the problem to the analysis of a stationary profile with s = 0, byintroducing a convenient change of variable and relabeling the flux function f accordingly Therefore,

we end up with a stationary solution (U, Q)(x) of the system

ap−1(u−) < 0 < ap(u−), (Lax entropy conditions), (H1)

(∇ap)⊤rp6= 0, for all u ∈ U, (genuine nonlinearity), (H2)

lp(u±)LB(u±)rp(u±) > 0, (positive diffusion) (H3)Remark 1.3 Systems of form (1.1) arise in the study of radiative hydrodynamics, for which theparadigmatic system has the form

ρt+ (ρu)x= 0,(ρu)t+ (ρu2+ p)x= 0,

ρ(e + 12u2)

t+ρu(e +12u2) + pu + q

pρ> 0, pθ6= 0, eθ> 0

Finally, q = ρχx is the radiative heat flux, where χ represents the radiative energy, and a, b > 0are positive constants related to absorption System (1.8) can be (formally) derived from a morecomplete system involving a kinetic equation for the specific intensity of radiation For this derivationand further physical considerations on (1.8) the reader is referred to [37, 20, 11]

The existence and regularity of traveling wave type solutions of (1.1) under hypotheses (S0) - (S2),(H0) - (H3) is known, even in the more general case of non-convex velocity fluxes (assumption (H2)does not hold) For details of existence, as well as further properties of the profiles such as mono-tonicity and regularity under small-amplitude assumption (features which will be used throughoutthe analysis), the reader is referred to [17, 18]

1.1 Main results In the spirit of [41, 22, 24, 25], we first consider the linearized equations of (1.1)about the profile (U, Q):

ut+ (A(U )u)x+ Lqx= 0,

with initial data u(0) = u0 Hence, the Laplace transform applied to system (1.9) gives

λu + (A(U ) u)x+ Lqx= S,

where source S is the initial data u0

As it is customary in related nonlinear wave stability analyses (see, e.g., [1, 33, 41, 38]), we start

by studying the underlying spectral problem, namely, the homogeneous version of system (1.10):

λu + (A(U ) u)x+ Lqx= 0,

An evident necessary condition for orbital stability is the absence of L2 solutions to (1.11) forvalues of λ in {Re λ ≥ 0}\{0}, being λ = 0 the eigenvalue associated to translation invariance Thisspectral stability condition can be expressed in terms of the Evans function, an analytic functionplaying a role for differential operators analogous to that played by the characteristic polynomial forfinite-dimensional operators (see [1, 33, 3, 41, 22] and the references therein) The main property

of the Evans function is that, on the resolvent set of a certain operator L, its zeroes coincide inboth location and multiplicity with the eigenvalues of L Thence, we express the spectral stabilitycondition as follows:

There exists no zero of the Evans function D on {Re λ ≥ 0} \ {0};

equiva-lently, there exist no nonzero eigenvalues of L with Re λ ≥ 0 (SS)Like in previous analyses [41, 38, 40], we define the following stability condition (or Evans functioncondition) as follows:

There exists precisely one zero (necessarily at λ = 0; see Lemmas 2.5 - 2.6)

of the Evans function on the nonstable half plane {Re λ ≥ 0}, (D)which implies the spectral stability condition (SS) plus the condition that D vanishes at λ = 0 atorder one Notice that just like in the scalar case [16], due to the degenerate nature of system (1.11)(observe that A(U ) vanishes at x = 0) the number of decaying modes at ±∞, spanning possibleeigenfunctions, depends on the region of space around the singularity Therefore the definition of

D is given in terms of the Evans functions D± in regions x ≷ 0, with same regularity and spectralproperties (see its definition in (2.23) and Lemmas 2.5 - 2.6 below)

Our main result is then as follows

Theorem 1.4 Assuming (1.5), (S0)–(S2), (H0)–(H3), and the spectral stability condition (D), thenthe Lax radiative shock profile (U, Q) with sufficiently small amplitude is asymptotically orbitallystable More precisely, the solution (˜u, ˜q) of (1.1) with initial data ˜u0 satisfies

Remark 1.5 The time-decay rate of q is not optimal In fact, it can be improved as we observethat |q(t)|L2 ≤ C|ux(t)|L2 and |ux(t)|L2 is expected to decay like t−1/2; however, we omit thedetail of carrying this out Likewise, assuming in addition a small L1 first moment on the initialperturbation, we could obtain by the approach of [32] the sharpened bounds | ˙α| ≤ C(1 + t)σ−1,and |α − α(+∞)| ≤ C(1 + t)σ−1/2, for σ > 0 arbitrary, including in particular the information that

α converges to a specific limit (phase-asymptotic orbital stability); however, we omit this again infavor of simplicity

We shall prove the following result in the appendix, verifying Evans condition (D)

Theorem 1.6 For ǫ := |u+− u−| sufficiently small, radiative shock profiles are spectrally stable.Corollary 1.7 The condition (D) is satisfied for small amplitudes

Proof In Lemmas 2.5 - 2.6 below, we show that D(λ) has a single zero at λ = 0 Together with

1.1.1 Discussion Prior to [16], asymptotic stability of radiative shock profiles has been studied inthe scalar case in [12] for the particular case of Burgers velocity flux and for linear g(u) = M u, withconstant M Another scalar result is the partial analysis of Serre [35] for the exact Rosenau model

In the case of systems, we mention the stability result of [21] for the full Euler radiating system underspecial zero-mass perturbations, based on an adaptation of the classical energy method of Goodman-Matsumura-Nishihara [4, 27] Here, we recover for systems, under general (not necessarily zero-mass)perturbations, the sharp rates of decay established in [12] for the scalar case

We mention that works [12, 16] in the scalar case concerned also large-amplitude shock profiles(under the Evans condition (D), automatically satisfied in the Burgers case [12]) At the expense

of further effort book-keeping– specifically in the resolution of flow near the singular point andconstruction of the resolvent– we could obtain by our methods a large-amplitude result similar tothat of [16] However, we greatly simplify the exposition by the small-amplitude assumption allowing

us to approximately diagonalize before carrying out these steps As the existence theory is only forsmall-amplitude shocks, with upper bounds on the amplitudes for which existence holds, known tooccur, and since the domain of our hypotheses in [16] does not cover the whole domain of existence

in the scalar case (in contrast to [12], which does address the entire domain of existence), we havechosen here for clarity to restrict to the small-amplitude setting It would be interesting to carryout a large-amplitude analysis valid on the whole domain of existence in the system case

1.2 Abstract framework Before beginning the analysis, we orient ourselves with a few simpleobservations framing the problem in a more standard way Consider now the inhomogeneous version

The generator L := −(A(U ) u)x−J u of (1.16) is a zero-order perturbation of the generator −A(U )ux

of a hyperbolic equation, so generates a C0 semigroup eLt and an associated Green distributionG(x, t; y) := eLtδy(x) Moreover, eLtand G may be expressed through the inverse Laplace transformformulae

eLt= 12πi

for all η ≥ η0, where Gλ(x, y) := (λ − L)−1δy(x) is the resolvent kernel of L

Collecting information, we may write the solution of (1.15) using Duhamel’s principle/variation

where G is determined through (1.18)

That is, the solution of the linearized problem reduces to finding the Green kernel for the equation alone, which in turn amounts to solving the resolvent equation for L with delta-functiondata, or, equivalently, solving the differential equation (1.10) with source S = δy(x) This we shall do

u-in standard fashion by writu-ing (1.10) as a first-order system and solvu-ing appropriate jump conditions

at y obtained by the requirement that Gλ be a distributional solution of the resolvent equations.This procedure is greatly complicated by the circumstance that the resulting (n + 2) × (n + 2)first-order system

is singular at the special point where A(U ) vanishes, with Θ dropping to rank n + 1 However,

in the end we find as usual that Gλ is uniquely determined by these criteria, not only for thevalues Re λ ≥ η0 > 0 guaranteed by C0-semigroup theory/energy estimates, but, as in the usualnonsingular case [7], on the set of consistent splitting for the first-order system (1.20), which includesall of {Re λ ≥ 0} \ {0} This has the implication that the essential spectrum of L is confined to{Re λ < 0} ∪ {0}

Remark 1.8 The fact (obtained by energy-based resolvent estimates) that L − λ is coercive for

Re λ ≥ η0 shows by elliptic theory that the resolvent is well-defined and unique in class of tions for Re λ large, and thus the resolvent kernel may be determined by the usual construction usingappropriate jump conditions That is, from standard considerations, we see that the constructionmust work, despite the apparent wrong dimensions of decaying manifolds (which happens for any

distribu-Re λ > 0)

To deal with the singularity of the first-order system is the most delicate and novel part ofthe present analysis It is our hope that the methods we use here may be of use also in othersituations where the resolvent equation becomes singular, for example in the closely related situation

of relaxation systems discussed in [22, 25]

2 Construction of the resolvent kernel2.1 Outline In what follows we shall denote ′ = ∂x for simplicity; we also write A(x) = A(U )and B(x) = B(U ) Let us now construct the resolvent kernel for L, or equivalently, the solution

of (1.10) with delta-function source in the u component The novelty in the present case is theextension of this standard method to a situation in which the spectral problem can only be written

as a degenerate first order ODE Unlike the real viscosity and relaxation cases [22, 23, 24, 25] (wherethe operator L, although degenerate, yields a non-degenerate first order ODE in an appropriatereduced space), here we deal with a system of form

ΘW′= A(x, λ)W,where

Θ =

A

I2

,

in the distributional sense for all x 6= y with appropriate jump conditions (to be determined) at

x = y The first entry of the three-vector Gλ is the resolvent kernel Gλof L that we seek

Namely Gλ, is the solution in the sense of distribution of system (1.10) (written in conservationform):

The asymptotic system thus can be written as

“slow” system (as |λ| → 0), eigenvalues are

where a±j are eigenvalues of A± = A(±∞) Thus, we readily conclude that at x = +∞, there are

p + 1 unstable eigenvalues and n − p + 1 stable eigenvalues The stable S+(λ) and unstable U+(λ)manifolds (solutions which decay, respectively, grow at +∞) have, thus, dimensions

dim U+(λ) = p + 1,

in Re λ > 0 Likewise, there exist n − p + 1 unstable eigenvalues and p stable eigenvalues so that thestable (solutions which grow at −∞) and unstable (solutions which decay at −∞) manifolds havedimensions

dim U−(λ) = p,

Remark 2.1 Notice that, unlike customary situations in the Evans function literature [1, 41, 3,

22, 23, 33], here the dimensions of the stable (resp unstable) manifolds S+ and S− (resp U+ and

U−) do not agree Under these considerations, we look at the dispersion relation

π±(iξ) = −iξ3− A−1± (λ + LB±)ξ2− iξ − A−1± = 0

For each ξ ∈ R, the λ-roots of the last equation define algebraic curves

λ±j(ξ) ∈ σ(1 + LB±ξ)−1(−ξ2+ iA±ξ(1 + ξ2)), ξ ∈ R,touching the origin at ξ = 0 Denote Λ as the open connected subset of C bounded on the left bythe rightmost envelope of the curves λ±j(ξ), ξ ∈ R Note that the set {Re λ ≥ 0, λ 6= 0} is properlycontained in Λ By connectedness the dimensions of U±(λ) and S±(λ) do not change in λ ∈ Λ Wedefine Λ as the set of (not so) consistent splitting [1], in which the matrices A±(λ) remain hyperbolic,with not necessarily agreeing dimensions of stable (resp unstable) manifolds

Lemma 2.2 For each λ ∈ Λ, the spectral system (2.8) associated to the limiting, constant cients asymptotic behavior of (2.4), has a basis of solutions

coeffi-eµ±j (λ)xVj±(λ), x ≷ 0, j = 1, , n + 2

Moreover, for |λ| ∼ 0, we can find analytic representations for µ±j and Vj±, which consist of 2n slowmodes

µ±j(λ) = −λ/a±j + O(λ2), j = 2, , n + 1,and four fast modes,

µ±1(λ) = ±θ±1 + O(λ),

µ±n+2(λ) = ∓θ±n+2+ O(λ)

where θ± and θ± are positive constants

In view of the structure of the asymptotic systems, we are able to conclude that for each initialcondition x0> 0, the solutions to (2.4) in x ≥ x0 are spanned by decaying/growing modes

Lemma 2.3 For |λ| sufficiently small, there exist growing and decaying solutions ψj±(x, λ), φ±j (x, λ),

in x ≷ 0, of class C1 in x and analytic in λ, satisfying

ψ±j(x, λ) = eµ±j (λ)xVj±(λ)(I + O(e−η|x|)),

φ±j(x, λ) = eµ±j (λ)xVj±(λ)(I + O(e−η|x|)), (2.15)where 0 < η is the decay rate of the traveling wave, and µ±j and Vj± are as in Lemma 2.2 above.Proof This a direct application of the conjugation lemma of [29] (see also the related gap lemma in

2.3 Solutions near x ∼ 0 Our goal now is to analyze system (2.4) close to the singularity x = 0

To fix ideas, let us again stick to the case x > 0, the case x < 0 being equivalent We introduce a

“stretched” variable ξ as follows:

ξ =

Z x 1

α(ξ) ≥ δ0> 0,for some δ0 and any ξ sufficiently large or x sufficiently near zero

The blocks −αI and 0 are clearly spectrally separated and the error is of order O(|ap(ξ)|) → 0

as ξ → +∞ By the pointwise reduction lemma (see Lemma B.1 and Remark B.2 below), we canseparate the flow into slow and fast coordinates Indeed, after proper transformations we separatethe flows on the reduced manifolds of form

˙

˙

Since −α ≤ −δ0< 0 for λ ∼ 0 and ξ ≥ 1/ǫ, with ǫ > 0 sufficiently small, and since ap(ξ) → 0 as

ξ → +∞, the Z1 mode decay to zero as ξ → +∞, in view of

e−R0ξα(z) dz e−(Re λ+12 δ 0 )ξ

.These fast decaying modes correspond to fast decaying to zero solutions when x → 0+ in theoriginal u-variable The Z2 modes comprise slow dynamics of the flow as x → 0+

Proposition 2.4 There exists 0 < ǫ0 ≪ 1 sufficiently small, such that, in the small frequencyregime λ ∼ 0, the solutions to the spectral system (2.4) in (−ǫ0, 0) ∪ (0, ǫ0) are spanned by fast modes

with bounded limits as x → 0±

Moreover, the fast modes (2.19) decay as

as x → 0±; here, α0 is some positive constant and uk p = (uk p 1, , uk p p, , uk p n)⊤

2.4 Two Evans functions We first define the following related Evans functions

D±(y, λ) := det(Φ+Wk∓pΦ−)(y, λ), for y ≷ 0, (2.23)where Φ± are defined as in (2.13), (2.14), and Wk±p = (u±kp, q±kp, p±kp)⊤ are defined as in (2.19) Notethat kp here is always fixed and equals to n − p + 2

We first observe the following simple properties of D±

Lemma 2.5 For λ sufficiently small, we have

D±(y, λ) = (det A)−1γ±(y)∆λ + O(|λ|2), (2.24)where

Proof By our choice, at λ = 0, we can take

I2



Thus, ∂λφ+1(x, λ) satisfies

Θ(∂λφ+1)x= A(x, 0)∂λφ+1(x, 0) + ∂λA(x, 0)φ+1(x, 0),which directly gives

(a∂λu+1)x= −L(∂λq+1)x− ¯ux (2.29)Likewise, ∂λφ−n+2(x, λ) = (∂λu−n+2, ∂λqn+2− , ∂λp−n+2) satisfies

(a∂λu−n+2)x= −L(∂λq−n+2)x− ¯ux (2.30)Integrating equations (2.29) and (2.30) from +∞ and −∞, respectively, with use of boundaryconditions ∂λφ+1(+∞) = ∂λφ−n+2(−∞) = 0, we obtain

A∂λu+1 = −L∂λq1+− ¯u + u+

A∂λu−n+2= −L∂λqn+2− − ¯u + u− (2.31)and thus

Using estimates (2.33) and (2.32), we can now compute the λ-derivative (2.27) of D± at λ = 0 as

r+2 · · · r+kp−1 r−kp+1 · · · r−n+1 −[u]

(2.34)

Lemma 2.6 Defining the Evans functions

we then have

where m is some nonzero factor

Proof Proposition 2.4 gives

3 Resolvent kernel bounds in low–frequency regions

In this section, we shall derive pointwise bounds on the resolvent kernel Gλ(x, y) in low-frequencyregimes, that is, |λ| → 0 For definiteness, throughout this section, we consider only the case y < 0.The case y > 0 is completely analogous by symmetry

We solve (2.3) with the jump conditions at x = y:

where Madj denotes the adjugate matrix of a matrix M Note that

Cjp±(y, λ) = ap(y)−1D−(y, λ)−1

where ()ij is the determinant of the (i, j) minor, and (A(y)−1)kl, l 6= p, are bounded in y

We then easily obtain the following

Lemma 3.1 For y near zero, we have

where v0([u]) is some constant vector depending only on [u] and

Ck+p(y, λ) = ap(y)−1|y|−α0O(1),

where γ−(y) and ∆ are defined as in (2.25), and ∆p,n+2 denotes the minor determinant Thus,recalling (2.24) and (3.5), we can estimate Cn+2,p− (y, λ) as

Cn+2,p− (y, λ) = ap(y)−1D−(y, λ)−1

uni-as |y| → 0, and the estimate (2.19) on w+kpp,

for some θ > 0 This together with the fact that φ−n+2≡ φ+

1 at λ = 0 yields the estimate for Ck+p asclaimed

We next estimate Cj+ (resp Cj−) for 1 < j < kp (resp kp< j < n + 2) We note that by view ofestimate (2.19) on Wk p,



Φ+ Wk+p Φ−pj

= O(λ)O(|y|α0ap(y))and for k 6= p,



Φ+ Wk+p Φ−kj

= O(λ)O(|y|α0)These estimates together with (3.9) and (3.6),(3.5) immediately yield estimates for Cj± as claimed

Proposition 3.2 (Resolvent kernel bounds as |y| → 0) For y near zero, there hold

Similar bounds can be obtained for the case y > 0

Proof For the case y < 0 < x, using (3.7) and recalling that φ+1(x) = ¯Wx+ O(λ)e−θ|x| and

Wk+p(x) ≡ 0, we have

Gλ(x, y) = Φ+(x)C+(y) =

kXp −1 j=1

k p −1

X

eµ+j x,

yielding (3.10); here, we recall that

µ±j = −λ/a±j + O(λ2)with a+j > 0 for j = 2, , kp− 1 and a−j < 0 for j = kp+ 1, , n + 1 (a±j are necessarily eigenvalues

of A± ) In the second case y < x < 0, from the formula (3.2), we have

Gλ(x, y) = Φ+(x, λ)C+(y, λ) + Wk+p(x, λ)Ck+p(y, λ)where the first term contributes λ−1v0([u]) ¯Wx+ O(1) as in the first case, and the second term isestimated by (3.8) and (2.21)

Finally, we estimate the last case x < y < 0 in a same way as done in the first case, noting that

Next, we estimate the kernel Gλ(x, y) for y away from zero Note however that the representations(3.2) and above estimates fail to be useful in the y → −∞ limit, since we actually need precise decayrates in order to get an estimate of form

|Gλ(x, y)| ≤ Ce−η|x−y|,which are unavailable from φ+j in the y → −∞ regime Thus, we need to express the (+)-bases interms of the growing modes ψj− at −∞, and the decaying mode φ−j where ψ−j, φ−j are defined as inLemma 2.3 Expressing such solutions in the basis for y < 0, away from zero, there exist analyticcoefficients djk(λ), ejk(λ) such that

˜

Ψ− Φ˜−

:= Ψ− Φ−
−1

We then obtain the following estimates

Lemma 3.3 For |λ| sufficiently small and |x| sufficiently large,

where µ−j are defined as in Lemma 2.3

Proof The proof is clear from the estimates of ψ−j , φ−j in (2.15) Lemma 3.4 We have

Cj+(y, λ) =X

Cj−(y, λ) =X

c−jk(λ) ˜ψk−(y, λ)∗+ ˜φ−j(y, λ)∗, (3.17)for meromorphic coefficients c±jk in λ

Proof The proof follows by using (3.13), definition (3.14), and property of computing determinants



We then have the following representation for G (x, y), for y large

Proposition 3.5 Under the assumptions of Theorem 1.4, for |λ| sufficiently small and |y| ciently large, we have

d±= 0 −In−k p



Φ+ Wk+p Φ−−1

Ψ−.Proof Using representation (3.2) of Gλ(x, y) together with (3.16) and (3.17), we easily obtain theexpansions (3.18) and (3.20), respectively For (3.19), again, using (3.16), (3.13), and (3.2), we canwrite

Proposition 3.6 (Resolvent kernel bounds as |y| → +∞) Make the assumptions of Theorem 1.4.Then, for |y| large, defining

Similar bounds can be obtained for the case y > 0.

Proof The proof follows directly from the representations of Gλ(x, y) derived in Proposition 3.5 andthe corresponding estimates on normal modes, noting that

|c+jk|, |d±jk| =

O(λ−1) j = 1,O(1) otherwise

Indeed, we recall, for instance, that

c+jk= D−−1 −Ik p 0


Φ+ Wk+p Φ−kj

Ψ−,where ()kj denotes the determinant of the (k, j) minors For the case j 6= 1, by using the fact that

we choose φ+1 ≡ φ−n+2≡ ¯Wxat λ = 0, determinant of the (k, j) minor therefore has the order one in

λ, which cancels out the λ−1 term coming from our spectral stability condition: |D−1− | ≤ O(λ−1)



4 Pointwise bounds and low-frequency estimates

In this section, using the previous pointwise bounds (Propositions 3.2 and 3.6) for the resolventkernel in low-frequency regions, we derive pointwise bounds for the “low-frequency” Green function:

GI(x, t; y) := 1

2πi

Z

Γ T {|λ|≤r}

where Γ is any contour near zero, but away from the essential spectrum

Proposition 4.1 Under the assumptions of Theorem 1.4, defining the effective diffusion β± :=(LpLBRp)±(see (2.5)), the low-frequency Green distribution GI(x, t; y) associated with the linearizedevolution equations may be decomposed as

!

− errfn y − a

−

ktp4β−t

!!