1001.0048.txt

arXiv:1001.0048v2  [math.AP]  7 Jan 2010Nonlinear stability of periodic traveling wave solutions o f
viscous conservation laws in dimensions one and two
Mathew A. Johnson∗Kevin Zumbrun†
November 12, 2018
Keywords : Periodic traveling waves; Bloch decomposition; modulate d waves.
2000 MR Subject Classification : 35B35.
Abstract
Extending results of Oh and Zumbrun in dimensions d≥3, we establish nonlin-
ear stability and asymptotic behavior of spatially-periodic traveling- wave solutions of
viscous systems of conservation laws in critical dimensions d= 1,2, under a natural
set of spectral stability assumptions introduced by Schneider in th e setting of reaction
diffusion equations. The key new steps in the analysis beyond that in d imensionsd≥3
are a refined Green function estimate separating off translation as the slowest decaying
linear mode and a novel scheme for detecting cancellation at the leve l of the nonlinear
iteration in the Duhamel representation of a modulated periodic wav e.
1 Introduction
Nonclassical viscous conservation laws arising in multiph ase fluid and solid mechanics ex-
hibit a rich variety of traveling wave phenomena, including homoclinic (pulse-type) and
periodic solutions along with the standard heteroclinic (s hock, or front-type) solutions
[GZ, Z6, OZ1, OZ2]. Here, we investigate stability of period ic traveling waves: specifi-
cally, sufficient conditions for stability of the wave. Our ma in result, generalizing results of
Oh and Zumbrun [OZ4] in dimensions d≥3, is to show that strong spectral stability in the
sense of Schneider [S1, S2, S3] implies linearized and nonli nearL1∩HK→L∞bounded
stability, for all dimensions d≥1, andasymptotic stability for dimensions d≥2.
∗Indiana University, Bloomington, IN 47405; matjohn@india na.edu: Research of M.J. was partially sup-
ported by an NSF Postdoctoral Fellowship under NSF grant DMS -0902192.
†Indiana University, Bloomington, IN 47405; kzumbrun@indi ana.edu: Research of K.Z. was partially
supported under NSF grants no. DMS-0300487 and DMS-0801745 .
11 INTRODUCTION 2
More precisely, we show that small L1∩Hsperturbations of a planar periodic solution
u(x,t)≡¯u(x1) (without loss of generality taken stationary) converge at Gaussian rate in
Lp,p≥2 to a modulation
(1.1) ¯ u(x1−ψ(x,t)),
of the unperturbed wave, where x= (x1,˜x), ˜x= (x2,...,xd), andψis a scalar function
whosex- andt-gradients likewise decay at least at Gaussian rate in all Lp,p≥2, but which
itself decays more slowly by a factor t1/2; in particular, ψis merely bounded in L∞for
dimensiond= 1.
The one-dimensional study of spectral stability of spatial ly periodic traveling waves of
systems of viscous conservation laws was initiated by Oh and Zumbrun [OZ1] in the “quasi-
Hamiltonian” case that the traveling-wave equation posses ses an integral of motion, and in
the general case by Serre[Se1]. An important contribution o f Serre was to point out a larger
connection between the linearized dispersion relation (th e functionλ(ξ) relating spectra to
wave number of the linearized operator about the wave) near z ero and the formal Whitham
averaged system obtained by slow modulation, or WKB, approx imation.
In [OZ3], this was extended to multi-dimensions, relating t he linearized dispersion rela-
tion near zero to
(1.2)∂tM+/summationdisplay
j∂xjFj= 0,
∂t(ΩN)+∇x(ΩS) = 0,
whereM∈Rndenotes the average over one period, Fjthe average of an associated flux,
Ω =|∇xΨ| ∈R1the frequency, S=−Ψt/|∇xΨ| ∈R1the speeds, andN=∇xΨ/|∇xΨ| ∈
Rdthe normal νassociated with nearby periodic waves, with an additional c onstraint
(1.3) curl (Ω N) = curl ∇xΨ≡0.
As an immediate corollary, similarly as in [OZ1], [Se1] in th e one-dimensional case, this
yieldedas anecessary condition formulti-dimensional sta bility hyperbolicityoftheaveraged
system (1.2)–(1.3).
The present study is informed by but does not directly rely on this observation relating
Whitham averaging and spectral stability properties. Like wise, the Evans function tech-
niquesusedin[Se1,OZ3]toestablishthisconnection play n oroleinouranalysis; indeed, the
Evans function makes no appearance here. Rather, we rely on a direct Bloch-decomposition
argument in the spirit of Schneider [S1, S2, S3], combining s harp linearized estimates with
subtle cancellation in nonlinear source terms arising from the modulated wave approxima-
tion. The analytical techniques used to realize this progra m are somewhat different from
those of [S1, S2, S3], however, coming instead from the theor y of stability of viscous shock
fronts through a line of investigation carried out in [OZ1, O Z2, OZ3, OZ4, HoZ]. In partic-
ular, the nonsmooth dispersion relation at ξ= 0 typical for convection-diffusion equations1 INTRODUCTION 3
requires different treatment from that of [S1, S2, S3] in the re action diffusion case; see Re-
mark 2.4. Moreover, we detect nonlinear cancellation in the physicalx-tdomain rather than
the frequency domain as in [S1, S2, S3]. The main difference bet ween the present analysis
and that of [OZ4] is the systematic incorporation of modulat ion approximation (1.1).
1.1 Equations and assumptions
Consider a parabolic system of conservation laws
(1.4) ut+/summationdisplay
jfj(u)xj= ∆xu,
u∈ U(open)∈Rn,fj∈Rn,x∈Rd,d≥1,t∈R+, and a periodic traveling wave solution
(1.5) u= ¯u(x·ν−st),
of periodX, satisfying the traveling-wave ODE ¯ u′′= (/summationtext
jνjfj(¯u))′−s¯u′with boundary
conditions ¯u(0) = ¯u(X) =:u0.Integrating, we obtain a first-order profile equation
(1.6) ¯ u′=/summationdisplay
jνjfj(¯u)−s¯u−q,
where (u0,q,s,ν,X )≡constant. Without loss of generality take ν=e1,s= 0, so that
¯u= ¯u(x1) represents a stationary solution depending only on x1.
Following [Se1, OZ3, OZ4], we assume:
(H1)fj∈CK+1,K≥[d/2]+4.
(H2) Themap H:R×U×R×Sd−1×Rn→Rntaking (X;a,s,ν,q)/mapsto→u(X;a,s,ν,q)−a
is a submersion at point ( ¯X;¯u(0),0,e1,¯q), whereu(·;·) is the solution operator for (1.6).
Conditions (H1)–(H2) imply that the set of periodic solutio ns in the vicinity of ¯ uform
a smooth (n+d+1)-dimensional manifold {¯ua(x·ν(a)−α−s(a)t)}, withα∈R,a∈Rn+d.
1.1.1 Linearized equations
Linearizing (1.4) about ¯ u(·), we obtain
(1.7) vt=Lv:= ∆xv−/summationdisplay
(Ajv)xj,
where coefficients Aj:=Dfj(¯u) are now periodic functions of x1. Taking the Fourier
transform in the transverse coordinate ˜ x= (x2,···,xd), we obtain
(1.8)ˆvt=L˜ξˆv= ˆvx1,x1−(A1ˆv)x1−i/summationdisplay
j/negationslash=1Ajξjˆv−/summationdisplay
j/negationslash=1ξ2
jˆv,
where˜ξ= (ξ2,···,ξd) is the transverse frequency vector.1 INTRODUCTION 4
1.1.2 Bloch–Fourier decomposition and stability conditions
Following [G, S1, S2, S3], we define the family of operators
(1.9) Lξ=e−iξ1x1L˜ξeiξ1x1
operating on the class of L2periodic functions on [0 ,X]; the (L2) spectrum of L˜ξis equal
to the union of the spectra of all Lξwithξ1real with associated eigenfunctions
(1.10) w(x1,˜ξ,λ) :=eiξ1x1q(x1,ξ1,˜ξ,λ),
whereq, periodic, is an eigenfunction of Lξ. By continuity of spectrum, and discreteness of
the spectrum of the elliptic operators Lξon the compact domain [0 ,X], we have that the
spectra ofLξmay be described as the union of countably many continuous su rfacesλj(ξ).
Without loss of generality taking X= 1, recall now the Bloch–Fourier representation
(1.11) u(x) =/parenleftBig1
2π/parenrightBigd/integraldisplayπ
−π/integraldisplay
Rd−1eiξ·xˆu(ξ,x1)dξ1d˜ξ
of anL2functionu, where ˆu(ξ,x1) :=/summationtext
ke2πikx1ˆu(ξ1+ 2πk,˜ξ) are periodic functions of
periodX= 1, ˆu(˜ξ) denoting with slight abuse of notation the Fourier transfo rm ofuin the
full variable x. By Parseval’s identity, the Bloch–Fourier transform u(x)→ˆu(ξ,x1) is an
isometry in L2:
(1.12) /ba∇dblu/ba∇dblL2(x)=/ba∇dblˆu/ba∇dblL2(ξ;L2(x1)),
whereL2(x1) is taken on [0 ,1] andL2(ξ) on [−π,π]×Rd−1. Moreover, it diagonalizes the
periodic-coefficient operator L, yielding the inverse Bloch–Fourier transform representation
(1.13) eLtu0=/parenleftBig1
2π/parenrightBigd/integraldisplayπ
−π/integraldisplay
Rd−1eiξ·xeLξtˆu0(ξ,x1)dξ1d˜ξ
relating behavior of the linearized system to that of the dia gonal operators Lξ.
Following [OZ4], weassumealongwith(H1)–(H2) the strong spectral stability conditions:
(D1)σ(Lξ)⊂ {Reλ<0}forξ/ne}ationslash= 0.
(D2) Reσ(Lξ)≤ −θ|ξ|2,θ>0, forξ∈Rdand|ξ|sufficiently small.
(D3)λ= 0 is a semisimple eigenvalue of L0of multiplicity exactly n+1.1
For each fixed angle ˆξ:=ξ/|ξ|, expandLξ=L0+|ξ|L1+|ξ|2L2. By assumption (D3)
and standard spectral perturbation theory, there exist n+1 smooth eigenvalues
(1.14) λj(ξ) =−iaj(ξ)+o(|ξ|)
1The zero eigenspace of L0is at least ( n+1)-dimensional by the linearized existence theory and (H2 ),
and hence n+ 1 is the minimal multiplicity; see [Se1, OZ3]. As noted in [O Z1, OZ3], minimal dimension
of this zero eigenspace implies that ( M,NΩ) of (1.2) gives a nonsingular coordinatization of the fami ly of
periodic traveling-wave solutions near ¯ u.1 INTRODUCTION 5
ofLξbifurcating from λ= 0 atξ= 0, where −iajare homogeneous degree one functions
given by |ξ|times the eigenvalues of Π 0L1|KerL0, with Π 0the zero eigenprojection of L0.
Conditions(D1)–(D3) areexactly thespectralassumptions of[S1,S2,S3], corresponding
to “dissipativity” of the large-time behavior of the linear ized system. As in [OZ4], we make
the further nondegeneracy hypothesis:
(H3) The eigenvalues λ=−iaj(ξ)/|ξ|of Π0L1
KerL0are simple.
The functions ajmay be seen to be the characteristics associated with the Whi tham av-
eraged system (1.2)–(1.3) linearized about the values of M,S,N, Ω associated with the
background wave ¯ u; see [OZ3, OZ4]. Thus, (D1) implies weak hyperbolicity of (1 .2)–(1.3)
(reality ofaj), while (H1) corresponds to strict hyperbolicity.
1.2 Main results
With these preliminaries, we can now state our main results.
Theorem 1.1. Assuming (H1)–(H3) and (D1)–(D3), for some C >0andψ∈WK,∞(x,t),
(1.15)|˜u−¯u(·−ψ)|Lp(t)≤C(1+t)−d
2(1−1/p)|˜u−¯u|L1∩HK|t=0,
|˜u−¯u(·−ψ)|HK(t)≤C(1+t)−d
4|˜u−¯u|L1∩HK|t=0,
|(ψt,ψx)|WK+1,p≤C(1+t)−d
2(1−1/p)|˜u−¯u|L1∩HK|t=0,
and
(1.16) |˜u−¯u|Lp(t),|ψ(t)|Lp≤C(1+t)−d
2(1−1
p)+1
2|˜u−¯u|L1∩HK|t=0
for allt≥0,p≥2,d= 1, for solutions ˜uof(1.4)with|˜u−¯u|L1∩HK|t=0sufficiently small.
In particular, ¯uis nonlinearly bounded L1∩HK→L∞stable for dimension d= 1.
Theorem 1.2. Assuming (H1)–(H3) and (D1)–(D3), for any ε >0, someC >0and
ψ∈WK,∞(x,t),
(1.17)|˜u−¯u(·−ψ)|Lp(t)≤C(1+t)−d
2(1−1/p)|˜u−¯u|L1∩HK|t=0,
|˜u−¯u(·−ψ)|HK(t)≤C(1+t)−d
4|˜u−¯u|L1∩HK|t=0,
|(ψt,ψx)|WK+1,p≤C(1+t)−d
2(1−1/p)+ε−1
2|˜u−¯u|L1∩HK|t=0,
and
(1.18)|˜u−¯u|Lp(t),|ψ(t)|Lp≤C(1+t)−d
2(1−1
p)+ε|˜u−¯u|L1∩HK|t=0,
|˜u−¯u|HK(t),|ψ(t)|HK≤C(1+t)−d
4+ε|˜u−¯u|L1∩HK|t=0,
for allt≥0,p≥2,d= 2, for solutions ˜uof(1.4)with|˜u−¯u|L1∩HK|t=0sufficiently small.
In particular, ¯uis nonlinearly asymptotically L1∩HK→HKstable for dimension d= 2.1 INTRODUCTION 6
Remark 1.1. In Theorem 1.2, derivatives in x∈R2refer to total derivatives. Moreover,
unless specified by an appropriate index, throughout this pa per derivatives in spatial variable
xwill always refer to the total derivative of the function.
In dimension one, Theorem 1.1 asserts only bounded L1∩HK→L∞stability, a very
weak notion of stability. The absence of decay in perturbati on ˜u−¯uindicates the delicacy
of the nonlinear analysis in this case. In particular, it is c rucial to separate off the slower-
decaying modulated behavior (1.1) in order to close the nonl inear iteration argument.
Remark 1.2. In dimension d= 1, it is straightforward to show that the results of Theorem
1.1 extend to all 1 ≤p≤ ∞using the pointwise techniques of [OZ2]; see Remark 3.3.
Remark 1.3. The slow decay of |˜u−¯u|Lp(t)∼ |ψ(t)|Lpin (1.16) is due to nonlinear
interactions; as shown in [OZ2, OZ4], the linearized decay r ate is faster by factor (1+ t)−1/2
(Proposition 2.1). In [OZ4], it was shown that for d≥3, where linear effects dominate
behavior, (1.16) may be replaced by the stronger estimate |˜u−¯u|Lp(t),|ψ(t)|Lp≤C(1+
t)−d
2(1−1
p)|˜u−¯u|L1∩HK|t=0.These distinctions reflect fine details of both linearized es timates
(Section 3) and nonlinear structure (Sections 4.1–4.2) tha t are not immediately apparent
from the formal Whitham approximation (1.2)–(1.3).
1.3 Discussion and open problems
Linearized stability under the same assumptions, with shar p rates of decay, was established
ford= 1 [OZ2] and for d≥1 in [OZ4], along with nonlinear stability for d≥3. Theorem
1.1 completes this line of investigation by establishing no nlinear stability in the critical
dimensions d= 1,2, a fundamental open problem cited in [OZ1, OZ4].
This gives a generalization of the work of [S1, S2, S3] for rea ction diffusion equations
to the case of viscous conservation laws. Recall that the ana lysis of [S1, S2, S3] concerns
also multiply periodic waves, i.e., waves that are either pe riodic or else constant in each
coordinate direction. It is straightforward to verify that the methods of this paper apply
essentially unchanged to this case, to give a corresponding stability result under the analog
of (H1)–(H3), (D1)–(D3), as we intend to report further in a f uture work. Likewise, the
extension from the semilinear parabolic case treated here t o the general quasilinear case is
straightforward, following the treatment of [OZ4].
On the other hand, as noted in [OZ2], condition (D3) is in the c onservation law setting
nongeneric, corresponding to the special “quasi-Hamilton ian” situation studied there; in
particular, it implies that speed is to first order constant a mong the family of spatially pe-
riodic traveling-wave solutions nearby ¯ u. In the generic case that (D3) is violated, behavior
is essentially different [OZ1, OZ2], and perturbations decay more slowly at the linearized
level. Nonlinear stability remains an interesting open pro blem in this setting.
Our approach to stability in the critical dimensions d= 1,2, as suggested in [OZ4], is,
loosely following the approach of [S1, S2, S3], to subtract o ut a slower-decaying part of the
solution describedby anappropriatemodulation equation a ndshowthat theresidualdecays2 BASIC LINEARIZED STABILITY ESTIMATES 7
sufficiently rapidly to close a nonlinear iteration. It is wor th noting that the modulated
approximation ¯ u(x1−ψ(x,t)) of (1.1) is not the full Ansatz
(1.19) ¯ ua(Ψ(x,t)),
Ψ(x,t) :=x1−ψ(x,t), associated with the Whitham averaged system (1.2)–(1.3) , where ¯ua
isthemanifoldofperiodicsolutions near ¯ uintroducedbelow(H2), butonlythetranslational
part not involving perturbations ain the profile. (See [OZ3] for the derivation of (1.2)–
(1.3) and (1.19).) That is, we don’t need to separate out all v ariations along the manifold
of periodic solutions, but only the special variations conn ected with translation invariance.
The technical reason is an asymmetry in y-derivative estimates in the parts of the Green
function associated with these various modes, something th at is not apparent without a
detailed study of linearized behavior as carried out here. T his also makes sense formally,
if one considers that (1.2) indicates that variables a,∇xΨ are roughly comparable, which
would suggest, by the diffusive behavior Ψ >>∇xΨ, thatais neglible with respect to Ψ.
However, note that in the case that (D3) holds, hence wave spe ed is stationary along the
manifold of periodic solutions, the final equation of (1.2) d ecouples to (Ψ x)t= (ΩN)t= 0,
and could be written as Ψ t= 0 in terms of Ψ alone. Hence, there is some ambiguity in this
degenerate case which of Ψ, Ψ xis the primary variable, and in terms of linear behavior, the
decay of variations aand Ψ are in fact comparable [OZ4]; in the generic case, aand Ψxare
comparable at the linearized level [OZ2]. It would be very in teresting to better understand
the connection between the Whitham averaged system (or suit able higher-order correction)
and behavior at the nonlinear level, as explored at the linea r level in [OZ3, OZ4, JZ1, JZB].
2 Basic linearized stability estimates
We begin by recalling the basic linearized stability estima tes derived in [OZ4]. We will
sharpen these afterward in Section 3. By standard spectral p erturbation theory [K], the
total eigenprojection P(ξ) onto the eigenspace of Lξassociated with the eigenvalues λj(ξ),
j= 1,...,n+1describedintheintroductioniswell-definedandanalyti cinξforξsufficiently
small, since these (by discreteness of the spectra of Lξ) are separated at ξ= 0 from the rest
of the spectrum of L0. Introducing a smooth cutoff function φ(ξ) that is identically one
for|ξ| ≤εand identically zero for |ξ| ≥2ε,ε >0 sufficiently small, we split the solution
operatorS(t) :=eLtinto low- and high-frequency parts
(2.1) SI(t)u0:=/parenleftBig1
2π/parenrightBigd/integraldisplayπ
−π/integraldisplay
Rd−1eiξ·xφ(ξ)P(ξ)eLξtˆu0(ξ,x1)dξ1d˜ξ
and
(2.2) SII(t)u0:=/parenleftBig1
2π/parenrightBigd/integraldisplayπ
−π/integraldisplay
Rd−1eiξ·x/parenleftbig
I−φP(ξ)/parenrightbig
eLξtˆu0(ξ,x1)dξ1d˜ξ.2 BASIC LINEARIZED STABILITY ESTIMATES 8
2.1 High-frequency bounds
By standard sectorial bounds [He, Pa] and spectral separati on ofλj(ξ) from the remaining
spectra ofLξ, we have trivially the exponential decay bounds
(2.3)/ba∇dbleLξt(I−φP(ξ))f/ba∇dblL2([0,X])≤Ce−θt/ba∇dblf/ba∇dblL2([0,X]),
/ba∇dbleLξt(I−φP(ξ))∂l
x1f/ba∇dblL2([0,X])≤Ct−l
2e−θt/ba∇dblf/ba∇dblL2([0,X]),
/ba∇dbl∂l
x1eLξt(I−φP(ξ))f/ba∇dblL2([0,X])≤Ct−l
2e−θt/ba∇dblf/ba∇dblL2([0,X]),
forθ,C >0, and 0 ≤m≤K(Kas in (H1)). Together with (1.12), these give immediately
the following estimates.
Proposition 2.1 ([OZ4]).Under assumptions (H1)–(H3) and (D1)–(D2), for some θ,
C >0, and allt>0,2≤p≤ ∞,0≤l≤K+1,0≤m≤K,
(2.4)/ba∇dbl∂l
xSII(t)f/ba∇dblL2(x),/ba∇dblSII(t)∂l
xf/ba∇dblL2(x)≤Ct−l
2e−θt/ba∇dblf/ba∇dblL2(x),
/ba∇dbl∂m
xSII(t)f/ba∇dblLp(x),/ba∇dblSII(t)∂m
xf/ba∇dblLp(x)≤Ct−d
2(1
2−1
p)−m
2e−θt/ba∇dblf/ba∇dblL2(x),
where, again, derivatives in the variable x∈Rdrefer to total derivatives.
Proof.The first inequalities follow immediately by (1.12) and (2.3 ). The second follows for
x1derivatives in the case p=∞,m= 0 by Sobolev embedding from
/ba∇dblSII(t)f/ba∇dblL∞(˜x;L2(x1))≤Ct−d−1
4e−θt/ba∇dblf/ba∇dblL2([0,X])
and
/ba∇dbl∂x1SII(t)f/ba∇dblL∞(˜x;L2(x1))≤Ct−d−1
4−1
2e−θt/ba∇dblf/ba∇dblL2([0,X]),
which follow by an application of (1.12) in the x1variable and the Hausdorff–Young in-
equality /ba∇dblf/ba∇dblL∞(˜x)≤ /ba∇dblˆf/ba∇dblL1(˜ξ)in the variable ˜ x. The result for derivatives in x1and general
2≤p≤ ∞then follows by Lpinterpolation. Finally, the result for derivatives in ˜ xfollows
from the inverse Fourier transform, equation (2.2), and the large|ξ|bound
|eLtf|L2(x1)≤e−θ|˜ξ|2t|f|L2(x1),|ξ|sufficiently large ,
which easily follows from Parseval and the fact that Lξis a relatively compact perturbation
of∂2
x−|ξ|2. Thus, by the above estimate we have
/ba∇dbleLt∂˜xf/ba∇dblL2(x)≤C/ba∇dbleLξt|˜ξ|ˆf/ba∇dblL2(x1,ξ)
≤Csup/parenleftBig
e−θ|˜ξ|2t|ξ|/parenrightBig
/ba∇dblˆf/ba∇dblL2(x1,ξ)
≤Ct−1/2/ba∇dblf/ba∇dblL2(x).
A similar argument applies for 1 ≤m≤K.2 BASIC LINEARIZED STABILITY ESTIMATES 9
2.2 Low-frequency bounds
Denote by
(2.5) GI(x,t;y) :=SI(t)δy(x)
the Green kernel associated with SI, and
(2.6) [ GI
ξ(x1,t;y1)] :=φ(ξ)P(ξ)eLξt[δy1(x1)]
the corresponding kernel appearing within the Bloch–Fouri er representation of GI, where
the brackets on [ Gξ] and [δy] denote the periodic extensions of these functions onto the
whole line. Then, we have the following descriptions of GI, [GI
ξ], deriving from the spectral
expansion (1.14) of Lξnearξ= 0.
Proposition 2.2 ([OZ4]).Under assumptions (H1)–(H3) and (D1)–(D3),
(2.7)[GI
ξ(x1,t;y1)] =φ(ξ)n+1/summationdisplay
j=1eλj(ξ)tqj(ξ,x1)˜qj(ξ,y1)∗,
GI(x,t;y) =/parenleftBig1
2π/parenrightBigd/integraldisplay
Rdeiξ·(x−y)[GI
ξ(x1,t;y1)]dξ
=/parenleftBig1
2π/parenrightBigd/integraldisplay
Rdeiξ·(x−y)φ(ξ)n+1/summationdisplay
j=1eλj(ξ)tqj(ξ,x1)˜qj(ξ,y1)∗dξ,
where∗denotes matrix adjoint, or complex conjugate transpose, qj(ξ,·)and˜qj(ξ,·)are right
and left eigenfunctions of Lξassociated with eigenvalues λj(ξ)defined in (1.14), normalized
so that/an}b∇acketle{t˜qj,qj/an}b∇acket∇i}ht ≡1, whereλj/|ξ|is a smooth function of |ξ|andˆξ:=ξ/|ξ|andqjand˜qj
are smooth functions of |ξ|,ˆξ:=ξ/|ξ|, andx1ory1, withℜλj(ξ)≤ −θ|ξ|2.
Proof.Smooth dependence of λjand ofq, ˜qas functions in L2[0,X] follow from standard
spectral perturbation theory [K] using the fact that λjsplit to first order in |ξ|asξis varied
along rays through the origin, and that Lξvaries smoothly with angle ˆξ. Smoothness of
qj, ˜qjinx1,y1then follow from the fact that they satisfy the eigenvalue eq uation forLξ,
which has smooth, periodic coefficients. Likewise, (2.7)(i) is immediate from the spectral
decomposition of elliptic operators on finite domains. Subs tituting (2.5) into (2.1) and
computing
(2.8) /hatwideδy(ξ,x1) =/summationdisplay
ke2πikx1/hatwideδy(ξ+2πke1) =/summationdisplay
ke2πikx1e−iξ·y−2πiky1=e−iξ·y[δy1(x1)],
where the second and third equalities follow from the fact th at the Fourier transform either
continuous or discrete of the delta-function is unity, we ob tain
GI(x,t;y) =/parenleftBig1
2π/parenrightBigd/integraldisplayπ
−π/integraldisplay
Rd−1eiξ·xφP(ξ)eLξt/hatwideδy(ξ,x1)dξ
=/parenleftBig1
2π/parenrightBigd/integraldisplayπ
−π/integraldisplay
Rd−1eiξ·(x−y)φP(ξ)eLξt[δy1(x1)]dξ,2 BASIC LINEARIZED STABILITY ESTIMATES 10
yielding (2.7)(ii) by (2.6)(i) and the fact that φis supported on [ −π,π].
Proposition 2.3 ([OZ4]).Under assumptions (H1)-(H3) and (D1)-(D3),
(2.9) sup
y/ba∇dblGI(·,t,;y)/ba∇dblLp(x),sup
y/ba∇dbl∂x,yGI(·,t,;y)/ba∇dblLp(x)≤C(1+t)−d
2(1−1
p)
for all2≤p≤ ∞,t≥0, whereC >0is independent of p.
Proof.From representation (2.7)(ii) and ℜλj(ξ)≤ −θ|ξ|2, we obtain by the triangle in-
equality
(2.10) /ba∇dblGI/ba∇dblL∞(x,y)≤C/ba∇dble−θ|ξ|2tφ(ξ)/ba∇dblL1(ξ)≤C(1+t)−d
2,
verifying the bounds for p=∞. Derivative bounds follow similarly, since derivatives fa lling
onqjor ˜qjare harmless, whereas derivatives falling on eiξ·(x−y)bring down a factor of ξ,
again harmless because of the cutoff function φ.
To obtain bounds for p= 2, we note that (2.7)(ii) may be viewed itself as a Bloch–
Fourier decomposition with respect to variable z:=x−y, withyappearing as a parameter.
Recalling (1.12), we may thus estimate
(2.11)sup
y/ba∇dblGI(x,t;y)/ba∇dblL2(x)=/summationdisplay
jsup
y/ba∇dblφ(ξ)eλj(ξ)tqj(·,z1)˜q∗
j(·,y1)/ba∇dblL2(ξ;L2(z1∈[0,X]))
≤C/summationdisplay
jsup
y/ba∇dblφ(ξ)e−θ|ξ|2t/ba∇dblL2(ξ)/ba∇dblqj/ba∇dblL2(0,X)/ba∇dbl˜qj/ba∇dblL∞(0,X)
≤C(1+t)−d
4,
where we have used in a crucial way the boundedness of ˜ qj; derivative bounds follow simi-
larly. Finally, bounds for 2 ≤p≤ ∞follow byLp-interpolation.
Remark 2.4. In obtaining the key L2-estimate, we have used in an essential way the
periodic structure of qj, ˜qj. For, viewing GIas a general pseudodifferential expression
rather than a Bloch–Fourier decomposition, we find that the s moothness of qj, ˜qjis not
sufficient to apply standard L2→L2bounds of H¨ ormander, which require blowup in ξ
derivatives at less than the critical rate |ξ|−1found here; see, e.g., [H] for further discussion.
Nor do the weighted energy estimate techniques used in [S1, S 2, S3] apply here, as these also
rely on the property of smoothness of λj,qj, ˜qjwith respect to ξat the origin ξ= 0. The
lack of smoothness of the linearized dispersion relation at the origin is an essential technical
difference separating the conservation law from the reaction diffusion case; see [OZ4] for
further discussion.
Remark 2.5. Underlying the above analysis, and also the technically rat her different
approach of [OZ2], is the fundamental relation
(2.12) G(x,t;y) =/parenleftBig1
2π/parenrightBigd/integraldisplayπ
−π/integraldisplay
Rd−1eiξ·(x−y)[Gξ(x1,t;y1)]dξ2 BASIC LINEARIZED STABILITY ESTIMATES 11
which, provided σ(Lξ) is semisimple, yields the simple formula
G(x,t;y) =/parenleftBig1
2π/parenrightBigd/integraldisplayπ
−π/integraldisplay
Rd−1eiξ·(x−y)/summationdisplay
jeλj(ξ)tqj(ξ,x1)˜qj(ξ,y1)∗dξ
resembling that of the constant-coefficient case, where λjruns through the spectrum of Lξ.
The basic idea in both cases is to separate off the principal pa rt of the series involving small
λj(ξ) and estimate the remainder as a faster-decaying residual.
Corollary 2.6 ([OZ4]).Under assumptions (H1)–(H3) and (D1)–(D3), for all p≥2,t≥0,
(2.13) /ba∇dblSI(t)f/ba∇dblLp,/ba∇dbl∂xSI(t)f/ba∇dblLp,/ba∇dblSI(t)∂xf/ba∇dblLp≤C(1+t)−d
2(1−1
p)/ba∇dblf/ba∇dblL1.
Proof.Immediate, from (2.9) and the triangle inequality, as, for e xample,
/ba∇dblSI(t)f(·)/ba∇dblLp=/vextenddouble/vextenddouble/vextenddouble/integraldisplay
RdGI(x,t;y)f(y)dy/vextenddouble/vextenddouble/vextenddouble
Lp(x)≤/integraldisplay
Rdsup
y/ba∇dblGI(·,t;y)/ba∇dblLp|f(y)|dy.
Proposition 2.1 ([OZ4]).Assuming (H1)-(H3), (D1)-(D3), for some C >0, allt≥0,
p≥2,0≤l≤K,
(2.14) /ba∇dblS(t)∂l
xu0/ba∇dblLp≤Ct−l
2(1+t)−d
2(1
2−1
p)+l
2t−d
4−l
2/ba∇dblu0/ba∇dblL1∩L2.
Proof.Immediate, from (2.4) and (2.13).
2.3 Additional estimates
Lemma 2.7. Assuming (H1)–(H3), (D1)–(D3), for all t≥0,0≤l≤K,
(2.15) /ba∇dbl∂l
xSI(t)f/ba∇dblLp(x),/ba∇dblSI(t)∂l
xf/ba∇dblLp(x)≤C(1+t)−d
2(1/2−1/p)/ba∇dblf/ba∇dblL2(x).
Proof.From boundedness of the spectral projections Pj(ξ) =qj/an}b∇acketle{t˜qj,·/an}b∇acket∇i}htinL2[0,X] and their
derivatives, another consequence of first-order splitting of eigenvalues λj(ξ) at the origin,
we obtain boundedness of φ(ξ)P(ξ)eLξtand thus, by (1.12), the global bounds
(2.16) /ba∇dbl∂l
xSI(t)f/ba∇dblL2(x),/ba∇dblSI(t)∂l
xf/ba∇dblL2(x)≤C/ba∇dblf/ba∇dblL2(x),
for allt≥0, yielding the result for p= 2. Moreover, by boundedness of ˜ q,qin allLp(x1),
we have
|φ(ξ)P(ξ)eLξtˆf(ξ,·)|L∞(x1)≤Ce−θ|ξ|2t|P(ξ)ˆf(ξ,·)|L∞(x1)≤Ce−θ|ξ|2t|ˆf(ξ,·)|L2(x1),3 REFINED LINEARIZED ESTIMATES 12
C, θ>0, yielding by SIf=/parenleftBig
1
2π/parenrightBigd/integraltextπ
−π/integraltext
Rd−1eiξ·xφ(ξ)P(ξ)eLξtˆf(ξ,x1)dξ1d˜ξthe bound
(2.17)/ba∇dblSI(t)f/ba∇dblL∞(x)≤/parenleftBig1
2π/parenrightBigd/integraldisplayπ
−π/integraldisplay
Rd−1|φ(ξ)P(ξ)eLξtˆf(ξ,·)|L∞(x1)dξ1d˜ξ
≤/parenleftBig1
2π/parenrightBigd/integraldisplayπ
−π/integraldisplay
Rd−1Cφ(ξ)e−θ|ξ|2t|ˆf(ξ,·)|L2(x1)dξ1d˜ξ
≤C|φ(ξ)e−θ|ξ|2t|L2(ξ)|ˆf|L2(ξ,x1)
=C(1+t)−d
4/ba∇dblf/ba∇dblL2([0,X]),
yielding the result for p=∞,l= 0. The result for p=∞, 1≤l≤Kfollows by a similar
argument. The result for general 2 ≤p≤ ∞then follows by Lpinterpolation between p= 2
andp=∞.
By Riesz–Thorin interpolation between (2.15) and (2.13), w e obtain the following, ap-
parently sharp bounds between various LqandLp.2
Corollary 2.8. Assuming (H0)–(H3) and (D1)–(D3), for all 1≤q≤2≤p,t≥0,
0≤l≤K,
(2.18) /ba∇dbl∂l
xSI(t)f/ba∇dblLp,/ba∇dblSI(t)∂l
xf/ba∇dblLp≤C(1+t)−d
2(1
q−1
p)/ba∇dblf/ba∇dblLq.
Proposition 2.2. Assuming (H1)-(H3), (D1)-(D3), for some C >0, allt≥0,1≤q≤
2≤p, and0≤l≤K,
(2.19) /ba∇dblS(t)∂l
xu0/ba∇dblLp≤C(1+t)−d
2(1
2−1
p)+l
2t−d
2(1
q−1
2)−l
2/ba∇dblu0/ba∇dblLq∩L2.
Proof.Immediate, from (2.4) and (2.8).
3 Refined linearized estimates
The bounds of Proposition 2.1 are sufficient to establish nonl inear stability and asymptotic
behavior in dimensions d≥3, as shown in [OZ4]. However, they are not sufficient in the
critical dimensions d= 1,2; see Remark 1, Section 7 of [OZ4]. Comparison with standard
diffusive stability arguments as in [Z7] show that this is due t o the fact that the full solution
operator |S(t)∂x|decays no faster than S(t), or, equivalently, Gyno faster than G.
Following the basic strategy introduced in [ZH, Z1, MaZ2, Ma Z4] in the context of vis-
cous shock waves, we now perform a refined linearized estimat e separating slower-decaying
translational modes from a faster-decaying “good” part of t he solution operator. This will
be used in Section 4 in combination with certain nonlinear ca ncellation estimates to show
convergence to the modulated approximation (1.1) at a faste r rate sufficient to close the
nonlinear iteration.
The key to this decomposition is the following observation.
2The inclusion of general p≥2 in Lemma 2.7 repairs an omission in [OZ4], where the bounds ( 2.8) were
stated but not used.3 REFINED LINEARIZED ESTIMATES 13
Lemma 3.1. Assuming (H1)–(H3), (D1)–(D3), let λj(ξ/|ξ|,ξ),qj(ξ/|ξ|,ξ,·),˜qj(ξ/|ξ|,ξ,·)
denote the eigenvalues and associated right and left eigenf unctions of Lξ, withqj,˜qjsmooth
functions of ξ/|ξ|and|ξ|as noted in Prop. 2.2. Then, without loss of generality, q1(ω,0,·)≡
¯u′, while˜qj(ω,0,·)forj/ne}ationslash= 1are constant functions depending only on angle ω=ξ/|ξ|.
Proof.Expanding Lξ=L0+|ξ|L1
ξ/|ξ|+|ξ|2L2
ξ/|ξ|as in the introduction, consider the con-
tinuous family of spectral perturbation problems in |ξ|indexed by angle ω=ξ/|ξ|. Then,
both facts follow by standard perturbation theory [K] using the observations that ¯ u′is in
the right kernel of L0and constant functions care in the left kernel of L0, with
/an}b∇acketle{tc,L1¯u′/an}b∇acket∇i}ht=/an}b∇acketle{tc,(ω1(2∂x1−A1)−/summationdisplay
j/negationslash=1ωjAj))¯u′/an}b∇acket∇i}ht=/an}b∇acketle{tc,ω1∂2
x1¯u−/summationdisplay
j/negationslash=1ωj∂x1fj(¯u)/an}b∇acket∇i}ht ≡0,
where/an}b∇acketle{t·,·/an}b∇acket∇i}htdenotesL2(x1) inner product on the interval x1∈[0,X], that the dimension
of kerL0by assumption is ( n+ 1), so that the orthogonal complement of ¯ u′in KerL0
is dimension nso exactly the set of constant functions, and that by (H3) the functions
qj(ω,0,·) and ˜qj(ω,0) are right and left eigenfunctions of Π 0L1|kerL0(Π0as earlier denoting
the zero eigenprojection associated with L0).
Remark 3.2. The key observation of Lemma 3.1 can be motivated by the form o f the
Whitham averaged system (1.2). For, recalling (Section 1.3 ) that (D3) implies that speed
sis stationary to first order at ¯ ualong the manifold of nearby periodic solutions, we find
that the last equation of (1.2) reduces to ( ∇xΨ)t= 0, i.e., the equation for the translational
variation Ψ decouples from the equations for variations in o ther modes. This corresponds
heuristically to the fact derived above that the translatio nal mode ¯u′(x1) decouples in the
first-order eigenfunction expansion.
Corollary 3.1. Under assumptions (H1)–(H3), (D1)–(D3), the Green functio nG(x,t;y)
of(1.7)decomposes as G=E+˜G,
(3.1) E= ¯u′(x)e(x,t;y),
where, for some C >0, allt>0,1≤q≤2≤p≤ ∞,0≤j,k,l,j+l≤K,1≤r≤2,
(3.2)/vextendsingle/vextendsingle/vextendsingle/integraldisplay+∞
−∞˜G(x,t;y)f(y)dy/vextendsingle/vextendsingle/vextendsingle
Lp(x)≤C(1+t)−d
2(1/2−1/p)t−1
2(1/q−1/2)|f|Lq∩L2,
/vextendsingle/vextendsingle/vextendsingle/integraldisplay+∞
−∞∂r
y˜G(x,t;y)f(y)dy/vextendsingle/vextendsingle/vextendsingle
Lp(x)≤C(1+t)−d
2(1/2−1/p)−1
2+r
2
×t−d
2(1/q−1/2)−r
2|f|Lq∩L2,
/vextendsingle/vextendsingle/vextendsingle/integraldisplay+∞
−∞∂r
t˜G(x,t;y)f(y)dy/vextendsingle/vextendsingle/vextendsingle
Lp(x)≤C(1+t)−d
2(1/2−1/p)−1
2+r
×t−d
2(1/q−1/2)−r|f|Lq∩L2.3 REFINED LINEARIZED ESTIMATES 14
(3.3)/vextendsingle/vextendsingle/vextendsingle/integraldisplay+∞
−∞∂j
x∂k
t∂l
ye(x,t;y)f(y)dy/vextendsingle/vextendsingle/vextendsingle
Lp≤(1+t)−d
2(1/q−1/p)−(j+k)
2|f|Lq.
Moreover,e(x,t;y)≡0fort≤1.
Proof.We first treat the simpler case q= 1. Recalling that
(3.4) GI(x,t;y) =/parenleftBig1
2π/parenrightBigd/integraldisplay
Rdeiξ·(x−y)φ(ξ)n+1/summationdisplay
j=1eλj(ξ)tqj(ξ,x1)˜qj(ξ,y1)∗dξ,
define
(3.5) ˜e(x,t;y) =/parenleftBig1
2π/parenrightBigd/integraldisplay
Rdeiξ·(x−y)φ(ξ)eλ1(ξ)t˜q1(ξ,y1)∗dξ,
so that
(3.6)
GI(x,t;y)−¯u′(x1)˜e(x,t;y) =/parenleftBig1
2π/parenrightBigd/integraldisplay
Rdeiξ·(x−y)φ(ξ)n+1/summationdisplay
j=2eλj(ξ)tqj(ξ/|ξ|,0,x1)˜qj(ξ,y1)∗dξ
+/parenleftBig1
2π/parenrightBigd/integraldisplay
Rdn+1/summationdisplay
j=1eiξ·(x−y)φ(ξ)eλj(ξ)tO(|ξ|)dξ.
Noting, by Lemma 3.1, that ∂y˜q(ω,0,y)≡constant for j/ne}ationslash= 1, we have therefore
(3.7)∂r
y(GI(x,t;y)−¯u′(x1)˜e(x,t;y)) =/parenleftBig1
2π/parenrightBigd/integraldisplay
Rdeiξ·(x−y)φ(ξ)n+1/summationdisplay
j=1eλj(ξ)tO(|ξ|)dξ,
which readily gives
(3.8) |∂r
y(GI(x,t;y)−¯u′(x1)˜e(x,t;y))|Lp≤C(1+t)−d
2(1−1/p)−1
2,
p≥2, by the same argument used to prove (2.9), and similarly
(3.9) |∂r
t(GI(x,t;y)−¯u′(x1)˜e(x,t;y))|Lp≤c(1+t)−d
2(1−1/p)−1
2.
These yield (3.2) by the triangle inequality.
Defininge(x,t;y) :=χ(t)˜e(x,t;y), whereχisasmoothcutofffunctionsuchthat χ(t)≡1
fort≥2 andχ(t)≡0 fort≤1, and setting ˜G:=G−¯u′(x1)e(x,t;y), we readily obtain the
estimates (3.2) by combining (3.9) with bound (2.4) on GII. Bounds (3.3) follow from (3.5)
by the argument used to prove (2.9), together with the observ ation thatx- ort-derivatives
bring down factors of |ξ|, followed again by an application of the triangle inequalit y.
Thecases1 ≤q≤2followsimilarly, bytheargumentsusedtoprove(2.15)and (2.8).4 NONLINEAR STABILITY IN DIMENSION ONE 15
Remark 3.3. Despite their apparent complexity, the above bounds may be r ecognized
as essentially just the standard diffusive bounds satisfied fo r the heat equation [Z7]. For
dimensiond= 1, it may be shown using pointwise techniques as in [OZ2] tha t the bounds
of Corollary 3.1 extend to all 1 ≤q≤p≤ ∞.
Note the strong analogy between the Green function decompos ition of Corollary 3.1
and that of [MaZ3, Z4] in the viscous shock case. We pursue thi s analogy further in the
nonlinearanalysisofthefollowingsections, combiningth e“instantaneous tracking” strategy
of [ZH, Z1, Z4, Z7, MaZ2, MaZ4] with a type of cancellation est imate introduced in [HoZ].
4 Nonlinear stability in dimension one
For clarity, we carry out the nonlinear stability analysis i n detail in the most difficult,
one-dimensional, case, indicating afterward by a few brief remarks the extension to d= 2.
Hereafter, take x∈R1, dropping the indices on fjandxjand writing ut+f(u)x=uxx.
4.1 Nonlinear perturbation equations
Given a solution ˜ u(x,t) of (1.4), define the nonlinear perturbation variable
(4.1) v=u−¯u= ˜u(x+ψ(x,t))−¯u(x),
where
(4.2) u(x,t) := ˜u(x+ψ(x,t))
andψ:R×R→Ris to be chosen later.
Lemma 4.1. Forv,uas in(4.1),(4.2),
(4.3) ut+f(u)x−uxx= (∂t−L)¯u′(x1)ψ(x,t)+∂xR+(∂t+∂2
x)S,
where
R:=vψt+vψxx+(¯ux+vx)ψ2
x
1+ψx=O(|v|(|ψt|+|ψxx|)+/parenleftBig|¯ux|+|vx|
1−|ψx|/parenrightBig
|ψx|2)
and
S:=−vψx=O(|v|(|ψx|).
Proof.To begin, notice from the definition of uin (4.2) we have by a straightforward
computation
ut(x,t) = ˜ux(x+ψ(x,t),t)ψt(x,t)+ ˜ut(x+ψ,t)
f(u(x,t))x=df(˜u(x+ψ(x,t),t))˜ux(x+ψ,t)·(1+ψx(x,t))4 NONLINEAR STABILITY IN DIMENSION ONE 16
and
uxx(x,t) = (˜ux(x+ψ(x,t),t)·(1+ψx(x,t)))x
= ˜uxx(x+ψ(x,t),t)·(1+ψx(x,t))+(˜ux(x+ψ(x,t),t)·ψx(x,t))x.
Using the fact that ˜ ut+df(˜u)˜ux−˜uxx= 0, it follows that
(4.4)ut+f(u)x−uxx= ˜uxψt+df(˜u)˜uxψx−˜uxxψx−(˜uxψx)x
= ˜uxψt−˜utψx−(˜uxψx)x
where it is understood that derivatives of ˜ uappearing on the righthand side are evaluated
at (x+ψ(x,t),t). Moreover, by another direct calculation, using the fact t hatL(¯u′(x)) = 0
by translation invariance, we have
(∂t−L)¯u′(x)ψ= ¯uxψt−¯utψx−(¯uxψx)x.
Subtracting, and using the facts that, by differentiation of ( ¯u+v)(x,t) = ˜u(x+ψ,t),
(4.5)¯ux+vx= ˜ux(1+ψx),
¯ut+vt= ˜ut+ ˜uxψt,
so that
(4.6)˜ux−¯ux−vx=−(¯ux+vx)ψx
1+ψx,
˜ut−¯ut−vt=−(¯ux+vx)ψt
1+ψx,
we obtain
ut+f(u)x−uxx= (∂t−L)¯u′(x)ψ+vxψt−vtψx−(vxψx)x+/parenleftBig
(¯ux+vx)ψ2
x
1+ψx/parenrightBig
x,
yielding (4.3) by vxψt−vtψx= (vψt)x−(vψx)tand (vxψx)x= (vψx)xx−(vψxx)x.
Corollary 4.2. The nonlinear residual vdefined in (4.1)satisfies
(4.7) vt−Lv= (∂t−L)¯u′(x1)ψ−Qx+Rx+(∂t+∂2
x)S,
where
(4.8) Q:=f(˜u(x+ψ(x,t),t))−f(¯u(x))−df(¯u(x))v=O(|v|2),
(4.9) R:=vψt+vψxx+(¯ux+vx)ψ2
x
1+ψx,
and
(4.10) S:=−vψx=O(|v|(|ψx|).
Proof.Taylor expansion comparing (4.3) and ¯ ut+f(¯u)x−¯uxx= 0.4 NONLINEAR STABILITY IN DIMENSION ONE 17
4.2 Cancellation estimate
Our strategy in writing (4.7) is motivated by the following b asic cancellation principle.
Proposition 4.3 ([HoZ]).For anyf(y,s)∈Lp∩C2withf(y,0)≡0, there holds
(4.11)/integraldisplayt
0/integraldisplay
G(x,t−s;y)(∂s−Ly)f(y,s)dyds=f(x,t).
Proof.Integrating the left hand side by parts, we obtain
(4.12)/integraldisplay
G(x,0;y)f(y,t)dy−/integraldisplay
G(x,t;y)f(y,0)dy+/integraldisplayt
0/integraldisplay
(∂t−Ly)∗G(x,t−s;y)f(y,s)dyds.
Noting that, by duality,
(∂t−Ly)∗G(x,t−s;y) =δ(x−y)δ(t−s),
δ(·) here denoting the Dirac delta-distribution, we find that th e third term on the righthand
side vanishes in (4.12), while, because G(x,0;y) =δ(x−y), the first term is simply f(x,t).
The second term vanishes by f(y,0)≡0.
Remark 4.1. Forψ=ψ(t), term (∂t−L)¯u′ψin (4.7) reduces to the term ˙ψ(t)¯u′(x)
appearing in the shock wave case [ZH, Z1, Z4, Z7, MaZ2, MaZ4].
4.3 Nonlinear damping estimate
Proposition 4.2. Letv0∈HK(Kas in (H1)), and suppose that for 0≤t≤T, theHK
norm ofvand theHK(x,t)norms ofψtandψxremain bounded by a sufficiently small
constant. There are then constants θ1,2>0so that, for all 0≤t≤T,
(4.13) |v(t)|2
HK≤Ce−θ1t|v(0)|2
HK+C/integraldisplayt
0e−θ2(t−s)/parenleftBig
|v|2
L2+|(ψt,ψx)|2
HK(x,t)/parenrightBig
(s)ds.
Proof.Subtracting from the equation (4.4) for uthe equation for ¯ u, we may write the
nonlinear perturbation equation as
(4.14) vt+(df(¯u)v)x−vxx=Q(v)x+ ˜uxψt−˜utψx−(˜uxψx)x,
where it is understood that derivatives of ˜ uappearing on the righthand side are evaluated
at (x+ψ(x,t),t). Using (4.6) to replace ˜ uxand ˜utrespectively by ¯ ux+vx−(¯ux+vx)ψx
1+ψx
and ¯ut+vt−(¯ux+vx)ψt
1+ψx, and moving the resulting vtψxterm to the lefthand side of
(4.14), we obtain
(4.15)(1+ψx)vt−vxx=−(df(¯u)v)x+Q(v)x+ ¯uxψt
−((¯ux+vx)ψx)x+/parenleftBig
(¯ux+vx)ψ2
x
1+ψx/parenrightBig
x.4 NONLINEAR STABILITY IN DIMENSION ONE 18
Taking the L2inner product in xof/summationtextK
j=0∂2j
xv
1+ψxagainst (4.15), integrating by parts, and
rearranging the resulting terms, we arrive at the inequalit y
∂t|v|2
HK(t)≤ −θ|∂K+1
xv|2
L2+C/parenleftBig
|v|2
HK+|(ψt,ψx)|2
HK(x,t)/parenrightBig
,
for someθ >0,C >0, so long as |˜u|HKremains bounded, and |v|HKand|(ψt,ψx)|HK(x,t)
remain sufficiently small. Using the Sobolev interpolation |v|2
HK≤ |∂K+1
xv|2
L2+˜C|v|2
L2for
˜C >0 sufficiently large, we obtain ∂t|v|2
HK(t)≤ −˜θ|v|2
HK+C/parenleftBig
|v|2
L2+|(ψt,ψx)|2
HK(x,t)/parenrightBig
from which (4.13) follows by Gronwall’s inequality.
4.4 Integral representation/ ψ-evolution scheme
By Proposition 4.3, we have, applying Duhamel’s principle t o (4.7),
(4.16)v(x,t) =/integraldisplay∞
−∞G(x,t;y)v0(y)dy
+/integraldisplayt
0/integraldisplay∞
−∞G(x,t−s;y)(−Qy+Rx+St+Syy)(y,s)dyds+ψ(t)¯u′(x).
Definingψimplicitly as
(4.17)ψ(x,t) =−/integraldisplay∞
−∞e(x,t;y)u0(y)dy
−/integraldisplayt
0/integraldisplay+∞
−∞e(x,t−s;y)(−Qy+Rx+St+Syy)(y,s)dyds,
following [ZH, Z4, MaZ2, MaZ3], where eis defined as in (3.1), and substituting in (4.16)
the decomposition G= ¯u′(x)e+˜Gof Corollary 3.1, we obtain the integral representation
(4.18)v(x,t) =/integraldisplay∞
−∞˜G(x,t;y)v0(y)dy
+/integraldisplayt
0/integraldisplay∞
−∞˜G(x,t−s;y)(−Qy+Rx+St+Syy)(y,s)dyds,
and, differentiating (4.17) with respect to t, and recalling that e(x,s;y)≡0 fors≤1,
(4.19)∂j
t∂k
xψ(x,t) =−/integraldisplay∞
−∞∂j
t∂k
xe(x,t;y)u0(y)dy
−/integraldisplayt
0/integraldisplay+∞
−∞∂j
t∂k
xe(x,t−s;y)(−Qy+Rx+St+Syy)(y,s)dyds.
Equations (4.18), (4.19) together form a complete system in the variables ( v,∂j
tψ,∂k
xψ),
0≤j≤1, 0≤k≤K, from the solution of which we may afterward recover the shif tψvia
(4.17). From the original differential equation (4.7) togeth er with (4.19), we readily obtain
short-time existence and continuity with respect to tof solutions ( v,ψt,ψx)∈HKby a
standard contraction-mapping argument based on (4.13), (4 .17), and and (3.3).4 NONLINEAR STABILITY IN DIMENSION ONE 19
4.5 Nonlinear iteration
Associated with the solution ( u,ψt,ψx) of integral system (4.18)–(4.19), define
(4.20)ζ(t) := sup
0≤s≤t|(v,ψt,ψx)|HK(s)(1+s)1/4.
Lemma 4.3. For allt≥0for whichζ(t)is finite, some C >0, andE0:=|u0|L1∩HK,
(4.21) ζ(t)≤C(E0+ζ(t)2).
Proof.By (4.9)–(4.10) and definition (4.20),
(4.22) |(Q,R,S)|L1∩L∞≤ |(v,vx,ψt,ψx)|2
L2+|(v,vx,ψt,ψx)|2
L∞≤Cζ(t)2(1+t)−1
2,
so long as |ψx| ≤ |ψx|HK≤ζ(t) remains small, and likewise (using the equation to bound t
derivatives in terms of x-derivatives of up to two orders)
(4.23) |(∂t+∂2
x)S|L1∩L∞≤ |(v,ψx)|2
H2+|(v,ψx)|2
W2,∞≤Cζ(t)2(1+t)−1
2.
Applying Corollary 3.1 with q= 1,d= 1 to representations (4.18)–(4.19), we obtain for
any 2≤p<∞
(4.24)|v(·,t)|Lp(x)≤C(1+t)−1
2(1−1/p)E0
+Cζ(t)2/integraldisplayt
0(1+t−s)−1
2(1/2−1/p)(t−s)−3
4(1+s)−1
2ds
≤C(E0+ζ(t)2)(1+t)−1
2(1−1/p)
and
(4.25)
|(ψt,ψx)(·,t)|WK,p≤C(1+t)−1
2E0+Cζ(t)2/integraldisplayt
0(1+t−s)−1
2(1−1/p)−1/2(1+s)−1
2ds
≤C(E0+ζ(t)2)(1+t)−1
2(1−1/p).
Using (4.13) and(4.24)–(4.25), we obtain |v(·,t)|HK(x)≤C(E0+ζ(t)2)(1+t)−1
4. Combining
this with (4.25), p= 2, rearranging, and recalling definition (4.20), we obtain (4.3).
Proof of Theorem 1.1. By short-time HKexistence theory, /ba∇dbl(v,ψt,ψx)/ba∇dblHKis continuous
so long as it remains small, hence ηremains continuous so long as it remains small. By
(4.3), therefore, it follows by continuous induction that η(t)≤2Cη0fort≥0, ifη0<1/4C,
yielding by (4.20) the result (1.15) for p= 2. Applying (4.24)–(4.25), we obtain (1.15) for
2≤p≤p∗for anyp∗<∞, with uniform constant C. Takingp∗>4 and estimating
|Q|L2,|R|L2,|S|L2(t)≤ |(v,ψt,ψx)|2
L4≤CE0(1+t)−3
45 NONLINEAR STABILITY IN DIMENSION TWO 20
in place of the weaker (4.22), then applying Corollary 3.1 wi thq= 2,d= 1, we obtain
finally (1.15) for 2 ≤p≤ ∞, by a computation similar (4.24)–(4.25); we omit the detail s of
this final bootstrap argument. Estimate (1.16) then follows using (3.3) with q=d= 1, by
(4.26)
|ψ(t)|Lp≤CE0+Cζ(t)2/integraldisplayt
0(1+t−s)−1
2(1−1/p)(1+s)−1
2ds≤C(1+t)1
2p(E0+ζ(t)2),
together with the fact that ˜ u(x,t)−¯u(x) =v(x−ψ,t)+(¯u(x)−¯u(x−ψ),so that|˜u(·,t)−¯u|
is controlled by the sum of |v|and|¯u(x)−¯u(x−ψ)| ∼ |ψ|. This yields stability for
|u−¯u|L1∩HK|t=0sufficiently small, as described in the final line of the theore m.
5 Nonlinear stability in dimension two
We now briefly sketch the extension to dimension d= 2. Given a solution ˜ u(x,t) of (1.4),
define the nonlinear perturbation variable
(5.1) v=u−¯u= ˜u(x1+ψ(x,t),x2,t)−¯u(x1),
where
(5.2) u(x,t) := ˜u(x1+ψ(x,t),t)
andψ:Rd×R→Ris to be chosen later.
Lemma 5.1. Forv,uas in(5.2),
(5.3)ut+d/summationdisplay
j=1fj(u)xj−d/summationdisplay
j=1uxjxj= (∂t−L)¯u′(x1)ψ(x,t)+d/summationdisplay
j=1∂xjRj+∂tS+T,
where
Rj=O((|v,ψt,ψx)||(v,vx,ψt,ψx)|), S:=−vψx1= (|v|(|ψx|), T:=O(|ψx|3+|(v,ψx)||ψxx|).
Proof.Similarly as in the proof of Lemma 4.1, it follows by a straigh tforward computation
Using the fact that ˜ ut+/summationtext
jdfj(˜u)˜uxj−/summationtext
j˜uxjxj= 0, it follows that
(5.4)ut+/summationdisplay
jdfj(u)uxj−/summationdisplay
juxjxj= ˜ux1ψt−˜utψx1+/summationdisplay
j/negationslash=1dfj(˜u)˜ux1ψxj
−/summationdisplay
j/negationslash=1˜uxjx1ψxj−/summationdisplay
j(˜ux1ψxj)xj,
where it is understood that derivatives of ˜ uappearing on the righthand side are evaluated
at (x+ψ(x,t),t). Moreover, by another direct calculation, using the fact t hatL(¯u′(x1)) = 0
by translation invariance, we have
(∂t−L)¯u′(x1)ψ= ¯ux1ψt−¯utψx1+/summationdisplay
j/negationslash=1dfj(¯u)¯ux1ψxj−/summationdisplay
j/negationslash=1¯uxjx1ψxj−/summationdisplay
j(¯ux1ψxj)xj.5 NONLINEAR STABILITY IN DIMENSION TWO 21
Subtracting, and using (4.5) and
(5.5)¯uxj+vxj= ˜uxj+ ˜ux1ψxj,
¯ut+vt= ˜ut+ ˜ux1ψt,
so that
(5.6)˜uxj−¯uxj−vxj=−(¯ux1+vx1)ψxj
1+ψx1,
˜ut−¯ut−vt=−(¯ux1+vx1)ψt
1+ψx1,
we obtain
ut+/summationdisplay
jdfj(u)uxj−/summationdisplay
juxjxj= (∂t−L)¯u′(x1)ψ+vx1ψt−vtψx1
+/summationdisplay
j/negationslash=1(dfj(˜u)˜ux1−dfj(¯u)¯ux1)ψxj
−/summationdisplay
j/negationslash=1(˜uxjx1−¯uxjx1)ψxj−/summationdisplay
j((˜ux1−¯ux1)ψxj)xj.
Usingvx1ψt−vtψx1= (vψt)x1−(vψx1)t,
dfj(˜u)˜ux1=f(u)x1−dfj(˜u)˜ux1ψx1=f(u)x1(1−ψx)−dfj(˜u)˜ux1ψ2
x1,
and ˜uxjx1= (˜uxj)x1−˜uxjx1ψx1= (˜uxj)x1(1−ψx1)+ ˜uxjx1ψ2
x1,and rearranging, we obtain
ut+/summationdisplay
jdfj(u)uxj−/summationdisplay
juxjxj= (∂t−L)¯u′(x1)ψ+(vψt)x1−(vψx1)t
+/summationdisplay
j/negationslash=1(fj(u)−fj(¯u))x1)ψxj
−/summationdisplay
j/negationslash=1f(u)x1ψx1ψxj−/summationdisplay
j/negationslash=1dfj(˜u)˜ux1ψ2
x1ψxj
−/summationdisplay
j/negationslash=1(˜uxj−¯uxj)x1ψxj+/summationdisplay
j/negationslash=1(˜uxj)x1ψx1ψxj
+/summationdisplay
j/negationslash=1˜uxjx1ψ2
x1ψxj
−/summationdisplay
j(vx1ψx1)xj−/summationdisplay
j/parenleftBig
(¯ux1+vx1)ψxjψx1
1+ψx1/parenrightBig
xj.
Noting that
(fj(u)−fj(¯u))x1)ψxj= ((fj(u)−fj(¯u)ψxj)x1−(fj(u)−fj(¯u))ψxjx1,5 NONLINEAR STABILITY IN DIMENSION TWO 22
f(u)x1ψx1ψxj= (f(u)ψx1ψxj)x1−f(u)(ψx1ψxj)x1,
and
(˜uxj−¯uxj)x1ψxj= ((˜uxj−¯uxj)ψxj)x1−(˜uxj−¯uxj)ψxjx1,
with|fj(u)−fj(¯u)|=O(|v|) and|˜uxj−¯uxj|=O(|v|),we obtain the result
Proof of Theorem 1.2. The result of Lemma 5.1 is the only part of the analysis that di ffers
essentially from that of the one-dimensional case. The canc ellation and nonlinear damping
arguments go through exactly as before to yield the analogs o f Propositions 4.3 and (4.2).
Likewise, we obtain a Duhamel representation analogous to ( 4.18)–(4.19), forming a closed
system in variables ( v,ψx,ψt).
To obtain the analog of Lemma 4.3, completing the proof of non linear stability, we can
carry out a somewhat simpler argument than in the one-dimens ional case, using Corollary
3.1 withd= 2,q= 2 for all estimates, not only the final bootstrap argument, g iving in
place of (4.24) the estimate
(5.7)
|v(·,t)|Lp(x)≤C(1+t)−(1−1/p)E0+Cζ(t)2/integraldisplayt
0(1+t−s)−(1/2−1/p)(t−s)−1
2(1+s)−1ds
≤C(E0+ζ(t)2)(1+t)−(1−1/p),
(5.8)|(ψx,ψt)(·,t)|Lp(x)≤C(1+t)−(1−1/p)−1
2E0
+Cζ(t)2/integraldisplayt
0(1+t−s)−(1/2−1/p)(t−s)−1
2(1+s)−1ds
≤C(E0+ζ(t)2)(1+t)ε−(1−1/p)−1
2
for divergence-form source terms, and
(5.9)|v(·,t)|Lp(x)≤Cζ(t)2/integraldisplayt
0(1+t−s)−(1/2−1/p)(1+s)−3
2ds
≤C(E0+ζ(t)2)(1+t)−(1−1/p),
(5.10)|(ψx,ψt)(·,t)|Lp(x)≤C(1+t)−(1−1/p)−1
2E0
+Cζ(t)2/integraldisplayt
0(1+t−s)−(1/2−1/p)(t−s)−1
2(1+s)−3
2ds
≤C(E0+ζ(t)2)(1+t)ε−(1−1/p)−1
2
for faster-decaying nondivergence-form source terms.
We omit the details, which are entirely similar to, but subst antially simpler than, those
of the one-dimensional case.REFERENCES 23
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