Abstract
We establish nonlinear stability and asymptotic behavior of traveling periodic waves of viscous conservation laws under localized perturbations or nonlocalized perturbations asymptotic to constant shifts in phase, showing that long-time behavior is governed by an associated second-order formal Whitham modulation system. A key point is to identify the way in which initial perturbations translate to initial data for this formal system, a task accomplished by detailed estimates on the linearized solution operator about the background wave. Notably, our approach gives both a common theoretical treatment and a complete classification in terms of “phase-coupling” or “-decoupling” of general systems of conservation or balance laws, encompassing cases that had previously been studied separately or not at all.
At the same time, our refined description of solutions gives the new result of nonlinear asymptotic stability with respect to localized perturbations in the phase-decoupled case, further distinguishing behavior in the different cases. An interesting technical aspect of our analysis is that for systems of conservation laws the Whitham modulation description is of system rather than scalar form, as a consequence of which renormalization methods such as have been used to treat the reaction-diffusion case in general do not seem to apply.
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Notes
Verifying formal predictions and rigorous spectral descriptions of [17].
Concerning the more tractable (since faster-decaying) three and higher dimensional case.
See Remark 1.2 below.
The needed structural conditions may indeed be deduced from assumptions (H1)–(H3) and (D1)–(D3) below.
These intermediate decay rates require some assumptions about characteristics speeds provided by assumption (H3) below; see Remark 1.21.
Henceforth, in our notation for Lebesgue and Sobolev spaces we will suppress the definition of the range; in particular, we will write \(L^{p}(\mathbb{R})\) for the equivalence class of p-integrable \(\mathbb{R}^{n}\)-valued functions \(L^{p}(\mathbb {R};\mathbb{R}^{n})\).
Mark that the important property we have used through these arguments and obtained from the introduction of Bloch symbols L ξ is compactness, which plays for this periodic setting the role of the finite dimensionality that one obtains with Fourier symbols associated to constant-coefficient operators.
A periodic Evans function computation showing that the (n+1)st derivative of the Evans function at λ=0 is proportional to det∂ α (M,k), hence \(\det\partial_{\alpha}(M,k)_{|(\bar{\alpha},\bar{\beta})}\neq0\) is necessary for (D3) (implicit also in the earlier work [62]).
Full hyperbolicity requiring of course also semisimplicity of a j as eigenvalues of \(\frac{\partial(F-\bar{c}M,-\omega-\bar{c}k)}{\partial(M,k)}|_{(\bar{M} ,\bar{k})}\).
These are not true distances, but rather measures associated with a seminorm.
The proof gives also a \(\delta_{L^{1}\cap H^{K}}-\delta_{H^{K}}\) asymptotic stability.
This issue does not arise in the related analysis [33] of the reaction-diffusion case, as M does not appear.
This follows from the general theory for parabolic systems of conservation laws, see for instance Proposition B.1.
See Remark 1.2.
See Remark 1.2.
At second order, a linear heat equation κ t =dκ xx , d>0.
Recall that \(\mathcal{M}\) does not appear in the Whitham equation for the reaction-diffusion case.
Up to the contribution of the critical part of the evolution not described by phase modulation but estimated in a similar way.
Here and elsewhere, we denote \(\|g\|_{L^{q}([-\pi,\pi ],L^{p}([0,1]))}:= (\int_{-\pi}^{\pi}\|g(\xi,\cdot)\| _{L^{p}([0,1])}^{q}d\xi )^{1/q}\).
The first space is the one of U-values, the second one is the one of modulation parameters \(({\mathcal{M}},\kappa)\).
In particular, for the three example systems considered in the introduction, the Whitham system does have large-amplitude solutions for data merely bounded in L 1∩L 2 since it has an associated convex entropy; see [22].
Compare to the similar but much simpler argument of [32, Proposition 4.1], in the reaction-diffusion case.
Compare to the argument of [28, Lemma 4.2], regarding localized perturbations in the decoupled case.
A situation that trivially occurs when some symmetry is present, see Remark 1.27.
Here and elsewhere, we identify as usual bilinear maps with vector-valued matrices, and in particular d 2 g(w ∗) with Hess(g)(w ∗). Moreover, for these vector-valued matrices, we use coordinate notations so that for instance \(\varGamma_{*}^{j}\in\mathbb {R}^{n\times n}\) satisfies \(w^{t}\varGamma_{*}^{j}w=(w^{t}\varGamma_{*}w)_{j}=d^{2}g_{j}(w,w)\).
Computing \(\|\theta_{j}(t)\|_{L^{p}(\mathbb{R})}= t^{-1/2}\|\bar{\theta}_{j} (\cdot/\sqrt{t})\|_{L^{p}(\mathbb{R})}\sim t^{-\frac{1}{2}(1-1/p)}\).
Linearized analysis suggests that the sharp condition is, rather, some averaged version of this one, which holds trivially by the fact that perfect derivatives have zero mean [4].
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Acknowledgements
K.Z. thanks Björn Sandstede for a number of helpful orienting discussions regarding modulation of periodic reaction-diffusion waves, Guido Schneider for bringing to our attention the treatment of Bénard–Marangoni cells in [20, 70], and Denis Serre for his interest in the subject of modulation of periodic solutions and his contributions through [62] and private and public communications. M.J., P.N., and M.R. thank Indiana University, and K.Z. thanks the École Normale Supérieure, Paris, the University of Paris 13, and the Foundation Sciences Mathématiques de Paris for their hospitality during visits in which this work was partially carried out. Special thanks to David Lannes for his careful reading of the manuscript and many helpful suggestions. Finally, thanks to the anonymous referees for suggestions improving the exposition.
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Research of M.J. was partially supported under NSF grant no. DMS-1211183 and by the University of Kansas General Research Fund allocation 2302278.
Research of P.N. was partially supported by the French ANR Project no. ANR-09-JCJC-0103-01.
Stay of M.R. in Bloomington was supported by French ANR project no. ANR-09-JCJC-0103-01.
Research of K.Z. was partially supported under NSF grant no. DMS-0300487.
Appendices
Appendix A: Algebraic relations
We record in this appendix some crucial relations obtained by differentiating the profile equations. In order to differentiate, we here consider variable parameters (M,k) rather than fixed values \((\bar{M},\bar{k})\), imposing dependence implicitly through the profile equations (denoting \(\langle a\rangle:=\int_{0}^{1} a\)):
We expand L ξ =L 0+ikξL (1)+(ikξ)2 L (2) with
Then, by differentiation of (A.1), we obtain
Accordingly, with ω=−kc, using L (1) U′=kU″, we have
Appendix B: The Whitham equations and asymptotic equivalence
In this appendix, we explain how to obtain the needed formal averaged modulation system for comparison to our analytical description of asymptotic behavior. This is performed in three steps.
-
1.
First, we develop a direct WKB-like formal approximation. At this stage we obtain a system that may contain harmless irrelevant terms.
-
2.
Next, we use known results about large-time asymptotic behavior of systems of conservation laws about constant states to get a canonical form for the averaged modulation system.
-
3.
Finally, we adapt the system taking into account the fact that the analysis of the main part of the paper is carried out after an implicit nonlinear change of coordinates.
2.1 B.1 Formal asymptotics
Though the full nonlinear analysis may be carried out without distinction between linearly coupled and linearly uncoupled cases, the formal derivation of averaged equations involves resolutions of systems of the form L 0 g=h and therefore requires knowledge of the kernel of L 0. We are thus compelled to provide two separate derivations.
Besides, there are at least two ways to obtain relevant averaged equations. The first one is to develop a full WKB-type expansion as in [48, 49], extending the procedure in [62] to get higher order equations. This method provides the hyperbolic part of the averaged system in a quick way and a nice form. Its main drawback is that it requires a knowledge of the kernel of L 0 for all waves close to the wave under study essentially reducing the scope of the method to the nondegenerate case or to a fully degenerate case where ∂ M c would vanish in a neighborhood of the studied wave.Footnote 29 The second method is designed to study dynamics about a given wave, so that it does not suffer from the same flaws; moreover, it is closer to our nonlinear analysis, and yields a semilinear system.
We derive the system for the generic case with the first method and the one for the linearly uncoupled case with the second one. Note that both methods provide averaged systems with diffusion matrices containing terms that are not relevant for our present analysis.
2.1.1 B.1.1 Generic case
To treat the linearly phase-coupled case, we essentially borrow the derivation of [49] for the Korteweg–de Vries/Kuramoto–Sivashinsky equation, a model for which linear phase-coupling is a consequence of assumptions (H1)–(H2) and (D3). In the present derivation, we assume that all the waves involved in the slow-modulation description satisfy (H1)–(H2) and (D3) and are linearly phase-coupled.
Since in this derivation there is no reference wave, thus no privileged frame, we go back to the original equation
We are looking for a formal expansion of a solution u of Eq. (B.1) according to the two-scale ansatz
where
with the functions U and U j being 1-periodic in the y-variable. We insert the ansatz (B.2), (B.3) into (B.1) and collect terms of the same order in ε.
First this yields, with Ω 0=∂ T ϕ 0 and κ 0=∂ X Ψ 0, \(\varOmega_{0} \partial_{y} U_{0}+\kappa_{0} \partial_{y}(f(U_{0})) =\kappa_{0}^{2} \partial _{y}^{2} U_{0}\), which is solved by
We have disregarded in (B.4) the possibility of a phase shift dependent on (X,T) since this is already encoded by Ψ 1. We will have to rule out similar problems of uniqueness in the following steps. At this stage the compatibility condition ∂ T ∂ X Ψ 0=∂ X ∂ T Ψ 0 already yields the first equation of a Whitham’s modulation system:
In the rest of the derivation, we will use the notations of Proposition 3.1 and Appendix A, with the convention that operators act in y and are associated to the wave profile \(U( \cdot ;{\mathcal{M}}_{0}(X,T),\kappa_{0}(X,T))\). To fix some of the uniqueness issues of the ansatz, we pick, for any (M,k), u adj(⋅;M,k) a generalized zero eigenfunction of \(L_{0}^{*}\) such that 〈u adj,∂ M U |(M,k)〉=0 and 〈u adj,U′(⋅;M,k)〉=1, set \(u_{0}^{\mathit{adj}}(y;X,T)=u^{\mathit{adj}}( \cdot ;{\mathcal{M}}_{0}(X;T),\kappa _{0}(X,T))\) and add to the ansatz the normalizing condition
The next step of the identification process gives, with Ω 1=∂ T Ψ 1 and κ 1=∂ X Ψ 1,
whose solvability condition reads
where F denotes the averaged flux F(M,k)=〈f(U(⋅;M,k))〉. To proceed, for arbitrary (M,k) we introduce g k(⋅;M,k), g M(⋅;M,k) solutions of
orthogonal to u adj(⋅;M,k) and set g=(g M g k) and \(g_{0}( \cdot ;X,T)= g( \cdot ;{\mathcal{M}}_{0}(X,T),\kappa_{0}(X,T))\). Then with (B.5)–(B.8) and (A.3) Eq. (B.7) reads
for any choice of \(\tilde{\mathcal{M}}_{1}\). Let us set \({\mathcal{M}}_{1}=\langle U_{1}\rangle\). Choosing \(\tilde{\mathcal{M}}_{1}\) to get
and normalizing the parametrization, as in Lemma 4.1, to get, for any (M,k),
Eq. (B.11) is reduced to
Then, compatibility condition ∂ T κ 1=∂ X Ω 1 yields
Returning to the identification process, we obtain an equation of the form
whose solvability condition is
To write the second order system in a compact form, let us introduce, for arbitrary (M,k),
With these notations, systems (B.5), (B.8), (B.14), (B.15) coincide with the first systems obtained in the formal expansion of a solution \(({\mathcal{M}},\kappa)\) of
according to the slow ansatz
We call system (B.16) a (second-order) Whitham’s modulation system.
2.1.2 B.1.2 Phase-decoupled case
For the phase-decoupled case, we propose an alternative derivation that would also work for the uncoupled case. We pick a wave of parameters \((\bar{M},\bar{k})\) and assume that it satisfies (H1)–(H2) and (D3) and is linearly phase-decoupled.
We again insert the ansatz (B.2), (B.3) into (B.1) and collect terms of the same order in ε but this time we specialize to \(({\mathcal{M}}_{0},\kappa_{0})=(\bar {M},\bar{k} )\). We keep (B.6) as ansatz normalization and (B.13) as parametrization normalization. The first nontrivial equation is with (Ω 1,κ 1)=(∂ T Ψ 1,∂ X Ψ 1)
which may also be written as
for any \({\mathcal{M}}_{1}\). Solvability yields
and with our normalization choices (B.18) reduces to
Compatibility condition ∂ T κ 1=∂ X Ω 1 already gives
At the next step of the identification, we get with (Ω 2,κ 2)=(∂ T Ψ 2,∂ X Ψ 2)
which may also be written
for any \({\mathcal{M}}_{1}\). Solvability then reads
Note that the latter equation yields
To proceed, we introduce \(\tilde{g}^{k}\), \(\tilde{g}^{M}\), the solutions of
mean free and orthogonal to \(u^{\mathit{adj}}( \cdot ;\bar{M},\bar{k})\) and set . With (B.21) and (B.22), setting \({\mathcal{M}}_{2}=\langle U_{2}\rangle\), Eq. (B.20) becomes
Finally, substituting (B.2), (B.3) into (B.1), and comparing terms of order ε 3, we obtain an equation of the form
whose solvability implies
To write the second order system in a compact form, let us introduce, for arbitrary (M,k),
With these notations, systems (B.19), (B.21), (B.23), (B.24) coincide with the first nontrivial systems obtained in the formal expansion of a solution \(({\mathcal{M}},\kappa)\) of
according to the slow ansatz
We call system (B.25), likewise, a (second-order) Whitham’s modulation system.
As should be clear from the formal derivations, there is some freedom in the choice of the diffusion matrices. This reflects the fact that many systems of conservation laws share the same asymptotic behavior about constant states. We recall next how to classify these systems according to their asymptotic behavior; this will provide a canonical modulation system for our nonlinear analysis.
We emphasize however that the classification below has been so far as we know verified only for symmetrizable hyperbolic-parabolic systems satisfying a Kawashima structural condition [35]. It is unclear to us whether our formally obtained Whitham systems satisfy such conditions. Yet, we know that, by applying to them, on formal grounds, the asymptotic-equivalence reduction, we do obtain a system satisfying such conditions and providing the correct asymptotic behavior. Hence we may safely ignore these technical details.
2.2 B.2 Asymptotic equivalence of systems of conservation laws
2.2.1 B.2.1 General theory
We now recall the notion of asymptotic equivalence and behavior of solutions of systems of conservation laws near a constant state, useful in our context since, being able to prove modulational behavior, we reduce the dynamics about a periodic wave to motion of parameters near a constant state. Given a general system of conservation laws
and a reference state w ∗ at which dg(w ∗) has distinct eigenvalues, so that L ∗ dg(w ∗)R ∗ is diagonal for some
define the quadratic approximant
and the decoupled quadratic approximant
about w ∗, whereFootnote 30
Assume the parabolicity condition, \({\rm diag }\{ L_{*} B(w_{*})R_{*}\}\) is positive, and define the self-similar nonlinear (resp. linear if γ j =0) diffusion waves \(\theta_{j}(x,t)=t^{-1/2}\bar{\theta}_{j}(x/\sqrt{t})\) to be the solutions of the Burgers equations (resp. heat equations if γ j =0)
with delta-function initial data \(l_{j}^{*} m_{0} \delta(\cdot)\), where m 0:=∫z 0(x)dx. Then, we have the following fundamental result describing behavior of (B.26)–(B.28) with respect to localized initial perturbations.
Proposition B.1
Let η>0. Let w and z be solutions of (B.26) and (B.28) with initial data w 0 and z 0=w 0−w ∗ such that \(E_{1}:=\|z_{0}\|_{L^{1}(\mathbb{R})\cap H^{4}(\mathbb{R})}+\||\cdot|z_{0}\| _{L^{1}(\mathbb{R})}\) is sufficiently small. Then, for 1≤p≤∞, \(m_{0}:=\int_{\mathbb{R}}z_{0}\), and θ j as in (B.31),
and
with \(a_{j}^{*}:=l_{j}^{*}A_{*}r_{j}^{*}\), \(b_{j}^{*}:=l_{j}^{*}B_{*}r_{j}^{*}\), whence,Footnote 31 if η<1/4,
Proposition B.1 asserts that (B.26) and (B.28) (hence also (B.27)) are asymptotically equivalent with respect to small localized initial data \(w_{0}-w_{*}=z_{0}\in L^{1}(\mathbb{R},(1+|x|)dx)\cap H^{3}(\mathbb{R})\), in the sense that the difference between solutions z(t) and w(t)−w ∗ decays at rate \((1+t)^{-\frac{1}{2}(1-1/p)-\frac{1}{4}+\eta}\) approximately \((1+t)^{-\frac{1}{4}}\) faster than the (Gaussian) rate \(|m_{0}| (1+t)^{-\frac{1}{2}(1-1/p)}\) at which either one typically (i.e., for data with small L 1 first moment) decays. Moreover, through (B.33), it gives a simple description of asymptotic behavior as the linear superposition of scalar diffusion waves θ j moving with characteristic speeds (eigenvalues \(a_{j}^{*}\)) in the characteristic modes (eigendirections \(r_{j}^{*}\)) of dg(w ∗), satisfying Burgers equations (B.31).
We have also the following more elementary result comparing to the full quadratic approximant.
Proposition B.2
([33]Footnote 32)
Let η>0. Let w and y be solutions of (B.26) and (B.27) with initial data w 0 and y 0=w 0−w ∗ such that \(E_{0}:=\|y_{0}\|_{L^{1}(\mathbb{R})\cap H^{4}(\mathbb{R})}\) is sufficiently small. Then, for 1≤p≤∞,
An important consequence of Proposition B.2 is that only the quadratic order quantities appearing in (B.27) need be taken into account in the study of asymptotic behavior of (B.26) to the order of approximation considered in Theorem 1.12. Finally, we note the following result following from a proof similar to but much simpler than the one for Proposition B.2 given in [33, Appendix A].
Lemma B.3
Let k satisfy k(0)=0 and
where a,γ,d are constant, d>0 and F is a given function such that \(\|F(t)\|_{L^{2}(\mathbb{R})}\le E_{0}(1+t)^{-\frac{1}{4}}\). Then, for any η>0, provided E 0 is small enough, for 1≤p≤∞,
For the sake of completeness, we recall the proof of the previous Proposition in Appendix C.
2.2.2 B.2.2 A first application
As an immediate application, we may now establish the improved decay bounds (1.22)–(1.21) of Corollary 1.18. We will use these tools again in establishing (1.18).
Proof of Corollary 1.18
Bound (1.21) follows from the assumption \(\bar{k}\partial _{x}h_{0}=0\). For, a solution (M W ,k W ), with an initial data (∗,0), of the decoupled approximating equations (B.28) to (1.7) satisfies k W (t)≡0, since the k equation decouples in (B.28) for the linearly phase-decoupled case. Comparing to the actual solution of (1.7) using (B.32), we obtain the result. Bound (1.22) goes similarly, observing that in the quadratically decoupled case, the k equation in the full quadratic approximating system (B.27) to (1.7), though it does not completely decouple, is of the form (B.34) with \(F=\mathcal{O}(M)\). □
Remark B.4
Analogous to (B.34) in the quadratically decoupled case, the rate-determining bound in the linearly decoupled case of Proposition B.1 is the key estimate
established by Liu [40] for quadratic coupling terms involving different modes, thus obeying k(0)=0, \(k_{t} + ak_{x} + (\gamma k^{2})_{x}- dk_{xx}= (\tilde{\theta}^{2})_{x}\), where \(\tilde{\theta}(x,t)=\theta(x-\tilde{a}t,\tilde{b} t)\) with \(\tilde{a}\ne a\) and θ a self similar solution of a Burgers equation (B.31). The anomalous rate \((1+t)^{\frac{1}{4}}\) is different from the powers of \((1+t)^{\frac{1}{2}}\) arising in scalar convection–diffusion processes, reflecting the additional complications present in the system case.
2.2.3 B.2.3 Quadratic approximants of modulation systems
For later reference, let us write, in the original frame (and not the co-moving one), as
the quadratic approximant of (1.7) (obtained as (B.16) and (B.25) above). As pointed out in Remark 4.11, it follows from Lemma 4.10 that this system is independent of the choices made in the course of the formal derivation.
From the general theory, we know that instead of comparing \(({\mathcal{M}},\kappa)\) in Theorem 1.12 to a solution \(({\mathcal{M}}_{W},\kappa_{W})\) of (1.7), we only need to compare it with \((\bar{M},\bar{k})+(M_{W},k_{W})\) with (M W ,k W ) a solution of (B.35) expressed in the co-moving frame.
2.3 B.3 Implicit change of variables
Our nonlinear analysis begins with an implicit nonlinear change of variable (2.5). We explain now how the modulation system is affected by this change of variables. We could have first performed this implicit change of variables then carried out the formal modulation process, but we find more enlightening to change the system a posteriori.
Since our diffeomorphism is close to identity, only nonlinear terms should be changed, and from the asymptotic equivalence theory we know that nonlinear terms are relevant only in the hyperbolic part. Therefore it is enough to investigate how (B.5), (B.8) is altered. Let us introduce Φ 0 such that Φ 0(Ψ 0(X,T),T)=X. Recall that \(\partial_{T}\varPsi_{0}=\omega({\mathcal{M}}_{0},\partial _{X}\varPsi_{0})\). Therefore if A,B are such that ∂ T A+∂ X B=0 then \((\tilde{A},\tilde{B})(X,T)=(A,B)(\varPhi_{0}(X,T),T)\) implies
also written \(\partial_{T} (\partial_{X}\varPhi_{0}\tilde{A} )+\partial_{X} (\tilde{B}-\partial_{T}\varPhi_{0}\tilde{A} )=0\) or
with \(\tilde{\mathcal{M}}_{0}(X,T)={\mathcal{M}}_{0}(\varPhi_{0}(X,T),T)\). Note that this kind of manipulation is completely similar to the ones needed to perform usual Lagrangian change of coordinates and of course closely related to the computations involved in the proof of Lemma 2.3. As expected, applying this to (B.5) leads to a trivial equation while an application on the trivial equation ∂ T (1)+∂ x (0)=0 gives
Equation (B.5) is changed into
At the hyperbolic level, we are thus lead to the system
whose quadratic expansion in
gives
Two main comments are in order: 1. We end up naturally with equations expressed in a co-moving frame thus no further change is needed. 2. The wavenumber equation remains unaltered at this level of description. This explains why the fact that the implicit change of variables could change the modulation equations was not revealed by previous studies [33, 57] focusing on situations where no other wave parameter is involved,
Remark B.5
Though we do not need it for the present semilinear analysis, let us describe for the sake of generality what would happen for a full quasilinear parabolic system. For
with
where A n+1=e n+1⋅A, D n+1=e n+1⋅D, the transformation Φ=Ψ −1, \(\tilde{\mathcal{M}}={\mathcal{M}}\circ\varPhi\) leads to
with
where
Collecting the results of this appendix, we find that to validate the formal Whitham modulation approximation, we only need to compare the couple \((M,\bar{k}\psi_{x})\) of Theorem 1.12 to a solution (M W ,k W ) of
where \(A_{*}=\partial_{(M,k)} (F-\bar{c}M,-\omega-\bar{c}k)|_{(\bar {M},\bar{k})}\). For writing convenience, we denote this system by
Likewise, ψ in Theorem 1.12 needs then to be compared with ψ W a solution of
Appendix C: Asymptotic equivalence of quadratic approximants
For completeness, we include here a proof of Proposition B.2 including the treatment of off-diagonal diffusion terms not arising in the scalar case considered in [32].
Proof of Proposition B.2
(Case \(B_{*}=\tilde{B}_{*}\).) We first review the case \(B_{*}=\tilde{B}_{*}\) treated in [32]. By the general results of [35], provided \(E_{0}:=\|y_{0}\|_{L^{1}\cap H^{3}(\mathbb{R})}\) is sufficiently small, we have for 1≤p≤∞
Setting δ:=w ∗+y−w, we have, subtracting and rearranging,
with δ| t=0=0 and A ∗ and B ∗ as in (B.29)–(B.30). By Duhamel’s formula,
where σ is the solution operator of the parabolic system of conservation laws u t +A ∗ u x −B ∗ u xx =0. Applying the standard bounds [41] \(\|\sigma(t)\partial_{x}^{r} h\|_{L^{p}(\mathbb{R})}\lesssim t^{-\frac{1}{2}(\frac{1}{q}-\frac{1}{p})-\frac{r}{2}}\|h\|_{L^{q}(\mathbb{R})}\), 1≤q≤p≤∞, together with
1≤q≤2, we find, defining \(\nu(t){:=} \sup_{0\le s\le t}\|\delta(s)\|_{L^{2}(\mathbb{R})}(1+s)^{\frac{1}{2}(1-1/p)+\frac{1}{2} -\eta}\), that, for all 1≤p≤∞,
whence ν(t)≤C η E 0(E 0+ν(t)). This implies that \(\nu(t)\le2C_{\eta}E_{0}^{2}\) for E 0<1/(2C η ), giving
(General case.) We treat now the general case that \(B(w)=B_{*}+\mathcal{O}(w-w_{*})\) with B ∗ constant but not equal to \(\tilde{B}_{*}\). Defining again δ:=w ∗+y−w, and denoting by \(\tilde{\sigma}(t)\) the solution operator of linear system \(u_{t} + A_{*} u_{x}-\tilde{B}_{*} u_{xx}=0\), we have by Duhamel’s principle
From [35], provided \(E_{0}:=\|y_{0}\|_{L^{1}\cap H^{4}(\mathbb{R})}\) is sufficiently small, we have for 1≤p≤∞
Applying the bounds [35, 41] \(\|(\tilde{\sigma}-\sigma)(t)\partial_{x}^{r} h\|_{L^{p}(\mathbb {R})}\lesssim t^{-\frac{1}{2}(\frac{1}{q}-\frac{1}{p})-\frac{r}{2}}(1+t)^{-\frac{1}{2}}\| h\|_{L^{q}(\mathbb{R})}+e^{-\theta t}\|\partial_{x}^{r}h\|_{L^{p}(\mathbb{R})}\), 1≤q≤p≤∞ for r=0,1 (and some θ>0), estimating
and other terms either similarly or similarly as in the previous case, we obtain the result. □
Appendix D: Generalizations
We conclude in this appendix by describing briefly extensions to more general types of equations arising in applications, and the modifications in our arguments that are needed to accomplish this, discussing also, when possible, the verification of (H1)–(H3) and (D1)–(D3) in specific cases.
4.1 D.1 Extensions in type: quasilinear and partially parabolic systems
Our analysis carries over in straightforward fashion to divergence-form systems of general quasilinear 2r-parabolic type. For example, the spectral preparation results of Lemma 1.5, Proposition 1.7, and Proposition 3.1 all go through essentially as written, depending on no special structure other than divergence form. From these low-frequency/Bloch number descriptions, we obtain the same linear bounds on the critical modes \(s^{\rm p}\) as described here in the 2-parabolic semilinear case. The high-frequency and or high Bloch number analysis also go through unchanged, the former depending again only on the spectral preparation results and the latter depending only (through Prüss’ Theorem) on high-frequency resolvent bounds following from (but not requiring) sectoriality of the linearized operator L about the wave. This completes the linear analysis.
Likewise, by Remark 2.4, we obtain the useful representation (2.6) of the nonlinear perturbation equations stated in Lemma 2.3, with sources \(\mathcal{Q}\), \(\mathcal{R}\), \(\mathcal{S}\) of quadratic order in v, ψ x , ψ t , and a finite number of their derivatives, which was all that was needed for our nonlinear arguments. To obtain the nonlinear damping estimate of Proposition 2.5, we note that (2.14) becomes
\(\tilde{U}=\bar{U}+v\). Thus, taking the \(L^{2}(\mathbb{R})\) inner product against \(\sum_{j=0}^{K} \frac{(-1)^{j}\partial_{x}^{2j}v}{1-\psi_{x}}\), integrating by parts, and rearranging, we obtain \(\frac{d}{dt} \|v\|_{H^{K}(\mathbb{R})}^{2}(t) \leq-\tilde{\theta} \|\partial_{x}^{K+r} \cdot v(t)\|_{L^{2}(\mathbb{R})}^{2} + \hbox{lower order terms} \), similarly as in the second-order semilinear case, leading thereby to
and (by Gronwall’s inequality) the result. See the proof of [3, Proposition 3.4], for full details in the fourth-order semilinear case.
Combining these ingredients, we obtain, modulo an appropriate increase in the integer K encoding regularity requirements, stability, as stated in Theorem 1.10, and refined stability, as stated in Proposition 4.5, yielding the first part (1.16) of description of asymptotic behavior in Theorem 1.12. By Remark 4.15, we get also a partial version of the second part (1.17)–(1.19) of Theorem 1.12, but describing comparisons not to the Whitham system, but only to a second-order hyperbolic-parabolic system agreeing with the Whitham system in its linearization about the constant state \((\bar{M},\bar{k})\). This in turn yields the conclusions of (1.21), Corollary 1.18, regarding decay with respect to localized perturbations for linearly phase-decoupled systems.
Finally, to recover the full result (1.17)–(1.19) of Theorem 1.12, comparing to the exact Whitham system, and thus the sharpened decay rate (1.22) for localized data in the quadratically decoupled case, we have only to observe that performing the same computations as in Appendix A (differentiating the traveling-wave ODE), and in the proof of Lemma 4.9 (pulling out quadratic order parts of nonlinear term \(\mathcal{N}\)) while carrying along the additional higher-order terms arising in the general case, we obtain a higher-order analog of Lemma 4.14, expressing the resulting quadratic coupling constants (means) in terms of derivatives of first-order terms arising in the Whitham system, after which computations go as before to yield the result; see the proof of Theorem 1.12, Sect. 4.6.
This completes the treatment of the quasilinear 2r-parabolic case. Reviewing the above discussion, but omitting algebraic considerations on which we focus in the next section, we find that the two ingredients needed to treat more general divergence-form systems are the nonlinear damping estimate used to control higher-derivative by lower-derivative norms, and the high-frequency linearized resolvent bounds used to apply Prüss’ Theorem. For, these were the only two places where we used the parabolic form of the equations; the rest of the argument was completely general, Moreover, the second, linearized, estimate can typically be obtained by a linearized version of the same energy estimate that is used to obtain the first, damping-type estimate. This allows us, in particular, to treat (partially parabolic) symmetric hyperbolic–parabolic equations such as arise in continuum mechanics, using “Kawashima-type” energy estimates as described in [35], and variants thereof. See, for example, Proposition 4.4 (proved in Appendix A) and Lemma B.1 in [30].
Remark D.6
The strategy of using a common energy estimate to get, simultaneously, damping high-frequency resolvent, and high-frequency decay estimates, with derivative gains in the first compensating for derivative losses in the third, originates in the study of viscous shock stability; see [73, Sect. 4.2.1]. For simpler, and somewhat sharpened, versions in this context, see [37, 45].
4.2 D.2 Extensions in form: an abstract continuum of models
Still more generally, we may treat the full class of systems
\(u,f, g\in\mathbb{R}^{n}\), \(B_{j}\in\mathbb{R}^{n\times n}\), with , , , , \(u_{2}\in\mathbb{R}^{r}\), including both divergence- and nondivergence-type equations. Note that this includes both reaction diffusion and conservation law cases as limits f≡0 and g≡0, but also many cases in between: for example, the viscous relaxation case n=2, r=1 occurring for the Saint-Venant equations (1.28), or the case n=3, r=1 occurring for the Bénard–Marangoni model (D.4) below.
For such models, integrating the conservative u 1 equation in the traveling-wave ODE, and writing as an N×N first-order system, we obtain from the requirement of periodicity N constraints, while we have N+n−r+2 degrees of freedom consisting of the initial condition u(0), the wave number \(k\in\mathbb{R}\), the speed \(c\in \mathbb{R}\), and the constant of integration \(q_{1}\in\mathbb{R}^{n-r}\) arising from integration of the u 1 equation; thus, we expect generically a manifold of periodic solutions of dimension n−r+2. In the reaction-diffusion case r=n, this returns the familiar value 2, or, up to translation, a one-dimensional family (generically) indexed by wave number k. In the conservation law case r=0, it returns the value n+2, leading, up to translation, to an (n+1)-dimensional family as in hypothesis (H2) of the introduction.
Substituting this value n−r+1 in hypotheses (H2) and (D3), therefore, we readily obtain by the same derivation as for (1.7) a modified Whitham system consisting of the (n−r+1)×(n−r+1) system of viscous conservation laws
where \(\omega({\mathcal{M}},\kappa)=-\kappa c({\mathcal{M}},\kappa)\) denotes time frequency, \({\mathcal{M}}:= \int_{0}^{1} U_{1}^{{\mathcal{M}},\kappa}(x) dx\) and
denote mean and mean “total flux” in the u 1 coordinate, and \(d_{ij}({\mathcal{M}},\kappa)\) are determined by higher-order corrections. (Note that, for B j≡constant, the terms involving B j are perfect derivatives, so disappear; this explains the fact that they were not present in the discussion of the second-order semilinear case.)
Likewise, we obtain in straightforward fashion analogs of the spectral preparation results of Lemma 1.5, Proposition 1.7, and Proposition 3.1, thus yielding corresponding linear bounds on critical modes \(s^{\rm p}\). Note that the slow decay rates that may arise at the linear level from a possible Jordan block will still be compensated by the special structure of the nonlinear terms, coming now in the form
See for example [48] for a careful derivation of the second-order derivative Whitham system up to linear and quadratic order in first-order derivative terms, and [30] for a proof of the needed spectral preparation results in the Saint-Venant case (1.28). Indeed, so long as the nonlinear structure of the equations permits a nonlinear damping estimate as in Proposition 2.5, and high-frequency linearized resolvent estimates as needed to apply Prüss’ Theorem in estimating high-frequency linearized behavior as in the proof of (3.18) and (3.19) above, we obtain again (modulo increase in the exponent of regularity K) the stability results of Theorem 1.10 and Proposition 4.5, and a partial version of Theorem 1.12 describing comparisons to a second-order hyperbolic-parabolic system agreeing with (D.2) in its linearization about the constant state \((\bar{M},\bar{k})\), yielding again the result (1.21), of Corollary 1.18 asserting decay with respect to localized perturbations for linearly phase-decoupled systems.
That is, we obtain in this case exactly the conclusions cited in the examples of the introduction, obtained by examination of the linearization of the first-order part of the Whitham equations.
To recover the full result (1.17)–(1.19) of Theorem 1.12 showing convergence to the exact Whitham system, one also needs an analog of Lemma 4.14. But, the only difference between the (formal) computations of the derivation in Sect. B.1.2 and the ones of Lemma 4.14 is that the former are carried out before the implicit change of variables, while the latter are carried out after. Thus, analogs of Lemma 4.14 essentially follow by commutation of an implicit change of variables and expansions to a desired order.
4.3 D.3 Verification of (H1)–(H3), (D1)–(D3)
Regarding verification of our stability hypotheses, we recall that, assuming the trivial regularity hypothesis (H1), hypothesis (H2) is implied by (D1)–(D3), by Lemma 1.6, while (H3) by Proposition 1.7 can generally be verified by the same spectral expansion process needed to verify (D2). Meanwhile, (D1)–(D3) can be verified numerically by Galerkin approximation or numerical Evans function computation (see, e.g., [3, 4, 18]), or, in some cases, analytically, using bifurcation theory (see, e.g., [59]) or singular perturbations (see, e.g., [31]). For general discussion, see [4, 5, 30].
4.4 D.4 Applications revisited
We now discuss previous examples and some new ones in a bit more depth.
4.4.1 D.4.1 The Korteweg–de Vries/Kuramoto–Sivashinsky equation
A more canonical form of (1.26) is
modeling phenomena from plasma and flame-front instabilities to inclined thin-film flow [38, 48, 54, 63]. As a fourth-order parabolic equation, this fits the framework of Sect. D.1, so that all of the results of this paper apply. Spectral stability has been studied in detail in [3, 12], indicating the existence of both spectrally stable and unstable waves; in particular, (H1)–(H3) and (D1)–(D3) have been shown in [3] to hold for a wide variety of waves. We note that stability under these hypotheses has been proven for localized perturbations in [3]; the new observations here are asymptotic behavior, and decay for nonlocalized perturbations.
4.4.2 D.4.2 The Saint-Venant equations
Recall, in Lagrangian coordinates, the Saint-Venant equations
where τ:=h −1, h is fluid height, u is fluid velocity, and x is a Lagrangian marker. These are not parabolic, yet nonlinear damping and high-frequency resolvent estimates can still be carried out, yielding by the discussion of Sect. D.2 all of the results of this paper. Specifically, nonlinear damping is established in [30, Proposition 4.4] (proved in Appendix A of the reference), under the “slope condition” \(\nu\bar{u}_{x}<F^{-1}\), where \(\bar{U}=(\bar{\tau}, \bar{u})\), a technical condition that appears to hold in most cases of interest, but which we expect can be dropped.Footnote 33 Again, the new observation here is asymptotic behavior, and also stability under nonlocalized perturbations, stability under localized data having been established in [30]. As noted earlier, (D.3) is a balance law rather than a conservation law, with nonconservative source term g.
4.4.3 D.4.3 The capillary Saint-Venant equations
With capillary pressure effects, (D.3) becomes
where σ>0 is the coefficient of capillarity. These equations can be reduced by Kotschote’s [36] method of auxiliary variables (introducing z:=τ x ) to a 3×3 second-order quasilinear parabolic system
to which standard techniques can be applied [69, 75]. This fits the framework of Sect. D.1, yielding all of the results of this paper, the only change being in the regularity assumptions on data, which must be incremented by one to accommodate the new variable z=τ x . Existence and spectral stability or instability of these waves is a topic of ongoing investigation [2].
4.4.4 D.4.4 Bénard–Marangoni flow
A qualitative model introduced in [20] for Bénard–Marangoni flow, or flow driven by temperature-induced surface tension variation, is
with f(u,v x ,w x )=−u 3+γ(uv x +uw x ), g 1(u,v,w)=−uv, g 2(u,v,w)=−uw. Though of mixed fourth-order parabolic/second-order parabolic form, it is readily seen that these equations are both sectorial and admit a nonlinear damping estimate; moreover, they are of the mixed conservative/nonconservative form (D.1). Thus, by the discussion of Sects. D.1 and D.2, the main results of this paper apply, giving stability and behavior in terms of a 3×3 hyperbolic–parabolic system agreeing with the Whitham system (D.2).
Let us now discuss existence, the form of the Whitham equations, and validation of (D1)–(D3). Setting v≡w≡0, we find that the equations reduce to the Swift–Hohenberg equation (1.23) for u, with bifurcation parameter r=ε 2 restricted to the positive side of the bifurcation point r=0 at which periodic solutions appear. Thus, we inherit from the Swift–Hohenberg equations a special class of periodic solutions with (v,w) vanishing. Up to translation, such solutions are given by the 2-parameter family of zero-speed \(\frac{ 2\pi}{1+\varepsilon \omega}\)-periodic Swift–Hohenberg solutions
where ε, recall, is the bifurcation parameter, a fixed constant in (D.4). However, there are many other solutions for which \((\bar{V},\bar{W}) \not\equiv(0,0)\), yielding an additional two parameters in the description of nearby periodic traveling waves. Moreover, though the Swift–Hohenberg solutions are zero speed as a result of reflection symmetry (see Remark 1.27), reflection symmetry of (D.4) is broken as soon as \((v,w)\not\equiv(0,0)\), and so in general these waves may have arbitrary speed. It is our expectation, therefore, that the Whitham system is not phase-decoupled even about such special waves.
Numerical experiment by Galerkin approximation in [70] indicate that solutions (D.5) satisfy stability conditions (D1)–(D3) for ω=0 and ε>0 in a moderate range. Here, we demonstrate the same conclusion for \(|\omega|< 1/2\sqrt{3}\) and ε≪1, using decoupling of the equations and known analytical results for the Swift–Hohenberg equation, at the same time obtaining the limiting ε→0 coefficients of the linearized Whitham system about \((\bar{M},\bar{k})=(0,\bar{k})\). It would be interesting to carry out a systematic numerical stability investigation as in [3, 7] on the entire parameter range, and in particular to determine phase-coupling or decoupling of the associated Whitham system.
Proof of (D1)–(D3)
About the special solutions (D.5), the linearized eigenvalue equations are
where L 0 is the linearized operator of the Swift–Hohenberg equation about \(\bar{U}\) and
By upper triangular form of (D.6), the eigenvalues of L ξ , counted by algebraic multiplicity, consist of the union of the eigenvalues of \(L^{0}_{\xi}\) and \(L^{\pm}_{\xi}\). Let us first consider the eigenvalues of the Swift–Hohenberg operator \(L^{0}_{\xi}\). In [13, 14, 17] (see also [44, 59]) it was analytically verifiedFootnote 34 that for ε≪1, solutions \(\bar{U}^{\omega,\varepsilon }\) in (D.5) are spectrally stable for
(in particular, for ω=0). From the fact that the waves are of speed c≡0, we find that the characteristic speed of the associated scalar Whitham equation is a 0≡0, and the associated critical mode has expansion λ 0(ξ)=−d 0 ξ 2. Turning to the operators L ±, and noting that \(\bar{U}^{\omega,\varepsilon }\to0\) uniformly in all derivatives as ε→0, we find that as ε→0 their eigenvalues approach uniformly the eigenvalues of the limiting constant-coefficient operators
which, by direct (discrete Fourier transform) computation, are
where the Fourier frequency j runs through the integers. By continuity, these are therefore spectrally stable for |ε|≪1, with approximate critical mode expansions (obtained at j=0) of ±iξ−ξ 2. Combining these facts, we find that the limiting linearized Whitham system has characteristic speeds a j =0,±1, with corresponding (diagonal) viscosity coefficients d, 1, 1. This verifies (D1)–(D3) and (by distinctness of a j ) (H3) for |ε| sufficiently small, yielding spectral stability by the discussion of Sect. D.3. □
4.4.5 D.4.5 Inclined Marangoni flow
The related inclined thin-film equation
models Marangoni flow driven by a thermal gradient up an inclined silicon wafer, where H denotes fluid height [10, 11]. As a cousin of the Kuramoto–Sivashinsky equation, it would be interesting to investigate whether this model too supports stable periodic traveling-waves solutions.
4.4.6 D.4.6 Surfactant-driven Marangoni flow
Finally, we mention the surfactant model [43]
modeling flow in a thin horizontal film driven by surfactant induced gradients in surface tension, where H is fluid height and Γ surface surfactant concentration, and \({\rm Pe}_{\rm s}\) is the modified Peclet number, a dimensionless constant, and σ(Γ)=1−Γ is an equation of state encoding the dependence of surface tension on surfactant density. Like (D.7), this appears to be an interesting example for study by the methods developed here and in [3, 25, 28, 30]. Note, as the second equation is conservative, that the associated Whitham approximation is indeed of system form.
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Johnson, M.A., Noble, P., Rodrigues, L.M. et al. Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations. Invent. math. 197, 115–213 (2014). https://doi.org/10.1007/s00222-013-0481-0
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DOI: https://doi.org/10.1007/s00222-013-0481-0