Florian Lapierre

Maxime Harnois

Yannick Coffinier

Rabah Boukherroub

Vincent Thomy
email: vincent.thomy@iemn.univ-lille1.fr

Split and flow: reconfigurable capillary connection for digital microfluidic

Liquid supplying to droplet-based microfluidic microsystems remains a delicate task facing the problems of coupling continuous to digital or macro to microfluidic systems. Here, we take advantage of superhydrophobic microgrids to address this problematic. Insertion of a capillary tube inside microgrid aperture leads to a simple and reconfigurable droplet generation set up.

In the last twenty years, microfluidic systems have attracted impressive interest in the fields of biotechnology industries for healthcare monitoring, clinical diagnostics, chemical or biological reagents handling. [1][2][3][4] These applications have common specificities and require new microsystems concepts for handling micro-to pico-volumes of reagents, detecting/analyzing at very high sensitivity rate, and sampling with minimal reagents consumption. To overcome these challenges, two main categories of microfluidic devices have been developed and can be classified as follow: Droplet-based MicroFluidics (DMF) and Continuous-based MicroFluidics (CMF). DMF is usually based on liquid droplet, handled via electrowetting for instance. First introduced by Lippman 140 years ago 5 , Berge et al. demonstrated the benefit of a dielectric layer for reversible droplet actuation. 6 As a result, moving, splitting and mixing of well-calibrated volume droplets could be easily performed with an electrode network. This multitasking microfluidics concept is nowadays one of the most used technologies for discrete liquid handling. 7 However, loading samples into the device requires a "buffer tank", which constitutes one of the main drawbacks of DMF, espe-cially for experiments that require large quantities of reagents and long operation times. The CMF technology uses welldefined microchannels for handling continuous flows of fluids in a closed device. Basic fluidic operations are accessible but require specific areas for each function. For instance, mixing can be generated by diffusion with or without turbulences; 8 splitting of continuous flow can be performed using non miscible fluids, but necessitates supplementary liquid access and additional liquid extraction step. 9 The combination of the DMF and CMF techniques within the same device results in powerful microfluidic systems that ensure easy fluids handling (i.e. splitting, calibrating, mixing) thanks to DMF and easy transfer of high amount of fluid using CMF. An analogy can be made with Digital to Analog electronic converters for data handling. But in the case of fluid handling, the first limitation concerns the connection between the macro and micro worlds. This question is of particular interest for dropletbased devices where parameters such as droplet volume and speed have to be controlled. How to feed microsystem with liquid in a simple and reconfigurable manner? Only a few recent studies deals with this concept. For instance, Kim et al. reported an Automated Digital Microfluidic system for DNA sequencing. 10 In this device, DMF was used to operate repeated steps for library preparation and CMF capillaries interface was combined to allow easy fluid transfer between the two interfaces. Wheeler's group also demonstrated hybrid digital-channel microfluidics for pre-processing and chemical separations. 11 Droplets of liquids are digitally displaced and mixed together prior to their loading in a PDMS channel by capillary action or electrokinetically. In 2010, the same group reported a modified architecture to interface a multilayer Digital-to-Channel design on glass. This configuration allowed droplets' dispensing from reservoirs and splitting droplets on chip. 12,13 In the same context, Wu et al. described recently the fabrication of "Analog-to-Digital" and "Digitalto-Analog" (AD/DA) microfluidic converters. 14 Using a series of microfabrication processes, they succeeded to integrate both CMF and DMF technologies in a single chip for liquidhandling and micro-analytical microsystems. All the systems reported in the literature describe the high potentialities of CMF/DMF converters, but suffer from the lack of reconfigurability and adaptability. In fact, each CMF inlet and each outlet are fixed at a specific location near the DMF system. As a consequence it is impossible to reconfigure the system without redesigning and fabricating again the whole device.

From this statement, we propose in this paper an original solution to address the liquid supplying problem within a droplet-based microfluidic system. This strategy deals with an electrowetting set up where a water based droplet is squeezed between a surface composed of a hydrophobic electrode matrix and a counter-electrode. The key point of this work is the use of superhydrophobic microgrid, as counter-electrode, prepared using micro/nanofabrication techniques. Any aperture of this grid can be used to host a capillary, localized with the naked eye and fixed in a reversible manner authorizing a huge reconfigurability. Once the liquid flows continuously through the capillary tube, a droplet is easily formed by electrowetting; the superhydrophobic character of the grid avoids droplet pinning from the capillary. The technique developed herein has several advantages such as reconfigurability, flexibility, simplicity, and reduced time of assembly, compared to the reported techniques.

Experimental

Reagents and materials

In order to fabricate the ElectroWetting-On-Dielectric Digital Microfluidic (EW-DMF) lab-on-chip system and silicon nanowires substrates, clean room reagents including AZnLOF2070, AZ4562 and SU-8 2002 photoresist, MIF-326 and SU-8 developers from Micro Chemicals (Germany), hydrophobic Cytop from AGCCE (Netherlands), glass substrates from ACM Verre Industrie (France), and p-type <100> crystalline highly doped silicon substrates (0.009 0.01 ohm/cm) from Siltronix (France), were used. Nickel deposition was performed with a PLASSYS MP 400S Electron Beam Evaporator (France). Hydrofluoric acid (50%), hydrogen peroxide (30%) were all purchased from Carlo Erba. Nitric acid (65%) was purchased from Merck. Acetone, isopropyl alcohol (i-PrOH) were obtained from Sigma Aldrich. The continuous flow was controlled via the Fluigent Microfluidic Flow Control System (MFCS).

Device fabrication

A similar EW-DMF system as previously reported by Lapierre et al. 15 and Jonsson et al. 16 was used. The electrode network and final device are illustrated in Fig. 2b. 16 Briefly, 20 nmthick Ni layer was sputtered on a transparent glass substrate and the electrodes network was fabricated using standard photolithography. Then, 2 µm of SU-8 2002 as dielectric layer followed by 30 nm of Cytop as hydrophobic layer were spin coated. This resulted in a contact angle and a contact angle The superhydrophobic microgrid counter-electrode was fabricated on a 380 µm thick, <100> silicon wafer. First, a 200 nm Si x N y layer was deposited by LPCVD on both sides of the substrate. Optical lithography was used to produce squared areas (from 120 µm to 250 µm). After removal of the photoresist, a CHF 3 /CF 4 plasma was used to etch the silicon nitride exposing the square patterns. Silicon nitride was also removed on the backside. The substrate was then cleaned with acetone, ethanol and dried under nitrogen. Backside substrate was first thinned to 100 µm in a KOH bath for 5h. Then, the grid was etched in the same bath; the Si x N y mask protects the top wafer and only the <111> oriented plan is etched at the naked squares. Anisotropic etching produced trapezoidal shape holes (Fig 1 .a). A more detailed explanation of the fabrication process can be found in Brunet et al. 17 In this study, the grids (2 cm x 2 cm) with 250 µm backside holes were selected. A 4 nm layer of Au was then sputtered on the top wafer. The metallized interface was introduced in a 500 • C CVD furnace chamber under SiH 4 flow. At this eutectic temperature, the gold film dewets to 10-100 nm nanodroplets which act as catalysts for silane decomposition. Due to oversaturation of the Au-Si droplets by silicon atoms, the formation of nuclei at the solid-liquid interface was obtained and nanowires elongated with the gold nanoparticle on top. Using 40 sccm

b).

A more detailed description of the nanowires growth can be found in Brunet et al. 18 and Lapierre et al. 19 To achieve the superhydrophobic character of the counter-electrode, 30 nm thick C 4 F 8 layer was deposited by plasma. The resulting contact angle and contact angle hysteresis were 160 • and 0 • , respectively, evidencing the super-repellent property of the surface. The final microgrid surface displays on its top layer the nanowire carpet which covers entirely the trapezoidal holes (Fig. 1.c). On the backside surface, the presence of the nanowires is limited only to the border of each hole due to the inexistence of gold film on this side during the sputtering process. The optimal superhydrophobic (SH) microgrid selected for this experiment presented final hole opening of ∼150 µm 2 for the insertion of the capillary tubes.

Robustness of the SH microgrid with droplet impalement and liquid spreading was tested under drop impact experiment. Similar experimental bench described by Lapierre et al. 19 allowed the measurement of the maximum dynamic pressure when the droplet passes through the SH microgrid. Video 1 † shows that for a drop impact at 0.469 m/s, a little amount of liquid crossed the other side of the grid; the rest of the droplet bounced off the surface and was never impaled. The dynamical pressure can be calculated using P ext = ρU 2 = 247Pa with ρ, the density of the liquid equal to 1126 kg.m 3 and U the speed of the droplet before impact.

Results and discussion

The base-plate and SH microgrid counter-electrode were placed adjacent at 300 µm gap. The electrode network was controlled via a LabView routine and electrically supplied by a signal generator (CENTRAD GF 265, ELC, France) coupled with a 50 dBm high-voltage amplifier (TEGAM, USA) set at 100 V T RMS @ 1 kHz. A silica capillary tube 75 µm inner diameter and 150 µm outer diameter was inserted through one of the superhydrophobic holes of the grid and connected to the Fluigent MFCS (see Fig. 2). The nanowire carpet bound the capillaries and prevented them to detach from the microgrid. This is advantageous as no epoxy or silicon glue was required to seal each tube. It is to be noted that the capillary was always placed at the vicinity of the SH surface (on the top of the V groove hole) when inserted. The precise location of the capillary was performed in a reproducible manner by 3 methods of visualisation:

• Depth of the capillary was checked thanks to an horizontal line previously drawn on the capillary (view from the bottom of the system),

• Location of the capillary on the electrode network (view from the top, through the base plate), • Depth of the capillary with the view from the side of the microsystem, Moreover, as the grid has a V-groove like shape, the nanowire carpet is protected from the insertion and removal of the capillary. Some of the nanowires at the bottom of the groove and backside of the silicon wafer could be destroyed by the capillary but this step does not affect the device functionalities. By selecting appropriate diameter, capillary tubes can be easily inserted into the grid just by eye sight, without any specific and complex tools such as microhandlers, and/or microscopes. Moreover capillaries can be relocated as desired along the electrode path. This major advantage allows extra reconfigurability of the lab-on-chip to allow various fluidic operations on demand, at any specific location. A continuous water flow rate (10 µL/min) was loaded into the EW-DMF system as shown in Fig. 3 droplet volume reached 1 µL and the distance electrode/SH microgrid was adjusted to 300 µm, the electrode network was sequentially turned on in order to first, split the droplet from the tube and second, to move the droplet along the electrode network (Fig 3 .c-e). Video 2 †, taken from the side of the microsystem, summarizes the fluidic sequence. The calibrated volume droplet was then available for other microfluidic tasks such as mixing, transporting, and merging also achieved by EW.

One of the most critical steps, the depinning of the droplet off the capillary, normally occurs during the droplet generation from the capillary. However, by using such a configuration, this challenge is overcome: the water repellency behaviour of the SH surface minimizes the wetted/sticky areas at the noozle part of the capillary. The liquid pinching mechanism is enhanced, leading to droplet splitting from the tubes. The high hole density allows multi-inlet/outlet accesses, but does not provoke liquid leakage out of the SH grid. Drop impact experiments showed that a dynamic external pressure of at least 250 Pa is necessary for the liquid to go through the SH grid. As the electrowetting is applied on the electrode base plate, no pressure is exerted on the SH surface. The droplet "sees" the holes as an air coffin where it can slide over. Furthermore, the superhydrophic property of the microgrid presents two crucial advantages regarding the aimed lab-on-chip applications. First, as detailed by Lapierre et al. 15 and Jonsson et al. 16 , water repellent nanowire carpet ensures antibiofouling properties: biomolecules adsorbed non specifically on these surfaces can be easily removed by a simple water droplet motion that resulted in a very efficient lift-off of the bioparticles. Second, as shown by Nguyen et al. 20 , by using an appropriate surface chemistry, silicon nanowire surfaces presenting super repellent properties for a wide range of liquids (from 33 mN/m to 72 mN/m) can be prepared. This is of prime interest for various applications knowing that the surface chemistry coatings are compatible with EW (no electrical field is applied to the counter-electrode, preserving the coating). By adding lateral walls on each side of the elec-

A C C E P T E D M A N U S C R I P T
trowetting electrodes, it is possible to fill this channel by oil. Due to the superomniphobic property of the micro-grid, this low surface tension liquid will be confined and biphasic electrowetting (water-based droplet in oil) should be easily realized. Moreover, by using these micro-grids coupled to a biphasic microfluidic setup, it should be possible to interact with water droplets without losing its sealing.

Conclusion

While connecting macroscopic world to microfluidic systems remains usually a question of experimenter patience and skills, we demonstrated in this work an original way to take advantage of the superhydrophobic microgrid interface to perform continuous to digital microfluidic operations. The technique proposed herein presents several advantages:

• Easy position of CMF capillaries at different locations on a microgrid. Capillaries can be located anywhere on the cover of the microfluidic system grid for reagent delivery into the digital microsystem.

• The use of a superhydrophobic grid is also advantageous as the capillary is fixed without requiring any glue or epoxy and it can be displaced reversibly from one grid aperture to the other.

• No leakage out of the microsystem occurs thanks to the liquid repellency behaviour of the SH microgrid. Furthermore, this solution can be employed for biphasic microfluidic authorizing droplet water based creation in an oil flux.

We do believe that the method developed in this work may represent a universal solution for coupling microfluidic microsystems to the macroscopic world.

a

  Institute of Electronics, Microelectronics and Nanotechnology (IEMN), UMR CNRS 8520, University Lille 1, F-59652 Villeneuve d'Ascq, France b Micro/Nanophysics Research Laboratory, RMIT University, Melbourne, Australia c UMR CNRS 6164, Institut d'Electronique et des Telecommunications de Rennes, Departement Microelectronique & Microcapteurs, Universite Rennes 1, Bat.11B, Campus de Beaulieu 35042, Rennes Cedex, France d Interdisciplinary Research Institute (IRI), USR CNRS 3078, University Lille 1, 50 Avenue Halley, F-59658 Villeneuve d'Ascq, France

Fig. 1

 1 Fig. 1 SEM images of (a) the silicon grid in V-groove shape with 117 µm 2 square holes at the top and 49 µm 2 at the backside; (b) Silicon nanowires grown via VLS technique. This robust nanowires carpet is 50 µm thick and ensures a complete superhydrophobic behaviour with Cassie-Baxter state; (c) Nanowires carpet surrounding one hole of the grid; (d) Backside of the grid with the presence of the nanowires carpet at the border of each hole. The backside hole was 250 µm 2 without nanowires and ∼150 µm 2 with the nanowire carpet.

of

  SiH 4 , total pressure of 0.532 mbar and growth time of 60 min, silicon nanowires of 50 µm in height were produced (Fig.1.

Fig. 2

 2 Fig.2(a) 3D view of the CMF/DMF microsystem. The SH microgrid acts as counter-electrode where tubing accesses are available on the whole EW-DMF electrode network. This setup allows easy reconfiguration of the microfluidic system for various applications. (b) Closer view of the EW-DMF coupled with a capillary tube. The base plate was composed of the transparent electrodes network, driving the droplet. The superhydrophobic microgrid (∼150 µm 2 ) counter-electrode allows capillary tube access (150 µm 2 ) without any use of epoxy or silicon rubber. The capillary tube can be located anywhere on the grid, depending on the user's need. The superhydrophobic character of the grid allows easy droplet detachment from the capillary, easy droplet motion and avoids leakage from the grid pores.

Fig. 3

 3 Fig. 3 Photographs of the CMF/DMF setup. (a-b) A 1 µL droplet is generated from the SH microgrid. (c-e)The DMF electrode network is sequentially activated to actuate the droplet. The grid holes are not "seen" by the droplet during its motion due to their SH character. No leakage is observed while generating the liquid droplet. Blue color was computationally added to differentiate the real droplet from its mirror on the base plate. Schematics of the grid, the electrode network and capillary were also added to help visualisation.

Acknowledgements

The Centre National de la Recherche Scientifique (CNRS) and the European Community Seventh Frame-work Programme (FP7/ 2007-2013) under grant agreement no. 227243 are gratefully acknowledged for financial support.


Luis Miguel

Miguel Rodrigues

L Miguel Rodrigues

SPACE-MODULATED STABILITY AND AVERAGED DYNAMICS

Keywords: periodic traveling waves, stability, modulation. 2010 MSC: 35B10, 35B35, 35K59, 35P05, 35Q53, 37K35

come

Introduction troduction

A classical approach to the qualitative analysis of dynamical behavior driven by partial differential equations starts with the identification of special classes of solutions that exhibit some simple internal structure. One hope is that when these coherent structures possess a nontrivial basin of attraction they could serve as elementary blocks to describe a richer complex dynamics.

However, in many situations, determining dynamical stability even for the simplest reference solutions appears as a daunting task. Hence the general strategy consisting in obtaining on one hand general theorems -or at least general accurate frameworks -to derive dynamical behavior from spectral information on linearized evolution and on the other hand in gathering separately the required pieces of spectral information. Fortunately it turns out that in many cases the latter may at least be obtained either analytically in some asymptotic regimes or numerically for (a large part of) the full family of background solutions. Interestingly enough, besides those classical ways of determining spectral stability, an approach based on interval arithmetics and leading to computer assisted proofs of spectral stability seems to have reached sufficient maturity to completely solve some stability issues; the reader is referred to Barker [START_REF] Barker | Numerical proof of stability of roll waves in the small-amplitude limit for inclined thin film flow[END_REF] for an instance of such a line of study.

With this general scheme in mind some considerable amount of effort has been devoted recently and is still directed towards the development of a general stability theory for periodic traveling wave solutions of systems of partial differential equations. The purpose of this note is to offer both a terse introduction to the field and an overview of the recent activity of the author, partly jointly with others.

As a preliminary warning we want to strongly emphasize that traveling waves under consideration are solutions of equations modeling extended systems. In particular our physical domains are unbounded, at least in the direction of propagation. This situation arises naturally when idealizing a phenomena where boundaries seems to play a minor role as in the coherent motion of waves over large distances. As a consequence in our set of problems periodicity in space is a feature of the internal structure of background waves and not of the physical domain.

From the point of view of stability this results in the fact that we are not interested in coperiodic or sub-harmonic perturbations but in stability under arbitrary smooth and localized ones. Consequently we aim also at taking into account the rich multiscale spatio-temporal dynamics emerging from the slow modulations of periodic waves. In contrast with the local analysis around patterns with a simpler internal spatial structure such as solitary waves, fronts, kinks or shocks, and even for simpler-looking systems as those modeling reaction-diffusion processes, this inherent complexity has precluded until very recently any completely satisfactory mathematical analysis of dynamical stability and large-time asymptotic behavior.

Nevertheless, for general parabolic systems, after the pioneering work of Schneider

Schneider-SH,Schneider-pro [START_REF] Schneider | Diffusive stability of spatial periodic solutions of the Swift-Hohenberg equation[END_REF][START_REF] Schneider | Nonlinear diffusive stability of spatially periodic solutions-abstract theorem and higher space dimensions[END_REF] at the end of the past century, an intense activity in the last few years -more or less culminating in JNRZ-conservation [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF] -has led to a clear picture of the relevant notion of spectral stability, called diffusive spectral stability, and of the asymptotic description of the dynamics near stable periodic waves. The general theory even includes systems that are only parabolic in the pointwise hypocoercive sense of Kawashima, thus effectively including at least small waves of viscous compressible models, and have been extended recently to some systems that are only symmetrizable in a suitable averaged sense RZ [START_REF] Rodrigues | Periodic-coefficient damping estimates, and stability of large-amplitude roll waves in inclined thin film flow[END_REF], in particular relaxing the smallness condition for roll-waves of the St. Venant system describing surface waves on fluid film flowing down an incline. Incidentally, we point out that the diffusive spectral stability of the latter roll-waves have been extensively studied by a combination of numerical methods and spectral perturbation analysis BJNRZ-KdV-SV,BJNRZ-KdV-SV-not [START_REF] Barker | Stability of St. Venant roll-waves: from onset to the large-Froude number limit[END_REF][START_REF] Barker | Note on the stability of viscous roll-waves[END_REF], leading -jointly with the abstract nonlinear theory -to a fairly comprehensive description of the dynamics near those primary instabilities.

In contrast, for dispersive equations, as far as the author knows, even a single example of a proof of nonlinear stability is missing. Yet the detailed description of the linear dynamics of solutions to the Korteweg-de Vries near cnoidal waves -recently obtained in R_linKdV [START_REF] Rodrigues | Linear asymptotic stability and modulation behavior near periodic waves of the Kortewegde Vries equation[END_REF] -shades some light on what kind of nonlinear dynamics is to be expected.

Based essentially on JNRZ-conservation,R_linKdV [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF][START_REF] Rodrigues | Linear asymptotic stability and modulation behavior near periodic waves of the Kortewegde Vries equation[END_REF] our goal here is to give both an overview of the well-developed parabolic theory and a glimpse at what could be expected of a dispersive theory, still to come. On the former the reader may also benefit from consulting R [31]. We warn the reader that, to keep the presentation as short as possible, in most of the present note the general tone is rather descriptive and informal.

General information

We shall restrict our discussion to planar traveling waves -as these are the objects for which the theory is more advanced -and we shall regard them as one-dimensional objects. The latter choice comes with two main consequences. On one hand nonlinear analysis is considerably harder in space dimension one since scattering mechanisms are weaker thus decay resulting from diffusive-like or dispersive-like behavior is slower1 . This is reflected by the fact that the first results of nonlinear stability of spectrally stable periodic waves of parabolic systems of conservation laws were first proved in dimension higher than two, then in dimension two and finally in dimension one Oh-Zumbrun-cons-low-freq-multiD,Oh-Zumbrun-cons-nonlinear,Oh-Zumbrun-cons-nonlinear-err [START_REF] Oh | Low-frequency stability analysis of periodic traveling-wave solutions of viscous conservation laws in several dimensions[END_REF][START_REF] Oh | Stability and asymptotic behavior of periodic traveling wave solutions of viscous conservation laws in several dimensions[END_REF][START_REF] Oh | Erratum to: Stability and asymptotic behavior of periodic traveling wave solutions of viscous conservation laws in several dimensions Oh-Zumbrun-cons-nonlinear [28[END_REF][START_REF] Johnson | Nonlinear stability of periodic traveling wave solutions of systems of viscous conservation laws in the generic case[END_REF],

Johnson-Zumbrun-cons-1D-2D [START_REF] Johnson | Nonlinear stability of periodic traveling-wave solutions of viscous conservation laws in dimensions one and two[END_REF], JNRZ-conservation [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF]. This is also responsible for the asymptotically linear behavior exhibited by localized perturbations of periodic waves in dimension higher than one. On the other hand, spectral stability may seem easier to met in dimension one since in some sense spectral stability in dimension one may be interpreted as a higher-dimensional spectral stability but restricted to co-planar perturbations. Obviously one may then rightfully wonder what could be the physical meaning of a situation where a planar wave would be stable as a one-dimensional object but unstable as a higher-dimensional object. An element of answer lies in the fact that our one-dimensional solutions sometimes correspond in higher dimension not to planar waves but to genuinely multidimensional objects resulting from some confinement mechanisms that may preserve stability 2 . In short, dimension one offers both a richer nonlinear dynamics -at the cost of a substantially more involved analysis -and better hope to find stable waves.

Moreover, in this brief note we discuss only two sets of equations, on one side the Korteweg-de Vries equation (KdV)

KdV KdV (2.1)

U t + 1 2 U 2
x + U xxx = 0 . and, on the other side, a general second-order semi-linear parabolic system of conservation laws nservation nservation (2.2)

U t + (f (U)) x = D U xx .
In the above, time variable t belongs either to R or R + and spatial variable x belongs to R.

The unknown of ( KdV 2.1) is scalar, U (t, x) ∈ R, whereas the unknown of ( conservation 2.2) is vector-valued, U(t, x) ∈ R d , for some d ∈ N * , the corresponding advection flux is given by a smooth function f : R d → R d and the diffusion matrix D is some d × d positive definite symmetric matrix.

To start being more specific let us add that here by a periodic wave we mean a special solution U (t, x) = U (kx+ωt) given by a periodic profile U and traveling with uniform velocity c = -ω/k. When normalizing the period of U to one -as we shall do -k accounts for spatial wavenumber and ω for time frequency.

The choice of ( conservation 2.2) is purely expository as it does not bring any significant simplification in statements and proofs, which apply almost word-by-word to a much larger class of parabolic systems. In contrast, the choice of ( KdV 2.1) is commanded by very specific reasons. Indeed, prior to R_linKdV [START_REF] Rodrigues | Linear asymptotic stability and modulation behavior near periodic waves of the Kortewegde Vries equation[END_REF], the author was ignorant of what could be the best notion of spectral "dispersive" stability that one may expect and which kind of spectral conditions could ensure some form of linear asymptotic stability. Hence the choice of ( KdV 2.1) that, beyond its own importance as a universal amplitude equation for long weakly-nonlinear phenomena, enjoys the advantage of being integrable so that in principle everything on its periodic waves and their spectra may be explicitly derived. We stress however that the main intention of R_linKdV [START_REF] Rodrigues | Linear asymptotic stability and modulation behavior near periodic waves of the Kortewegde Vries equation[END_REF] is not to study the Korteweg-de Vries equation specifically but to develop a strategy, first illustrated on ( KdV 2.1), that transfers spectral knowledge to linear bounds. This opens possible applications to cases where the spectrum is not known explicitly but where required spectral conditions may be either proved in an asymptotic regime, for instance for small waves, or investigated numerically.

Since we aim at discussing dynamics near a specific wave described by some profiles U and with wavenumber k and frequency ω it is convenient 3 to switch to the moving frame of this specific wave. For instance, defining implicitly W through U (t, x) = W (t, k x + ω t), equation ( KdV

2.1) becomes

KdV-move KdV-move (2.3)

W t + ω W x + k 1 2 W 2 x + k 3 W xxx = 0 and U is a stationary solution of ( KdV-move

2.3). A (naive) linearization of (

KdV-move 2.3) about U leads to the consideration of W t -L W = 0 where linop linop (2.4)

L W = -ω W x -k U W x -k 3 W xxx .
2 Think about a fluid flowing down a channel. 3 But not absolutely mandatory at least for some of the arguments. See for instance KR [START_REF] Kabil | Spectral validation of the Whitham equations for periodic waves of lattice dynamical systems[END_REF] where translational invariance is essentially broken by discreteness of the spatial domain and no obvious way to turn traveling waves in steady solutions is available.

2.1. The Bloch transform. This naturally suggests the introduction of an integral transform specially tailored to analyze differential operators with periodic coefficients, here normalized to period one. To motivate its definition, let us observe that in this context the simplest relevant class of functions that may span the full set of arbitrary functions is given by so-called Bloch waves that have a simple behavior under translation by one period as they are simply multiplied by some number of modulus one, say e iξ where ξ is a Floquet exponent. Equivalently those are the functions that may be written as x → e iξ x ǔ(x) for some Floquet exponent ξ ∈ [-π, π) and some function of period one ǔ.

One rationale behind the definition of the Bloch transform -sometimes called Floquet-Bloch transform -is that one may indeed achieve a decomposition erse-Bloch erse-Bloch (2.5)

g(x) = π -π e iξx ǧ(ξ, x) dξ,
with each ǧ(ξ, •) periodic of period one, by summing modes of the Fourier decomposition sharing the same Floquet exponent

Bloch Bloch (2.6) ǧ(ξ, x) := j∈Z e i 2jπx g(ξ + 2jπ).
where

g(ξ) := 1 2π R e -iξx g(x) dx.
An equivalent approach consists in introducing a periodization of x → e -iξx g(x)

ǧ(ξ, x) := ∈Z e -iξ(x+ ) g(x + )
which is readily seen to coincide with the above formula through the Poisson summation formula. Bloch analysis has essentially the same flavor as classical Fourier analysis 4 . Observe for instance that -modulo a multiplicative constant -the Bloch transform is a total isometry between L 2 (R) and L 2 ((-π, π), L 2 per ((0, 1))) and that interpolating between Parseval identity and triangle inequalities leads to the following Hausdorff-Young inequalities, for 2 ≤ p ≤ ∞,

g L p (R) ≤ (2π) 1/p ǧ L p ([-π,π],L p ((0,1))) , ǧ L p ([-π,π],L p ((0,1))) ≤ (2π) -1/p g L p (R)
where p denotes conjugate Lebesgue exponent, 1/p + 1/p = 1. Going on with the former point it may be rather instructive 5 to regard g ∈ L 2 (R) as an element of 2 (Z, L 2 ((0, 1))) providing the L 2 ((0, 1))-valued Fourier series of ξ → e iξ• ǧ(ξ,

•) ∈ L 2 ((-π, π), L 2 ((0, 1))).
As expected, differential operators L with coefficients of period one and acting on a space of functions defined on the whole line, say L 2 (R), may then be thought on the Bloch side as operator-valued multipliers acting Floquet exponent by Floquet exponent, that is (Lg)ˇ(ξ, •) = L ξ ǧ(ξ, •) where for each ξ ∈ [-π, π) the operator L ξ acts on a space of functions of period one, say L 2 per ((0, 1)). This follows readily from applying L to ( inverse-Bloch 2.5), a process that also gives

L ξ = e -iξ• L e iξ•
where L, L ξ are here used as formal operators. This "diagonalization" has direct counterparts for spectra and generated semi-groups. Observe in particular that for operators L obtained from linearization (in a suitable frame) of either ( KdV 2.1) or ( conservation 2.2) each L ξ has compact resolvents (when acting say on L 2 per ((0, 1))) hence has a spectrum made of a discrete set of eigenvalues with finite multiplicity and the associated 4 That is why we have been rather loose with convergence issues in the above definitions since they are solved in the classical way. 5 This is probably the point of view that generalizes in the most straight-forward way to the spatially discrete case, where, for any N ∈ N, a suitable discrete Bloch transform provides an isometry between 2 (Z) and

L 2 ((-π, π), 2 per ([[0, N ]])). See KR [18].
spectral map ξ → σ(L ξ ) is continuous. Therefore the spectrum of L consists locally in curves parametrized by Floquet exponents and

σ L 2 (R) (L) = ξ∈[-π,π) σ L 2 per ((0,1)) (L ξ ) .
As an illustration we mention that accordingly the minimal notion of spectral stability, that is

σ(L) ⊂ {λ | Re(λ) ≤ 0}, is also written as σ(L ξ ) ⊂ {λ | Re(λ) ≤ 0} for any ξ ∈ [-π, π
). Yet obviously such a requirement is too weak to yield any form of dynamical stability even at the linear level.

2.2.

Stability. On the other hand, concerning the stability of traveling waves, the strongest notion of spectral stability, that requires a spectral gap σ(L) ⊂ {λ | Re(λ) ≤ -η} for some η > 0, is precluded by the presence of a full family of similar traveling waves near any given wave of a specific type. Indeed at the very least translation invariance of the equations yields that any translated copy of a traveling wave is a traveling wave. Then for traveling waves with localized variations -fronts, solitary waves... -taking derivatives along this family of waves provides elements of the generalized kernel of L. However, for localized waves of parabolic systems one may still expect that σ(L) ⊂ {λ | Re(λ) ≤ -η} ∪ {0} for some η > 0 and that 0 is an eigenvalue of finite multiplicity corresponding to the dimension of the family of neighboring traveling waves. In this case one may prove by adapting the arguments providing the central manifold theory that such waves are (orbitally) asymptotically stable and that at leading order nonlinear dynamics is reduced to a finite-dimensional evolution describing the time evolution of parameters encoding traveling waves. In short, the main dynamics occurs through modulation in time of wave parameters. Let us observe already that some changes in parameters of the waves -for instance phase parameters, encoding the position of the wave -have a dramatic impact on norm comparisons and the notion of dynamical stability has to be accommodated accordingly. This leads to the classical notion of orbital stability that asks for a control of the proximity of orbits of functions under the action of symmetries of the equation, which when the only relevant symmetry is space translation amounts to a bound on inf

Ψ uniform translation U • Ψ -U X
(for some chosen functional norm • X ). See

Henry,KapitulaPromislow_book [START_REF] Henry | Geometric theory of semilinear parabolic equations[END_REF][START_REF] Kapitula | Spectral and dynamical stability of nonlinear waves[END_REF] for some related results, references and discussions.

The situation differs significantly when periodic waves are considered. First when differentiating a family of periodic waves along a parameter independent of the period one obtains not elements of the generalized kernel of L -which is reduced to {0} -but of L 0 , since obtained elements do not belong to L 2 (R) but to L 2 per ((0, 1)). This seems to be a minor difference but then by varying ξ this implies that no matter how small η > 0 is chosen σ(L) \ {λ | Re(λ) ≤ -η} contains curves of spectrum passing through zero. Therefore the best one can reasonably expect is that when η is sufficiently small then the former set is indeed reduced to those curves, that the number of such critical curves -which is the algebraic dimension of zero as an eigenvalue of L 0 -is indeed the dimension of the family of periodic waves minus one 6 and that those curves are non degenerate at ξ = 0. The corresponding notion of spectral stability is now well-known as diffusive spectral stability. We recall its precise definition in Section s:parabolic 3.1. In such a situation there is no hope for a reduction even at the leading-order to a finite-dimensional evolution that could not account for such dispersion relations. Moreover there is also no hope for some exponential decay and the decay mechanism of the critical part of the evolution is truly infinite-dimensional as it is due to scattering of solutions by a diffusive spreading. We stress that with this respect the assumption of non degeneracy of critical curves is crucial since this is the non vanishing at 6 We remind the reader that derivatives corresponding to variations of the period do not generate elements of the generalized kernel of L0. ξ = 0 of the second order derivatives of the real part of spectral curves that encodes at the spectral level the diffusive character of the spectral stability. Note in particular that no matter what the order of parabolicity of the original system is the deduced decay is of heat-like type.

The issue that even a small perturbation may modify spatial position and therefore have a dramatic effect on direct norm comparison is even stronger for periodics as it occurs cell by cell for an unbounded number of cells and thus can not be compensated by some spatially uniform transformation, nor more generally by any element of a prescribed finite-dimensional set of transformations. For the sake of clarification and unification, a specially-designed notion of nonlinear stability was formalized in JNRZ-conservation [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF] and called there space-modulated stability. It goes as follows. Given a functional space X, instead of measuring at some prescribed time the proximity of a function U to a background periodic wave U directly with U -U X , one tries to control m_distance m_distance (2.7)

δ X (U, U ) = inf Ψ one-to-one U • Ψ -U X + ∂ x (Ψ -Id) X .
This allows, before comparison, for a resynchronization of cells through a phase change Ψ that is in turn forced to remain locally in space (and time) close to uniform translations. As the notion of orbital stability does for unimodal waves, this provides a global way to quantify preservation of shapes in a periodic context. Besides being a generalization of orbital stability, it is also closely related to the Skohokhod metric on functions with discontinuities that allows for a near-identity synchronization of jumps.

As we recall in Section s:parabolic 3.1 abstract stability results for parabolic systems, as proved in JNRZ-conservation [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF], show that if a periodic wave is diffusively spectrally stable it is also nonlinear stable, in a spacemodulated sense, from L 1 ∩ H s to H s provided s is large enough. However for Hamiltonian systems as ( KdV 2.1), and this even in finite dimension, diffusive spectral stability is excluded by Hamiltonian symmetry of the spectrum which implies that in a stable situation the whole spectrum lies on the imaginary axis. For finite dimensional Hamiltonian systems no decay mechanism is available and one instead prove bounded -and not asymptotic -stability by a Lyapunov functional argument. This actually does use a spectral gap argument but not on the dynamical generator L but on the variational Hessian of the Hamiltonian. Generalizations of this argument also apply for waves with localized variations of Hamiltonian partial differential equations. But for the same reason expounded above on L, a spectral gap argument on the variational Hessian is precluded by the Floquet structure of its spectrum and another argument is needed -even at the linear level -to analyze the periodic wave problem. See however Angulo-Pava,KapitulaPromislow_book,Ben [START_REF] Pava | Nonlinear dispersive equations[END_REF][START_REF] Kapitula | Spectral and dynamical stability of nonlinear waves[END_REF][START_REF] Benzoni-Gavage | Co-periodic stability of periodic waves in some Hamiltonian PDEs[END_REF] for examples of what may be obtained when one restricts to co-periodic or sub-harmonic perturbations and hence enforce discreteness of Floquet exponents.

Contrary to what happens in finite dimension, note that for localized waves of Hamiltonian partial differential equations it is actually possible to go further and prove in some cases asymptotic stability by using decay provided by scattering of dispersive type. In Section s:KdV

we show how the spreading argument is successfully generalized in

R_linKdV [START_REF] Rodrigues | Linear asymptotic stability and modulation behavior near periodic waves of the Kortewegde Vries equation[END_REF] to prove linear asymptotic stability of cnoidal waves of ( KdV 2.1). Observe that while in the diffusive context a similar argument is used to deal with critical spectral curves passing through zero, here the whole spectrum is critical and part of the difficulty is to apply uniform arguments to spectral curves containing an infinite number of Floquet eigenvalues. We stress that linear stability also needs to be understood in a space-modulated sense that we exhibit now.

To prove nonlinear space-modulated stability of U for ( KdV-move 2.3), one should introduce some (V, ψ) related to the expected solution W through mod-ansatz mod-ansatz (2.8)

W • Ψ = U + V , Ψ = Id -ψ .
Stability would then be proved if one were able to choose (W, ψ) in such a way that W is indeed a solution and V and the derivatives of ψ remain small if they are sufficiently small initially. With this in mind, we observe that in terms of (V, ψ) equation ( KdV-move 2.3) takes the form

(V + U x ψ) t -L (V + U x ψ) = N [V, ψ t , ψ x ]
where L is the operator defined above, and N is nonlinear in (V, ψ t , ψ x ) and their derivatives, and, locally, at least quadratic. One important point is that the nonlinear part does not involve ψ itself but only its derivatives. This is crucial since ψ itself is not expected to decay but to become locally constant. Then the linearized dynamics of interest is given by LinKdV LinKdV (2.9)

(V + U x ψ) t -L (V + U x ψ) = 0 .
As a result, at the linear level, one may indeed focus on the evolution generated by L as we have done up to now, but instead of aiming at controlling the evolved W through W X one should aim at bounding sm_norm sm_norm (2.10)

N X (W ) = inf W =V +U x ψ ψ(∞)=-ψ(-∞) V X + ψ x X .
Observe that we have added the constraint ψ(∞) = -ψ(-∞) to ensure that ψ = ∂ -1

x ψ x , where ∂ -1

x is defined as a principal value on the Fourier side. It is intended to ensure that a knowledge of ψ x does provide some control on ψ. Mark that this constraint does not restrict potential applications of linear estimates to a nonlinear analysis since the centering constraint may be achieved initially by translating U by 1 2 (ψ 0 (∞) + ψ 0 (-∞)) and is then preserved by the evolution.

2.3. Averaged dynamics. We now turn to the question of determining a leading-order description of the dynamics. As we have already pointed out, since studied phenomena are genuinely multi-scale in time and space, one can not expect a reduction to a finite-dimensional time evolution as for localized patterns. As we have also suggested, in dimension one -in contrast with what happens for higher dimensional cases -decay rates are too slow to expect that the nonlinear dynamics of ( conservation 2.2) will be asymptotically linear. So we must seek another set of nonlinear partial differential equations providing an asymptotically equivalent simpler dynamics. Note also that even when the original equation is scalar, as ( KdV 2.1) is, perturbations of periodic waves may split in different parts traveling at distinct characteristic velocities -usually denominated group velocities -, a phenomenon more easily captured by a system of equations, so that in general 7 no reduction in the number of equations may be expected either.

What is the expected gain then ? The gain is in separation of scales. For instance, we may enumerate typical scales for ( conservation 2.2) as being of size one for oscillations of the background wave, t for the hyperbolic-like behavior encoded by group velocities and √ t for the heat-like diffusive character leading to diffusive spectral stability. One may hope to capture the evolution at large scales t, √ t by some form of averaged dynamics that does not contain any oscillation anymore. In this homogenized description the slow evolution then occurs near a constant solution -and not near a periodic wave. In turn once the slow part of the evolution is known one expects to be able to recover an asymptotic description of the full evolution including oscillations by using solutions of one-cell problems provided by profiles of periodic waves.

To implement this reduction -even arguing on formal grounds -one needs a way to derive averaged equations on one hand and on the other hand a way to translate initial data for the original system into effective initial data for the averaged system. The former may be achieved by inserting a slow/oscillatory ansatz, that is, by studying a suitable family of solutions evolving from a family of well-prepared data. In contrast the main difficulty in answering the latter comes from the fact that one needs to understand what is the effective initial datum emerging from an ill-prepared datum. This problem may nevertheless be solved explicitly here by carrying out the analysis.

In this way we prove in JNRZ-conservation [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF] that in the large-time regime the dynamics of parabolic systems near stable periodic waves follows at main order the slow modulation scenario that one may derive formally by some second-order version of the averaging method of Whitham. As a byproduct this refined asymptotic description proves that the notion of space-modulated stability is sharp for generic parabolic systems and identifies what are the null conditions -called phase uncoupling in JNRZ-conservation [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF] -that a given wave must satisfy to recover classical orbital stability from space-modulated stability. In R_linKdV [START_REF] Rodrigues | Linear asymptotic stability and modulation behavior near periodic waves of the Kortewegde Vries equation[END_REF] we also show that a similar scenario occurs for the linearized dynamics of ( KdV 2.1).

Specific results

s:results

We now state precise results concerning ( conservation 2.2) and ( KdV 2.1), as proved in JNRZ-conservation [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF] and R_linKdV [START_REF] Rodrigues | Linear asymptotic stability and modulation behavior near periodic waves of the Kortewegde Vries equation[END_REF].

:parabolic 3.1. Periodic waves of parabolic systems. We first give a precise definition of the notion of diffusive spectral stability. To do so we fix a periodic wave of (

U t + ωU x + k (f (U)) x = k 2 D U xx
and denote by L the generator of the corresponding linearized dynamics

LV = -ωV x -k (df (U)(V)) x + k 2 D V xx .
We shall say that the wave U is diffusively spectrally stable if the four conditions (

D1 D1 (D1) σ L 2 (R) (L) ⊂ { λ ∈ C | Re(λ) < 0 } ∪ {0} D2 D2 (D2)
There exists θ > 0 such that for any ξ ∈ [-π, π)

σ L 2 per ((0,1)) (L ξ ) ⊂ λ ∈ C Re(λ) ≥ -θ ξ 2 . D3 D3 (D3) 0 is an eigenvalue of L 0 of algebraic dimension d + 1.
H H (H) First-order derivatives of critical curves with respect to the Floquet exponent are distinct at ξ = 0.

It is not immediately clear that assumption ( H H) makes sense since 0 is in general not semisimple and generally speaking spectral perturbation of a Jordan block does not yield differentiable spectral curves. Nevertheless under assumption ( D3 D3) one may indeed prove that those spectral curves are differentiable at ξ = 0 so that assumption ( D2) amounts to the non vanishing at ξ = 0 of the second-order derivative of the real part of critical spectral curves with respect to the Floquet exponent. See Noble-Rodrigues [START_REF] Noble | Whitham's modulation equations and stability of periodic wave solutions of the Korteweg-de Vries-Kuramoto-Sivashinsky equation[END_REF] for detailed proofs and further discussions. As emphasized by the choice in notation, assumption ( H H) plays a role distinct in the spectral stability and is not expected to be sharp for the obtained results. Indeed it plays a role similar to the assumption of strict hyperbolicity in the analysis of hyperbolic systems and could likely be replaced with something analogous to an assumption of symmetrizability. See R [31] for further discussions in this direction.

As is implicit in assumption ( D3 D3) one expects the family of periodic waves to be of dimension d + 2 -counting phase translations. Indeed when analyzing the profile system arising from ( conservation 2.2), it is immediate that all equations are conservative and hence may be integrated once by introducing d constants of integration, besides the already-introduced phase speed. Once those d+1 parameters are fixed one does expect that around a given periodic orbit in the phase portrait of the reduced profile system no other periodic orbit co-exist. Hence in this case follows a local 8 The most instructive proof of this leads to a connection at the spectral level with averaged dynamics. Yet alternatively classical arguments proving Rellich's theorem do show that this is also a consequence of ( D1 D1).

parametrization of periodic wave profiles by d + 2 parameters. When a similar parametrization is available then ( D3 D3) really is an assumption of minimal multiplicity. In turn assuming ( D3 D3) one may prove that a parametrization by d + 2 parameters holds and even identify wavenumber, averages over a period (of components corresponding to conservation laws) and phase shift as a set of suitable parameters. This essentially follows from an Evans function computation in the spirit of Serre [START_REF] Serre | Spectral stability of periodic solutions of viscous conservation laws: large wavelength analysis[END_REF]. See detailed discussions in JNRZ-conservation,Benzoni-Noble-Rodrigues,R [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF][START_REF] Benzoni-Gavage | Slow modulations of periodic waves in Hamiltonian PDEs, with application to capillary fluids[END_REF]31].

nservation Theorem 3.1 ( JNRZ-conservation [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF]). Let K ≥ 3. For any U a periodic traveling wave profile of ( conservation 2.2) that is diffusively spectrally stable there exist positive 0 and C such that for any U 0 such that

E 0 := δ L 1 (R)∩H K (R) (U 0 , U) < 0
there exists U a global classical solution of ( conservation-move 3.1) with initial data U 0 such that for any t ≥ 0

δ H K (R) (U(t), U) ≤ C E 0 and any 2 ≤ p ≤ ∞ δ L p (R) (U(t), U) ≤ C (1 + t) - 1 2 1- 1 p E 0 .
The proof of the foregoing theorem also includes decay in δ H K but with non sharp decay rates. Indeed our proof is not optimized in regularity -neither in decay of the high derivatives nor in the threshold K ≥ 3 -but in robustness. In particular the proof does not rely directly on analytic semigroup techniques and applies almost word-by-word to quasilinear cases satisfying pointwise Kawashima conditions. See detailed discussion in JNRZ-conservation,R [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF]31]. By similar techniques one does not expect to recover sharp decay rates in Ẇ k,p for 1 2 K ≤ k ≤ K ; see Rodrigues-compressible [START_REF] Rodrigues | Vortex-like finite-energy asymptotic profiles for isentropic compressible flows[END_REF] for instance. The restriction 2 ≤ p ≤ ∞ has a distinct technical origin. It follows from our will to prove most of the required linear estimates on the spectral side by relying on Hausdorff-Young estimates as this leads to relatively simpler proofs than those relying on finer multiplier theorems or pointwise bounds of Green functions. See however Jung-pointwise-RD,Jung-pointwise-conservation [START_REF] Jung | Pointwise asymptotic behavior of modulated periodic reaction-diffusion waves[END_REF][START_REF] Jung | Pointwise stability estimates for periodic traveling wave solutions of systems of viscous conservation laws[END_REF] for examples of latter estimates. In any case decay in L 1 should not be expected when the main decay mechanism is -as here -due to spatial dispersion of the solution. In this case, the trade-off encoded by estimates is localization against (regularity and) decay. The situation is in some sense parallel to the case where the decay mechanism is mixing and the trade is regularity against decay.

More generally one should not expect to be able to prove decay rates without loss -in localization or in regularity or... -when optimal decay rates are not of exponential type. At the linearized level the fact that this is indeed impossible is essentially a consequence of the Datko-Pazy theorem (t, x) → U (M,κ)(t,x) (Ψ(t, x))

with κ = ∂ x Ψ, where (M, κ) are slowly evolving according to some system whitham whitham (3.3)

M t + (F(M, κ)) x = (d 11 (M, κ)M x + d 12 (M, κ)κ x ) x , κ t -(ω(M, κ)) x = (d 21 (M, κ)M x + d 22 (M, κ)κ x ) x .
Here F is simply the averaged flux

F(M, k) = 1 0 f (U (M,k) (x)) dx
and d i,j are determined in a more complicated way as usually expected from higher-order corrections in averaging processes. Though we do not expound the full derivation of ( whitham 3.3) we emphasize now two key-points of this derivation. The first one is that the last equation of ( that is then differentiated with respect to the space variable. It is important to note that a completely similar scenario occurs at the spectral level, in particular when desingularizing the Jordan block structure, as the generalized kernel is spanned by [START_REF] Noble | Whitham's modulation equations and stability of periodic wave solutions of the Korteweg-de Vries-Kuramoto-Sivashinsky equation[END_REF][START_REF] Benzoni-Gavage | Slow modulations of periodic waves in Hamiltonian PDEs, with application to capillary fluids[END_REF][START_REF] Kabil | Spectral validation of the Whitham equations for periodic waves of lattice dynamical systems[END_REF]. The second point is that for our purpose of describing to main order the large-time dynamics near (M, k) system ( whitham 3.3) is far from being uniquely determined but there is a canonical choice that also ensures that though the system has been derived by fitting slow evolutions -a low-Fourier requirement -the obtained system is a well-posed hyperbolic-parabolic system -a high-Fourier feature. See Noble-Rodrigues [START_REF] Noble | Whitham's modulation equations and stability of periodic wave solutions of the Korteweg-de Vries-Kuramoto-Sivashinsky equation[END_REF] and

U x , ∂ M 1 U, • • • , ∂ M d U,
JNRZ-conservation [13, Appendix B].
To compare directly to solutions of ( conservation-move nservation Theorem 3.2 ( JNRZ-conservation [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF]). Let K ≥ 4. For any U a periodic traveling wave profile of ( conservation 2.2) that is diffusively spectrally stable there exist positive 0 and C such that for any U 0 such that there exists ψ 0 such that

E 0 := U 0 • (Id -ψ 0 ) -U L 1 (R)∩H K (R) + ∂ x ψ 0 L 1 (R)∩H K (R) < 0
there exist U a global classical solution of ( conservation-move 3.1) with initial data U 0 , a local phase shift ψ with initial data ψ 0 and local averages M such that for any t ≥ 0 and any 2 ≤ p ≤ ∞ modulation modulation (3.5)

U(t, • -ψ(t, •)) -U M+M(t,•), k (1-ψx(t,•)) ( • ) L p (R) ≤ C E 0 ln(2 + t) (1 + t) -3 4 (M, k ψ x )(t, •) L p (R) ≤ C E 0 (1 + t) -1 2 (1-1/p) ψ(t, •) -ψ 0 (-∞)+ψ 0 (+∞) 2 L ∞ (R) ≤ C E 0 ; and setting Ψ(t, •) = (Id -ψ(t, •)) -1 , κ = k ∂ x Ψ, M(t, •) = (M + M(t, •)) • Ψ(t, •), and letting (M W , κ W ) solve (W ) with initial data data data (3.6) κ W (0, •) = k ∂ x Ψ(0, •) , M W (0, •) = M + U 0 -U • Ψ(0, •) + 1 ∂ x Ψ(0, •) -1 (U • Ψ(0, •) -M) ,
we have

(3.7) (M, κ)(t, •) -(M W , κ W )(t, •) L p (R) ≤ C E 0 (1 + t) -1 2 (1-1/p)-1 2 +η ; at last recovering Ψ W from (M W , κ W ) though (W ) phase with Ψ W (0, •) = Ψ(0, •) (3.8) Ψ(t, •) -Ψ W (t, •) L p (R) ≤ C E 0 (1 + t) -1 2 (1-1/p)+η .
The upshot of estimates ( est:slow-modulation 3.5) is that U is indeed very well approximated by modulating U in all its parameters, showing that asymptotically U essentially takes the form ( slow-modulation 3.2). The remaining estimates prove that the evolution of slow parameters is indeed well-captured by ( whitham 3.3). Observe that in order to achieve relevant comparisons one needs to stick to the description ( slow-modulation

3.2)

and effectively perform all modulations on U hence to invert Id -ψ(t, •). For a typical cnoidal wave, part of the spectrum of L with positive imaginary part : Im(λ) vs ξ such that λ ∈ σ L 2 per ((0,1)) (L ξ ). There are infinitely many red branches going to infinity.

spectrum-sk

In contrast in order to derive asymptotic stability we do use a huge wealth of information. Indeed our argument uses a non degeneracy assumption on the Floquet parametrization of the spectrum similar to ( D2 D2) but here the whole spectrum is critical hence instead of requiring such a condition at one point, zero, as in ( D2 D2) we need such a piece of information everywhere. To be a bit less pessimistic let us precise that actually both at infinity and near zero the required pieces of information may be obtained by robust techniques so that what remain is really a sign control on smooth functions defined on compact intervals 10 .

Before giving a precise statement of this non degeneracy condition we give more details concerning the structure of the spectrum of L. First, real and Hamiltonian symmetries yield that λ ∈ σ L 2 per ((0,1)) (L ξ ) if and only if -λ ∈ σ L 2 per ((0,1)) (L -ξ ). As a by-product of the proof of Theorem th:stab-KdV 3.3 we also know on one side that the large spectrum consists in two symmetric branches that by gluing together various Floquet parametrizations may be parametrized by two symmetric semi-unbounded intervals, with a parametrization equivalent to ξ → -(ikξ) 3 and whose third order derivative of the imaginary part converges to 6k and on the other side that near zero there are three branches of spectrum going through zero with second-order derivative vanishing at zero but third order derivatives non zero. By the techniques used in Bottman-Deconinck [START_REF] Bottman | KdV cnoidal waves are spectrally stable[END_REF] one may actually derive a complete parametrization of (λ, ξ) in terms of an auxiliary parameter -a spectral Lax parameter -and conclude that this picture extends as follows: the spectrum may be split in two parts, one infinite line covering the imaginary axis, and a symmetric loop covering twice a neighborhood of zero in the imaginary axis 11 ; see Figure The picture is of course consistent with descriptions obtained by more robust techniques in asymptotic regimes. In particular, in the solitary wave limit Gardner-large-period [START_REF] Gardner | Spectral analysis of long wavelength periodic waves and applications[END_REF] the line coincides with the essential spectrum of solitary waves and the loop arises from the embedded eigenvalue zero.

Despite the relatively explicit description of the spectrum of L, up to now the author has not been able to prove the following condition

A A (A)
Along the line the third-order derivative with respect to Floquet exponent does not vanish. Along the loop the second-order derivative with respect to Floquet exponent does not vanish except at zero. 10 This is typically the kind of verification amenable to computer-assisted proofs relying on interval arithmetics as in Barker [START_REF] Barker | Numerical proof of stability of roll waves in the small-amplitude limit for inclined thin film flow[END_REF]. 11 The presence of this triply-covered area near zero does not contradict the simplicity of Floquet eigenvalues as the passage at the same point of the spectrum occurs at distinct Floquet parameters.

However for all cnoidal waves picked randomly by the author an eye inspection of corresponding graphs seems to show that condition ( In contrast with most of results introduced so far our proof of Theorem th:asymptotic-KdV 3.4 is not carried out on the Bloch side but proceeds by pointwise estimates of suitable Green functions through oscillatory integral estimates. As in the parabolic case one may actually go further and prove at this linearized level that the large-time dynamics is at main order of slow-modulation type and that the involved local parameters evolve essentially according to a third-order hyperbolicdispersive system. For the sake of brevety we do not detail this here but add two comments. First, as the reader may have already deduced from the structure of the spectrum of L, the dimension of the family of cnoidal waves is 4 (counting phase shift), the extra parametercompared to phase shift, wavenumber, average of U -, which arises from the Hamiltonian structure and invariance by space translation, may be chosen as the average of the Benjamin impulse; see Benzoni-Noble-Rodrigues,Benzoni-Mietka-Rodrigues [START_REF] Benzoni-Gavage | Slow modulations of periodic waves in Hamiltonian PDEs, with application to capillary fluids[END_REF][START_REF] Benzoni-Gavage | Co-periodic stability of periodic waves in some Hamiltonian PDEs[END_REF]. Second, we stress that to capture the large-time effects encoded by the form of spectral curves near zero one does need to include a third-order correction to the classical first-order Whitham system.

At last we warn the reader that it is very unlikely that decay rates expounded here could serve directly in a nonlinear large-time study since they seem by far too slow for such a purpose, and that the lack of normality of L prevents a direct use of the classical T -T * argument to deduce from dispersive estimates of Theorem

conservation 2 . 2 )

 22 , U(t, x) = U(kx + ωt). Then we write ( conservation 2.2) in the corresponding moving frame ation-move ation-move(3.1)

H

  H) makes perfect sense8 . Moreover assumption ( H H) then implies that those critical curves are analytic with respect to the Floquet exponent. In particular, under assumptions (

vanNeerven [ 24 , 1 0U[ 13 ,

 24113 Theorem 3.1.5 & Corollary 3.1.6]. We now turn to large-time asymptotics. First we point out that assumption ( D3 D3) implies the existence of a family of nearby periodic waves with profiles U (M,k) ( • + φ) and phase speed c(M, k), where k is the corresponding wavenumber, φ is a phase shift and M ∈ R d are averagedvalues over a period M = (M,k) . As a consequence corresponding time frequency is ω(M, k) = -k c(M, k). Note carefully that what was U in Theorem th:stab-conservation 3.1 is now U (M,k) for some M and that ω = ω(M, k). Following the extended version of the original strategy of Whitham Whitham Appendix B] -, one may derive arguing on formal grounds that two-scale slow/oscillating solutions of ( conservation 2.2) should behave as modulation modulation (3.2)

whitham 3 . 3 )

 33 is first obtained aspsieq psieq (3.4) Ψ t = ω(M, κ) + d 21 (M, κ)M x + d 22 (M, κ)κ x

3. 1 )

 1 (and not ( conservation 2.2)) we now denote by (W ) and (W ) phase system ( whitham 3.3) and equation ( psieq 3.4) written in the frame of U (M,k) .

Figure 1 .

 1 Figure1. For a typical cnoidal wave, part of the spectrum of L with positive imaginary part : Im(λ) vs ξ such that λ ∈ σ L 2 per ((0,1)) (L ξ ). There are infinitely many red branches going to infinity.

A

  A) is satisfied. ptotic-KdV Theorem 3.4. For any cnoidal wave of ( KdV 2.1) satisfying ( A A), there exists C such that for any W 0 and any time t∈ R * N L ∞ (R) (S(t) W 0 ) ≤ C |t| -1/3 N L 1 (R) (W 0 ) and N L ∞ (R) (S(t) W 0 ) ≤ C (1 + |t|) -1/3 N L 1 (R)∩H 1 (R) (W 0 )where (S(t)) t∈R is the group of operator solutions of the linearized dynamics.

  th:asymptotic-KdV 3.4 bounds of Strichartz type.

  but one needs to unravel the role of ∂ k U by transforming a phase-like component in a wavenumber-like component through a suitable multiplication by iξ ; see

	Noble-Rodrigues,Benzoni-Noble-Rodrigues,KR

Think about decay rate t -N

in L ∞ -norms for solutions to the heat or the Schrödinger equation in spacedimension N starting from integrable initial data.

In particular this is always the case when the system contains only conservation laws as in the present note.

Research of L.Miguel Rodrigues was partially supported by the ANR project BoND ANR-13-BS01-0009-01.

Observe also how ( data 3.6) prescribes data in a non trivial way. In particular the last term in M W (0, •) incorporates contributions of the high frequencies of ψ 0 and hence cannot be included in a formal slow/oscillatory ansatz.

Relying then on a fine description of near constant dynamics -see Liu_Zeng [START_REF] Liu | Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws[END_REF] and JNRZ-conservation [13, Appendices B & C] -, one may then use Theorem th:asymptotic-conservation 3.2 to obtain a very precise description of the dynamics in terms of weakly interacting diffusion-waves in local parameters. This yields that decay rates in Theorem th:stab-conservation 3.1 are sharp for general systems but also enables us to identify various sharp cancellation conditions, coined as phase uncoupling conditions in JNRZ-conservation [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF], that leads when ψ 0 is constant (or when ∂ x ψ 0 is mean-free) to extra decay hence to classical (and not space-modulated) asymptotic orbital stability and to an asymptotically linear behavior.

s:KdV 3.2. Cnoidal waves of the Korteweg-de Vries equation. We now discuss similar results for the linearized dynamics of ( KdV 2.1) near cnoidal waves. First select such a cnoidal wave U and consider the operator L introduced in ( linop 2.4). One readily checks that L does generates a group (S(t)) t∈R on L 2 (R) (or on H s (R) with s ∈ N, or...) by using energy estimates which also lead to the crude bound

That energy estimates yield such a poor bound is a manifestation of the fact that L is not a normal operator. Nevertheless space-modulated bounded linear stability holds in H s (R) for any s ∈ N.

h:stab-KdV Theorem 3.3. For any cnoidal wave of ( KdV 2.1) and any s ∈ N, there exists C such that for any W 0 and any time

where (S(t)) t∈R is the group of operator solutions of the linearized dynamics.

The foregoing theorem is non trivial even when s = 0. Yet there does not seem to be an easy way to deduce the general case from the case s = 0. Since the emphasis here is in developing robust techniques rather than in the result itself, we now give some hints about its proof 9 .

The starting point is that one knows that the spectrum of L lies on the imaginary axis -see for instance

Bottman-Deconinck,R_linKdV [START_REF] Bottman | KdV cnoidal waves are spectrally stable[END_REF][START_REF] Rodrigues | Linear asymptotic stability and modulation behavior near periodic waves of the Kortewegde Vries equation[END_REF]. Yet by itself this is insufficient to derive any suitable bound. To go beyond one may build for each Floquet ξ a spectral decomposition of L ξ . This is essentially a consequence of the theory pioneered by Keldyš Keldysh-1,Keldysh-2 [START_REF] Keldyš | On the characteristic values and characteristic functions of certain classes of non-self-adjoint equations[END_REF][START_REF] Keldyš | The completeness of eigenfunctions of certain classes of nonselfadjoint linear operators[END_REF] -see Markus,Yakubov,Gohberg-Goldberg-Kaashoek-1 [START_REF] Markus | Introduction to the spectral theory of polynomial operator pencils[END_REF][START_REF] Yakubov | Completeness of root functions of regular differential operators[END_REF][START_REF] Gohberg | Classes of linear operators[END_REF] -on spectral decompositions of relatively compact perturbations of self-adjoint operators with compact resolvents. It relies on the fact that as far as spectrum at infinity is concerned L ξ may be thought as a perturbation of the skew-adjoint -k 3 (∂ x + iξ) 3 . With this decomposition in hands and knowing that except zero as an eigenvalue of L 0 all Floquet eigenvalues are simple, after a desingularization of the zero eigenvalue in a way similar to what was required in the analysis of parabolic cases -see Benzoni-Noble-Rodrigues,R_linKdV [START_REF] Benzoni-Gavage | Slow modulations of periodic waves in Hamiltonian PDEs, with application to capillary fluids[END_REF][START_REF] Rodrigues | Linear asymptotic stability and modulation behavior near periodic waves of the Kortewegde Vries equation[END_REF] -one may then define Sobolev-like norms • X s (R) for which the evolution generated by L is unitary. What is left then is to show that • X s (R) is indeed equivalent to N H s (R) . This is actually the core of the proof but relies, as above arguments, mostly on fine analyses of large and small spectrum.

While some proofs are easier to carry out by using explicit formulas derived from Lax pair representation, most of them may be obtained in a robust way. Notable exceptions are the knowledge that the spectrum indeed lies on the imaginary axis -a minimal requirement for linear stability -and that non zero Floquet eigenvalues are indeed simple. 9 In contrast we have chosen not to comment proofs of Theorems th:stab-conservation 3.1 & th:asymptotic-conservation 3.2 since they are already sketched in the introduction of JNRZ-conservation [START_REF] Johnson | Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations[END_REF] itself, further commented in R

[31] and one may gain some insight about their key-features by looking first at the analysis of a simpler sub-case treated in JNRZ-RD1,JNRZ-RD2 [START_REF] Johnson | Nonlocalized modulation of periodic reaction diffusion waves: nonlinear stability[END_REF][START_REF] Johnson | Nonlocalized modulation of periodic reaction diffusion waves: the Whitham equation[END_REF]. Incidentally we mention that this sub-case also exhibits the very specific feature that at the averaged level it is asymptotically self-similar and hence is amenable to a resolution by the renormalization techniques initially used by Schneider; see SSSU [START_REF] Sandstede | Diffusive mixing of periodic wave trains in reactiondiffusion systems[END_REF].

Université de Rennes 1, IRMAR, UMR CNRS 6625, 263 avenue du General Leclerc; F-35042 Rennes Cedex, FRANCE E-mail address: luis-miguel.rodrigues@univ-rennes1.fr