Corner defects in almost planar interface propagation De´fauts faibles en propagation d’interfaces planes MarianaHaragus a &ArndScheel b aUniversite´deFranche-Comte´,De´partementdeMathe´matiques, 16routedeGray,25030Besançoncedex,France bUniversityofMinnesota,SchoolofMathematics, 206ChurchSt. S.E.,Minneapolis,MN55455,USA Abstract We studyexistenceandstabilityofinterfacesinreaction-diffusionsystemswhichareasymptoti- cally planar. The problemofexistenceofcorners isreducedto an ordinarydifferentialequation thatcanbeviewedasthetravelling-waveequationtoaviscousconservationlaworvariantsofthe Kuramoto-Sivashinskyequation. Thecornertypicallybutnotalwayspointsinthedirectionoppo- siteto thedirectionofpropagation. Fortheexistenceandstabilityproblem, werelyonaspatial dynamicsformulationwithanappropriateequivariantparameterizationforrelativeequilibria. Re´sume´ Nouse´tudionsl’existenceetlastabilite´desinterfacesasymptotiquementplanesdansdessyste`mes dereáction-diffusion. Leproble`mede l’existencedesde´fautsestre´duita` l’e´tuded’unee´quation diffe´rentielle ordinaire qui est, selon le cas, approcheé par l’e´quation stationnaire d’une loi de conservation scalaire ou d’une variante de l’e´quation de Kuramoto-Sivashinsky. Typiquement, les de´fauts pointent dans la direction opposeé a` la direction de propagation. Pour l’analyse des proble`mesd’existenceet destabilite´, nousutilisons uneformulationde typedynamiquespatiale combineéavecuneparame´trisationade´quated’e´quilibresrelatifs. Runninghead: Cornersininterfacepropagation Correspondingauthor: ArndScheel Keywords: interfaces,stability,reaction-diffusionsystems,Burgersequation,Kuramoto-Sivashinsky equation,quadraticsystems. 1 Introduction Characterizingpropagationof interfaces in spatiallyextendedsystemsis amajorchallengeintheap- plied sciences. Flame fronts in solid and gaseous combustion have stimulated a variety of different approaches to interface formation and propagation [17, 44, 64]. Experimental observation and theoretical predictions range from rigid plane front propagation over periodically oscillating speeds and cellular patternson the interface, to seemingly chaotic motionof the interface. In a slightly different context, frontand pulsepropagationturnsout to be crucialforthe dynamicsof many self-organizing chemicalreactions,suchasthecarbon-monoxideoxidationonplatinumsurfaces[30]ortheBelousov- Zhabotinsky reaction [66]. One-dimensional interfaces have been observed in spiral wave patterns, whereinterfacial cornersnaturallyariseatthedomainboundariesbetweendifferentspiralcores[66]. More recently, oscillatoryfront propagation[34] and interfaces betweenhomogeneousand patterned states [22] have been studied in the Belousov-Zhabotinsky reaction. Propagation and reflective or annihilationcollisionof 2-dimensionalpulse trains has also beenobserved in the CO-oxidation[25]. In the CIMAreaction, famous forexhibitingstationaryTuringpatterns, propagationand propagation failure of a one-dimensional interface separating a region occupied by a hexagonal lattice built with isolated Turing spots, into an unpatterned region governs the later stage of spot replication; see [9] for an experimental survey and [46] for (mostly one-dimensional) theoretical approaches. Interfaces betweenpatternedstatesariseinmanyotherappliedareas. Wementionsemiconductors[58],viscous shockwaves[41],Rayleigh-Beńardconvection[48]orcertainmodelsforextendedlasers[40]. In asingularperturbationapproach tointerface propagation,spatialvariablesarescaledsuchthatthe interface becomesasharpline, forwhichageometricevolutionequationcanbederived froma inner andouterexpansionsattheinterfacialregion;see[15,61]foravarietyofapplicationsofthismethod. In many cases, the formal asymptotics can be justified, either in a general dynamical setup [6], or in specificcontexts[29]. Themostgeneralresultsareavailablewhenacomparisonprincipleisavailable [2]. Morerecently, ainterface propagationhasbeenaddressedfromadifferent perspective. Thecommon featureisthatexistenceandstabilityareconsideredinunboundeddomains,correspondingtotheinner expansion in the sharp interface limit. For various reasons, however, a scaling cannot be rigorously justified such that interfaces have to be studied in the original equations. We mention recent work on propagation of fronts in discrete two-dimensional lattices with possible pinning of interfaces [7], stability of plane viscous shocks [28, 36], existence and stability of conically shaped fronts in scalar reaction-diffusion modelsforcombustion[3,20,21]. We refer the reader to the beautiful review [63] as a guide to the tremendous amount of work on (mostlyone-dimensional)frontpropagationinthephysicsliterature. 2 Inthisarticle,wefocusonexistenceandstabilityofalmostplanarinterfaces. Almostplanarhererefers to the angle of the interface at each point, relative to a fixed planar interface. Most of the interfaces that we constructwill be planar at infinity, withpossibly different orientationsat + and in an 1 (cid:0)1 arclengthparameterization. Werefertoallthesetypesofinterfacesascornerdefects. Differentangles at resultinconicallyshapedinterfaces,likeforexamplethetravellingwavesconstructedin[20]. (cid:6)1 In an isotropicsharpinterface scaling, conicalinterfaces correspond to corners. Equalangles at (cid:6)1 may result in infinitesimalstep discontinuities, when the position of the interface differs at . We (cid:6)1 constructcornerdefectsasperturbationsofaplanarinterface. Assumptionsaresolelyontheexistence of a primary planar travelling-wave solution and spectral properties of the linearization at the planar wave. Allinterfaces thatweconstructinthepresentarticlearestationaryortime-periodicpatternsin an appropriatelycomoving frame. However, wegive stabilityresultswhichshow that“open”classes of initial conditions actually converge to the corner-shaped interfaces we constructed before. The results are stated for reaction-diffusion systems but the method is sufficiently general to cover most applicationsmentionedabove. Inparticularwedonotrelyonmonotonicityargumentsorcomparison principles such that we can naturally include the case of interfaces separating patterned states from spatiallyhomogeneousstates. The methodweuseis based onthe(essentiallyone-dimensional)dynamicalsystems approachto the existenceofboundedsolutionstoellipticequationsincylindersintroducedbyKirchga¨ssner[39]. Later thisapproachhasbeenusedtoconstructnontrivialtransversemodulationsofone-dimensionalwaves, suchas pulseor periodicsolutions,aphenomenontheauthorsreferredto as dimensionbreaking[23, 24]. Themainideaistoconsideranellipticequation,posedonthe(x;y)-planeinaneighborhoodofan x-independentwaveq (y)asadynamicalsysteminthex-variableandrelyondynamicalsystemstools (cid:3) suchascenter-manifoldreductionandbifurcationtheorytoconstructboundedsolutionstotheelliptic equation in a neighborhood of the original wave. Nontrivial, that is non-equilibrium, x-“dynamics” thencorrespondtonontrivialx-profiles. In the present work, we extend these ideas, incorporating the shift of the y-profile q (y) into the re- (cid:3) duced dynamics. We then respect this affine action of the symmetry group in the construction and parameterization of the center-manifold such that the reduced equations take a skew-product form. Theconceptof an equivariant reductionand skew-product descriptionof bifurcationsin thepresence of non-compact, non-smooth group actions has been introduced in [14, 55] in order to describe me- andering anddrift motionof spiral waves under the presenceof the Euclideangroup of rotationsand translationsin the plane. The constructionof the center-manifold is “semi-global”in the sensethat a neighborhoodof all translates of the primary solution is describedby the reduced equations. For example,constantdrift(cid:24)0 = (cid:11)inx-dynamicsalongthetranslatesq (y+(cid:24)),correspondstotheoriginal (cid:3) frontinclinedbyanangle#= arctan(cid:11). The methods and results are related to recent work in [11, 54] on dynamics of defects in oscillatory 3 media. The common feature between the present work and the study of wave trains is the presence of a neutraleigenvalue inducedbythetranslationof a primaryprofile, the Goldstonemode. A major differenceliesinthefactthatwavetrainspossessanon-compactisotropy generatedbytranslationsof one period, suchthat the resultingsymmetry action is isomorphicto the circlegroup, whereas in our case, the isotropy of the travelling wave is trivial. As a consequence, we have to study bifurcations from a non-compact group orbit, isomorphic to R, whereas the group orbit in [11, 54] is compact, a circle. Wedevelop the idea,prove areductiontheorem,and describethemostbasicshock-typecorner solu- tions in Section 2. We prove asymptotic stability of these structures in Section 3. We then consider more complicated scenarios, where the front undergoes a transverse long-wavelength instability, in Section 4, and when the primary front is pulsating, in Section 5. We conclude with a discussion, pointingoutpossiblegeneralizationsandopenquestionsinSection6. Acknowledgments M.HaraguswishestothanktheSchoolofMathematics,UniversityofMinnesota, forhospitalityprovidedduringthepreparationofpartofthispaper. A.Scheelwaspartiallysupported bytheNSFthroughgrantDMS-0203301. 2 Existence of corners Weintroducethegeneralsetupfortravellingwavesinreaction-diffusionsystemsanddefinethetypical types of corners one might expect to find. We then state and prove the first main result on existence andnonexistenceofcornerdefectsforinterfacesseparatingtwohomogeneousstates. Weconcludethe sectionwithseveralexamplesandpossibleextensions. Throughoutthepaper,weconsiderthereaction-diffusion system u = D(cid:1) u+c@ u+f(u); (2.1) t x;y y whereu RN isavectorofN chemicalspecies,D = diag(D ;:::;D )>0isapositive,diagonal 1 N 2 diffusion matrix, and (x;y) Rn R. The reaction kinetics f are assumed to be smooth. The 2 (cid:2) Laplacian is assumed to be isotropic (cid:1) = (cid:1) +@ . The speed c > 0 is assumed to be positive, x;y x yy suchthatboundedsolutionstothestationaryequation D(cid:1) u+c@ u+f(u)= 0; (2.2) x;y y are(right-)travelling-wave solutionsu(x;y ct)ofthereaction-diffusion systeminthesteadyframe (cid:0) u = D(cid:1) u+f(u); (2.3) t x;y withdirectionofpropagationinthepositivey-direction. Wewillassumen= 1suchthat(x;y) R2, 2 throughout. WebrieflycommentonhigherspacedimensionsinSection6. 4 Throughoutthissection,wewillassumeexistenceofaplanartravellingwaveconnectingtwohomo- geneousequilibria. Hypothesis2.1(Existence) We assume that there exists c > 0 and asymptotic states q such that (cid:3) (cid:6) thereexistsanx-independentplanartravelling-wavesolutionq (y)of(2.2) (cid:3) Dq00+c q0 +f(q ) = 0; (2.4) (cid:3) (cid:3) (cid:3) (cid:3) connectingq andq ,i.e. (cid:0) + q (y) q fory + ; q (y) q fory : (2.5) (cid:3) + (cid:3) (cid:0) ! ! 1 ! !(cid:0)1 Weemphasizethatweallowforthepossibilityofpulses,q = q . + (cid:0) Thesecondassumptioninthissectionisconcernedwithstabilityoftheabovetravellingwavesolution. Therefore,considerthelinearizedoperator :H2(R;RN) L2(R;RN) L2(R;RN); u @ u D(cid:0)1(c @ u+f0(q ())u): (2.6) (cid:3) yy (cid:3) y (cid:3) L (cid:26) ! 7!(cid:0) (cid:0) (cid:1) Notice that under suitable decay assumptions, q0 belongs to the kernel of due to the translation (cid:3) L(cid:3) invarianceiny. Hypothesis2.2(Zero-Stability) Weassumethat (cid:21)idisinvertibleforall(cid:21) < 0andthat(cid:21) = 0 (cid:3) L (cid:0) isanisolatedeigenvaluewithalgebraicmultiplicityone. Although this might not seem obvious, Hypothesis 2.2 is intimately related to stability properties of thetravellingwaveq (). Considerthelinearizedoperator (cid:3) (cid:1) : H2(R2;RN) L2(R2;RN) L2(R2;RN); u D(cid:1) u+c @ u+f0(q ())u; (2.7) (cid:3) x;y (cid:3) y (cid:3) M (cid:26) ! 7! (cid:1) anditsFourierconjugates : H2(R;RN) L2(R;RN) L2(R;RN); u D(@ k2)u+c @ u+f0(q ())u: (2.8) k yy (cid:3) y (cid:3) M (cid:26) ! 7! (cid:0) (cid:1) Hypothesis2.3(Transverseasymptoticstability) Assumethatthetravellingwaveisasymptotically stable in one space dimension, that is, the essential spectrum of is strictly contained in the left 0 M complex halfplaneandzero istheonlyeigenvalue in theclosedrighthalfplane, withalgebraic mul- tiplicityone. Furthermore, assumethatthespectra of ,fork = 0arestrictlycontainedintheleft k M 6 halfplaneandthattheuniqueeigenvalue (cid:21) (k),k 0with(cid:21) (0) = (cid:21)0 (0) = 0satisfies(cid:21)00(0) < 0; d (cid:24) d d d seeFigure2.1. 5 (cid:21)2C (cid:21)2R k Figure2.1: Totheleftthespectrumof andtotherightthecriticalspectraofthe parameterized 0 k M M byk. Remark2.4 (i) This hypothesis and, in particular, the quadratic tangency of the dispersion relation(cid:21)00(0) < 0impliesasymptoticstabilityofthetravellingwavewithrespecttoperturbations d thataresufficientlylocalizedinthetransverse, x-direction[28,35,36]. (ii) Forequaldiffusionconstants,D = d id,thesecondpartofHypothesis2.3isaconsequenceof 1 the first part, on the spectrum of . However, this is not always the case when the diffusion 0 M constantsarenotequal;seealsoSection4. Lemma2.5 Hypothesis2.3impliesthatHypothesis2.2onstabilityholds. Proof. The proof is similar to [57, Lemma 2.11, Remark 2.12]. Since is invertible for all k M k = 0, we immediately conclude, upon multiplying by the inverse of the diffusion matrix D, that 6 +k2 isinvertibleforallk = 0. Similarly,thekernelsof and coincide. Denotebyu(k)the (cid:3) 0 (cid:3) L 6 M L (normalized)uniquefamilyofeigenvectorstotheeigenvalue (cid:21) (k)of d k M D(@ k2)u(k)+c @ u(k)+f0(q ())u(k) = (cid:21) (k)u(k): yy (cid:3) y (cid:3) d (cid:0) (cid:1) Differentiatingthisequalitytwice,andevaluatingink = 0withu(0) = q0,u0(0) =0,(cid:21) (0) = 0,and (cid:3) d (cid:21)0(0) = 0,wefind d D@ u00(0)+c @ u00(0)+f0(q ())u00(0) = (cid:21)00(0)q0 +2Dq0; (2.9) yy (cid:3) y (cid:3) (cid:1) d (cid:3) (cid:3) where(cid:21)00(0)istheuniqueLagrangemultipliersuchthat(2.9)possessesanontrivialsolution. Avector d u~inthegeneralizedkernelof solves (cid:3) L @ u~+D(cid:0)1(c @ u~+f0(q ())u~) = q0: (2.10) yy (cid:3) y (cid:3) (cid:1) (cid:3) Upon comparing (2.10) and (2.9), where (cid:21)00(0) = 0, we conclude that (2.10) does not possess a d 6 solutionand(cid:21)= 0isalgebraicallysimpleasstatedinHypothesis2.2. 6 Remark2.6 (i) The converse implicationgenerally fails. For example, after a temporal Hopf bi- furcation ofaone-dimensionalpropagating front,causedbypointspectrumcrossing theimag- inary axis, Hypothesis 2.2 would still hold, whereas Hypothesis 2.3 would fail. However, Hy- pothesis 2.2 doesimply the quadratic tangency of the dispersion relation(cid:21)00(0) < 0. This fact d willbeusedlaterinTheorem1. (ii) It is straightforward to verify that in the case of a fourth order tangency (cid:21) (k) (cid:21)(d4)(0)k4, d (cid:24) (cid:0) 4! (cid:21) = 0 is algebraically double as an eigenvalue of ; see [57, Remark 2.12]. This fact will (cid:3) L becomerelevantinSection4.2. Thefollowingclassificationofcornerdefectsismuchinspiredbytheclassificationofdefectsinoscil- latorymedia;see[54]. Definition2.7(Cornerdefects) Asolutionu(x;y)ofthetravelling-waveproblemiscalledanalmost planartravelling-wave solution(cid:14)-closetoq ,ifuisoftheform (cid:3) u(x;y) = q (y+(cid:24)(x))+w(x;y); (2.11) (cid:3) with(cid:24) C2(R)and 2 supj(cid:24)0(x)j < (cid:14); supjw(x;(cid:1))jH1(R;RN) < (cid:14); jc(cid:0)c(cid:3)j < (cid:14): (2.12) x x Wesayuistrivialifuisarotatedplanarinterfaceu = q ((cos#)x+(sin#)y),forsome# R. (cid:3) 2 Wesayuisacornerdefectifitisoftheform(2.11)and(cid:24)0(x) (cid:17) R,asx . Wedistinguish (cid:6) ! 2 !(cid:6)1 cornerdefectsaccordingtothefollowinglist: if(cid:17) < 0 <(cid:17) wesaythatthecornerdefectisaninteriorcorner; + (cid:0) (cid:15) if(cid:17) = (cid:17) = 0wesaythatthecornerdefectisahole; + (cid:0) (cid:15) if(cid:17) = (cid:17) = 0wesaythatthecornerdefectisastep; + (cid:0) (cid:15) 6 if(cid:17) > 0 >(cid:17) wesaythatthecornerdefectisanexteriorcorner; + (cid:0) (cid:15) seeFigure2.2. Remark2.8 If we think of individual points on the interface evolving with the normal speed c , we (cid:3) noticethatinterfaceisconsumedonbothsidesof interiorcorners and interfaceis generated at exte- riorcorners. Atastep,interfaceisconsumedononesideandgeneratedontheotherside,whereasat ahole,interfacialpointsneitherenternorleavethedefect. Inadditiontothegeometriccharacteriza- tion in Definition 2.7 we therefore suggest the following dynamic characterization, building a closer analogy to[54]: 7 (a) (b) (c) (d) # # # Figure 2.2: Schematic plot of the four different types of corner defects, interior corner (a), hole (b), step (c), and exteriorcorner(d). The middlearrows indicate thespeed of the defect, whereasthe left and right (smaller) arrows indicate the normal speed of propagation of the interface. The angle # is givenbytan#= (cid:17) (cid:0) interiorcorner sink ! step transmissiondefect ! hole contactdefect ! exteriorcorner source ! Thinkingintermsofinterfacialenergy,energyislostatasink,generatedatasource,transmittedata transmissiondefectandpreservedatacontactdefect. Thereisyetanothermotivationforthisterminology. Wewilllaterseethatalldefectspossessanatural characterization as heteroclinic and homoclinic orbits. In this terminology, they coincide with the localizeddefectsinspatiallyone-dimensionaloscillatorymedia,whichhavebeenpreviouslyclassified in the terminology of sink, contact, transmission, and source in [54]. To make the analogy clearer, transportofpointsontheinterfacehastobephrasedintermsofgroupvelocities. Sincethedispersion relation (cid:21) (k) at the interfaceis symmetric in k, the group velocity (cid:21)0(0) in the tangential direction d d vanishes and transport is generated solely by geometry. In oscillatory media, transport is induced by group velocities of wave trains and described at small amplitudes by a viscous Burgers equation. Defects are then classified in [54] according to the relative slope of characteristics with respect to the speed of the defect. The correspondence actually goes much further, since spectra of linearized operators atcornerdefectsandatdefectsinoscillatorymediaqualitativelyagree. Remark2.9 Throughout the paper, we consider propagation in a direction normal to the primary interface. Most of the defects we find are actually symmetric with respect to x x. A slightly 7! (cid:0) more general characterization of almost planar interfaces would allow for a propagation in the x- direction, aswell. Thiswouldcontribute a termc @ uin theequation,with anadditionalparameter x x c . Equivalently, we can rotate the plane by an angle ’ such that the speed of propagation points x againinthey-direction. Wearenowreadytostateourmainresultofthissection. 8 c(cid:3) c(cid:3) y y c(cid:3) c(cid:3) c c y y x x Figure2.3: One-dimensionalpulsesandfronts(top)generatetwo-dimensionalcornerdefectsin line- andininvasionpatterns,respectively (bottom). Theorem1 Assumeexistenceandzero-stabilityofaplanartravellingwaveq (),Hypotheses2.1and (cid:3) (cid:1) 2.2. Thenthereis(cid:14) > 0suchthatforeachcwith c c < (cid:14), c > c ,thereexistsaninteriorcorner (cid:3) (cid:3) j (cid:0) j defect. Thedefectisuniqueintheclassofnontrivialalmostplanarcornerdefectsuptotranslationin x. Moreover,itisinvariantunderreflectionx x xforanappropriatex andistoleadingorder 0 0 7! (cid:0) giventhrough q(x;y;c) = q (y+(cid:24)(x))+O(c c ); (cid:3) (cid:3) j (cid:0) j 2(c c ) 2c (c c ) (cid:24)0(x) = (cid:0) (cid:3) tanh((cid:12)x)+O(c c e(cid:0)2j(cid:12)xj); (cid:12) = (cid:3) (cid:0) (cid:3) <0: (2.13) s c(cid:3) j (cid:0) (cid:3)j p (cid:21)0d0(0) Forc c ,therearenonontrivialalmostplanarcornerdefects. (cid:3) (cid:20) NotethatthetheoremdoesnotrequireHypothesis2.3;seeRemark2.6(i). We give a sketch of these interior corner defects in Figure 2.3; see also Figure 2.4. Note that the speedofpropagationoftheasymptoticallyplanarinterface withangle# = arctan(cid:17) isgiven bythe (cid:6) simplegeometric conditionc = c =cos#; see also [3] and the references therein. The existence and (cid:3) nonexistence part in Theorem 1 coincide withthe resultsin [3] for scalar reaction-diffusion systems. Weemphasize,however,thattheresultstherecoverlargeangles#,aswell. WeoutlinefirsttheproofofTheorem1. Werewritethetravelling-wave equation(2.2)asadynamical systeminthedirectionx,perpendiculartothedirectionofpropagation.Wethenparameterizesolutions similarly to (2.11) exploiting the translation invariance y y + (cid:24) of (2.2). The main step then is 7! a dynamic center-manifold reduction to a two-dimensional center-manifold diffeomorphic to a strip ((cid:24);(cid:17)) R ( (cid:14);(cid:14))andflowgivenby 2 (cid:2) (cid:0) 2 c (cid:24)0 = (cid:17)+O(c c (cid:17) + (cid:17) 3); (cid:17)0 = (c c (cid:3)(cid:17)2)+O(c c 2 + (cid:17) 4); (2.14) j (cid:0) (cid:3)jj j j j (cid:21)00(0) (cid:0) (cid:3) (cid:0) 2 j (cid:0) (cid:3)j j j d 9 c c(cid:3) # Figure 2.4: Interface plotted at time t = 0 and time t = 1. The normal speed of propagationc and (cid:3) thespeedofthedefectformarectangulartrianglewithangle#. Inparticular,therighthandsidesoftheseequationsdonotdependon(cid:24). Moreover,theycommutewith thereversibilitysymmetryx x,(cid:24) (cid:24),(cid:17) (cid:17). Boundedsolutionsarethewell-knownBurgers 7! (cid:0) 7! 7! (cid:0) shocks. Notethatthereducedequationreflectssteady-stateprofilesoftheviscousBurgers’equation (cid:17) = (cid:21)00(0)(cid:17) c ((cid:17)2) : (2.15) t (cid:0) d xx (cid:0) (cid:3) x This equation has been derived formally as a modulation equation for planar interfaces, previously, but we are not aware of any rigorous results in this direction; see however [11] for a justification in a different context. The Rankine-Hugoniotconditionfor the jump of the shock reducesto the purely geometric condition that the speed of the corner is determined by the condition that the orthogonal projectionof thevelocityofthecorneronthe(exterior)normaloftheinterfaces onbothsidesequals c ;seeFigure2.4. (cid:3) Proof. [ofTheorem1]Werewritethetravelling-wave equation(2.2)asafirst-ordersysteminx u = v x v = @ u D(cid:0)1(c@ u+f(u)); (2.16) x yy y (cid:0) (cid:0) ontheHilbertspaceY = (H1 L2)(R;RN),or,inshortnotation, (cid:2) u = (c)u+ (u); (2.17) x A F whereu= (u;v)T, 0 id 0 (c) = ; (u) = : (2.18) A @yy D(cid:0)1c@y 0 ! F D(cid:0)1f(u) ! (cid:0) (cid:0) (cid:0) Notethat (c)isclosedonY withdomainofdefinitionY1 = (H2 H1)(R;RN). Thenonlinearity A (cid:2) issmoothasamapfromY toY. F 10

Corner defects in almost planar interface propagation Défauts faibles en propagation d'interfaces

Similar Corner defects in almost planar interface propagation Défauts faibles en propagation d'interfaces