Decay of Solutions to a Linear Viscous Asymptotic Model for Water Waves

Chin. Ann. Math.31B(6), 2010, 841-854Chinese Annals ofDOI: 10.1007/s1 1401-010-0615-2Mathematics, Series BG The Editorial Ofice of CAM andSpringer-Verlag Berlin Heidelberg 2010Decay of Solutions to a Linear Viscous AsymptoticModel for Water Waves***Olivier GOUBET* Gullaume WARNAULT**(Dedicated to Professor Roger Temam on the Occasion of his 70th Birthday)Abstract The authors discu8s a linear viscous asymptotic model for water waves and thedecay rate of solutions towards the equilibrium.Keywords Water waves, Viscous asymptotic models, Nonlocal operators,Long-time asymptotics2000 MR Subject Classfication 35Q35, 35Q53, 76B151 IntroductionModeling the effect of viscosity on the propagation of long waves in shallow water has re~ceived much renewed interest in the last decade. Without viscosity effects, it is now a standardprocedure to derive asymptotic models for one-way wave propagation. The most encounteredmodels in the literature are Boussinesq systems and Korteweg de Vries equation, whose deriva-tion was first performed in the 19th century. More general models for two way waves wereintroduced in [2]. The derivation starts from Euler equation and proceeds through fine asymp-totic analysis to obtain an equation for the horizontal velocity at the top of the fuid, or asystem of equations for this velocity and the height of the wave. Taking into account viscosityeffects is a challenging issue, since we have to deal with Navier-Stokes equations that providesthe flow with a viscous layer at the bottom of the fuid. Some finer asymptotic analysis has tobe performed.The pioneering work for this issue is due to T. Kakutani and K. Matsuuchi [9] who havepointed out that the asymptotic model for viscous water waves is a dispersive PDE supple-mented with a difusion and a nonlocal pseudo-differential operator that features both a dis-persive and a difusive effect. For the physics, this means that the viscous layer in the fuidprovides difusion (this was expected), but also dispersion. Independently, P. Liu and T. Orfila[11], and D. Dutykh and F. Dias [7], have recently derived viscous asymptotical models thatfeature also nonlocal operators, and that possess the same dispersive properties of those in [9)],but with diferent mathematical properties. These models are Boussinesq type systems withviscous terms. A one way reduction of these models was addressed in [6].Manuscript received May 31, 2010. Published online Octob*LAMFA UMR 6140 CNRS UPJV, 33 rue Saint-Leu, Amien中国煤化工E-mail: olivier.goubet@u-picardie.fr**LMPT UMR CNRS 6083, Universite Francois Rabelais ParTYHCNMH Gunce.E-mail: gillaume. warnault@u-picardie.fr***Project supported by the CNRS, research program "waterwaves".8420. Gorbet and G. WarmaultIn a previous work [4], we were concerned with computing both theoretically and numericallythe decay rate of solutions to a water wave model with a nonlocal viscous dispersive term. Thismodel reads as followsut+ux+βuzx+ds + Wur = Yuza,(1.1)where u is the horizontal velocity of the fAuid. This equation requires some comments: theusual difusion is - YUuxx, while Burx is the geometric dispersion and兴Jo vds stands forthe nonlocal diffusive-dispersive term. Here β and v,γ≥0 are parameters dedicated to balanceor unbalance the effects of viscosity and dispersion against nonlinear effects. The dispersionanalysis for the linear part of this equation was also addressed in [4]. In the same work,assuming that the effect of the geometric dispersion is less important that viscosity efects (i.e,considering β = 0 in the equation), we were able to prove that for small initial data, the decayrates of solutions compare to those of solutions to KdV-Burgers equations.Computing the decay rate for solutions to dispersive difusive equations has a long historytoo. The pioneering work is due to C. Amick, J. Bona and M. Schonbek [1] where the authorshandle the decay rate of solutions of KdV-Burgers solutions for any initial data, i.e, withoutassuming any smallness assumption on the initial data. For a large review of methods forcomputing the decay rates for solutions to dissipative evolution PDEs, we refer to [8]. Amongrecent works concerned with dissipative Boussinesq systems, we mention [3], {5] and [12]. Thislist is by no mean exhaustive.In the present article, we are interested in computing theoretically the decay rate for so-lutions to an asymptotic linear viscous model for water waves similar to (1.1), but withoutthe main difusive term, i.e., considering γ = 0. In the linear case, our equation reads then(normalizing the other constants)1u +-=(.4()-d8+Uxxx +ux= 0,(1.2)(t-8)室supplemented with initial data 1uo in L' (R). For this purpose, we follow the method advocatedin [4]. Our result compares for large times with the corresponding result for the heat equationand state as follows.Theorem 1.1 There exrists a constant C such that the following estimate holds true:min(t年,境)(L(x) + tl()x)≤ClullL().(1.3)Actually, the method is to compute a representation of the kernel K(t, x) defined as u(t, ) =K(t,.)* uo if and only if u solves (1.2) with initial data uo. This representation is indeed anoscillatory integral. As we will see in the sequel, the estimates on this kernel are much moreinvolved than those for the kernel corresponding to the heat equation or the nonlocal viscousequation as in [4]; it turns out that since we do not have the diffusion term -Uxr in the equation,the diffusion is weaker since high frequencies are not exponentially damped. We are concernedhere with a viscosity that vanishes for high frequencipoint anddiscuss the drawbacks of this fact in the sequel. It is中国煤化工at, due to thepresence of the non-local term, we do not have anyC N M H Ghe trajectoriesand that the famous Schonbek 's splitting method does not apply.Linear Viscous Asymptotic Model for Water Waves843We do believe that these inconveniences are only due to mathematical technicalities. For thephysics, the validity of the model holds true for long waves, i.e, for initial data whose energy isconcentrated for small frequencies. Hence only small frequencies monitor the flow of solutionsand the drawback of the vanishing viscosity for high eddics do not account. Thesc topics willbe discussed, both from the theoretical and from the numerical point of view in a forthcomingwork.This article is organized as follows. In Section 2, we compute a representation for the kernelK(t,x) as an oscillatory integral. The idea is to solve the equation in Laplace-Fourier variablesand then to come back to the (t, x) variables. For this purpose, we need some estimates onsolutions to a polynomial equation whose proof will appear in an annex in the last section.In Section 3, we provide some decay estimates on the kernel using van der Corput lemma;statement and short proofs of this well-known result are postponed to the annex in Section 5.Hence we complete the proof of Theorem 1.1 and we provide the reader with a short conclusion.We cormplete this introduction by outlining some notations. Consider two numerical func-tions h(t, x), g(t, x) which take values in R. Hence we write h(t,x) S g(t,x) if there exists anumerical constant c that does not depend on t and x such that h(t, x)≤cg(t, x). We also writeg(t,x)≈h(t,x) if and only if h(t,x) S g(t,x) and g(t,x) S h(t, x). For complex valued func-tions, we write g(t,x)元h(t, x) if their moduli compare, i.e, lg(t, x)|元|h(t, x)|. The Fouriertransform of a function u in L'(R) is defined as勾() = fx u(x) exp( - ixξ)dx and the Laplacetransform of a bounded function U is defined, for any complex number T such that Rer> 0asv(r)= st∞o(t)exp(- -tr)dt. We set (x) = V1+正. For any complex number T which doesnot belong to R- , we define vF as v斤= |r位exp( jarg r), where argT belongs to (-π π); thisfunction T→F is analytic.2 Computing K(t, x) as an Oscillatory Integral2.1 Fourier-Laplace transformWe follow here [4]. Introduce the Fourier. Laplace transform of a function u as(T,6)=/f*(u(t,z)exp(- ixε - tr)dx )dt.(2.1)We apply the Fourier Laplace transform to (1.2) and obtain(+√F+i- ()(,)=(1+))u0(),(2.2)where钻(ξ) is the Fourier transform of the initial data u(0) = 40. Solving for i, we have(r,) = K(t,)2o()with(,()= (1+-))F+V中国煤化工(2.3)At this stage, to invert the Fourier transform in theTYHCNMHGthefollowinglemma.844O. Goubet and C. WarmnaultLemma 2.1 LetS= {z∈C such that Rez> 0}. Forz∈几, the equation x3+X = zadmits three branches of solutions a(z2), ar(z), a2(z), which vary analytically with respect to z,such that Rea(z)> 0, Rear(z) <0fori= 1,2, Ima2(z)<0 < lmar(z).Proof For z = 2, the equation admits a unique positive solution X = 1. The derivative3X2+ 1 does not vanish on the set x3 + X = z. Hence the equation owns three differentsolutions that we can follow continuously with the Implicit Function Theorem. On the otherhand, the equation x3 + X = z cannot have a solution that belongs either to the imaginaryaxis or the axis {x < 0}. Then the result is proved.Remark 2.1 Actually this result is true for z≠0 belonging to{z∈C;Rez> 0}∪{z∈C;|z|≤} which contains 0 in its interior.More results about the behavior of these solutions will appear in Annex (see Section 5). Wenow proceed to the inverse Fourier transform. We set(后(r,)= F(r.)= (+1-ε+eLemma 2.2 Let a(r) be the unique solution with a positive real part and a1(r), az(r) thesolutions with a negative real part ofX3+ X =r+√F. Thene-a(r)=P(r,x)=1+32万，Vx∈R+,(2.4)二e~a()x_Vx∈R~ .(2.5)所四)=一Ei+3历'Proof For any positive x, we haveieixξ(r,x)=;2.,低+VA-ξ+ed(2.6)Note that if T∈S, then T+√T∈2. We apply Lemma 2.1 withξ = iX, then ia(r) is theunique solution of i(r +√F)-ξ+ζ3 = 0 whose imaginary part is positive.Let r be a lace who is constituted by the segment [-R, R] and the semicircle z = Rei0 whereθ∈[0, rx]. We apply the Residue Theorem to f(ξ) = F(r,ξ) and we obtainieixε .ReifeirRele,r(f()e*df =.Jm.; i(T +VT)-ξ+d-1。++ v7)-ielE= 2Re(i(y)0+vF)-ξ+e)Sincex>0, we have lerRe°≤1, hence when R→∞, .RelegixRe-dA， i(r+vh)-Re09+F中国煤化工It fllows (2.4) by computing the residue. We use th{TYHC N M H G..5) choosing asuitable lace.Linear Viscous Asymptotic Model for Water Waues845We now proceed to the final estimate forx > 0. By the inverse Laplace transform, we have,for any positive ε,1K(t,x)=;2i元/'t. (1+-V7)1+ 3a2(介)dr.(2.7)Since the singularity is integrable in 0, we pass to the limitε→0 and obtaineitseits-a(is)xK(,2)=2]0 (1+ )i+sa2ids.(2.8)We cut (2.8) in four parts:∞1 eits-a(ia)x1 p0 1 eit8-a(is)xKt(t,x)=元√is1 + 3a2(is)ls， K(t,x)= ;πJ√is1+ 3a2(is)ds,K&(t,x)= ;1 f+∞eit8-a(is)x-ds，Kz(t,x)= ;1 p0 eits-a(i8)xAs.2π Jo1 + 3a2(is)2πJ_。1 + 3a2(is)Remark 2.2 Analogous formulae hold true for x < 0, substituting (2.5) to (2.4) in the .computations; in this case we have eight integrals to handle.3 Proof of Theorem 1.13.1 Strategy for the proofLet u8 describe our strategy on the simple example of the heat equation whose kernelis Khea(t,x) 心方exp(-示). It is an exercise to prove that |Kheat(t,x)| S min(t- ，,x 1).Therefore, |Kheat(t, .)|L∞(R)St-专andKee(t, )i2xa)≤。min(t-1,x- 2)dx ~+ (+∞=t-1.Here some extra dificulty occurs. To begin with, due to the drift Ux + Urrx the kernel isnot symmetric with respect tox= 0; we expect, as for the Airy equation, the kernel to havebetter decay properties for positive x. .On the other hand, we are interested in the long time behavior of solutions. By a rule ofthumbs, we know that the large time behavior of the kernel K relies on the behavior of theoscillatory integral for small 8. Hence we surmise that the kernel K1 will monitor the decay ratefor solutions. It transpires from the computations below that the other part of the kernel K2is more dificult to handle; we do believe that this drawback is only due to the mathematicalsetting and is not relevant for the physics.Let us go a lttle further. Let a be any solution to x3+ X = is +√is. Then the modulusof the integrand in the very definition of K2(t,x) is exptan. As stated in Annex, it turnsout that for large 8, |Rea1|≈|Rea|元g while |Re中国煤化工case, we havean exponential decay which smashes down the high: case, we havea vanishing viscosity at the infinity. This explains wh:YHCNMHGwilloccurforx<0.846O. Goubet and G. WarmnaultRemark 3.1 For the sake of comparison, it is worth to point out that the heat kernel readsas the sum of oscillatory integral (for positive x) asexp(-号x)Ktea(t,x)心exp--ds.J(x-影))√83.2 Estimates on K1(t, ax)We begin with the term which monitors the decay rate for large times. We now state andprove a result that asserts a heat kernel decay rate for this term.Theorem 3.1 Consider K1 defined as above. For anyt> 0, for anyx in R the followinginequality holds true:|K(t,x)| S min玉'网)Proof We just focus on KT (t, x), the other integral being similar. We perform the changeof variable s H s2 in the integral and we bound Jt∞ eis^t-Bxs8 where β is any root ofx3+X = is2 +s√i. For this purpose, we apply van der Corput Lemma 5.5 (see Annex for theprecise staterment) with the phase ψ(s)= s2t and A(8)=e-3xtgr, withb=0 andd=+oo.We then have, observing ψ"(0) = 2t,1+ 352|St(A|x∞ + Il).(3.1)On one hand |A(s)| S-IRe别)2 S 1. On the other hand,1+392T|'|x|61|11l3||'|(x)|A'(8)1≤(I+39+1+ 3B2)xp(-IRe |])≤+ 3891 x(-1ReBlpx).(3.2)We now divide the computation according to the case 8< 1 ors > 1. For small frequencies,we then have, due to Remark 5.1 below|β'|(x)|1+ 3β32|; exp(- Relx)dsS$ [" (&(x)>exp(- clx)dsS1;(3.3)indeed for |x|≤1, we just use exp(-cs|x)≤1, while for |x| ≥1, we perform the change ofvariable y = cxs. For high frequencies, and for the worst case that isx < 0 and for β = a2(is2)the root that has a vanishing real part for large 8, we havee-RePx_|B'( 1. For the former case,d:1es2t-Bz_exp( - cs|x)ds心(3.5)1+332a|sj。|x['For the latter中国煤化工”1+352|CNMHG(3.6)The proof of the theorem is then completed.Linear Viscous Asymptotic Model for Water Waves8473.3 Estimates on K2(t,x),x < 0Here we encounter the main mathematical dificulties, for small t. We state and prove thefollowing result.Theorem 3.2 Consider K2 defined as above. For anyt> 0, for anyx < 0 the followinginequality holds true:|x(,x)l≤min(t- ,国-1)+ min(xI-1,t,tt1-+).Proof It is worth to point out that K2 is actually the sum of many integrals, some witha1, the other ones with a2. Since the first case is similar to the case x > 0, and then sincethese integrals can be bounded as in Theorem 3.3 below, we skip the details for the sake ofconciseness. We focus here on the integrals with Q2, which have a vanishing viscosity at theinfinity. Set a = a2(is) for the sake of convenience. By symmetry we only consider 8> 0 thatis K2 (t,x).To begin with, we divide the integral into two parts, according to the case8< 1or8> 1. Theformer stands for small frequencies, the latter for high ones. Set K古(t,x) = L(t,x) + H(t,x), .where(3.7)(.)=云61+3azds.We first have, for small frequencies,Lemma 3.1|[(t,x) S min(,1x[-2,t-1)≤min(t-,|x1-).(3.8)Proof On one hand, due to Lemma 5.1 there exists a constant c that is independent of x .and t such that[C(,21S f'; expa-cD≤min (1司)(3.9)ito-asOn the other hand, using Lemma 5.4 with ψ(8)= st and A(s) = 1+3a, we have[(,x)I≤(AIL∞+ I1'ILr).(3.10)Then using once again Lemma 5.1, we obtain |A(8)I≤1 and6|aa'|)ds1+ 3a2≤)ds≤1;(3.11)actually for |x≤1 we have used exp(- cs年|x|)(粤)≤8-3 while for |x|≥1 we have performedthe change of variable y = c|x|s气.We gather the previous computations in the follo中国煤化工min(1,主,永).The proof of the lemma is then completed by interpold:YHCNMHGFor high frequencies, we state and prove the following lemma.8480. Goubet and G. WaraultLemma 3.2 .|H(, x)|≤min(xI-',t-I,t-,tx|-). .(3.12)Proof The proof consists in establishing precise estimates on different regions of the quar-ter planet> 0,x< 0.First step t≥1or tl]x|≥1.Consider S≥1 to be precise in the sequel. On one hand, due to Lemma 5.3,ex(-(qxls-) s min (S$s -1),分);(3.13)actually exp(-c|x|s-刮) ≤1 provides us with the upper bound St- 1, while the estimatey'exp(-y)S1 for y= c|x|s-t provides us with条On the other hand, using Lemma 5.4 with ψ(s)= st and A(s)= +3at, we haveeits-ax(3.14)Using Lemma 5.3, we first have|A()I≤xp(-(lx]1-1)o-8≤min(S-t,x-).We also infer from Lemma 5.3|a'x||aa'|)exp(- clReaxI)≤(二+=)ex(-c|x]s-3). (3.15)|"()( (1+3a1 ++ ap)On one hand, performing the change of variable y = c|x|s-言,1pt∞x2Js s8°! exp(-c|x|s~ #)ds下|x\Js. exp(- -c|x|s~ B)ds1 pluls-t(3.16). |x|JoOn the other hand,、dp+∞exp(-(lx|s-+)exp(-(x1l-)g号|x|S Jsg。8.(exp(-c|x|s~ ))ds|x|S$ Js.1- exp(-c|x|S-台)ss-i.(3.17)|x|StUsing the inequality y3 exp( -2y)≤exp(-y), we also haveds_sexp(-c|x|8-8)3号≤x4。exp( _ Clr1o-t\ |xlds中国煤化工=x-4a,(IMYHCNMHG4.(3.18)JsLinear Viscous Asymptotic Model for Water Waves849We gather the computations above in the formula: for any S≥1,)+min((脂())(3.19)Choosing S = 1 provides us with the upper bound主min(1,南). This completes the proofof Lemma 3.2 whilet≥1 or t|x\≥1. It remains to establish the result for small t.Second step t≤|x|≤培≤1.Setting S= t≥1 in inequality (3.19), we obtain|H(t,xS五+ .(3.20)This completes the result if moreover |x| S ts. At this stage, we have proved the lemma ifeithert≥1or |x|St3.Third step t≤t培 ≤|x|.In this region, we shall use the Stationary Phase method to enforce some decay, since inthis region the difusion has not enough strength. Assume in the sequel t≤ts S |x| and then里large enough. To begin with, we use the following trickH(t,x)元-0。(e-“)x1+3a2yds.(3.21)This equality is valid in the sense of oscillatory integrals. At this stage, let us observe thata(1 +3a2)=i+兴= i(1 + e(8)) is almost constant. We then haveH(,x)=-O,(e-atit)ds+sftx e-a+itds_= H(t,x) + H2(t,x). (3.22))1+ε + x1 +εOn one hand, integrating by parts,r+∞_e')ds1|H:(,)| Sexp(-lRe a|x)p1+ε|2|x|(3.23)On the other hand, we estimate the second integral by van der Corput Lemma 5.5. We setthe phase to be 4(8) = st - Ima(s)x and A(s)= exp(-Reax). We have ψ'(s) = t- Ima'(8)xwhich converges towards t when 8 goes to the infinity and ψ(1) = t- |Im a(1)|x|; we maywork on a region such that ψ(1) < 0 (i.e,里> imayT), the complementary region beinghandled by the computations above. Moreover, we may assume without loss of generality thatψ"(8) = -Imol"x does not vanish for s≥1 (if this is not true, we know that there exists θlarge enough such that Imal' does not vanish on (0,+∞) and we cut K$ in L+ H accordinglyto the case 8< θ and 8 > 0). Therefore, with this assumption, there exists a unique 8* suchthat ψ(8+) = 0 and moreover since -Imd'(s)s-3, we haves;元().We now write H2(t,x)= (ST + fto)(ei4 A)ds and compute each integral by using Lemma5.5. We then have中国煤化工|/,*。er Ads|s|AIlL∞|4"($.).MHCNMHG(3.24)850O. Goubet and G. WarnaultOn one hand, ("(.)|心(s. )x心On the other hand, |A(8)| S 1 and|x||A'(3)| S (Rea'x| + |')exp(-IReax)S(图+寺)exp(-c|x|s8- t)。(3.25)Then |Il|lrI S 1. We gather the previous computations to writet |x|操1|H(t,x)\≤口+(3.26)xt丽语~(xI)'sincetSts≤||. Then the proof of the lemma is completed.The proof of Theorem 3.2 follows by gathering the results of Lemmas 3.1 and 3.2.3.4 Estimates on K2(t,x), x > 0We state and prove the following result.Theorem 3.3 Consider K2 defined as above. For anyt> 0, for anyx > 0 the followinginequalities hold true:|K2(t,x)| S min活国)一+ min(i)≤min(Proof We focus on K2 , the other integral being similar. As in the casex < 0 we divideK$ = L+ H in two parts, accordingly tos< 1 ors> 1. It is worth to point out that L(t, x)satifies the same estimate than in Lemma 3.1. For the high frequency part, we set a = a(is)the root with positive real part of X3+X = is+√is. Set S≥1. On one hand, due to Lemma5.2,dsexp(-cx8当s min(st-1.);(3.27)indeed, the estimate exp(-cxs+) ≤1 provides us with the upper bound St - 1, while thechange of variable y= cxs李provides us with x-1. On the other hand, using Lemma 5.4 withψ(s)= st and A(s) = exprar), we haveeits-az(3.28)Using once again Lemma 5.2, we have\a'llx||aa'|e-xRe °ds.1\1+ 3a21|1 + 3a2|2S,/.* (号+1)e-czs dsp+∞exp(-cxss).≤min(武);(3.29)actually, the estimate exp(-cxs言) s 1 leads to the, npmr hnund S善while the change ofvariabley= cxs李leads to x -2. We also have中国煤化工0HCNMHG|A(8)| S exp(- cxs$)g-号≤hua(s ,1小(3.30)Linear Viscous Asymptotic Model for Water Waves851We gather the computations above as follows|H(t,刘)≤min (s$ - 1,)+ min(S-号,x-2).(3.31)Choosing S = 1 provides us with the upper bound t, while choosing for t≤1, S= t-1provides the bound t-s. For S = +∞, we have the bound 1 that completes the proof of thetheorem.3.5 Completing the proof of Theorem 1.1We begin with the L∞etimates on K(t, ):xH K(t, x). The upper bound |K()ILx≤t-主is a straightforward consequence of Theorems 3.1- 3.3. For the L2 estimate, it is worth topoint out that the upper bound min( t，1) provides us with a upper bound in L经that behavesas t-t, as for the heat equation. It remains to consider the high frequencies part of K2, forx< 0andt≤1; we focus on H(t, x) which satisfies the estimates in Lemma 3.2. We then have|H(, .)归z≤。min(旨邮丽)lxdx(3.32) .Therefore |K(t, .)Iz≤亦+木We conclude using cassial results about convolution: forany p≥1, it is well-known that|K(t, .)* ull≤|(IIol(3.33)This completes the proof of Theorem 1.1.4 ConclusionIn this article, we have addressed the issue of the decay rate of solutions to a linear viscousasymptotic model for water waves. In a previous work, where the equation features difusionterm as -Uxx， we have proved that the decay of solutions compare to the decay of solutionsto the heat equation. In this article, we have addressed the mathematical challenge of dealingwithout this diffusion term. Actually, the non-local term in the equation provides us with amuch weaker dissipation. Despite this fact, we were able to prove that solutions to the linearequation converges towards the equilibrium when time goes to the infinity, and with a goodrate of convergence for large time. Indeed, it turns out that the decay of solutions fortS 1 andx < 0 is much more weaker than for the solutions to the heat equation. At this stage, we donot know how to handle the full nonlinear equation. This will be the purpose of a forthcomingwork.5 Annex: Preliminary Material5.1 Behavior of the solutions to x3 + X = is .中国煤化工Here 8 isa real umber and√is = V8[ exp(io;fYHCN M H G only disecusthe case 8 > 0. As stated before, the equation possesses three branches of solutions that vary0. Goubet and G. Warnaultanalytically with respect to s. Singularity will occur either for 8=0or 8= +∞. Let a(is),a1(is), a2(is) as defined in Lemma 2.1.For small s, we can easily prove that a(is) ~√is, a1(i) ~i, a2(is) ~ -i. Actually, thereal part of each root vanishes and the decay rate of this real part monitors the decay rate ofthe kernel for large x. For 8 going to the infinity, we have that a(is) ~eigs, a1(is)~ei警g$,a2(is) ~ -ig3. We observe that the two first root provide an exponential damping of thehigh frequencies (with a speed that is s$; the same computation for the heat kernel providesdamping with speed si), while the real part of a2 vanishes at the infinity; this explains whythe proof of the main result is more involved than the similar proof for the heat equation.We now quantify these remarks first for small 8, then for large 8.Lemma 5.1 Leta be one of the roots of x3+X =is+√is. Then fors≤1,1≤|1 + 3a2l,(5.1)gt≤|Rea|,(5.2)|a(s)|≤1, .(5.3)|a(3)|≈g-$.(5.4)Proof The proof of the first and the third assertion are easy. For the second assertion, weuse the identity√2s= (a+ a)(a2 +σ2 - |a|P + 1) = 2Rea(4(Rea)2 - 3|al|2 +1).For the last assertion we use the identity a(1 + 3a2)=i+兴.Remark 5.1 We also need some properties of the solutions toX3+ X =is2 +8√i. Set βfor any root of this equation. Then, with exactly the same proof (and no singularity at 8= 0), .1≤|1 + 3β2|,(5.5)sS |Re BI,(5.6)|B(s)|≤1,(5.7)|B'(3)|≈1.(5.8)Let us now state the result for high frequencies, that is large 8.Lemma 5.2 Leta be the root ofX3+X = is+√is with positive real part. Then for1S8,1≤|1 + 3a2|,(5.9)s寺S |Real,(5.10)|a(s)|心s言,(5.11)|a(3)|心s-3.(5.12)中国煤化工Analogous results hold true for a1. For the other r.YHCNMHGomplicated.Lemma 5.3 Let a2 be the root ofx3+ X = i8+√is with negative real part and imaginaryLinear Viscous Asymptotic Model for Water Waves853part. Then for1Ss,1S|1 + 3a2|,(5.13)g-8心|IRea2|, .(5.14)|a2(s)\|心g李,(5.15)|a2(8)|心s-号. .(5.16)Proof The proofs are easy. We omit the proofs for the sake of conciseness; we just pointout that the proof of the second statement of this lemma comes from the identity √28 =2Rea2(4(Rea2)2 - 3|a|2 + 1).5.2 Around van der Corput lemmaWe recall some preliminary material. See [10] for instance. We recall first some Nonstation-ary Phase Lemma.Lemma 5.4 Consider a positive convex function ψ : [b,d]→R. Assume that ψ' is positiveon [b, d]. Consider a complex valued function A which is bounded and whose derivative is inL'. Thenexp()4(8)ds| s4(|AI|L∞+ IAI'I心)|ψ(6)|Proof WriterdA(8)i exp(ib(6))A(8)ds= -，:lexp(i)())中(可)A(s) nd=-exp(i()1)」。.。" epiv()( 0 + A().(司)ds.(5.17) .Therefore, using that 8。(y) is positive, we have(5.18)ψ(6)+ ψ(6)+Il-[(0)间and the proof is over.It is worth to point out that the bounds do not depend on b,d except through A. Let usnow state the van der Corput lemma.Lemma 5.5 Consider a positive strictly convex functionψ:(b,d→R. Assume ψ(b) = 0and ψ"(b) > 0. Consider a compler valued function A which is bounded and whose derivativeisinL'. Then' xp0)()≤8(141(*)|4"(6)|Proof Introduce δ > 0 to be specifed in the sequel中国煤化工|/.*exp(i()(s)ds|THCNMHG(5.19)8540. Goubet and G. WarmnaultOn the other hand, using the previous Lemma 5.4, we haveexp(i())A(s)ds{≤4(AlL∞+ I1心)(5.20)ψ(b + 6)|Using the estimate ψ(b+ 8)≥ψ"(b)8, we then have(5.21)Choosing δ= t completes the proof.Acknowledgements The authors are glad to present this article in the special issue inhonor of Roger Temam, for his great infuence in the world of applied mathematics and scientificcomputing. Merci beaucoup Roger. This work was initiated when the authors were enjoyingthe hospitality of the Department of Mathematics in Purdue University, with the support ofthe CNRS program“waterwaves".References[1] Amick, C. J, Bona, J. L. and Schonbek, M. E, Decay of solutions of some nonlinear wave equations, J.Dif. Eq8, 81, 1989, 1-49.[2] Bona, J. 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