This work deals with an example of class of ordinary differential equations which are singular near a codimension 2 set, with an homogeneous singularity of degree 0. Under some structural assumptions, we prove that for almost all initial data there exists a unique global solution and study the evolution of the Lebesgue measure of a transported set of initial data. The analysis is motivated by the Low Mach number asymptotics of compressible fluid models in the case of non isentropic flows, which involves such dynamical systems

Bresch, Didier

Desjardins, Benoit

Grenier, Emmanuel

Desjardins, Benoît

Hal - Université Grenoble Alpes

Singular ordinary differential equations homogenenous of degree 0 near a codimension 2 set

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Singular ordinary differential equations homogeneous of degree 0 near a codimension 2 set

HAL Université de Savoie

Singular ordinary differential equations homogeneous ofdegree 0 near a codimension 2 setDidier Bresch, Benoit Desjardins, Emmanuel GrenierTo cite this version:Didier Bresch, Benoit Desjardins, Emmanuel Grenier. Singular ordinary differential equationshomogeneous of degree 0 near a codimension 2 set. 2009. <hal-00408619>HAL Id: hal-00408619https://hal.archives-ouvertes.fr/hal-00408619Submitted on 31 Jul 2009HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Singular ordinary differential equationshomogeneous of degree 0 near a codimension 2 setDidier Bresch 1, Benoˆıt Desjardins 2, Emmanuel Grenier 3AbstractThis note deals with an example of class of ordinary differential equationswhich are singular near a codimension 2 set, with an homogeneous singularity,of degree 0. Under some structural assumptions, we prove that for almost allinitial data there exists a unique global solution and study the evolution of theLebesgue measure of a transported set of initial data.Keywords : Singular ODE’s, co-dimension 2 singularity, global existenceand uniquenessAMS subject classification : 47305, 37N10, 35A05, 74H35.1 IntroductionThis note deals with a simple example of a class of ordinary differentialequations which are singular on a manifold of codimension 2, the singularitybeing homogeneous of degree 0 near this singularity.Let H be an Hilbert space. Let Π denotes the orthogonal projection on agiven plane P . Let xh = Πx and let xv = x−Πx. Note that x = xh + xv. By aslight abuse of notation we shall say that x = (xh, xv), xh and xv referring tothe ”horizontal” and ”vertical” components of x. We then identify the plane Pwith C by choosing arbitrarily a basis on P . This allows to use polar coordinateson P and thus to define the modulus r and argument θ of xh, and to use thenotation xh = r exp(iθ).Let φ be a smooth function defined on H × S1 (S1 being the unit circle).Thenx˙ = φ(x,xh|xh|)(1)is a dynamical system, singular on P⊥, orthogonal of P , which is of codimension2. Moreover, this system is homogeneous of degree 0 in any direction orthogonalto P⊥.Under some assumptions on φ, we show global existence and uniquenessof solution for almost every initial data. Note that we only take care of thebehavior of t 7→ x(t) near P⊥ and by a slight abuse of language we shall saythat a solution is global if it does not reach the singularity P⊥ in finite time.1Didier.bresch@univ-savoie.fr, LAMA, UMR5127 CNRS, Universite´ de Savoie, 73376 LeBourget du lac, France2Benoit.Desjardins@mines.org, ENS Ulm, D.M.A., 45 rue d’Ulm, 75230 Paris cedex 05,France, and Mode´lisation Mesures et Applications S.A., 66 avenue des Champs Elyse´es, 75008Paris, France3egrenier@umpa.ens-lyon.fr, U.M.P.A., E´cole Normale Supe´rieure de Lyon, 46, alle´e d’Ita-lie, 69364 Lyon Cedex 07, France.12 MotivationThe toy model (1) mimics phenomena occurring in the low Mach numberlimit problem for non-isentropic flows in a periodic box for instance. Such flowsare described through a velocity field u, a density ρ implicitly given by a statelaw depending on the entropy S and the pressure p, see [4]). After some changeof variables, namely introducing q defined by p = p¯eεq with p a prescribedconstant and ε the low Mach number, the adimensionnalized system readsa(∂tq + u · ∇q) + 1εdivu = 0, (2)r(∂tu+ u · ∇u) + 1ε∇p = 0, (3)∂tS + u · ∇S = 0, (4)where a and r are two smooth functions of S and ε q. The singular limit consistsin letting the Mach number ε go to 0. In the ill prepared-data case, namelywhen the initial data do not satisfy the conditiondiv u0 = 0, ∇p0 = 0. (5)an oscillatory limit with changing eigenvalues occurs. Indeed the eigenvaluesand the spectrum of the singular operator A whereA =( 0 a−1divr−1∇ 0). (6)will depend on the solution itself. This leads to complex problem, since eigen-values may cross. The wave equation, related to the operator A, may also bewritten in this caseε∂ttψ − div (S−1∇ψ) = 0 (7)where S is the entropy quantity, see [4] for more details.In the very special case of only one spatial dimension, the limit can be bothcalculated completely and justified, see [4]. In the multi-dimensional case theformal calculation of the extra term in the limit, which once again involves thespectral decomposition of the fast operator, assumes that the spectrum of thatfast operator is simple and non-resonant, see [3] for the viscous case and [4]for the inviscid one. For certain finite-dimension al truncations of the equationsthose assumptions can be shown to be generic and to ensure convergence to thelimit equations. This has been done in the paper [4].The main difficulty in the general study is to prove, after defining appropriateinfinite dimension measures, that for almost all initial data, the limit flow doesnot meet double eigenvalues and crosses the resonance set transversally. Diffi-culties occur since the flow is singular across the resonant set which may beshown to define a codimension 2 set. This explain the necessity to study ODEsof the form (1). Our result will be used to treat oscillatory limits with changingeigenvalues and particularly the low Mach number limit in a forthcoming paper[2]. Note that another type of singular ordinary differential equation occuringin fluid mechanics has been studied recently, see for instance [1].23 Setup and main result3.1 Notations.Let us first introduce some notations. We define ψ(xh, xv, θ) as the argumentof Πφ((xh, xv), exp(iθ)), and ψ˜(xh, xv, θ) its modulus in such a way thatΠφ((xh, xv), exp(iθ))= ψ˜(xh, xv, θ) exp(iψ(xh, xv, θ)).Structural properties. We will assume that there exists an integer N0 > 0 andN0 smooth functions of xv, Θ1(xh, xv), . . ., ΘN0(xh, xv), satisfying the followingproperties :(H1) For all x = (xh, xv) ∈ H, the equation in θψ(xh, xv, θ) ∈ θ + piZ,has exactly N0 solutions Θ1(xh, xv) . . .ΘN0(xh, xv).(H2) For every j, the following sign condition holds for all x = (xh, xv) ∈ H∂θψ(xh, xv,Θj(xh, xv)) < 1.Note that this implies that the solutions Θj are all simple.(H3) ψ˜ does not vanish.Note that (H1) implies that{(τ exp(iΘj(0, xv)), xv), τ > 0}is a trajectoryforx˙h = Πφ((0, xv),xh‖xh‖). (8)This trajectory goes to the singularity or leaves it, depending on its orientation.In particular for some initial data we reach the singularity in finite time. Theflow is not defined everywhere and we can only get almost everywhere results.3.2 Main result and consequencesThe main result of this paper is :Theorem 3.1 Stable and unstable manifolds.Let us assume that (H1), (H2) and (H3) hold true. Let x0 ∈ P⊥ and let ρ > 0.There exists a finite number of manifolds Vk, of codimension 1, with boundaryΣ, such that :– For any initial data x1 in one of the manifolds Vk, the correspondingsolution of (9) reaches Σ in finite time (in the past or in the future).– For any initial data x1 in B(x0, ρ) outside all these manifold Vk, the cor-responding solution of (9) reaches the boundary of B(x0, ρ) before Σ.3With this result one can define the flow Ψ(t) of this equation, flow whichis defined everywhere except on the manifolds Vk. Let A be some set of initialdata. We now want to bound the measure of Ψ(t)A in terms of the measure ofA. This of course depends on the measure of H, which is not canonical. In orderto fix the ideas we will restrict ourselves to a finite dimensional space H, withthe Lebesgue measure µ, in order to proveTheorem 3.2 Control of the divergence.Let x0 ∈ P⊥. Let ρ > 0. There exists a constant C0(t) such that for any borelianset A of H,µ(Ψ(t)(A ∩B(x0, ρ)))≤ C0(t)µ(A ∩B(x0, ρ)),for any t > 0.The outline of the paper is as follows. First we study the simplified equationx˙ = φ( xh|xh|)(9)and then extend it by perturbation arguments to equations of the generalform (1). More precisely, in Section 4, we will study trajectories near the singu-lar set. The last section concerns measure type estimate of a transported set bythe flow which is defined everywhere except on the manifolds Vk of codimensionone.4 Study of systems of the form (9)We first prove Theorem 2.1 in the particular case of systems of the form(9). Note that (9) is a two dimensional dynamical system, which takes places inxh + P . Changing the notations, (9) is of the formy˙ = φ( y‖y‖)where y ∈ C. First we turn to polar coordinates, define y = r exp(iθ) and makethe change of time defined by dτ/dt = 1/r to getdrdτ= rψ˜(θ) cos(ψ(θ)− θ) (10)dθdτ= ψ˜(θ) sin(ψ(θ)− θ). (11)Note that (11) does not involve r, which greatly simplifies the analysis of (9).Note also that ψ˜ is always positive. Therefore the dynamics of (11) is given byAssumption (H1). There exists N0 fixed points Θj(xv) which are stable provided(ψ′(Θj)− 1) cos(ψ(Θj)−Θj) < 04and unstable if (ψ′(Θj)− 1) cos(ψ(Θj)−Θj) > 0 (the null case being ruled outby (H2)).Moreover the Θj are the only fixed points of (11) and the dynamics of solu-tions t 7→ θ(t) of (11) is very simple : t 7→ θ(t) goes in a monotonic way fromsome Θj(xv) (unstable equilibrium, limit value as τ goes to −∞) to Θj−1(xv)or Θj+1(xv) as τ goes to +∞ (stable equilibria). All the solutions of (11) areglobal in the τ variable and go from an unstable Θj to a close stable one.It then remains to solve (10). The behavior depends on the sign ofψ˜(θ) cos(ψ(θ)− θ).As ψ˜(θ) > 0 and as ψ′(Θj) < 1, ψ˜(θ) cos(ψ(θ)− θ) is positive if Θj is stable andnegative if Θj is unstable.Therefore solutions of (9) are global (except if θ constantly equals some ofthe Θj where the solution goes to the singularity in finite time in the future orin the past), and are asymptotic in +∞ to some stable Θj and in −∞ to someinstable Θj .The phase portrait can be described as follows :– There exists N0 particular solutions which are straight lines, going to orcoming from the origin in finite positive or negative times.– All the other trajectories are global in time and are asymptotic to two ofthe particular solutions as time goes to +∞ or −∞.Theorem 2.1 is then straightforward. Note that Hypothesis (H2) is crucial. If we assume ψ′(Θ′) > 1, then theconclusion is completely changed : all the trajectories come from the singularityand go back to the singularity in finite time, except for θ = Θj . In this case,almost all the trajectories blow up in finite time.5 Trajectories near the singular setIn this section we will describe the behavior of solutions near the singularset. Let x0 = (0, x0v) be a point of P⊥. Locally the geometry of the flow isdescribed by the angles Θj(x0v) which split the space into angular sectorsΩj = {Θj(x0v) < θ < Θj+1(x0v)}.If there were no xh dependence of the flow, the angles Ωj would be invariantunder the flow as in the previous section. This is not the case here and we haveto be more precise in the spatial description.5.1 Domain decomposition.Let α > 0 and η > 0 be small. Then by continuity there exists angles θ+jand θ−j such that ∣∣∣ψ˜(xh, xv, θ) sin(ψ(xh, xv, θ)− θ)∣∣∣ ≥ α (12)5when |xh|+ |xv − x0v| < 2η and Θj(x0v) < θ+j < θ < θ−j+1 < Θj+1(x0v).Let us now introduce the following sets, for ε chosen later on with ε < η :Ωj(ε, η) ={(xh, xv, θ), | |xh| < ε, |xv − x0v| < η, θ−j < θ < θ+j},Σj(ε, η) ={(xh, xv, θ), | |xh| < ε, |xv − x0v| < η, θ+j < θ < θ−j+1},andΩv(ε, η) ={(xh, xv, θ), | |xh| < ε, |xv − x0v| < η}.Note that Ωv(ε, η) is the union of the Ωj(ε, η) and Σj(ε, η) and of their bounda-ries. The Ωj(ε, η) will be the neighborhood of the stable and unstable manifolds.5.2 Study of the trajectory.Let us assume to fix the ideas that Θj is unstable and Θj+1 is stable (thediscussion is similar if the stabilities are interchanged). Let x(t) be a trajectorywith x(0) ∈ Σj(ε, η/2). We claimClaim : If ε is small enough, the trajectory x(t) is contained in Ωj(η, η) fort < t− and is contained in Ωj+1(η, η) for t > t+. Moreover, for |t| large enough,it exits Ωv(η).Proof of the claim : Let x(t) be a solution of (1), such that x(0) = (xh(0), xv(0))with xh(0) = r(0) exp(iθ(0)). Using polar coordinates and introducing again thechange of time variable, we getx˙v = r(1−Π)φ((xh, xv),xh‖xh‖), (13)withr˙ = rψ˜(xh, xv, θ) cos(ψ(xh, xv, θ)− θ) (14)θ˙ = ψ˜(xh, xv, θ) sin(ψ(xh, xv, θ)− θ). (15)Let ]a, b[ be the maximal time interval containing 0 such that x(t) ∈ Ωv(η)for t ∈]a, b[. By definition of θj and θj+1, θ(t) is increasing as long as it be-longs to Ωv(η, η)−Ωj(η, η)−Ωj+1(η, η). Let ]a1, b1[ the maximum time intervalcontaining 0 such that θj < θ(t) < θj+1. Of course a ≤ a1 < b1 ≤ b.If a < a1 then for t = a1, x(t) ∈ Ωj ∩ Ωv and therefore x(t) ∈ Ωj ∩ Ωv forany a < t < a1. Similarly, if b > b1 then for any b1 < t < b, x(t) ∈ Ωj+1 ∩ Ωv.Note that, using (15) and (12), we can bound b1 − a1, byb1 − a1 ≤ θ3 − θ2α. (16)This implies that b1 and −a1 are less than (θ3 − θ2)/α Moreoverddτlog r = ψ˜(xh, xv, θ) cos(ψ(xh, xv, θ)− θ) (17)6Note that on Ωv, ψ˜ is bounded, by some constant C0. Hence for a1 < τ < b1,integrating (17) with respect to τ and using (16), we getexp(−C0α−1(θ3 − θ2)) ≤ r(τ)R0(x)≤ exp(C0α−1(θ3 − θ2)). (18)Therefore ifε <η1 + 2C0exp(−2C0α−1(θ3 − θ2)), (19)r(τ) remains smaller than η for a1 < τ < b1. Hence the solution can not leaveΩv at τ = a1 or τ = b1 at the boundary |xh| = η.Now using (13), we see that as, using (18)–(19), r(τ) remains smaller thanη/2C0, the trajectory can not leave Ωv at its horizontal boundary. Thereforethe trajectory goes from Ωj(η) to Ωj+1(η), which ends the proof of the claim.Bounds on r(t) : Let us go back to the genuine time t. On Ωj(η) and Ωj+1(η),provided α is small enough, | cos(ψ(xh, xv, θ)− θ)| is larger than 1/2. Thereforeddt|xh(t)|2 = 2r(t)r˙(t)is bounded away from 0 by r(t) min ψ˜/2. Therefore, as long as r(t) remains inΩv, for t > b1 we haver(b1) + γ1(t− b1) ≤ r(t) ≤ r(b1) + γ2(t− b1)for some non negative constants γ1, γ2. A similar result is true for t < a1.Dynamics on Ωj : As r˙ > 0 we change time to getx˙v =rcos(ψ(xh, xv, θ)− θ) (1−Π)φ((xh, xv),xh‖xh‖), (20)r˙ = r, (21)θ˙ = tan(ψ(xh, xv, θ)− θ). (22)Note that (20,21,22) is a standard dynamical system, studied near the equi-librium set {(xv, 0,Θj)|xv ∈ P⊥} which appears to be unstable. It thereforeadmits an unstable manifold, which is precisely the manifold Vj .6 Measure of a transported setThis section is divided in three parts. The first one concerns a control ofthe integral of the divergence along a trajectory x(t) passing through a pointx0 in terms of its minimum distance R0(x) to the singularity. The second partconcerns the local intergrability of log(R−10 ) that will be used in the third partto prove Theorem 2.2 through the bound established in the first part.7Control of the divergence. Let us first bound the divergence of our vector field.This divergence D equalsD =1r(cos(ψ(xh, xv, θ)− θ)(∂θψ − 1)ψ˜ + ∂θψ˜ sin((ψ(xh, xv, θ)− θ))+D1where D1 is a sum of terms which are bounded as r goes to 0. If α and η aresmall enough (see the domain decomposition part for their definitions), the sineis nearly 0 and much smaller than the cosine, hence D has the sign of cos(Θj−θ)and, as ∂θψ 6= 1 and ψ˜ is bounded away from 0, D is bounded in modulus by|D| ≤ c02r(23)provided α is small enough.Let us now evaluate the integral of the divergence along a trajectory. Letx(t) with t1 < t < t2 be a part of trajectory included in Ωj+1. Note that r(t) isincreasing andr(t) ≥ r(t1) + C0(t− t1)for some constant C0 depending only on ψ˜. Therefore,∫ t2t11r(t)dt ≤∫ t2t11r(t1) + C0(t− t1)dt =1C0log(r(t1) + C0(t2 − t1)r(t1))(24)≤ 1C0log( ηR0(x))since r(t2) + C0(t2 − t1) ≤ η. A similar result is true for parts of trajectoriesincluded in Ωj . Note that the integral behaves like log(1/R0(x)).Local integrability and minimum distance to the singularity set. It remains toprovide a local integrability property related to log(R−10 ), where R−10 denotes theminimum distance between the singularity and a trajectory x(t) passing troughx0. Let us look at the formulation (13,14,15), and let us study a trajectoryx(t). Up to a shift in time, we may assume that x(t) reaches R0 at t = 0. Letus study x(t) when t becomes negative and large. First x(t) enters Ωj(ε, η) atsome time t1 < 0 with |t1| = O(1) as we consider sequences of trajectories withcorresponding R0 which go to 0.For t < t1, when d(x(t), P⊥) = δ (δ being fixed) then, using (14) and integra-ting from 0 to t, t is of order C1 log(δ/R0). Since the trajectory x(t) approachesVj with an exponential speed, we get that d(x(t), Vj) is of order C2(R0/δ)γ forsome positive constants C2 and γ.Therefore the measure of{x0 ∈ Ωj(ε, η)|d(x(t), P⊥) = δ, the trajectory x(t) passes at a distance smaller than r0}is of order C3(r0/δ)γ for some positive constants C3 and γ. Thusµ(x : R0(x) ≤ r0 and d(x, P⊥) = δ)≤ C(r0/δ)γ .8Hence log(R−10 ) is locally integrable.Bound of the measure of a transported set. Let F (t0, t1) denotes the resolventof the flow, which is defined outside the manifolds Vj , and thus defined almosteverywhere. Let x0 ∈ P⊥. Let ρ > 0, then for any measurable set A,µ(F (t0, t1)(A ∩B(x0, ρ))= µ(A) exp(∫ t1t0∫F (t0,τ)(A∩B(x0,ρ))D(τ, x)dτdx)= µ(A ∩B(x0, ρ)) exp(∫A∩B(x0,ρ)∫ t1t0D(τ, F (t0, τ)x)dτdx)≤ µ(A ∩B(x0, ρ)) exp(C−10∫A∩B(x0,ρ)log(ηR−10 (x))dx)≤ Cµ(A ∩B(x0, ρ))for some constant C since log(R−10 ) is locally integrable. This ends the proof ofTheorem 2.2. Acknowledgments. The first author would like to thank the French ANRProject ”MathOcean” (2008-2011) managed by D. Lannes.Re´fe´rences[1] S. Bianchini, L.V. Spinolo. Invariant manifolds for a singular ordinarydifferential equations. Submitted (2008).[2] D. Bresch, B. Desjardins, E. Grenier. Oscillatory Limits with Chan-ging Eigenvalues. In preparation (2009).[3] D. Bresch, B. Desjardins, E. Grenier, C.–K. Lin. Low Mach numberlimit of viscous polytropic flows : formal asymptotics in the periodic case.Stud. Appl. Math. 109 (2002), no. 2, 125–149.[4] G. Me´tivier, S. Schochet. Averaging theorems for conservative systemsand the weakly compressible Euler equations. J. Differential Equations 187(2003), no. 1, 106–183.9

HAL Id: hal-00408619https://hal.archives-ouvertes.fr/hal-00408619Preprint submitted on 31 Jul 2009HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Singular ordinary differential equations homogeneous ofdegree 0 near a codimension 2 setDidier Bresch, Benoit Desjardins, Emmanuel GrenierTo cite this version:Didier Bresch, Benoit Desjardins, Emmanuel Grenier. Singular ordinary differential equations homo-geneous of degree 0 near a codimension 2 set. 2009. ￿hal-00408619￿Singular ordinary differential equationshomogeneous of degree 0 near a codimension 2 setDidier Bresch 1, Benôıt Desjardins 2, Emmanuel Grenier 3AbstractThis note deals with an example of class of ordinary differential equationswhich are singular near a codimension 2 set, with an homogeneous singularity,of degree 0. Under some structural assumptions, we prove that for almost allinitial data there exists a unique global solution and study the evolution of theLebesgue measure of a transported set of initial data.Keywords : Singular ODE’s, co-dimension 2 singularity, global existenceand uniquenessAMS subject classification : 47305, 37N10, 35A05, 74H35.1 IntroductionThis note deals with a simple example of a class of ordinary differentialequations which are singular on a manifold of codimension 2, the singularitybeing homogeneous of degree 0 near this singularity.Let H be an Hilbert space. Let Π denotes the orthogonal projection on agiven plane P . Let xh = Πx and let xv = x−Πx. Note that x = xh + xv. By aslight abuse of notation we shall say that x = (xh, xv), xh and xv referring tothe ”horizontal” and ”vertical” components of x. We then identify the plane Pwith C by choosing arbitrarily a basis on P . This allows to use polar coordinateson P and thus to define the modulus r and argument θ of xh, and to use thenotation xh = r exp(iθ).Let φ be a smooth function defined on H × S1 (S1 being the unit circle).Thenẋ = φ(x,xh|xh|)(1)is a dynamical system, singular on P⊥, orthogonal of P , which is of codimension2. Moreover, this system is homogeneous of degree 0 in any direction orthogonalto P⊥.Under some assumptions on φ, we show global existence and uniquenessof solution for almost every initial data. Note that we only take care of thebehavior of t 7→ x(t) near P⊥ and by a slight abuse of language we shall saythat a solution is global if it does not reach the singularity P⊥ in finite time.1Didier.bresch@univ-savoie.fr, LAMA, UMR5127 CNRS, Université de Savoie, 73376 LeBourget du lac, France2Benoit.Desjardins@mines.org, ENS Ulm, D.M.A., 45 rue d’Ulm, 75230 Paris cedex 05,France, and Modélisation Mesures et Applications S.A., 66 avenue des Champs Elysées, 75008Paris, France3egrenier@umpa.ens-lyon.fr, U.M.P.A., École Normale Supérieure de Lyon, 46, allée d’Ita-lie, 69364 Lyon Cedex 07, France.12 MotivationThe toy model (1) mimics phenomena occurring in the low Mach numberlimit problem for non-isentropic flows in a periodic box for instance. Such flowsare described through a velocity field u, a density ρ implicitly given by a statelaw depending on the entropy S and the pressure p, see [4]). After some changeof variables, namely introducing q defined by p = p̄eεq with p a prescribedconstant and ε the low Mach number, the adimensionnalized system readsa(∂tq + u · ∇q) +1εdivu = 0, (2)r(∂tu+ u · ∇u) +1ε∇p = 0, (3)∂tS + u · ∇S = 0, (4)where a and r are two smooth functions of S and ε q. The singular limit consistsin letting the Mach number ε go to 0. In the ill prepared-data case, namelywhen the initial data do not satisfy the conditiondiv u0 = 0, ∇p0 = 0. (5)an oscillatory limit with changing eigenvalues occurs. Indeed the eigenvaluesand the spectrum of the singular operator A whereA =( 0 a−1divr−1∇ 0). (6)will depend on the solution itself. This leads to complex problem, since eigen-values may cross. The wave equation, related to the operator A, may also bewritten in this caseε∂ttψ − div (S−1∇ψ) = 0 (7)where S is the entropy quantity, see [4] for more details.In the very special case of only one spatial dimension, the limit can be bothcalculated completely and justified, see [4]. In the multi-dimensional case theformal calculation of the extra term in the limit, which once again involves thespectral decomposition of the fast operator, assumes that the spectrum of thatfast operator is simple and non-resonant, see [3] for the viscous case and [4]for the inviscid one. For certain finite-dimension al truncations of the equationsthose assumptions can be shown to be generic and to ensure convergence to thelimit equations. This has been done in the paper [4].The main difficulty in the general study is to prove, after defining appropriateinfinite dimension measures, that for almost all initial data, the limit flow doesnot meet double eigenvalues and crosses the resonance set transversally. Diffi-culties occur since the flow is singular across the resonant set which may beshown to define a codimension 2 set. This explain the necessity to study ODEsof the form (1). Our result will be used to treat oscillatory limits with changingeigenvalues and particularly the low Mach number limit in a forthcoming paper[2]. Note that another type of singular ordinary differential equation occuringin fluid mechanics has been studied recently, see for instance [1].23 Setup and main result3.1 Notations.Let us first introduce some notations. We define ψ(xh, xv, θ) as the argumentof Πφ((xh, xv), exp(iθ)), and ψ̃(xh, xv, θ) its modulus in such a way thatΠφ((xh, xv), exp(iθ))= ψ̃(xh, xv, θ) exp(iψ(xh, xv, θ)).Structural properties. We will assume that there exists an integer N0 > 0 andN0 smooth functions of xv, Θ1(xh, xv), . . ., ΘN0(xh, xv), satisfying the followingproperties :(H1) For all x = (xh, xv) ∈ H, the equation in θψ(xh, xv, θ) ∈ θ + πZ,has exactly N0 solutions Θ1(xh, xv) . . .ΘN0(xh, xv).(H2) For every j, the following sign condition holds for all x = (xh, xv) ∈ H∂θψ(xh, xv,Θj(xh, xv)) < 1.Note that this implies that the solutions Θj are all simple.(H3) ψ̃ does not vanish.Note that (H1) implies that{(τ exp(iΘj(0, xv)), xv), τ > 0}is a trajectoryforẋh = Πφ((0, xv),xh‖xh‖). (8)This trajectory goes to the singularity or leaves it, depending on its orientation.In particular for some initial data we reach the singularity in finite time. Theflow is not defined everywhere and we can only get almost everywhere results.3.2 Main result and consequencesThe main result of this paper is :Theorem 3.1 Stable and unstable manifolds.Let us assume that (H1), (H2) and (H3) hold true. Let x0 ∈ P⊥ and let ρ > 0.There exists a finite number of manifolds Vk, of codimension 1, with boundaryΣ, such that :– For any initial data x1 in one of the manifolds Vk, the correspondingsolution of (9) reaches Σ in finite time (in the past or in the future).– For any initial data x1 in B(x0, ρ) outside all these manifold Vk, the cor-responding solution of (9) reaches the boundary of B(x0, ρ) before Σ.3With this result one can define the flow Ψ(t) of this equation, flow whichis defined everywhere except on the manifolds Vk. Let A be some set of initialdata. We now want to bound the measure of Ψ(t)A in terms of the measure ofA. This of course depends on the measure of H, which is not canonical. In orderto fix the ideas we will restrict ourselves to a finite dimensional space H, withthe Lebesgue measure µ, in order to proveTheorem 3.2 Control of the divergence.Let x0 ∈ P⊥. Let ρ > 0. There exists a constant C0(t) such that for any borelianset A of H,µ(Ψ(t)(A ∩B(x0, ρ)))≤ C0(t)µ(A ∩B(x0, ρ)),for any t > 0.The outline of the paper is as follows. First we study the simplified equationẋ = φ( xh|xh|)(9)and then extend it by perturbation arguments to equations of the generalform (1). More precisely, in Section 4, we will study trajectories near the singu-lar set. The last section concerns measure type estimate of a transported set bythe flow which is defined everywhere except on the manifolds Vk of codimensionone.4 Study of systems of the form (9)We first prove Theorem 2.1 in the particular case of systems of the form(9). Note that (9) is a two dimensional dynamical system, which takes places inxh + P . Changing the notations, (9) is of the formẏ = φ( y‖y‖)where y ∈ C. First we turn to polar coordinates, define y = r exp(iθ) and makethe change of time defined by dτ/dt = 1/r to getdrdτ= rψ̃(θ) cos(ψ(θ)− θ) (10)dθdτ= ψ̃(θ) sin(ψ(θ)− θ). (11)Note that (11) does not involve r, which greatly simplifies the analysis of (9).Note also that ψ̃ is always positive. Therefore the dynamics of (11) is given byAssumption (H1). There exists N0 fixed points Θj(xv) which are stable provided(ψ′(Θj)− 1) cos(ψ(Θj)−Θj) < 04and unstable if (ψ′(Θj)− 1) cos(ψ(Θj)−Θj) > 0 (the null case being ruled outby (H2)).Moreover the Θj are the only fixed points of (11) and the dynamics of solu-tions t 7→ θ(t) of (11) is very simple : t 7→ θ(t) goes in a monotonic way fromsome Θj(xv) (unstable equilibrium, limit value as τ goes to −∞) to Θj−1(xv)or Θj+1(xv) as τ goes to +∞ (stable equilibria). All the solutions of (11) areglobal in the τ variable and go from an unstable Θj to a close stable one.It then remains to solve (10). The behavior depends on the sign ofψ̃(θ) cos(ψ(θ)− θ).As ψ̃(θ) > 0 and as ψ′(Θj) < 1, ψ̃(θ) cos(ψ(θ)− θ) is positive if Θj is stable andnegative if Θj is unstable.Therefore solutions of (9) are global (except if θ constantly equals some ofthe Θj where the solution goes to the singularity in finite time in the future orin the past), and are asymptotic in +∞ to some stable Θj and in −∞ to someinstable Θj .The phase portrait can be described as follows :– There exists N0 particular solutions which are straight lines, going to orcoming from the origin in finite positive or negative times.– All the other trajectories are global in time and are asymptotic to two ofthe particular solutions as time goes to +∞ or −∞.Theorem 2.1 is then straightforward. Note that Hypothesis (H2) is crucial. If we assume ψ′(Θ′) > 1, then theconclusion is completely changed : all the trajectories come from the singularityand go back to the singularity in finite time, except for θ = Θj . In this case,almost all the trajectories blow up in finite time.5 Trajectories near the singular setIn this section we will describe the behavior of solutions near the singularset. Let x0 = (0, x0v) be a point of P⊥. Locally the geometry of the flow isdescribed by the angles Θj(x0v) which split the space into angular sectorsΩj = {Θj(x0v) < θ < Θj+1(x0v)}.If there were no xh dependence of the flow, the angles Ωj would be invariantunder the flow as in the previous section. This is not the case here and we haveto be more precise in the spatial description.5.1 Domain decomposition.Let α > 0 and η > 0 be small. Then by continuity there exists angles θ+jand θ−j such that ∣∣∣ψ̃(xh, xv, θ) sin(ψ(xh, xv, θ)− θ)∣∣∣ ≥ α (12)5when |xh|+ |xv − x0v| < 2η and Θj(x0v) < θ+j < θ < θ−j+1 < Θj+1(x0v).Let us now introduce the following sets, for ε chosen later on with ε < η :Ωj(ε, η) ={(xh, xv, θ), | |xh| < ε, |xv − x0v| < η, θ−j < θ < θ+j},Σj(ε, η) ={(xh, xv, θ), | |xh| < ε, |xv − x0v| < η, θ+j < θ < θ−j+1},andΩv(ε, η) ={(xh, xv, θ), | |xh| < ε, |xv − x0v| < η}.Note that Ωv(ε, η) is the union of the Ωj(ε, η) and Σj(ε, η) and of their bounda-ries. The Ωj(ε, η) will be the neighborhood of the stable and unstable manifolds.5.2 Study of the trajectory.Let us assume to fix the ideas that Θj is unstable and Θj+1 is stable (thediscussion is similar if the stabilities are interchanged). Let x(t) be a trajectorywith x(0) ∈ Σj(ε, η/2). We claimClaim : If ε is small enough, the trajectory x(t) is contained in Ωj(η, η) fort < t− and is contained in Ωj+1(η, η) for t > t+. Moreover, for |t| large enough,it exits Ωv(η).Proof of the claim : Let x(t) be a solution of (1), such that x(0) = (xh(0), xv(0))with xh(0) = r(0) exp(iθ(0)). Using polar coordinates and introducing again thechange of time variable, we getẋv = r(1−Π)φ((xh, xv),xh‖xh‖), (13)withṙ = rψ̃(xh, xv, θ) cos(ψ(xh, xv, θ)− θ) (14)θ̇ = ψ̃(xh, xv, θ) sin(ψ(xh, xv, θ)− θ). (15)Let ]a, b[ be the maximal time interval containing 0 such that x(t) ∈ Ωv(η)for t ∈]a, b[. By definition of θj and θj+1, θ(t) is increasing as long as it be-longs to Ωv(η, η)−Ωj(η, η)−Ωj+1(η, η). Let ]a1, b1[ the maximum time intervalcontaining 0 such that θj < θ(t) < θj+1. Of course a ≤ a1 < b1 ≤ b.If a < a1 then for t = a1, x(t) ∈ Ωj ∩ Ωv and therefore x(t) ∈ Ωj ∩ Ωv forany a < t < a1. Similarly, if b > b1 then for any b1 < t < b, x(t) ∈ Ωj+1 ∩ Ωv.Note that, using (15) and (12), we can bound b1 − a1, byb1 − a1 ≤θ3 − θ2α. (16)This implies that b1 and −a1 are less than (θ3 − θ2)/α Moreoverddτlog r = ψ̃(xh, xv, θ) cos(ψ(xh, xv, θ)− θ) (17)6Note that on Ωv, ψ̃ is bounded, by some constant C0. Hence for a1 < τ < b1,integrating (17) with respect to τ and using (16), we getexp(−C0α−1(θ3 − θ2)) ≤r(τ)R0(x)≤ exp(C0α−1(θ3 − θ2)). (18)Therefore ifε <η1 + 2C0exp(−2C0α−1(θ3 − θ2)), (19)r(τ) remains smaller than η for a1 < τ < b1. Hence the solution can not leaveΩv at τ = a1 or τ = b1 at the boundary |xh| = η.Now using (13), we see that as, using (18)–(19), r(τ) remains smaller thanη/2C0, the trajectory can not leave Ωv at its horizontal boundary. Thereforethe trajectory goes from Ωj(η) to Ωj+1(η), which ends the proof of the claim.Bounds on r(t) : Let us go back to the genuine time t. On Ωj(η) and Ωj+1(η),provided α is small enough, | cos(ψ(xh, xv, θ)− θ)| is larger than 1/2. Thereforeddt|xh(t)|2 = 2r(t)ṙ(t)is bounded away from 0 by r(t) min ψ̃/2. Therefore, as long as r(t) remains inΩv, for t > b1 we haver(b1) + γ1(t− b1) ≤ r(t) ≤ r(b1) + γ2(t− b1)for some non negative constants γ1, γ2. A similar result is true for t < a1.Dynamics on Ωj : As ṙ > 0 we change time to getẋv =rcos(ψ(xh, xv, θ)− θ)(1−Π)φ((xh, xv),xh‖xh‖), (20)ṙ = r, (21)θ̇ = tan(ψ(xh, xv, θ)− θ). (22)Note that (20,21,22) is a standard dynamical system, studied near the equi-librium set {(xv, 0,Θj)|xv ∈ P⊥} which appears to be unstable. It thereforeadmits an unstable manifold, which is precisely the manifold Vj .6 Measure of a transported setThis section is divided in three parts. The first one concerns a control ofthe integral of the divergence along a trajectory x(t) passing through a pointx0 in terms of its minimum distance R0(x) to the singularity. The second partconcerns the local intergrability of log(R−10 ) that will be used in the third partto prove Theorem 2.2 through the bound established in the first part.7Control of the divergence. Let us first bound the divergence of our vector field.This divergence D equalsD =1r(cos(ψ(xh, xv, θ)− θ)(∂θψ − 1)ψ̃ + ∂θψ̃ sin((ψ(xh, xv, θ)− θ))+D1where D1 is a sum of terms which are bounded as r goes to 0. If α and η aresmall enough (see the domain decomposition part for their definitions), the sineis nearly 0 and much smaller than the cosine, hence D has the sign of cos(Θj−θ)and, as ∂θψ 6= 1 and ψ̃ is bounded away from 0, D is bounded in modulus by|D| ≤ c02r(23)provided α is small enough.Let us now evaluate the integral of the divergence along a trajectory. Letx(t) with t1 < t < t2 be a part of trajectory included in Ωj+1. Note that r(t) isincreasing andr(t) ≥ r(t1) + C0(t− t1)for some constant C0 depending only on ψ̃. Therefore,∫ t2t11r(t)dt ≤∫ t2t11r(t1) + C0(t− t1)dt =1C0log(r(t1) + C0(t2 − t1)r(t1))(24)≤ 1C0log( ηR0(x))since r(t2) + C0(t2 − t1) ≤ η. A similar result is true for parts of trajectoriesincluded in Ωj . Note that the integral behaves like log(1/R0(x)).Local integrability and minimum distance to the singularity set. It remains toprovide a local integrability property related to log(R−10 ), where R−10 denotes theminimum distance between the singularity and a trajectory x(t) passing troughx0. Let us look at the formulation (13,14,15), and let us study a trajectoryx(t). Up to a shift in time, we may assume that x(t) reaches R0 at t = 0. Letus study x(t) when t becomes negative and large. First x(t) enters Ωj(ε, η) atsome time t1 < 0 with |t1| = O(1) as we consider sequences of trajectories withcorresponding R0 which go to 0.For t < t1, when d(x(t), P⊥) = δ (δ being fixed) then, using (14) and integra-ting from 0 to t, t is of order C1 log(δ/R0). Since the trajectory x(t) approachesVj with an exponential speed, we get that d(x(t), Vj) is of order C2(R0/δ)γ forsome positive constants C2 and γ.Therefore the measure of{x0 ∈ Ωj(ε, η)|d(x(t), P⊥) = δ, the trajectory x(t) passes at a distance smaller than r0}is of order C3(r0/δ)γ for some positive constants C3 and γ. Thusµ(x : R0(x) ≤ r0 and d(x, P⊥) = δ)≤ C(r0/δ)γ .8Hence log(R−10 ) is locally integrable.Bound of the measure of a transported set. Let F (t0, t1) denotes the resolventof the flow, which is defined outside the manifolds Vj , and thus defined almosteverywhere. Let x0 ∈ P⊥. Let ρ > 0, then for any measurable set A,µ(F (t0, t1)(A ∩B(x0, ρ))= µ(A) exp(∫ t1t0∫F (t0,τ)(A∩B(x0,ρ))D(τ, x)dτdx)= µ(A ∩B(x0, ρ)) exp(∫A∩B(x0,ρ)∫ t1t0D(τ, F (t0, τ)x)dτdx)≤ µ(A ∩B(x0, ρ)) exp(C−10∫A∩B(x0,ρ)log(ηR−10 (x))dx)≤ Cµ(A ∩B(x0, ρ))for some constant C since log(R−10 ) is locally integrable. This ends the proof ofTheorem 2.2. Acknowledgments. The first author would like to thank the French ANRProject ”MathOcean” (2008-2011) managed by D. Lannes.Références[1] S. Bianchini, L.V. Spinolo. Invariant manifolds for a singular ordinarydifferential equations. Submitted (2008).[2] D. Bresch, B. Desjardins, E. Grenier. Oscillatory Limits with Chan-ging Eigenvalues. In preparation (2009).[3] D. Bresch, B. Desjardins, E. Grenier, C.–K. Lin. Low Mach numberlimit of viscous polytropic flows : formal asymptotics in the periodic case.Stud. Appl. Math. 109 (2002), no. 2, 125–149.[4] G. Métivier, S. Schochet. Averaging theorems for conservative systemsand the weakly compressible Euler equations. J. Differential Equations 187(2003), no. 1, 106–183.9

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Singular ordinary differential equations homogeneous of degree 0 near a codimension 2 set

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