Étienne Bernard
email: etienne.bernard@ign.fr

Marie Doumic
email: marie.doumic@inria.fr

Pierre Gabriel
email: pierre.gabriel@uvsq.fr

Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts

Keywords: growth-fragmentation equation, self-similar fragmentation, longtime behaviour, general relative entropy, periodic semigroups, non-hypocoercivity MSC 2010: (Primary) 35Q92, 35B10, 35B40, 47D06, 35P05 ; (Secondary) 35B41, 92D25, 92B25

We study the asymptotic behaviour of the following linear growthfragmentation equation

and prove that under fairly general assumptions on the division rate Bpxq, its solution converges towards an oscillatory function, explicitely given by the projection of the initial state on the space generated by the countable set of the dominant eigenvectors of the operator. Despite the lack of hypocoercivity of the operator, the proof relies on a general relative entropy argument in a convenient weighted L 2 space, where well-posedness is obtained via semigroup analysis. We also propose a non-diffusive numerical scheme, able to capture the oscillations.

Introduction

Over the last decades, the mathematical study of the growth-fragmentation equation and its linear or nonlinear variants has led to a wide literature.

Several facts explain this lasting interest. First, variants of this equation are used to model a wide range of applications, from the internet protocol suite to cell division or polymer growth; it is also obtained as a useful rescaling for the pure fragmentation equation (then the growth rate is linear). Second, despite the relative simplicity of such a one-dimensional equation, the study of its behaviour reveals complex and interesting interplays between growth and division, and a kind of dissipation even in the absence of diffusion. Finally, the underlying stochastic process has also -and for the same reasons -raised much interest, and only recently have the links between the probabilistic approach and the deterministic one begun to be investigated.

In its general linear form, the equation may be written as follows where upt, xq represents the concentration of individuals of size x ě 0 at time t, gpxq ě 0 their growth rate, Bpxq ě 0 their division rate, and kpy, xq ě 0 the quantity of individuals of size x created out of the division of individuals of size y.

B
The long-time asymptotics of this equation has been studied and improved in many successive papers. Up to our knowledge, following the biophysical pioneering papers [START_REF] Bell | Cell growth and division: I. a mathematical model with applications to cell volume distributions in mammalian suspension cultures[END_REF][START_REF] Bell | Cell growth and division: III. conditions for balanced exponential growth in a mathematical model[END_REF][START_REF] Sinko | A model for populations reproducing by fission[END_REF], the first mathematical study was carried out in [START_REF] Diekmann | On the stability of the cell size distribution[END_REF], where the equation was considered for the mitosis kernel / binary fission (kpy, xq " 2δ x" y 2 ) in a compact set x P rα, βs. The authors proved the central behaviour of the equation, already conjectured in [START_REF] Bell | Cell growth and division: I. a mathematical model with applications to cell volume distributions in mammalian suspension cultures[END_REF]: under balance and regularity assumptions on the coefficients, there exists a unique dominant eigenpair pUpxq ě 0, λ ą 0q such that upt, xqe ´λt Ñ Upxq in a certain sense, with an exponential speed of convergence. In [START_REF] Diekmann | On the stability of the cell size distribution[END_REF], the proofs were based on semigroup methods and stated for the space of continuous functions provided with the supremum norm. Many studies followed: some of them, most notably and recently [START_REF] Mischler | Spectral analysis of semigroups and growthfragmentation equations[END_REF], relaxing the previous assumptions in the context of semigroup theory [START_REF] Arino | Some spectral properties for the asymptotic behavior of semigroups connected to population dynamics[END_REF][START_REF] Banasiak | Perturbations of positive semigroups with applications[END_REF][START_REF] Banasiak | The discrete fragmentation equation: Semigroups, compactness and asynchronous exponential growth[END_REF][START_REF] Banasiak | Asynchronous exponential growth of a general structured population model[END_REF][START_REF] Engel | One-parameter Semigroups for Linear Evolution Equations[END_REF][START_REF] Greiner | Growth of cell populations via one-parameter semigroups of positive operators[END_REF]; others deriving explicit solutions [START_REF] Hall | Functional-differential equations determining steady size distributions for populations of cells growing exponentially[END_REF][START_REF] Zaidi | A model for asymmetrical cell division[END_REF][START_REF] Zaidi | Solutions to an advanced functional partial differential equation of the pantograph type[END_REF] or introducing new methods -one of the most elegant and powerful being the General Relative Entropy [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF], leading to convergence results in norms weigthed by the adjoint eigenproblem. However, though in some cases the entropy method may lead to an explicit spectral gap in some integral norm [START_REF] Perthame | Exponential decay for the fragmentation or cell-division equation[END_REF][START_REF] Laurençot | Exponential decay for the growthfragmentation/cell-division equation[END_REF][START_REF] Pakdaman | Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation[END_REF], or when the coefficients are such that an entropy-entropy dissipation inequality exists [START_REF] Cáceres | Rate of convergence to an asymptotic profile for the self-similar fragmentation and growthfragmentation equations[END_REF][START_REF] Balagué | Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates[END_REF], in general it fails to provide a rate of convergence.

On the margins of this central behaviour, some papers investigated nonuniqueness [START_REF] Banasiak | On a non-uniqueness in fragmentation models[END_REF] or other kinds of asymptotics, happening for instance when the balance or mixing assumptions between growth and division fail to be satisfied: e.g. when the fragmentation dominates the growth [START_REF] Banasiak | The discrete fragmentation equation: Semigroups, compactness and asynchronous exponential growth[END_REF][START_REF] Bertoin | The asymptotic behavior of fragmentation processes[END_REF][START_REF] Bertoin | Probabilistic aspects of critical growthfragmentation equations[END_REF][START_REF] Haas | Asymptotic behavior of solutions of the fragmentation equation with shattering: an approach via self-similar Markov processes[END_REF][START_REF] Doumic | Time asymptotics for a critical case in fragmentation and growth-fragmentation equations[END_REF]. A stronger "memory" of the initial behaviour may then be observed, contrary to the main case, where the only memory of the initial state which remains asymptotically is a weighted average. Among these results, the case when the growth rate is linear, i.e. gpxq " x, and the mother cell divides into two equal offspring, i.e. kpy, xq " 2δ x" x 2 , holds a special place, both for modelling reasons -it is the emblematic case of idealised bacterial division cycle, and also the rescaling adapted to the pure fragmentation equation -and as a limit case where the standard results fail to be true. The equation is then

$ ' & ' % B Bt upt, xq `B Bx `xupt, xq ˘`Bpxqupt, xq " 4Bp2xqupt, 2xq, x ą 0, up0, xq " u in pxq. (1)
In 1967, G. I. Bell and E. C. Anderson already noted [START_REF] Bell | Cell growth and division: I. a mathematical model with applications to cell volume distributions in mammalian suspension cultures[END_REF]:

"If the rate of cell growth is proportional to cell volume, then (...) a daughter cell, having just half the volume of the parent cell, will grow at just half the rate of the parent cell. It follows that if one starts with a group of cells of volume V , age r, at time 0, then any daughter cell of this group, no matter when formed, will always have a volume equal to half the volume of an undivided cell in the group. There will then be no dispersion of cell volumes with time, and the population will consist at any time of a number of cell generations differing by just a factor of 2 in volume. For more general initial conditions, the population at late times will still reflect the initial state rather than simply growing exponentially in time. "

After [START_REF] Bell | Cell growth and division: I. a mathematical model with applications to cell volume distributions in mammalian suspension cultures[END_REF], the reason for this specific behaviour was stated in [START_REF] Diekmann | On the stability of the cell size distribution[END_REF][START_REF] Heijmans | An eigenvalue problem related to cell growth[END_REF]: instead of a unique dominant eigenvalue, there exists a countable set of dominant eigenvalues, namely 1 `2iπ log 2 Z. O. Diekmann, H. Heijmans and H. Thieme explain in [START_REF] Diekmann | On the stability of the cell size distribution[END_REF]:

The total population size still behaves like re t s but convergence in shape does not take place. Instead the initial size distribution turns around and around while numbers are multiplied. (...) The following Gedanken experiment illustrates the biological reason. Consider two cells A and B with equal size and assume that at some time instant t0 cell A splits into a and a. During the time interval rt0, t1s, a, a and B grow and at t1 cell B splits into b and b. If gpxq " cx, the daughter cells a and b will have equal sizes just as their mothers A and B. ln other words, the relation "equal size" is hereditary and extends over the generations. The growth model behaves like a multiplicating machine which copies the size distribution.

In [START_REF] Greiner | Growth of cell populations via one-parameter semigroups of positive operators[END_REF], G. Greiner and R. Nagel are the first to prove this long-time periodic behaviour. They use the theory of positive semigroups combined with spectral analysis to get the convergence to a semigroup of rotations. The method relies on some compactness arguments, which force the authors to set the equation on a compact subset of p0, 8q (x P rα, βs with α ą 0).

In the present paper, we extend the result to the equation set on the whole R `. Additionally, we determine explicitly the oscillatory limit by the means of a projection of the initial condition on the dominant eigenfunctions. Our method relies on General Relative Entropy inequalities (Section 1), which unexpectedly may be adapted to this case and which are the key ingredient for an explicit convergence result (Theorem 2 in Section 2, which is the main result of our study). We illustrate our results numerically in Section 3, proposing a non-diffusive scheme able to capture the oscillations.

Eigenvalue problem and Entropy

To study the long time asymptotics of Equation ( 1), we elaborate on previously established results concerning the dominant positive eigenvector and general relative entropy inequalities.

Dominant eigenvalues and balance laws

The eigenproblem and adjoint eigenproblem related to Equation (1) are λUpxq ``xUpxq ˘1 `BpxqUpxq " 4Bp2xqUp2xq,

(2)

λφpxq ´xφ 1 pxq `Bpxqφpxq " 2Bpxqφ ´x 2 ¯. (3)
Perron eigenproblem consists in finding positive solutions U to (2), which in general give the asymptotic behaviour of time-dependent solutions, which align along e λt Upxq. Recognizing here a specific case of the eigenproblem studied in [START_REF] Doumic | Eigenelements of a general aggregationfragmentation model[END_REF], we work under the following assumptions:

$ ' ' ' ' & ' ' ' ' % B : p0, 8q Ñ p0, 8q is locally integrable, Dz 0 , γ 0 , K 0 ą 0, @x ă z 0 , Bpxq ď K 0 x γ0 , Dz 1 , γ 1 , γ 2 , K 1 , K 2 ą 0, @x ą z 1 , K 1 x γ1 ď Bpxq ď K 2 x γ2 . (4)
We then have the following result, which is a particular case of [START_REF] Doumic | Eigenelements of a general aggregationfragmentation model[END_REF]Theorem 1].

Theorem 1. Under Assumption (4), there exists a unique positive eigenvector U P L 1 pR `q to (2) normalised by ş 8 0 x Upxqdx " 1. It is related to the eigenvalue λ " 1 and to the adjoint eigenvector φpxq " x solution to (3).

Moreover, x α U P L 8 pR `q for all α P R, and U P W 1,1 pR `q.

As already noticed in [START_REF] Diekmann | On the stability of the cell size distribution[END_REF], though 1 is the unique eigenvalue related to a positive eigenvector, here it is not the unique dominant eigenvalue: we have a set of eigentriplets pλ k , U k , φ k q with k P Z defined by

λ k " 1 `2ikπ log 2 , U k pxq " x ´2ikπ log 2 Upxq, φ k pxq " x 1`2 ikπ log 2 . (5)
This is the first difference with the most studied case, where the Perron eigenvalue happens to be the unique dominant one: here all these eigenvalues have a real part equal to 1, so that they all belong to the peripheral spectrum. The natural questions which emerge are to know whether this set of dominant eigenvectors is attractive, as it is the case when it is formed by a unique function; and if so, where the proofs are different. First we notice an important property: the family `pU k q kPZ , pφ k q kPZ ˘is biorthogonal for the bracket xf, ϕy :"

ż 8 0 f pxqϕpxq dx, which means that @pk, lq P Z 2 , xU k , φ l y " δ kl . (6)
This is a direct consequence of the normalization of the Perron eigenvectors which writes xU, φy " 1 and the fact that λ k ‰ λ l for k ‰ l.

Even though we are interested in real-valued solutions to Equation (1), due to the fact that the dominant eigenelements have nonzero imaginary part, we have to work in spaces of complex-valued functions. Of course real-valued solutions are readily obtained from complex-valued solutions by taking the real or imaginary part. From now on when defining functional spaces we always consider measurable functions from R `to C.

The biorthogonal property (6) can be extended into balance laws for general solutions to Equation (1). For u in P L 1 pφpxqdxq and u P CpR `, L 1 pφpxqdxqq solution to (1) we have the conservation laws @k P Z, @t ě 0, xupt, ¨q, φ k y e ´λk t " xu in , φ k y.

General Relative Entropy inequalities

Additionally to the conservation laws above, we have a set of entropy inequalities. In this section, we remain at a formal level. Rigorous justification of the stated results will appear once the existence and uniqueness results are established.

Lemma 1 (General Relative Entropy Inequality). Let B satisfy Assumption (4), U be the Perron eigenvector defined in Theorem 1 and upt, xq be a solution of Equation [START_REF] Arendt | One-parameter semigroups of positive operators[END_REF]. Let H : C Ñ R `be a positive, differentiable and convex function. Provided the quantities exist, we have

d dt 8 ż 0 x UpxqH ´upt, xq Upxqe t ¯dx " ´DH ruptqe ´ts ď 0,
with D H defined by

D H rus :" 8 ż 0 xBpxq Upxq " H ´up x 2 q Up x 2 q ¯´H ´upxq Upxq ∇H ´upxq Upxq ¯¨´u p x 2 q Up x 2 q ´upxq Upxq ¯dx,
where ∇H is the gradient of H obtained by identifying C with R 2 and ¨stands for the canonical inner product in R 2 . Moreover, for H strictly convex, u : R `Ñ C satisfies D H rus " 0 iff it is such that upxq Upxq " up2xq Up2xq , a.e. x ą 0.

In particular, for all k P Z, D H rU k s " 0.

The proof is immediate and now standard, carried out by calculation term by term and use of the equations ( 1), ( 2) and ( 3), see for instance [33, p.92].

In the cases where the Perron eigenvector is a unique dominant eigenvector, the entropy inequality is a key step to obtain the convergence of upt, xqe ´t towards xu in , φy Upxq. The idea is to prove that upt, xqe ´t tends to a limit u 8 such that D H ru 8 s " 0, which in general implies that u 8 is proportional to U; the conservation law then giving the proportionality constant.

Here however, since any function vpxq " f plog xqUpxq with f log 2-periodic satisfies D H rvs " 0, the usual convergence result does not hold. This is due to the lack of hypocoercivity in our case. It is known from [START_REF] Cáceres | Rate of convergence to an asymptotic profile for the self-similar fragmentation and growthfragmentation equations[END_REF][START_REF] Balagué | Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates[END_REF][START_REF] Gabriel | Exponential relaxation to self-similarity for the superquadratic fragmentation equation[END_REF] that the general form (GF) of the growth-fragmentation equation is coercive for particular choices of the coefficients, in the sense that the differential inequality d dt }upt, ¨q} ď ´ν}upt, ¨q} holds for some positive constant ν and a well-chosen norm } ¨}, when u in is such that xu in , φy " 0. As already noticed in [START_REF] Laurençot | Exponential decay for the growthfragmentation/cell-division equation[END_REF] such an inequality cannot be valid for an entropic norm in the case of equal mitosis. Indeed if for some time t ě 0 (for instance t " 0) the solution satisfies upt, xq{Upxq " upt, 2xq{Up2xq, then the time derivative of the norm vanishes. However in this case the equation can be hypocoercive in the sense (see [36]) that }upt, ¨q} ď Ce ´νt }u in } holds for some positive constants C, ν and any initial distribution satisfying xu in , φy " 0. This result is proved in [START_REF] Mischler | Spectral analysis of semigroups and growthfragmentation equations[END_REF][START_REF] Bernard | Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate[END_REF] for a class of weighted L 1 norms in the case of a constant growth rate g. Roughly speaking this situation of a non-coercive but hypocoercive equation appears when the dissipation of entropy can vanish for a nontrivial set of functions, but this set is unstable for the dynamics of the equation. In our case the equation is not hypocoercive because the set of functions with null entropy dissipation is invariant under the flow, as expressed by the following lemma.

Lemma 2. Consider a strictly convex function H and let upt, xq be the solution to Equation (1) with initial condition up0, xq " u in pxq. We have the invariance result D H ru in s " 0 ùñ D H rupt, ¨qs " 0, @t ě 0.

As Lemma 1, Lemma 2 is valid in a space where the existence and uniqueness of a solution is proved, as for instance in the space L 2 pR `, x{Upxqdxq, see Section 2.

Proof. Let u in such that D H ru in s " 0 and denote upt, ¨q the solution to Equation [START_REF] Arendt | One-parameter semigroups of positive operators[END_REF].

We have already seen in Lemma 1 that for any u : R `Ñ C, we have

D H rus " 0 ðñ upxq Upxq " up2xq Up2xq
, a.e. x ą 0, so that by assumption u in pxq U pxq " u in p2xq U p2xq for almost every x ą 0. To prove Lemma 2 we thus want to prove that upt,xq U pxq " upt,2xq U p2xq for almost every x ą 0, t ą 0. To do so, we notice that if we have a solution ũ of Equation [START_REF] Arendt | One-parameter semigroups of positive operators[END_REF] which satisfies this property, then the ration vpt, xq " ũpt, xq{Upxqe ´t is solution of the following simple transport equation B t vpt, xq `xB x vpt, xq " 0, so that vpt, xq " vp0, xe ´tq. We are led to define a function u 1 by u 1 pt, xq :" u in pxe ´tq Upxq e t Upxe ´tq .

We easily check that u 1 pt, xq{Upxq " u 1 pt, 2xq{Up2xq for all t and almost all x, and that u 1 is solution to Equation (1). We conclude by uniqueness that we have u " u 1 and so D H rupt, ¨qs " 0 for all t ě 0.

For Hpzq " |z| p the entropy corresponds to the p-power of the norm in

E p :" L p pR `, φpxqU 1´p pxq dxq.
Define also the space

E 8 :" u : R `Ñ C measurable, DC ą 0, |u| ď CU a.e. ( ,
which
# |z| ´C if |z| ě C 0 if |z| ď C with C " }u in } E8 .
Lemma 4. Let 1 ď p ď q ď 8 and u P E q . Then u P E p and }u} Ep ď }u} Eq .

Proof. It is clear if q " `8. For q ă `8, since φpxq Upxqdx is a probability measure the Jensen's inequality ensures that

}u} q Ep " ˆż ˇˇu U ˇˇpφ U ˙q{p ď ż ˇˇu U ˇˇqφ U " }u} q Eq .
2 Convergence in the quadratic norm

Equipped with the General Relative Entropy inequalities, we now combine them with Hilbert space techniques to prove the convergence to periodic solutions. The Hilbert space formalism provides an interpretation of the periodic limit in terms of Fourier decomposition, and allows us to give the main ingredients of the proof while avoiding too many technicalities. We first introduce the Hilbert space (Section 2.1), in which we prove the well-posedness of Equation ( 1) (Section 2.2). We state our main result in Theorem 2.

The Hilbert space

As we will see below, working in a Hilbert setting is very convenient for our study. Drawing inspiration from the General Relative Entropy with the convex quadratic function Hpzq " |z| 2 , we work in the Hilbert space E 2 " L 2 pR `, x{Upxq dxq endowed with the inner product pf, gq :"

ż 8 0 f pxqgpxq x Upxq dx.
We denote by } ¨} the corresponding norm defined by }f } 2 " pf, f q.

In this space, the normalization we have chosen for U means }U} " }U k } " 1 and the biorthogonality property (6) reads

pU k , U l q " xU k , φ l y " δ k,l ,
meaning that pU k q kPZ is an orthonormal family in E 2 . As a consequence the family pU k q kPZ is a Hilbert basis of the Hilbert space X :" spanpU k q kPZ and the orthogonal projection on this closed subspace of E 2 is given by

P u :" `8 ÿ k"´8 pu, U k qU k , @u P E 2 .
Additionally, we have the Bessel's inequality

}P u} 2 " `8 ÿ k"´8 |pu, U k q| 2 ď }u} 2 .
As it is stated in the following lemma, there is a crucial link between X and the quadratic dissipation of entropy (i.e. D H rus for Hpzq " |z| 2 ), which can be written in a simpler way as

D 2 rus " ż 8 0 xBpxq Upxq ˇˇˇu pxq Upxq ´upx{2q Upx{2q ˇˇˇ2 dx. ( 8
)
Lemma 5. We have X " tu P E 2 , D 2 rus " 0u.

Proof. Since |z| 2 is strictly convex, we have already seen in Lemma 1 (and it is even clearer in the case of D 2 ) that tu P E 2 , D 2 rus " 0u " tu P E 2 , upxq{Upxq " up2xq{Up2xq, a.e. x ą 0u Ą X. f p´kqU k pxq P X.

We also deduce that f pkq " pu, U ´kq. It is proved in [START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF] (see also [START_REF] Bernard | Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate[END_REF]) that the problem ( 9) is well-posed in E 1 and admits an associated C 0 -semigroup pT t q tě0 which is positive, meaning that for any u in P E 1 there exists a unique (mild) solution v P CpR `, E 1 q to (9) which is given by vptq " T t u in , and vptq ě 0, t ě 0, for u in ě 0. From Lemma 3 we have that all subspaces E p with p P r1, 8s are invariant under pT t q tě0 . Additionally, the restriction of T t to any E p is a contraction, i.e.

Well-posedness of the Cauchy problem

@u P E p , @t ě 0, }T t u} Ep ď }u} Ep . (10)
To get the well-posedness of ( 9) in E 2 , it only remains to check the strong continuity of pT t q tě0 in E 2 .

Lemma 6. The semigroup pT t q tě0 restricted to E 2 is strongly continuous.

Proof. We use the subspace E 8 Ă E 2 and the contraction property [START_REF] Bernard | Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate[END_REF] to write for any u P E 8

}T t u ´u} 2 E2 " ż 8 0 |T t u ´u| 2 pxq x Upxq dx ď 2}u} E8 ż 8 0 |T t u ´u|pxqx dx " 2}u} E8 }T t u ´u} E1 .
The strong continuity of pT t q tě0 in E 1 ensures that }T t u ´u} E1 Ñ 0 and so }T t u ´u} E2 Ñ 0 when t Ñ 0. By density of E 8 Ă E 2 we get the strong continuity of pT t q tě0 in E 2 .

We denote by A the generator of the semigroup pT t q tě0 in E 2 . For any u in the domain DpAq we have in the distributional sense Aupxq " ´pxupxqq 1 ´upxq ´Bpxqupxq `4Bp2xqup2xq.

The eigenpairs pλ k , U k q are defined by A U k " pλ k ´1q U k and we easily prove the following properties.

Proposition 7. For all t ě 0 we have

1. @k P Z, T t U k " e 2ikπt log 2 U k , 2. @u P E 2 , @k P Z, pT t u, U k q " pu, U k qe 2ikπt log 2 , 3. @u P E 2 , P T t u " T t P u " ř kPZ pu, U k qe 2ikπt log 2 U k , 4. T t X Ă X and for all u P X, T t u " ř kPZ pu, U k qe 2ikπt log 2 U k ,
5. X Ă DpAq and for all u P X, Au "

ř kPZ pu, U k q 2ikπ log 2 U k .
The second property is nothing but a rewriting of the conservation laws [START_REF] Banasiak | Asynchronous exponential growth of a general structured population model[END_REF]. The fourth point makes more precise and proves more rigorously the invariance property in Lemma 2.

Convergence

We are now ready to state the asymptotic behaviour of solutions to Problem 1.

Theorem 2. Assume that B satisfies Hypothesis (4) and define U k by [START_REF] Banasiak | Perturbations of positive semigroups with applications[END_REF]. Then for any u in P E 2 , the unique solution upt, xq P C `R`, E 2 ˘to Equation (1) satisfies

8 ż 0 ˇˇˇu pt, xqe ´t ´`8 ÿ k"´8 pu in , U k qe 2ikπ log 2 t U k pxq ˇˇˇ2 x dx Upxq Ý ÝÝÝ Ñ tÑ`8 0.
Remark 1. This convergence result can also be formulated in terms of semigroups. Set R t u :" T t P u " P T t u "

8 ÿ k"´8 pu, U k qe 2ikπ log 2 t U k .
This defines a semigroup, R t`s u " T t`s P u " T t`s P 2 u " T t T s P 2 u " T t P T s P u " R t R s u, which is log 2-periodic. The result of Theorem 2 is equivalent to the strong convergence of pT t q tě0 to pR t q tě0 , i.e.

@u P E 2 , }T t u ´Rt u} Ý ÝÝÝ Ñ tÑ`8 0.
It is also equivalent to the strong stability of pT t q tě0 in X K " Ker P @u P X K , }T t u} Ý ÝÝÝ Ñ tÑ`8 0.

Remark 2. We may use the Poisson summation formula to reinterpret the limit function in terms of only u in pxq : we recall that this formula states that, under proper assumptions on f and its Fourier transform Ff pξq " `8 ş

´8 f pyqe ´iyξ dy, we have .

Taking a " log 2, f pyq " u in pe ´y qe ´2y , we apply it to the limit function taken in y " t ´log x

8 ÿ k"´8 pu in , U k qU k pxqe 2ikπt log 2 " Upxq 8 ÿ "´8 2 ´2 x 2 e ´2t u in p2 ´ xe ´tq.
This formula is reminiscent of a similar one found in [START_REF] Doumic | Time asymptotics for a critical case in fragmentation and growth-fragmentation equations[END_REF], Theorem 1.3. (b), for the limit case B constant.

Proof of Theorem 2. We follow here the classical proof of convergence, pioneered in [START_REF] Michel | General entropy equations for structured population models and scattering[END_REF][START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF]. Though the limit is now an oscillating function, this strategy may be adapted here, as shown below. ´hnpt,2xq U p2xq Ñ 0 in the distributional sense. We deduce from the convergence h n Ñ g that gpt,xq U pxq ǵpt,2xq U p2xq " 0, and so D 2 rgpt, ¨qs " 0 for all t ě 0. By Lemma 5 this means that gpt, ¨q P X for all t ě 0. But for all n P N and all t ě 0 we have h n pt, ¨q P X K " KerP by construction of h, and since X K is a linear subspace, the weak limit g of h n also satisfies gpt, ¨q P X K for all t ě 0. Finally gpt, ¨q P X X X K " t0u for all t ě 0, so g " 0 and the proof is complete.

The result in Theorem 2 is in contrast to the property of asynchronous exponential growth which states that the solutions behave like upt, xq " xu in , φy Upxqe t when t Ñ `8. This property is satisfied for a large class of growth-fragmentation equations [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF], but the lack of hypocoercivity in our case prevents it to hold. However we can deduce from Theorem 2 a "mean asynchronous exponential growth" property, in line with probabilistic results, e.g. [START_REF] Doumic | Statistical estimation of a growth-fragmentation model observed on a genealogical tree[END_REF].

Corollary 1. Under Assumption (4), the semigroup pT t q tě0 generated by pA, DpAqq is mean ergodic, i.e.

@u P E

P m t " `8 ÿ k"´8 pm t , U k qU k .
By the conservation laws [START_REF] Banasiak | Asynchronous exponential growth of a general structured population model[END_REF] we have for k ‰ 0

pm t , U k q " 1 t ż t 0 pT s u, U k q ds " pu, U k q 1 t ż t 0 e 2ikπ log 2 s ds " pu, U k q log 2 2ikπ
e 2ikπ log 2 t ´1 t and pm t , U 0 q " pu, U 0 q. This gives

P m t " P 0 u `1 t ÿ k‰0 pu, U k q log 2 2ikπ U k `e 2ikπ log 2 t ´1˘. Since › › › › ÿ k‰0 pu, U k q log 2 2ikπ U k `e 2ikπ log 2 t ´1˘› › › › 2 " ÿ k‰0 ˇˇpu, U k q log 2 2ikπ `e 2ikπ log 2 t ´1˘ˇˇˇ2 ď ´log 2 π ¯2 ÿ kPZ |pu, U k q| 2 " ´log 2 π ¯2}P u} 2
we conclude that

}P m t ´P0 u} ď 1 t log 2 π }P u} Ý ÝÝÝ Ñ tÑ`8 0.
3 Numerical solution

A first-order non diffusive numerical scheme

Another way to understand the origin of the oscillatory behaviour is to consider the underlying Piecewise Deterministic Markov Process (PDMP), see e.g. [START_REF] Bertoin | Probabilistic aspects of critical growthfragmentation equations[END_REF][START_REF] Cloez | Limit theorems for some branching measure-valued processes[END_REF][START_REF] Doumic | Statistical estimation of a growth-fragmentation model observed on a genealogical tree[END_REF]. If we follow a given cell of size x at time 0, it is of size 2 ´nxe t at time t if it has divided n times before t; hence, any of its descendants has to remain exactly in the countable set t2 ´ xe t , P Nu at any time. We can say that we need a "non-diffusive" numerical scheme: if the transport rate is not exactly linear but approximately linear, or if the splitting into two cells does not give rise to two exactly equally-sized but to approximately two equally-sized daughters, then the numerical scheme computes the solution of an approximate equation, which is proved, after renormalization, to converge exponentially fast toward a steady behaviour. Looking at the descendants, it means that instead of remaining in the countable set t2 ´ xe t , P Nu, they will disperse around, and progressively fill in the space p0, xe t q. This exponential convergence toward a steady state will give rise only to some damped oscillations.

The numerical scheme thus needs to satisfy the two following conditions:

1. the discretization of the transport equation B Bt u `B Bx pxuq must be non diffusive. If we use a standard upwind scheme, we would thus like to have a Courant-Friedrichs-Lévy (CFL) condition equal to 1. This means that any point of the grid at time t is transported by the transport equation B Bt u `B Bx pxuq to another point of the grid at time t `∆t. 2. The discretization of the fragmentation term 4Bp2xqupt, 2xq´Bpxqupt, xq must ensure that if x is a point of the grid, then so is x{2 and 2x -at least inside the computational domain rx min , x max s -so that there is no approximation when applying the fragmentation operator.

The condition 2 leads us to define the following geometric grid, for given n, N P N ˚:

δx :" 2

1 n ´1, x k :" p1 `δxq k´N , 0 ď k ď 2N. (11)
Then, for any k P N, 0 ď k ď 2N, 2x k " x k`n is in the grid. The computational domain is rx 0 , x 2N s " r2 ´1 n , 2

N n s. Thanks to the properties of the eigenvector U established in [START_REF] Balagué | Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates[END_REF][START_REF] Doumic | Eigenelements of a general aggregationfragmentation model[END_REF], we have U quickly vanishing toward 0 and infinity, so that the truncation does not lead to an important error. We consider the semi-implicit scheme with splitting given by

w l`1 2 k ´wl k δt `xk w l k ´xk´1 w l k´1 x k ´xk´1 `Bk w l`1 2 k " 0, 1 ď k ď 2N, w l`1 k ´wl`1 2 k δt " 2B k`n w l`1 2 k`n , 1 ď k ď 2N,
where B k :" Bpx k q, and the influx boundary condition chosen to keep the conservation property at the discrete level

w l 0 " x 2N x 0 w l 2N `1 x 0 n ÿ k"1 px k ´xk´1 qB k w l`1 2 k
. Lemma 8. The numerical scheme is conservative in the sense that for all l ě 0

2N ÿ k"1 px k ´xk´1 qw l`1 k " 2N ÿ k"1 px k ´xk´1 qw l k .
In order to avoid diffusivity of the numerical scheme we choose the CFL condition δt " δx 1 `δx .

Indeed under this condition the discretization of the transport term sends exactly a point of the grid on the next point of the grid (the discrete transport follows the characteristics). Under this CFL condition, the first step of the scheme can be written as

w l`1 2 k " 1 1 `δtB k δt δx w l k´1
which leads to the condensed form of the full scheme

w l`1 k " 1 1 `δtB k x k´1 x k w l k´1 `2δtB k`n 1 `δtB k`n δt δx w l k`n´1 , 1 ď k ď 2N, (12)
and

w l 0 " p1 `δtB 1 q " x 2N x 0 w l 2N `n´1 ÿ k"1 x k x 0 δtB k`1 1 `δtB k`1 w l k  . ( 13
)
This scheme is clearly positive. Together with the discrete conservation law we deduce that it is a contraction for the discrete L 1 norm } ¨}1 defined for a vector u " pu k q 1ďkď2N by }u} 1 :"

2N ÿ k"1 px k ´xk´1 q|u k | " δx 1 `δx 2N ÿ k"1 x k |u k |.
Theorem 3 (Convergence in the L 1 norm). Consider that B is continuous. Let u in P E 8 such that Au in P E 8 , and assume that the associated solution wpt, xq belongs to C 2 b pr0, `8q ˆp0, `8qq. Let w l k be the numerical solution obtained by the iteration rule (12)-( 13) and with the initial data u 0 k " u in px k q. Then for all r ą 0 there exists a constant C r ą 0 such that for all T ą 0 sup

t l ďT }e l } 1 ď C r T `2 N n ´log n log 2 `2´r N n ˘,
where e is the "error" vector defined by e l k " w l k ´wpt l , x k q.

This is a convergence result since if n and N {n tend to infinity in such a way that log n log 2 ´N n Ñ `8, then the error tends to zero. For instance if we take N " t ε log 2 n log nu with ε P p0, 1q we get a speed of convergence of order n ε´1 `n´rε . Choosing r " 1 ε ´1 we obtain an order n ε´1 for any ε P p0, 1q, meaning that the scheme is "almost" of order 1 in n.

Proof. We write the scheme in a condensed form w l`1 " Aw l where A is the iteration matrix. The contraction property reads }A} ď 1, and it implies the stability of the scheme. Now we prove the consistency. Taylor expansions give wpt l`1 2 , x k q " wpt l , x k q `δt 2 B t wpt l`1 2 , x k q `Opδt 2 q

x k wpt l , x k q " x k´1 wpt l , x k´1 q `px k ´xk´1 qB x pxwqpt l , x k q `Oppx k ´xk´1 q 2 q and so wpt l`1 2 , x k q " x k´1

x k wpt l , x k´1 q `δt " 1 2 B t wpt l`1 2 , x k q `Bx pxwqpt l , x k q ı `Opδt 2 `pδtqpx k ´xk´1 qq.

We get

wpt l`1 2 , x k q " 1 1 `δtB k x k´1 x k wpt l , x k´1 q `δt 1 `δtB k " 1 2 B t wpt l`1 2 , x k q `Bx pxwqpt l , x k q `Bk wpt l`1 2 , x k q ı `Opδt 2 `pδtqpx k ´xk´1 qq. Now from wpt l`1 , x k q " wpt l`1 2 , x k q `δt 2 B t wpt l`1 2 , x k q `Opδt 2 q we deduce wpt l`1 , x k q " 1 1 `δtB k x k´1 x k wpt l , x k´1 q `2δtB k`n 1 `δtB k`n δt δx wpt l , x k`n´1 q `δt 1 `δtB k " 1 2 B t wpt l`1 2 , x k q `Bx pxwqpt l , x k q `Bk wpt l`1 2 , x k q ı `δt B t wpt l`1 2 , x k q ´2δtB k`n 1 `δtB k`n δt δx wpt l , x k`n´1 q `Opδt 2 `pδtqpx k ´xk´1 qq.
It remains to estimate

ε l k :" 1 2 " 1 1 `δtB k `1ı B t wpt l`1 2 , x k q `1 1 `δtB k B x pxwqpt l , x k q `Bk 1 `δtB k wpt l`1 2 , x k q ´2B k`n 1 `δtB k`n 1 1 `δx wpt l , x k`n´1 q
and the boundary condition

ε l 0 :" wpt l , x 0 q ´p1 `δtB 1 q " x 2N x 0 wpt l , x 2N q `n´1 ÿ k"1 x k x 0 δtB k`1 1 `δtB k`1 wpt l , x k q  .
Using that B x pxwqpt l , x k q " B x pxwqpt l`1 2 , x k q `Opδtq wpt l , x k`n´1 q " wpt l`1 2 , x k`n q `Opδt `px k ´xk´1 qq and B t wpt l`1 2 , x k q `Bx pxwqpt l`1 2 , x k q `Bk wpt l`1 2 , x k q ´2B k`n wpt l`1 2 , x k`n q " 0 we get

|ε l k | ď 1 1 `δtB k " δtB k 2 |B t wpt l`1 2 , x k q| `ˇˇ1 `δtB k 1 `δtB k`n 1 1 `δx ´1ˇˇˇ2 B k`n wpt l`1 2 , x k`n q  `Opδt `px k ´xk´1 qq ď 1 1 `δtB k " δtB k 2 }Au in } E8 x k Upx k q `δtB k `δx `δtB k`n `δtδxB k`n p1 `δtB k`n qp1 `δxq 2B k`n }u in } E8 x k`n Upx k`n q  `Opδt `px k ´xk´1 qq ď δt " }Au in } E8 2 max k px k B k Upx k qq `2}u in } E8 max k pB k`n pB k `1 `2B k`n qx k`n Upx k`n qq  `Opδt `px k ´xk´1 qq " Opδt `px k ´xk´1 qq,
where we have used that |wpt, xq| " |xT t u in pxq| ď }u in } E8 x Upxq and

|B t wpt, xq| " |xB t T t u in pxq| " |xT t Au in pxq| ď }Au in } E8 x Upxq.
The boundedness of x Þ Ñ xBpxqUpxq and x Þ Ñ 2Bp2xqp1`Bpxq`2Bp2xqqx Up2xq is a consequence of the assumptions (4) on the continuous function B and the estimates on U available in [START_REF] Doumic | Eigenelements of a general aggregationfragmentation model[END_REF][START_REF] Balagué | Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates[END_REF]. Similarly for the boundary condition we have

|ε l 0 | ď }u in } E8 x 0 Upx 0 q `p1 `B1 q " x 2 2N }u in } E8 x 2N Upx 2N q `δt }u in } E8 x 0 n´1 ÿ k"1 x 2 k B k`1 Upx k q  which is small when N n is large since x 0 " 2 ´N n , x n " 2x 0 , x 2N " 2
N n and we know from [START_REF] Doumic | Eigenelements of a general aggregationfragmentation model[END_REF] that for all r P R, when x Ñ 0, Upxq " Opx r q and from [3] that when x Ñ `8,

Upxq " Ope ´K1 γ 1 x γ 1 q.
More precisely for all r P R we have We see the oscillatory behaviour for U 1 and U 2 .

|ε l 0 | " O ´2´p1`rq N n `23 N n exp `´K1 γ1 2 γ1 N n ˘`2 ´p1`γ0`rq N n ¯" Op2 ´r N n q.
R t u in as pR t u in q n " 5 ř k"´5 pu in , U n k qe 2ikπ log 2 t U n k
, and define the two error terms in the discrete norm E n 2 defined as E 2 :

Error 2

E n 2 :" › › › › u n pt, xqe ´t ´pR t u in q n › › › › 2 E n 2 , Error Mean 2 E n 2 :" › › › › 1 log 2 t`log 2 ş t u n ps, xqe ´sds ´Un 0 › › › › 2 E n 2 , ( 14
)
where u n is the numerical approximation of u. We observe several phases, which illustrate exactly the theory. First, a very fast decay of the quantity Error E n 2 , linked to its initial very high value since the constant C such that u in ď CU 0 is very large. Then we have a phase of exponential decay for both Error E n 2 and Error Mean E n 2 (the linear decay in Figure 5 Right), corresponding to a spectral gap, as proved in [START_REF] Greiner | Growth of cell populations via one-parameter semigroups of positive operators[END_REF] for the assumption of a compact support, which is satisfied here due to the truncation. Of note, this phase lasts much more for Error Mean E n 2 than for Error E n 2 , most probably due to averaging errors in the non-oscillatory solution. The final phase is either a plateau for Error Mean E n 2 (linked to our definition of U n 0 ) or a quadratic increase for Error E n 2 , which is linked to the fact that the convergence constant in Theorem 3 depends linearly on the final time.

Discussion

We studied here the asymptotic behaviour of a non-hypocoercive case of the growth-fragmentation equation. In this case, the growth being exponential and Left: for the peak as initial condition. Right: for the smooth initial condition. the division giving rise to two perfectly equal-sized offspring, the descendants of a given cell all remain in a countable set of characteristics. This results in a periodic behaviour, the solution tending to its projection on the span of the dominant eigenvectors. Despite this, we were able to adapt the proofs based on general relative entropy inequalities, which provide an explicit expression for the limit.

Our result could without effort be generalised to the conservative case, where only one of the offspring is kept at each division: in Equation ( 1), the term 4Bp2xqup2xq is then replaced by 2Bp2xqup2xq. The consequence is then simply that the dominant eigenvalue is zero, a simple calculation shows that the dominant positive eigenvector is x Upxq, and all the study is unchanged.

Equation (1) may also be viewed as a Kolmogorov equation of a piecewise deterministic Markov process, i.e. as the equation satisfied by the expectation of the empirical measure of this process, see [START_REF] Cloez | Limit theorems for some branching measure-valued processes[END_REF][START_REF] Haas | Asymptotic behavior of solutions of the fragmentation equation with shattering: an approach via self-similar Markov processes[END_REF]. Our study corresponds exactly to the case without variability in the growth rate studied in [START_REF] Doumic | Statistical estimation of a growth-fragmentation model observed on a genealogical tree[END_REF]. In [START_REF] Doumic | Statistical estimation of a growth-fragmentation model observed on a genealogical tree[END_REF], a convergence result towards an invariant measure for the distribution of new-born cells is proved (this measure being xBpxqUpxq up to a multiplicative constant). However, this does not contradict the above study, because the convergence result concerns successive generations and not a time-asymptotics. A deterministic equivalent corresponds to studying the behaviour of a time-average of the equation. Corollary 1 confirms that if we rescale the solution by e ´t and average it over a time-period, it does converge towards U.

Our result could also easily be extended to the case where the division kernel is self similar, i.e. kpy, dxq " 1 y k 0 p dx y q, and is a sum of Dirac masses specifically linked by the following relation (see Condition H in [START_REF] Doumic | Time asymptotics for a critical case in fragmentation and growth-fragmentation equations[END_REF]): Supppk 0 q " Σ where Σ is such that DL P N ˚Y t`8u, Dθ P p0, 1q, Dpp q PN, ďL Ă N, 0 ă p ă p `1 @ P N, ď L ´1, Σ " tσ P p0, 1q; σ " θ p u , pp q 0ď ďL are setwise coprime.

This condition expresses the fact that all the descendants of a given individual evolve permanently on the same countable set of characteristic curves. The case of binary fission into two equal parts corresponds to L " 1 and p L " 1 2 . Note also that for the same reason, an oscillatory behaviour also happens for the coagulation equation in the case of the so-called diagonal kernel [START_REF] Laurençot | Oscillatory dynamics in Smoluchowski's coagulation equation with diagonal kernel[END_REF].

Other generalisations may also be envisaged, for instance to enriched equations, or to other growth rate functions satisfying gp2xq " 2gpxq, see [START_REF] Diekmann | On the stability of the cell size distribution[END_REF], that is, functions of the form gpxq " xΦplogpxqq where Φ a logp2q periodic function.

An interesting future work could consist in strengthening the convergence result in Theorem 2. Indeed this result does not provide any rate of decay and the speed of convergence may depend on the initial data u in . A uniform exponential convergence would be ensured for instance by [1, Proposition C-IV.2.13], provided that one can prove that 0 is pole of A. This appears to be a difficult question, which is equivalent to the uniform stability of pT t q tě0 in X K or to the uniform mean ergodicity of pT t q tě0 in E 2 .

We chose to work in weighted L 2 spaces because the theory may be developed both very simply and elegantly in this framework, where the terms are interpreted in terms of scalar product and Fourier decomposition. The asymptotic result of Theorem 2 could most probably be generalised to weighted L 1 spaces, by considering the entropy inequality with an adequate convex functional. Developing a theory in terms of measure-valued solutions, in the same spirit as in [START_REF] Gwiazda | Generalized entropy method for the renewal equation with measure data[END_REF], would also be of interest, since the equation has no regularising effect: an initial Dirac mass leads to a countable set of Dirac masses at any time.

For 8 0wpt, xq dx " ż 8 0

 88 the numerical scheme, it is more convenient to consider the function wpt, xq :" xupt, xqe ´λt " xvpt, xq, which is solution to the conservative equation B Bt wpt, xq `B Bx pxwpt, xqq `Bpxqwpt, xq " 2Bp2xqwpt, 2xq. The conservation law reads ż wp0, xq dx and we also have the contraction property }wpt, ¨q} L 1 ď }wp0, ¨q} L 1 .

Figure 1 :

 1 Figure 1: The real part for the three first eigenvectors U 0 , U 1 , U 2 for Bpxq " x 2 .We see the oscillatory behaviour for U 1 and U 2 .

Figure 2 :

 2 Figure 2: Two different initial conditions. Left: peak in x " 2. Right: u in pxq " x 2 expp´x 2 {2q.

Figure 3 :

 3 Figure 3: Time evolution of max xą0 upt, xqe ´t.

Figure 4 :

 4 Figure 4: Size distribution upt, xqe ´t at five different times (each time is in a different grey). Left: for the peak as initial condition. Right: for the smooth initial condition.

Figure 5 : 2 (

 52 Figure 5: Left: initial distribution (full blue line) and dominant eigenvector (doted red line), for Bpxq " x 3 . We see that the constant such that u in ď U 0 is very large. Right: time evolution of Error E n 2 (doted red line) and Error Mean E n 2

  Introduce the sequence of functions h n pt, ¨q " hpt `n, ¨q. Since h and B t h are uniformly bounded in the Hilbert space E 2 , the Ascoli and Banach-Alaoglu theorems ensure that ph n q nPN is relatively compact in Cpr0, T s, E w 2 q where E w 2 is E 2 endowed with the weak topology. After extracting a subsequence, still denoted h n , we have h n Ñ g in Cpr0, T s, E w 2 q. Additionally since

			ş 8 0 D 2 rhpt, ¨qs dt ă `8,
	we have	ż T	ż T `n
		D 2 rh n pt, ¨qs dt "	D 2 rhpt, ¨qs dt Ñ 0.
		0	n
	and it ensures, using the definition (8) of D 2 , that hnpt,xq U pxq
	By contraction we get
		}qpt, ¨q} ď }Ahp0, ¨q}.

Define hpt, xq :" upt, xqe ´t ´`8 ÿ k"´8 pu in , U k qe 2ikπ log 2 t U k pxq " pI ´P qT t u in which is solution to Equation (2.2). Lemma 3 with p " 2 ensures that d dt }hpt, ¨q} ď 0, so that it decreases through time. Since it is a nonnegative quantity, it means that it tends toward a limit L ě 0 and it remains to show that L " 0. Let us adapt to our case the proof in B. Perthame's book [33, p.98]. Because of the contraction property, it is sufficient to do so for u in P DpAq which is a dense subspace of E 2 . Recall that for u in P DpAq the solution to Equation (1) can be understood as a classical solution, upt, ¨q belonging to DpAq for all time. The last property in Proposition 7 ensures that X Ă DpAq, so hp0, ¨q P DpAq. Define qpt, xq " B t hpt, xq which is clearly a mild solution to Equation (1) with initial datum qpt " 0, xq " Ahpt " 0, xq.

Acknowledgments

M.D. has been supported by the ERC Starting Grant SKIPPER AD (number 306321). P.G. has been supported by the ANR project KIBORD, ANR-13-BS01-0004, funded by the French Ministry of Research. We thank Odo Diekmann and Rainer Nagel for their very useful suggestions.

We conclude by the standard argument of Lax which deduces convergence from stability and consistency. By definition we have

we conclude by iteration, using that e 0 " 0.

Illustration

We illustrate here first the case Bpxq " x 2 : in Figure 1, we draw the real part of the first eigenvectors, taken for k " 0, 1, 2. The oscillatory behaviour will depend on the projection of the initial condition on the space generated by pU k q: it will be stronger if the coefficients for k ‰ 0 are large compared to the projection on U 0 . We show two results for two different initial condition (Figure 2, Left and Right respectively), one a peak very close to the Dirac delta in x " 2 and the other very smooth. In both cases, the solution oscillates, as showed in Figures 3 and4, though since the projections on X are very different (with a much higher projection coefficient on the positive eigenvector for the smooth case than for the sharp case) these oscillations take very different forms. In the second case, they are so small that for any even slightly diffusive numerical scheme they are absorbed by the diffusion, leading to a seemingly convergence towards the dominant positive eigenvector. We also see that the equation is no more regularizing: discontinuities remain asymptotically for the Heaviside case.

To explore the speed of convergence in the Theorem 2 and Corollary 1, we choose Bpxq " x 3 , take a very smooth initial condition (Figure 5 Left), with 8 ş 0 u in pxqxdx " pu in , U 0 q " 1, for which the coefficients pu in , U k q decrease rapidly with k, so that R t u in is very well estimated by the series truncated for k " 5. We then take a refined grid with n " 500 and N " t n log n 2 log 2 u " 2241. To estimate U 0 , we use that 1 log 2 t`log 2 ş t ups, xqe ´sds tends to U 0 , and take this limit value to define U n 0 " U 0 and accordingly U n k " U k . We then define the estimate for


Luc-Sy Tran

Olivier Herbinet

Yuyang Li

Julia Wullenkord

Meirong Zeng

Eike Bräuer

Fei Qi

Katharina Kohse-Höinghaus

Frédérique Battin-Leclerc

Low-temperature gas-phase oxidation of diethyl ether: Fuel reactivity and fuel-specific products

Keywords: Biofuel, Diethyl ether, Low-temperature oxidation, Dual NTC zone phenomenon, Detailed kinetic model

Diethyl ether (DEE) has been recently suggested as a potential biofuel for compression-ignition engines that are known to be significantly controlled by low-temperature (LT) chemistry. However, the LT oxidation of DEE has not fully been understood in term of the formation of LT fuel-specific products. We have thus studied the oxidation of DEE by examining detailed profiles of its oxidation products under LT conditions (400-1100 K). To this end, we have used a dedicated experimental setup including a nearly-atmospheric jet-stirred reactor (JSR) coupled to online gas chromatography (GC). The experiments were complemented by measurements made with a JSR coupled to tunable synchrotron vacuum ultraviolet (SVUV) photoionization (PI) molecular-beam mass spectrometry (MBMS) for a cross-validation of the identification of important LT species.

Experimental results indicate that DEE is very reactive; it starts to react around 425 K. DEE exhibits an unusual oxidation behavior with two negative temperature coefficient (NTC) zones in the JSR study. Because of this two-NTC observation, additional experiments were performed with a plug flow reactor (PFR) combined with electron ionization (EI)-MBMS, confirming this behavior in the two types of reactor. Moreover, about 20 oxidation species in C1-C4 range were detected with several intermediates containing 2-3 O-atoms. Acetic acid is found to peak at 525 K with a very large amount, suggesting that it is a key species in the early stage of DEE's LT oxidation. Possible DEE-consumption paths leading to acetic acid formation could play an important role in the oxidation mechanism of DEE. A new model is proposed based on the present experimental observations to include new primary LT reaction paths. The model reproduces the experimental phenomena quite well and enhances the understanding of the two-NTC-zone occurrence and of intermediates containing 2-3 O-atoms during the LT oxidation of DEE.

Introduction

Diethyl ether (DEE, C4H10O) has been proposed as a promising biofuel for compression-ignition engines [1] and as an excellent ignition improver for homogeneous charge compression ignition engines [2]. These engines are known to be significantly controlled by low-temperature (LT) chemistry. A deeper understanding of DEE's LT oxidation chemistry is therefore desirable. Several experimental studies were recently performed regarding the high-temperature (HT) oxidation of DEE, e.g., ignition delay times (IDTs) [3,4], flame speeds [5,6], and species profiles [3,5,7,8]. However, experimental studies with detailed species profiles for its LT gas-phase oxidation chemistry are surprisingly scarce. Early work of Salooja [9] has reported limited species data at different pre-ignition stages in a quartz chamber for a fuel-rich DEE-air mixture. Oscillatory cool flames in DEE/O2/N2 mixtures were studied in [10], without providing species profiles. Slow oxidation (< 425 K) of DEE addressing storage-safety problems has also been studied [11][START_REF] Waddington | Proc. R. Soc. Lond. A[END_REF][START_REF] Di Tommaso | [END_REF]. Recently, IDTs of DEE were measured [4] in a rapid compression machine at 253-1317 kPa for various equivalence ratios. Moreover, the LT oxidation of DEE was increasingly investigated by model predictions [14][15][16][17]. Sakai et al. [14] developed a model including LT reactions using rate coefficients calculated in their previous work [18] for unimolecular reactions derived from the first O2 addition, whereas rate coefficients for reactions from second O2 addition were estimated by analogy to those of the first O2 addition or to those of species with similar structures. Hu et al. [15] used this LT reaction subset in their model without modification. Tang et al. [16] improved a model of Yasunaga et al. [3] by mostly including important LT reactions proposed in [14,18]. Independently, Eble et al. [17] proposed a model using kinetic data empirically estimated or based on dimethyl ether reactions.

Experimental LT data of DEE, however, especially detailed species profiles, remain very scarce. Here, we have thus measured detailed profiles of oxidation products of DEE in a temperature range of 400-1100 K, providing useful data to analyze and understand details of the reaction mechanism in this technically relevant temperature regime.

Experiments

In the present experiments, we used two types of reactors, i.e., a jet-stirred reactor (JSR) and a plug flow reactor (PFR), and three analytical techniques: online gas chromatography (GC), tunable synchrotron vacuum ultraviolet photoionization molecular-beam mass spectrometry (SVUV-PI-MBMS), and electron ionization (EI)-MBMS.

JSR-GC experiment

Species mole fraction profiles were measured in an atmospheric JSR directly connected to GCs at the LRGP-Nancy that was previously described in [19]. The reactor (81.2 cm 3 volume) is made of fused silica and heated electrically. The GCs were fitted with different columns (Carbosphere, PlotQ, and HP-5 columns) and detectors, i.e., a thermal conductivity detector (TCD), a flame ionization detector (FID)/methanizer for quantification, and a mass spectrometric detector (MS) with electron ionization at 70 eV for identification. The GC-loop/injector temperature was reduced from the usual value of 523 K to 423 K to avoid DEE pre-reactions. Calibrations were performed using cold-gas mixtures when available. For other species, calibration factors were assumed relying on hydrogenation by the methanizer, to be identical to those of the alkanes with the same number of carbon atoms, with an uncertainty < 15%.

Flow rates of liquid DEE (Sigma-Aldrich, purity ≥ 99.7%) were controlled using a Coriolis flow controller followed by an evaporator set to 400 K. Experiments were performed for a mixture DEE/O2/He with equivalence ratio ϕ = 1, initial mole fraction 0.01/0.06/0.93, residence time of 2 s, temperature range of 400-1100 K, and pressure of 106.7 kPa.

JSR-SVUV-PI-MBMS experiment

To cross-validate the GC identification of species containing 2-3 oxygens (key LT intermediates), photoionization efficiency (PIE) spectra were measured with SVUV-PI-MBMS coupled to a JSR, located at the Hefei National Synchrotron Radiation Laboratory. Similar operating conditions were used, but replacing He by Ar that served as a reference in the SVUV-PI-MBMS data evaluation. The apparatus has been described elsewhere [20]. Gas samples were extracted from the JSR by a quartz sampling cone. The resulting molecular beam was then intersected and ionized by the tunable synchrotron VUV light. The ions were transferred by an ion guide to a home-made reflectron time-of-flight mass spectrometer (TOF-MS) with a mass-resolving power (m/Δm)∼2500. The PIE spectra were measured in a photon energy range of 8-11.5 eV, with uncertainties in the determination of ionization energies of ± 0.05 and ± 0.10 eV for strong and poor signal-to-noise ratio, respectively.

PFR-EI-MBMS experiment

Because of the unusual oxidation behavior with two negative-temperature-coefficient (NTC) zones observed in the JSR experiments, additional experiments were performed in a PFR coupled with EI-MBMS at Bielefeld University. The setup has been described in [21]. The reactor consists of a fused silica tube (8 mm inner diameter, 1.40 m length) and was electrically heated. Oxidation products were sampled at the reactor exit with a quartz nozzle (∼50 µm orifice diameter). The sampled gases were directed through a copper skimmer to the ion source of the mass spectrometer. Chemical species were ionized with an electron beam of 17 eV and accelerated into the reflectron TOF-MS with a mass resolution of m/Δm∼2200. Evaluation procedures were previously reported in [21] and are associated with uncertainties for major species of ∼15% and factors of 2-4 for intermediate species.

A mixture DEE/O2/Ar (0.005/0.030/0.965) was used with ϕ = 1, total flow of 0.5 slm, initial gas velocity of 0.23 m/s at 373 K, and pressure of 97 kPa (temperature-dependent residence time of ∼2-4 s). Liquid DEE (VWR Chemicals, purity ≥ 99.7%) was delivered with a dialysis pump and evaporated at 373 K. Carbon balances for both JSR and PFR experiments were checked to be close to 100%.

Model development and simulation method

Recent DEE models [14][15][16][17] cannot satisfactorily reproduce the present experimental data. To contribute to the interpretation of this data, we have therefore attempted to develop a new LT oxidation sub-mechanism of DEE. It includes usual LT reaction classes (as considered in the above literature models) and new reaction classes that were suggested from the present experimentally-detected species. This LT sub-mechanism was coupled to the model developed previously only for HT DEE chemistry [5], and implemented into the most recent core model of the NUI-Galway group [19]. The complete model consists of 3540 reactions involving 748 species; it is available in CHEMKIN format in Supplemental Material 1 (SM1).

The newly-proposed LT sub-mechanism of DEE includes 31 main reaction classes with details available in SM2. Only main features are presented here. The addition of O2 to the fuel radicals (first O2 addition) is the first step in the LT chain-branching process to form ethoxyethylperoxy radicals (ROO); the rate coefficients of this reaction class were determined using the original kinetic data (including thermochemistry) proposed by Sakai et al. [14], expecting thus a good consistency with the original source. The order of magnitude of these rate coefficients is in the range of those proposed for DME and n-pentane chemistry (compare the core mechanism). The kinetic data of bimolecular reactions following this first step were estimated based on reactions proposed for LT oxidation of di-n-butyl ether (DBE) [22]. Note that the effect of ether function in DBE is quite similar to that in DEE, e.g., close bond dissociation energies of Cα-H (∼95 kcal mol -1 ) and Cα-O (∼87 kcal mol -1 ) bonds (Cα is the carbon bonded to O atom) [5,22]. Kinetic data of ROO isomerization to hydroperoxyl-fuel radicals (QOOH) and of unimolecular reactions of these latter radicals were taken from calculations of Sakai et al. [18]. However, to improve the prediction of DEE reactivity at 500-550 K, we used averaged rate coefficients between those calculated by Sakai et al. [18] and those proposed empirically by Eble et al. [17] for the reaction CH3CHOCH(OOH)CH3→CH3CHO+CH3CHO+OH.

Rate coefficients of the second O2 addition forming the OOQOOH radicals were estimated based on the first O2 addition. Kinetic data of OOQOOH isomerization followed by a fast O-O β-scission to form ketohydroperoxides, and decomposition of the latter species were determined following Thion et al. [22]. Indications from the present experiments (see Section 4) led us to consider new reaction classes for LT DEE oxidation illustrated in Figure 1. The rate-determining step in the Korcek decomposition process proposed by Jalan et al. [23] was used for path i in Figure 1. As a first approximation, kinetic data of path ii was taken from theoretical calculations by Wang et al. [24] for a structurally-similar species (OCH2OCH=O) with an activation energy reduced by 2 kcal mol -1 to consider the difference in bond dissociation energies between tertiary-and secondarycarbon positions, and without any modification of the proposed pre-exponential factor. This data was estimated at the high-pressure limit to keep it consistent with those of competitive paths. More accurate determination of these rate coefficients remains an interesting subject needing theoretical investigations that are beyond the scope of the present study. Note that the path ii supplies the reactive radical CH3CO, increasing also the system reactivity. Rate coefficients of reactions in path iii were estimated mainly based on rate rules [25]. These reaction classes were not included in the DEE literature models [14][15][16][17].

Most thermochemical data of fuel, fuel radicals, and newly-involved species in LT reactions were obtained from previous quantum chemistry calculations [5,14]. Otherwise, such data were calculated using the group additivity method implemented in the RMG program [25].

Simulations of JSR data were performed using the "Perfectly_Stirred_Reactor" module of OpenSMOKE++ [26] with isothermal-constant pressure approach, whereas a multi-zone nonisothermal solution was applied following [21] for PFR simulations. The present model, after including LT reactions, was successfully examined against HT data [3][4][5]7] (details given in SM2). ) were not included, but their prediction performance is comparable to that of the Sakai model [14] because kinetic data of LT reactions in these models are from the same source. Figure 2a,b shows that DEE starts to react at quite low temperature (∼425 K). Importantly, it exhibits an unusual oxidation behavior with two NTC zones observed in both experiments. The two-NTC behavior is also evident in the profile shape of some products (including CO and CH2O) show two peaks in the two highest LT fuel conversion zones (further details given in SM2). The DEE conversion profile is composed of five zones. DEE is quickly consumed up to 70% at very low temperature, 425-550 K in the JSR (PFR: 425-500 K). The first NTC occurs until 600 K (PFR: 550 K), whereafter DEE's reactivity increases again until ∼650 K. The second NTC and a decrease in fuel reactivity is seen until ∼750 K. Finally, the reactivity rises again, and DEE is totally consumed above 950-1000 K. All three presented models show a two-NTC-zone behavior under JSR conditions, but only the present model can quantitatively predict the DEE conversion for the studied conditions. Model analyses with these models indicate the same origin of the two-NTC behavior, which is extensively explained in the coming paragraph. Under PFR conditions, all three models significantly under-predict DEE conversion, especially between ∼600-800 K, while the experiment shows the unambiguous presence of intermediate species (see SM2). The present model still predicts two-NTC behavior and well reproduces the first NTC, however. To further inspect the PFR discrepancies, we performed simulations assuming an uncertainty of ∼5% in temperature (by reducing the heat transfer coefficient by a factor of 2), 5% in pressure, or a factor of 2 for rate coefficients of H-abstractions from DEE by CxHyO2 radicals. The latter reaction class was chosen for the PFR test because it is important for the DEE conversion in the range of 600-800 K, and its estimated rate coefficients have potentially high uncertainties, and modifications of these rate coefficients affect only moderately the JSR simulations. Results (profiles i, ii, iii in Figure 2b, respectively) show that the individual effect of the tested uncertainties on DEE consumption is limited (∼5-12% at ∼625 K). Including all three uncertainty sources (profile iv) significantly enhances DEE conversion, especially in the second NTC zones (up to 90% at ∼625 K), improving the model-experiment agreement, showing the delicate temperature-sensitive reaction behavior. Remaining discrepancies at higher temperatures indicate influences of other error sources. Note that the tested uncertainties do not significantly affect the first NTC zone.

Figure 2c aims to explain the occurrence of the two NTC zones based on the fuel's structure. Only reaction paths via H-abstractions at Cα of DEE are shown because they are main consumption routes of DEE (see Section 4.2), and mostly control the fuel reactivity. The second NTC of DEE results from the competition between the first O2 addition to the fuel radicals (reaction r1 in Figure 2c) and the decomposition of these radicals (reaction r2). The first NTC is unusual, mainly resulting from a competition between the second O2 addition (reaction r3) and the C-O/O-O β-scission succession of the hydroperoxyl-fuel radical (Q1OOH) (reaction r4). Reaction r3 is favored at LT, and its subsequent reactions are chain-branching steps because they lead to formation of reactive species, such as an OH radical and ketohydroperoxide, which decomposes easily to form a second OH radical and other small radicals such as CH3 and CH3CO. In contrast, reaction r4 inhibits the system's reactivity because this reaction produces only one OH radical and two stable acetaldehyde molecules. The first NTC is significantly suppressed when reaction r4 is excluded in the simulation with the three presented models (thin lines in Figure 2a). For further demonstration, sensitivity analyses for JSR DEE consumption at 560 K (in the first NTC zone) and at 700 K (in the second NTC zone) were performed using these three models, showing the important role of reactions r4 and r2 in the first and second NTC zones, respectively (more details available in SM2). With increasing temperature (∼575-625 K), the formed CH3CO radicals can react and supply reactive species increasing again the system's reactivity. Indeed, near 600 K, CH3CHO is significantly consumed and produces CH3CO that reacts then with O2 leading to CH3CO3.

The latter radical reacts with DEE (increasing the system reactivity) and other species (e.g., HO2, CH3CHO) producing CH3CO3H (peroxyacetic acid) that decomposes to form OH (increasing the system reactivity). Moreover, temperature-dependent OH production rates show that their sum is consistent with the reactivity described above. OH production increases with temperature between 450 and 550 K mainly by formation/decomposition paths of DEE-specific ketohydroperoxides, between 600 and 650 K by decomposition of CH3CO3H, CH3O2H and C2H5O2H, and after a transition to HT chemistry above 750 K by H2O2 decomposition (further details in SM2).

Note that no first NTC zone has been observed in the LT oxidation of the corresponding five-heavyatom hydrocarbon (n-pentane) because the succession of β-scissions (similar to r4) in the case of n-pentane oxidation is energetically less favored due to the absence of the ether function, and therefore cannot compete with the second O2 addition (similar to r3) in the LT range. Furthermore, the first NTC zone was also not observed for dimethyl ether (DME). DME only has primary carbon atoms, inducing stronger C-H bonds. It is significantly less reactive and starts to react at much higher temperatures. For further demonstration, simulations were performed for DME and n-pentane (using the present core model originated from [19]) at the same conditions, showing that both DME and n-pentane only react at higher temperatures (≥ 550 K) (see SM2). The occurrence of a two-NTC-zone behavior is therefore only likely for a fuel such as DEE that presents simultaneously an ether function and high reactivity at LT. Note that Thion et al. [22] have also observed two-NTC-zone behavior for DBE, a molecule satisfying both abovementioned criteria. They explained the first NTC zone by a difference in temperature dependence of formation/consumption reactions of a DBE-specific ketohydroperoxide, C3H7C(=O)OCH(OOH)C3H7, produced through the second O2 addition process, and propylhydroperoxide, C3H7OOH, produced via some subsequent steps in the ketohydroperoxide's decomposition. Their explanation differs from our analysis for DEE. Because of differences in the used conditions and in the species distribution between DEE and DBE, the origin of the unusual chemistry with two-NTC behavior could be different for these two ethers. This highly interesting oxidation behavior thus needs further comprehensive studies.

Fuel-specific intermediate species

For an overview, peak mole fractions of selected intermediates measured in the oxidation of DEE are summarized in Table S2 (SM2); they are reported together with the temperatures at which they occur, typically in the range ∼480-900 K. Several categories of species are observed based on experimental data. First, species reaching maximum mole fraction above 800 K are mostly hydrocarbons including methane (CH4), ethylene (C2H4), ethane (C2H6), and propene (C3H6), produced via either HT chemistry or secondary reactions. Second, species with peaks in the intermediate temperature range of 550-750 K include formaldehyde (CH2O), acetaldehyde (CH3CHO), ethanol (C2H5OH), ethyl vinyl ether (C4H8O, EVE), and 2-methyl-1,3-dioxolane (C4H8O2-cy). Finally, several species exhibit peak mole fraction below 550 K including intermediates with 2-3 O-atoms such as acetic acid (CH3COOH), methyl formate (CH3OCHO), ethyl formate (C3H6O2, EF), and acetic anhydride (C4H6O3, AA). Note that some of these species exhibit corrosive or toxic potential such as acetic acid and acetic anhydride. Besides complete JSR and PFR datasets provided in SM3, full profiles of selected species are presented here. Simulations of the experimental species profiles are overall acceptable in both reactors, even in the PFR at temperatures outside the second NTC zone in which deviations are related to the previously discussed significant under-prediction of the fuel consumption.

Figure 3a-c presents intermediates containing 0-1 O-atom. While profiles of CH4, C2H6 (a), EVE, C3H6 (c) show mostly a unimodal behavior, of several species, including C2H5OH (a), CH2O, C2H4, and CH3CHO (b) have bimodal or multimodal behavior, indicating their more complex formation. The present model captures most of these trends well although some discrepancies between model and experiment are noted for a few species, such as EVE. A model analysis shows that EVE is formed via HO2 elimination of ethoxyethylperoxy radicals or by H-abstractions from fuel radicals by O2 or HO2. The estimated rate coefficients for the latter reaction class could contain high uncertainties, potentially resulting in EVE under-prediction. C2H5OH (a well-known product of HT DEE decomposition) was also detected at low temperature (Figure 3a), indicating LT-chemistry contributions to C2H5OH formation. Toxic aldehydes (CH2O and CH3CHO) are formed not unexpectedly in high amounts over a large range of temperatures. This latter undesirable trend was also previously observed under flame conditions [5]. 5c), clearly indicating production of these species in DEE oxidation. Note that in the absence of literature data, the IE of 2-methyl-1,3-dioxolane was theoretically calculated using the procedure described in [28]. Simulations with the present model included in Figure 4 indicate satisfactory prediction of the formation of intermediates containing 2-3 O-atoms, especially for the most abundant species CH3COOH. However, our model under-predicts the formation of CH3OCHO (a low mole fraction species, Figure 4a), indicating that possible unknown additional pathways leading to CH3OCHO might be missing in the mechanism.

For better understanding the LT chemistry of DEE and the formation of experimentally-observed species, rate of production analyses using the present model at different temperatures (500, 560, and 700 K) for the studied JSR conditions are presented in Figure 6. Here, H-abstraction reactions by radicals R' (OH, CH3O, H, etc.) are responsible for DEE consumption. These reactions mainly occur at Cα of DEE because of its lowest C-H bond energy (95.7 kcal mol -1 [5]), yielding the R1 radical (see Figure 6). H-abstractions from Cβ (C-H bond energy of 103.2 kcal mol -1 ) producing the R2 radical account for only 6-8%. The branching ratio of consumption paths of species from R1 and R2 significantly depends on temperature. At HT (700 K), a large fraction of fuel radicals (R1 and R2) is consumed by C-O β-scission forming CH3CHO + C2H5 and C2H4 + C2H5O, respectively. At lower temperatures both fuel radicals react largely by O2 addition leading to the formation of 1-ethoxyethylperoxy (R1OO) and 2-ethoxyethylperoxy (R2OO) radicals, respectively. Under LT conditions, the main consumption path of R1OO leads to the formation of acetic acid (CH3COOH) through some intermediate steps, explaining why we detected this species with very high amounts. Furthermore, along this consumption path of R1OO, acetic anhydride (AA) is produced via H-abstraction/O-O β-scission from ketohydroperoxide HOOQ1=O (Figure 6). Ethyl formate (EF) and ethyl acetate (EA) are formed via reactions of two R1OO radicals or R1OO with CXHYO2 radicals. Several paths following H-abstractions from the Cα positions of DEE lead to the formation of CH3CHO that was measured with high mole fraction (Table S2, Figure 3). While most detected species are found to be products of H-abstractions at Cα, 2-methyl-1,3-dioxolane (C4H8O2-cy) is mainly formed through H-abstractions at Cβ, by the cyclic ether formation process from the Q2OOH radical.

Figure 1 :

 1 Figure 1: Examples of the newly-considered reaction classes (in dashed box) for the DEE LT submechanism, leading to the formation of experimentally-detected species, e.g., acetic acid (CH3COOH) and acetic anhydride (AA).

4 .

 4 Figure 2a,b presents the mole fraction profiles of DEE obtained in JSR-GC (a) and PFR-EI-MBMS (b) experiments as function of temperature, together with predictions by three models (present, Eble [17], and Sakai [14] models). For clarity, other models (Hu et al. [15], Tang et al.[16]) were not included, but their prediction performance is comparable to that of the Sakai model[14] because kinetic data of LT reactions in these models are from the same source.

Figure 2 :

 2 Figure 2: DEE conversion as a function of reaction temperature in two reactor types: JSR (a) and PFR (b), and proposed mechanism resulting in two NTC zones (c). In (a) and (b): symbols: experiments; lines: present (solid), Eble [17] (dotted), Sakai [14] (dash-dotted) models. In (a): thin lines: model without reaction r4 in (c). In (b): PFR simulations using the present model with including individual uncertainties of temperature (i), pressure (ii), rate coefficients of DEE+CxHyO2(iii), and simultaneous inclusion of these uncertainties (iv).

Figure 3 :

 3 Figure 3: Mole fraction profiles of selected intermediates containing 0-1 O-atom. Symbols: JSR-GC experiment; lines: present model. An analogous figure for those from the PFR-EI-MBMS experiment is available in SM2.

Figure 4 :

 4 Figure 4: Profiles of intermediates containing 2-3 O-atoms. Symbols: experiment; lines: present model. Top: JSR-GC experiment (a-c); bottom: PFR-EI-MBMS experiment (d-f). In (e) and (f), left axis: simulated mole fraction; right axis: signal intensities (quantification of C4H8O2 and C4H6O3 was not possible because of lacking EI cross sections).

Figure 5 :

 5 Figure 5: PIE curves of m/z = 60 (a), 88 (b), and 102 (c) from the JSR-PI-MBMS experiment at 500 K.

Figure 6 :

 6 Figure 6: Reaction pathway analysis for the consumption of DEE under the studied JSR conditions using the present model. The experimentally-quantified species are highlighted by shadowed squares. The numbers are percent contribution to the consumption of the species on the source side at different temperatures, coded with different fonts: bold: 500 K, normal: 560 K, italic: 700 K. The thick arrow highlights the main route of acetic acid formation.

5 .

 5 The present paper reports new experimental data for species profiles measured in the oxidation of DEE at ϕ = 1 in two nearly-atmospheric reactor types (JSR and PFR) using GC and MBMS techniques. Two NTC zones have been observed. The first NTC zone is an unusual one and mainly results from a competition of the second O2 addition and the succession of C-O/O-O β-scissions of the hydroperoxyl-fuel radical (Q1OOH). Several intermediates containing 2-3 O-atoms were quantified in the present work and identified as important primary intermediates of the LT oxidation of DEE. A new model for DEE LT oxidation was established including new reaction classes. Results indicate that the main consumption path of DEE at LT leads mostly to acetic acid and CH3CO via one of the newly-added reaction classes, contributing to a new picture regarding the LT oxidation of DEE compared to available studies in the literature.

Acknowledgments

The authors wish to thank J.C. Lizardo Huerta for helpful discussions regarding the kinetic model. LST thanks the COST Action CM1404 and the Alexander von Humboldt-Foundation for financial support. Partial support from the Deutsche Forschungsgemeinschaft under project KO1363/31-1 is gratefully acknowledged. The research at SJTU was supported by the National Natural Science Foundation of China (51622605, 91541201) and the Shanghai Science and Technology Committee (No. 17XD1402000).