Vincent Bacot

Sander Wildeman

Surabhi Kottigegollahalli Sreenivas

Maxime Harazi

Xiaoping Jia

Arnaud Tourin

Mathias Fink

Emmanuel Fort
email: emmanuel.fort@espci.fr

Dissipation driven time reversal for waves

Dissipation is usually associated with irreversibility. Here we present a counter-intuitive concept to perform wave time reversal using a dissipation impulse. A sudden and strong modification of the damping, localized in time, in the propagating medium generates a counter-propagating timereversed version of the initial wave. In the limit of a high dissipation shock, it amounts to a 'freezing' of the wave, where the initial wave field is retained while its time derivative is set to zero at the time of the impulse. The initial wave then splits into two waves with identical profiles, but with opposite time evolution. In contrast with other time reversal methods, the present technique produces an exact time reversal of the initial wave field, compatible with broad-band time reversal. Experiments performed with interacting magnets placed on a tunable air cushion give a proof of concept. Simulations show the ability to perform time reversal in 2D complex media.

Dissipation in physical systems introduces a time-irreversibility by breaking the time symmetry of their dynamics [1] [2] [3] [4] [5]. Similarly damping in wave propagation deteriorates timereversal (TR) operations. However, for short times compared to the characteristic dissipation time, the dissipation-free wave-propagation model remains a good approximation of the wave dynamics [6]. Thus, increasing dissipation can generally be seen as effectively reducing the reversibility of the system. On the other hand, dissipation can be seen as the time reverse of gain. This is exemplified in the concept of coherent perfect absorbers (CPA), combining an interplay between interferences and absorption to produce the equivalent of a time-reversed laser [7] [8].

CPA addresses the TR of sources into sinks [9].

In this paper, we present a concept in which dissipation produces a time reversed wave. The generation process is similar to that employed in the Instantaneous Time Mirror (ITM) approach [10] in which a sudden change in the wave propagation speed in the entire medium produces a time reversed wave. Here, we consider instead a dynamical change of the dissipation coefficient. We will show that in this case also the production mechanism can be interpreted as a modification of the initial conditions characterizing the system at a given instant by virtue of Cauchy's theorem [START_REF] Hadamard | Lectures on Cauchy's Problem in Linear Partial Differential Equations[END_REF]. In the following we will refer to this new, dissipation based, TR concept as DTR, for Dissipation Time Reversal. DTR can, in principle, be implemented for any type of wave propagating in a (dissipative) medium. For example, for acoustic waves, through a modulation of the viscosity, and for EM waves, through a sudden change in conductivity in the medium. Because the source term involved is a first order time derivative, this new TR technique is able to create an exact TR wave, i.e. precisely proportional to the initial wave field. This results in a higher fidelity and enhanced broadband capabilities compared to other methods [10] [START_REF] Fink | [END_REF]. In the following, we demonstrate the validity of DTR concept performing a proof-of-concept 1D experiment using a chain of coupled magnets. In addition, we show the performance of DTR in a complex 2D highly scattering media using computer simulations.

Theory of Dissipation driven Time Reversal (DTR)

Waves in homogeneous non-dissipative medium are usually governed by d'Alembert wave equation [START_REF] Crawford | Berkeley Physics Course[END_REF]. More generally, the wave fields can be described by an equation of the same structure, but in which the Laplacian operator is replaced by a more complex spatial operator. This is the case for instance when describing acoustic waves in a non-homogeneous medium or with gravity-capillary waves at the surface of deep water. This type of equations can be written in a general manner in the spatial Fourier space for the wave vector 𝒌 [START_REF] Benjamin | [END_REF] [15]:

! ! ! !! ! 𝒌, 𝑡 + 𝜔 ! ! 𝒌 𝜙 𝒌, 𝑡 = 0, (1)
where 𝜔 ! 𝒌 is the dispersion relation of the waves. The time reversal symmetry is a direct consequence of the second order of the time derivative: if 𝜙 𝒌, 𝑡 is a solution of the equation, 𝜙 𝒌, -𝑡 obviously satisfies the same equation. The damping effect induced by dissipation is usually described by an additional first order derivative term in this equation:

! ! ! !! ! 𝒌, 𝑡 + 𝜁 𝒌, 𝑡 !! !" 𝒌, 𝑡 + 𝜔 ! ! 𝒌 𝜙 𝒌, 𝑡 = 0, (2)
where 𝜁 𝒌, 𝑡 is time-dependent damping coefficient. This additional dissipation term breaks the time symmetry of the equation which is precisely why dissipation is generally associated with irreversibility. For simplicity, we remove the dissipation k-dependence in the following notations.

If 𝜁 𝑡 remains small compared to 𝜔 ! , an approximate reversibility is retained for times smaller than 1/𝜁 𝑡 [10]. In the following, we consider a medium where the damping coefficient is initially small or negligible, i.e. 𝜁~0. At a time 𝑡 ! , the dissipation coefficient is set to a very high value over the entire medium 𝜁 ≫ 𝜔 ! and stays at this value until a later time

𝑡 ! = 𝑡 ! + 𝛥𝑡,
where it is set back to its original value. 𝜁 𝑡 can thus be written as 𝜁 𝑡 = 𝜁. Π(𝑡), where Π(𝑡) is a unit rectangle function spanning from 𝑡 ! to 𝑡 ! . An initial wave 𝜙 ! with a Fourier transform 𝜙 ! is originally present in the medium. During the dissipation impulse, the last term of equation ( 2) is negligible and one may write the approximate expressions for the wave field and its time derivative at times t after the damping impulse at time 𝑡 ! as:

𝜙 𝑡 = 𝜙 ! 𝑡 ! + ! ! !! ! !" 𝑡 ! 1 -𝑒 !! !!! ! !! !" 𝑡 = !! ! !" 𝑡 ! 𝑒 !! !!! ! . (3)
Taking 𝜁 towards infinity, we obtain that the field remains approximately constant 𝜙 𝑡 ≈ 𝜙 ! 𝑡 ! during this dissipation impulse. The system is described as an overdamped harmonic oscillator that returns very slowly to a steady equilibrium state without oscillating. In this regime, the characteristic oscillations corresponding to wave motion are stopped and the time it takes for the system to relax increases with dissipation so that in the high dissipation limit we are considering, amplitude damping does not have time to occur. A more detailed calculation shows that in the long run the amplitude decreases as exp -

! ! !
!! 𝑡 -𝑡 ! (see supplemental material). For the strong dissipation limit to hold and for the wave amplitude to be retained, the duration 𝛥𝑡 of the dissipation pulse should thus satisfy 1/𝜁 < 𝛥𝑡 < 𝜁/𝜔 ! ! . If the damping is strong enough, the duration of the dissipation phase may be large compared to the period of the original wave.

At time 𝑡 ! , when the dissipation ends, the wave field starts evolving again according to equation

(2), with the initial conditions:

𝜙 𝑡 ! , !! !" 𝑡 ! = 𝜙 ! 𝑡 ! , 0 . (4)
The DTR process can be interpreted as a change of the initial Cauchy conditions which characterizes the future evolution of the wave field from this initial time 𝑡 ! . As in the case of ITM [10], this state can be decomposed into two counter-propagative wave components using the superposition principle:

𝜙 𝑡 ! , !! !" 𝑡 ! = ! ! 𝜙 ! 𝑡 ! , !! ! !" 𝑡 ! + ! ! 𝜙 ! 𝑡 ! , - !! ! !" 𝑡 ! . ( 5
)
The first term is associated (up to a factor one half) to the exact state of the incident wave field before the DTR, it corresponds to the same wave shifted in time:

𝜙 ! 𝑡 = ! ! 𝜙 ! 𝑡 -𝑡 ! + 𝑡 ! .
The second term is associated to a wave whose derivative has a minus sign. It corresponds to the time reversed wave:

𝜙 ! 𝑡 = ! ! 𝜙 ! 𝑡 ! + 𝑡 ! -𝑡 .
Figure 1 shows the principle of DTR with a chirped pulse containing several frequencies. The pulse created at time 𝑡 = 0 propagates in a dispersive medium undergoing spreading (see figure 1a). At time 𝑡 !"# , a damping impulse is applied and the pulse is frozen (see figure 1b). The damping is then removed resulting in the creation of two counter-propagating pulses with half the amplitude of the initial wave field. The forward propagating pulse is identical to the initial propagating pulse as if no DTR was applied apart from the amplitude factor. The backward propagating pulse is the TR version of the initial pulse. It thus narrows as it propagates, reversing the dispersion, until at time 2𝑡 !"# it returns to the initial profile (with a factor of one half in the amplitude).

1D proof-of-concept experiment using a chain of magnets

We have performed a 1D experiment using a chain of magnets as a proof-of-concept for DTR.

Figure 2a shows the experimental set-up composed of a circular chain of 41 plastic disks. A small magnet oriented vertically with their North pole upward is glued on each 1 cm disk. The disks orientation being constrained, it induces a repulsive interaction between them. The disks are confined horizontally on a circle of 30 cm diameter by an underlying circle of fixed magnets with their North pole oriented upward. In addition, the friction of the disks on the table is drastically reduced by the use of an air cushion system. This allows waves to propagate in this coupled oscillator chain. A computer-controlled electromagnet is used to trigger the longitudinal waves in the chain. The change in dissipation is obtained by a sudden stop the air flow using the controlled valve which results in "freezing" the disk chain. Thus, in that case the damping coefficient 𝜁 can be considered infinite.

For small amplitude oscillations, the chain can be modeled has a system of coupled harmonic oscillators with a rigidity constant κ which can be fitted from the dispersion curve measurements.

When the air pressure is slightly increased, the disks are set in the random motion as shown in Figure 2b, which presents the displacements of the disks as a function of time in gray scale. It is possible to retrieve from this noise motion the dispersion curve of the chain by performing a space-time Fourier transform. The color scale is given in degree along the circle. One degree is approximately equivalent to 2.6 mm in length. Figure 2c shows the resulting experimental dispersion curve together with the fit of the harmonic model (dashed line). The good agreement of the fit confirms the validity of and harmonic interaction and permits to give a value for the rigidity constant of κ=0.13 Ν.m -1 .

Figure 3a shows the propagation of an initially localized perturbation of the magnetic chain (the magnet #37 is acting as a source). A wave packet is launched in the magnetic chain. At time t 0 the chain is suddenly damped by turning off the air flow resulting in a freezing of the disks motion.

At time t 1 the air flow is restored resulting in the creation of two counter propagating wave packets: i) one which resembles the initial wave packet as if no DTR had been applied apart from a decrease of the global amplitude by a factor of approximately 2; ii) a time reversed wave packet refocusing on the source. Figure 3b shows the disk displacements simulated using the harmonic interaction model with the fitted coupling constant. The initial magnet displacement is also taken from the experiment at time t init . This signal processing enables one to remove the forward propagating wave packet after the DTR. The resulting displacement pattern clearly shows the two counter propagating waves after the DTR impulse. Its similarity with the experimental data also

shows the validity of the model.

Simulations of DTR in 2D disordered medium

We performed computer simulations for a 2D disordered system to show the robustness of TDR technique in complex media. The simulations are based on the mass-spring model of Harazi et al. [START_REF] Harazi | Multiple Scattering and Time Reversal of Ultrasound in Dry and Immersed Granular Media[END_REF] [17] introduced to simulate wave propagation in disordered stressed granular packings.

The model consists of a two-dimensional percolated network of point particle with mass m, connected by linear springs of random stiffness as shown in the schematics in Figure 4a. The masses are able to move in the plane of the network and are randomly placed on a 70x70 square lattice with a filling factor of 91%. Each mass of the network is subjected to the Newton's equation:

! ! 𝒓 ! !! ! = 𝜔 !" ( 𝒓 !" -𝑎 ! ) 𝒓 !" 𝒓 !" !∈! ! , ( 6
)
where r i is the vector position of particle i, V i the set of neighboring particles connected to this particle, 𝜔 !" is the angular frequency and and 𝑎 ! the rest length of the spring, 𝒓 !" and 𝒓 !" are the vector between the two particles i and j and its norm respectively. The angular frequencies 𝜔 !" are uniformly distributed between 0.5 𝜔 !" and 1.5 𝜔 !" , where 𝜔 !! is the average angular frequency of the spring-mass systems.

Before launching the wave, the network is submitted to a static stress by pulling the four walls of the domain (strain equal to 0.2) in order to ease the propagation of transverse waves. After this phase, the boundaries of the domain are fixed (zero displacement). The network is then excited by a horizontal displacement of one of the particle during a finite time. The profile of the excitation is given by 𝑢(𝑡) = W 𝑡 cos 𝜔 ! 𝑡 , where W 𝑡 is a temporal window restricting the oscillation to a single period and 𝜔 ! is the driving source pulsation chosen at 0.35 of the average angular frequency 𝜔 !" of the spring-mass systems. Figure 4b and4c show the evolution of the horizontal displacement with time of the source and the map of the displacements of the particles at various times respectively. At t=0, the displacement is confined to the source particle. Then, the displacement of the source decreases rapidly at a noise level of approximately one tenth of its initial value. The initial perturbation propagates and is strongly scattered in the inhomogeneous mass-spring network. At time 𝑡 = 95 𝑇 ! , where 𝑇 ! = 2𝜋/𝜔 ! is the excitation period, the perturbation is spread over the network and the particle displacements become randomly distributed. The network acts as a complex medium due to the random spring stiffnesses and the random vacancies in the squared network.

The DTR freezing is applied at 𝑡 !"# = 300 𝑇 ! , all the particle velocities are set to zero, as if an infinite damping was applied instantaneously, keeping only the potential energy in the system.

Right after this instant, the masses are released from their frozen positions with zero velocity (see third panel Fig. 4d), and evolve according to equation (1). After a complex motion of the particles, a coherent field appears refocusing back to the initial source around time 𝑡 = 2𝑡 !"# =600 𝑇 ! (see panel four Fig. 4d). The horizontal displacement of the source undergoes a very sharp increase reaching approximately 80% of its initial value (see Fig. 4b). Right after this time, the converging wave diverges again, yielding to a new complex displacement field (see the movie of the propagation in supplemental material). In addition to the spatial focusing on the initial source, a time refocusing is also observed showing a temporal shortening of the initial impulsion at time 𝑡 = 2𝑡 !"# (see Fig. 4c). The time width of the refocusing is approximately 6𝑇 ! (FWHM) i.e. twice the one of the source signal.

Discussion

The DTR process is associated with a decrease of the amplitude of the initial wave field. The kinetic energy associated to the time derivative of the field 𝜕𝜙 ! 𝜕𝑡 vanishes during the dissipation impulse leaving the potential energy, associated to the wave field 𝜙 ! unaffected. In the case of an initially propagating wave, the energy of the wave is equally partitioned between potential and kinetic energy. Thus, half of the initial energy is lost in the DTR process resulting in a quarter of the initial energy being TR while an other quarter is retained in the initial wave. It is interesting to note that, for standing waves, the effect of the DTR depends on its relative time phase relative to the impulse since wave energy alternates between kinetic and potential.

The Cauchy analysis in terms of initial conditions to determine the wave field evolution enable one to make a link with Loschmidt's Gedankenexperiment for particle evolution. Loschmidt imagined a deamon capable of instantaneously reversing the velocity of the particles of a gas while keeping their position unaffected and thus time reversing the gas evolution [18] [19].

Although this scheme is impossible in the case of particles due to the extreme sensitivity to initial conditions, it is more amenable for waves because they can often be described with a linear operator and any error in initial conditions will not suffer from chaotic behavior. The wave analogue of this Loschmidt daemon in terms of Cauchy's initial condition 𝜙 ! , 𝜕𝜙 ! 𝜕𝑡 is to change the sign of the wave field time derivative 𝜙 ! , -𝜕𝜙 ! 𝜕𝑡 . Because of the superposition principle, the DTR is thus acting as a Loschmidt daemon by decoupling the wave field from its derivative.

The DTR concept is generic and applies in the case of complex inhomogeneous materials as shown by the 2D simulations (see Fig. 3). This can be shown directly from the Cauchy theorem.

After the freezing the wave field initial condition are reset with a time derivative of the wave the wave field equal to zero. The superposition principle given in Eq. 5 holds.

In contrast with the ITM approach based on wave velocity changes [10] and standard time reversal cavities [START_REF] Fink | [END_REF], the backward propagating wave is directly proportional to the TR of the original wave and not to the time reversal of its time derivative or antiderivative. From that perspective, DTR has thus no spectral limitations and can be applied for the TR of broadband wave packets removing one of the limitation of the existing TR techniques. The limitation in the TR spectral range comes from the ability to freeze the field sufficiently rapidly compared with the phase change in the wave packet, resulting in the maximum value for the time pulsation

𝜁 ≫ 𝜔 ! .

Conclusion

This paper presents a new way to perform an instantaneous time mirror using a dissipation modulation impulse. This concept is generic and could be applied to other type of waves. In optics, DTR could be induced by changing abruptly the conductivity of the medium as in graphene [20], in acoustics it could be obtained by changing electrorheological medium [21]. The excitation is initially a localized perturbation of the magnetic chain, the magnet #37 is acting as a source. At time t 0 the chain is suddenly damped by turning off the air flow resulting in a freezing of the disks motion. At time t 1 the air flow is restored resulting in the creation of two counter propagating wave packets; b) Simulations of the magnet displacements using the harmonic interaction model with the fitted coupling constant. The initial magnet displacement is also taken from the experiment at time t init .

Supplemental Material

For 𝑡 ∈ 𝑡 ! , 𝑡 ! :

! ! ! !! ! 𝒌, 𝑡 + 𝜁 𝒌 !!
!" 𝒌, 𝑡 + 𝜔 ! ! 𝒌 𝜙 𝒌, 𝑡 = 0, (S1)

We consider the regime of high dissipation, so that we assume ζ 𝒌 > 2𝜔 ! 𝒌 . (S1) is thus the equation of a damped harmonic oscillator in the overdamped regime, whose solutions are given by:

𝜙 𝑡 = 𝐴𝑒 ! ! 𝒌 ! !! !! ! ! ! 𝒌 ! ! 𝒌 (!!! ! ) + 𝐵𝑒 ! ! 𝒌 ! !! !! ! ! ! 𝒌 ! ! 𝒌 (!!! ! ) , ( S2
)
with 𝐴 and 𝐵 two constants. Given the initial conditions of continuity of the field and its time derivative, we obtain:

𝜙 𝒌, 𝑡 = !! ! ! ! 𝒌 ! ! 𝒌 !! ! ! 𝒌,! ! ! ! ! 𝒌 !! ! !" 𝒌,! ! ! !! ! ! ! 𝒌 ! ! 𝒌 𝑒 ! ! 𝒌 ! !! !! ! ! ! 𝒌 ! ! 𝒌 (!!! ! ) + !! ! ! ! 𝒌 ! ! 𝒌 !! ! ! 𝒌,! ! ! ! ! 𝒌 !! ! !" 𝒌,! ! ! !! ! ! ! 𝒌 ! ! 𝒌 𝑒 ! ! 𝒌 ! !! !! ! ! ! 𝒌 ! ! 𝒌 (!!! ! ) . ( S3
)
Developing at first order in

! ! 𝒌
! 𝒌 → 0 in front of and (at order 2) inside the exponential terms:

𝜙 𝑡 = - ! ! 𝒌 !! ! !" 𝒌, 𝑡 ! 𝑒 ! ! 𝒌 ! !! ! ! ! ! ! 𝒌 ! ! 𝒌 !! ! ! ! 𝒌 ! ! 𝒌 !!! ! + 𝜙 ! 𝒌, 𝑡 ! + ! ! 𝒌 !! ! !" 𝒌, 𝑡 ! 𝑒 ! ! 𝒌 ! ! ! ! ! ! 𝒌 ! ! 𝒌 !! ! ! ! 𝒌 ! ! 𝒌 !!! ! , ( S4

Figure captions:Figure 1 :

 1 Figure captions: Figure 1: Principle of DTR mirror: a) Propagation of a chirped pulse in a dispersive medium at 3

Figure 2 :

 2 Figure 2: a) Schematics of the experimental set-up composed of a circular chain of 41 plastic

Figure 3 :

 3 Figure 3: a) Displacement amplitudes of the magnets as a function of time represented. The

Figure 4 :

 4 Figure 4: a) Schematic view of the 2D spring model. The model consists of a 2D percolated

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1:

Figure 4 :

 4 Figure 4:

  ) where we used the fact that! ! 𝒌 !! ! !" 𝒌, 𝑡 ! is of order one in 𝜔 ! 𝒌 /ζ 𝒌 .Taking the zero th order inside the exponential yields equation (4) of the main text. Equation (S4) also reveals that the wave amplitude decreases like 𝑒! ! ! ! 𝒌 !! 𝒌!!! ! in the long run.

Acknowledgements:

We are very grateful to Y. Couder for fruitful and stimulating discussions. We thank A.

Fourgeaud for their help in building the experimental set-up. S. K. S. acknowledge the French Ambassy in India for a Charpak scholarship. The authors acknowledge the support of the AXA research fund and LABEX WIFI (Laboratory of Excellence ANR-10-LABX-24) within the French Program 'Investments for the Future' under reference ANR-10-IDEX-0001-02 PSL*


Prasanta Kumar Barik

Ankik Kumar Giri

Philippe Laurençot

Mass-conserving solutions to the Smoluchowski coagulation equation with singular kernel

Keywords: Coagulation, Singular coagulation kernels, Existence, Mass-conserving solutions MSC (2010): Primary: 45J05, 45K05, Secondary: 34A34, 45G10

 Cueto Camejo & Warnecke (2015). In particular, linear growth at infinity of the coagulation kernel is included and the initial condition may have an infinite second moment. Furthermore, all weak solutions (in a suitable sense) including the ones constructed herein are shown to be mass-conserving, a property which was proved in Norris (1999) under stronger assumptions. The existence proof relies on a weak compactness method in L 1 and a by-product of the analysis is that both conservative and non-conservative approximations to the SCE lead to weak solutions which are then mass-conserving.

Introduction

The kinetic process in which particles undergo changes in their physical properties is called a particulate process. The study of particulate processes is a well-known subject in various branches of engineering, astrophysics, physics, chemistry and in many other related areas. During the particulate process, particles merge to form larger particles or break up into smaller particles. Due to this process, particles change their size, shape and volume, to name but a few. There are various types of particulate processes such as coagulation, fragmentation, nucleation and growth for instance. In particular, this article mainly deals with the coagulation process which is governed by the Smoluchowski coagulation equation (SCE). In this process, two particles coalesce to form a larger particle at a particular instant. The SCE is a nonlinear integral equation which describes the dynamics of evolution of the concentration g(ζ, t) of particles of volume ζ > 0 at time t ≥ 0 [START_REF] Smoluchowski | Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen[END_REF]. The evolution of g is given Here ∂g(ζ,t) ∂t represents the time partial derivative of the concentration of particles of volume ζ at time t. In addition, the non-negative quantity Ψ(ζ, η) denotes the interaction rate at which particles of volume ζ and particles of volume η coalesce to form larger particles. This rate is also known as the coagulation kernel or coagulation coefficient. The first and last terms B c (g) and D c (g) on the right-hand side to (1.1) represent the formation and disappearance of particles of volume ζ due to coagulation events, respectively. Let us define the total mass (volume) of the system at time t ≥ 0 as:

M 1 (g)(t) := ∞ 0 ζg(ζ, t)dζ. (1.5)
According to the conservation of matter, it is well known that the total mass (volume) of particles is neither created nor destroyed. Therefore, it is expected that the total mass (volume) of the system remains conserved throughout the time evolution prescribed by (1.1)-(1.2), that is, M 1 (g)(t) = M 1 (g in ) for all t ≥ 0. However, it is worth to mention that, for the multiplicative coagulation kernel Ψ(ζ, η) = ζη, the total mass conservation fails for the SCE at finite time t = 1, see [START_REF] Leyvraz | Singularities in the kinetics of coagulation processes[END_REF]. The physical interpretation is that the lost mass corresponds to "particles of infinite volume" created by a runaway growth in the system due to the very high rate of coalescence of very large particles. These particles, also referred to as "giant particles" [START_REF] Aldous | Deterministic and stochastic model for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists[END_REF] are interpreted in the physics literature as a different macroscopic phase, called a gel, and its occurrence is called the sol-gel transition or gelation transition. The earliest time T g ≥ 0 after which mass conservation no longer holds is called the gelling time or gelation time.

Since the works by Ball & Carr [START_REF] Ball | The discrete coagulation-fragmentation equations: Existence, uniqueness and density conservation[END_REF] and Stewart [START_REF] Stewart | A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels[END_REF], several articles have been devoted to the existence and uniqueness of solutions to the SCE for coagulation kernels which are bounded for small volumes and unbounded for large volumes, as well as to the mass conservation and gelation phenomenon, see [START_REF] Dubovskii | Existence, uniqueness and mass conservation for the coagulationfragmentation equation[END_REF][START_REF] Escobedo | Gelation in coagulation and fragmentation models[END_REF][START_REF] Escobedo | Gelation and mass conservation in coagulation-fragmentation models[END_REF][START_REF] Giri | Weak solutions to the continuous coagulation with multiple fragmentation[END_REF][START_REF] Ph | From the discrete to the continuous coagulation-fragmentation equations[END_REF][START_REF] Norris | Smoluchowski's coagulation equation: uniqueness, non-uniqueness and hydrodynamic limit for the stochastic coalescent[END_REF][START_REF] Stewart | A uniqueness theorem for the coagulation-fragmentation equation[END_REF], see also the survey papers [START_REF] Aldous | Deterministic and stochastic model for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists[END_REF][START_REF] Ph | Weak compactness techniques and coagulation equations[END_REF][START_REF] Ph | On coalescence equations and related models[END_REF] and the references therein. However, to the best of our knowledge, there are fewer articles in which existence and uniqueness of solutions to the SCE with singular coagulation rates have been studied, see [START_REF] Camejo | Regular solutions to the coagulation equations with singular kernels[END_REF][START_REF] Camejo | The singular kernel coagulation equation with multifragmentation[END_REF][START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF][START_REF] Escobedo | Dust and self-similarity for the Smoluchowski coagulation equation[END_REF][START_REF] Norris | Smoluchowski's coagulation equation: uniqueness, non-uniqueness and hydrodynamic limit for the stochastic coalescent[END_REF]. In [START_REF] Norris | Smoluchowski's coagulation equation: uniqueness, non-uniqueness and hydrodynamic limit for the stochastic coalescent[END_REF], Norris investigates the existence and uniqueness of solutions to the SCE locally in time when the coagulation kernel satisfies

Ψ(ζ, η) ≤ φ(ζ)φ(η), (ζ, η) ∈ (0, ∞) 2 , (1.6)
for some sublinear function φ : (0, ∞) → [0, ∞), that is, φ enjoys the property φ(aζ) ≤ aφ(ζ) for all ζ ∈ (0, ∞) and a ≥ 1, and the initial condition g in belongs to L 1 ((0, ∞); φ(ζ) 2 dζ). Massconservation is also shown as soon as there is ε > 0 such that φ(ζ) ≥ εζ for all ζ ∈ (0, ∞).

In [START_REF] Escobedo | Dust and self-similarity for the Smoluchowski coagulation equation[END_REF][START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF], global existence, uniqueness, and mass-conservation are established for coagulation rates of the form

Ψ(ζ, η) = ζ µ 1 η µ 2 + ζ µ 2 η µ 1 with -1 ≤ µ 1 ≤ µ 2 ≤ 1, µ 1 + µ 2 ∈ [0, 2]
, and (µ 1 , µ 2 ) = (0, 1). Recently, global existence of weak solutions to the SCE for coagulation kernels satisfying ), and k * > 0, is obtained in [START_REF] Camejo | Regular solutions to the coagulation equations with singular kernels[END_REF] and further extended in [START_REF] Camejo | The singular kernel coagulation equation with multifragmentation[END_REF] to the broader class of coagulation kernels

Ψ(ζ, η) ≤ k * (1 + ζ + η) λ (ζη) -σ , (ζ, η) ∈ (0, ∞) 2 , with σ ∈ [0, 1/2], λ -σ ∈ [0, 1
Ψ(ζ, η) ≤ k * (1 + ζ) λ (1 + η) λ (ζη) -σ , (ζ, η) ∈ (0, ∞) 2 , (1.7)
with σ ≥ 0, λ -σ ∈ [0, 1), and k * > 0. In [START_REF] Camejo | The singular kernel coagulation equation with multifragmentation[END_REF], multiple fragmentation is also included and uniqueness is shown for the following restricted class of coagulation kernels

Ψ 2 (ζ, η) ≤ k * (ζ -σ + ζ λ-σ )(η -σ + η λ-σ ), (ζ, η) ∈ (0, ∞) 2 ,
where σ ≥ 0 and λ -σ

∈ [0, 1/2].
The main aim of this article is to extend and complete the previous results in two directions. We actually consider coagulation kernels satisfying the growth condition (1.6) for the non-negative function

φ β (ζ) := max ζ -β , ζ , ζ ∈ (0, ∞),
and prove the existence of a global mass-conserving solution of the SCE (1.1)-(1.2) with initial conditions in L 1 ((0, ∞); (ζ -2β + ζ)dζ), thereby removing the finiteness of the second moment required to apply the existence result of [START_REF] Norris | Smoluchowski's coagulation equation: uniqueness, non-uniqueness and hydrodynamic limit for the stochastic coalescent[END_REF] and relaxing the assumption λ < σ + 1 used in [START_REF] Camejo | The singular kernel coagulation equation with multifragmentation[END_REF] for coagulation kernels satisfying (1.7). Besides this, we show that any weak solution in the sense of Definition 2.2 below is mass-conserving, a feature which was enjoyed by the solution constructed in [START_REF] Norris | Smoluchowski's coagulation equation: uniqueness, non-uniqueness and hydrodynamic limit for the stochastic coalescent[END_REF] but not investigated in [START_REF] Camejo | Regular solutions to the coagulation equations with singular kernels[END_REF][START_REF] Camejo | The singular kernel coagulation equation with multifragmentation[END_REF]. An important consequence of this property is that it gives some flexibility in the choice of the method to construct a weak solution to the SCE (1.1)-(1.2) since it will be mass-conserving whatever the approach. Recall that there are two different approximations of the SCE (1.1) by truncation have been employed in recent years, the so-called conservative and non-conservative approximations, see (4.4) below. While it is expected and actually verified in several papers that the conservative approximation leads to a mass-conserving solution to the SCE, a similar conclusion is not awaited when using the nonconservative approximation which has rather been designed to study the gelation phenomenon, in particular from a numerical point of view [START_REF] Filbet | Numerical simulation of the Smoluchowski coagulation equation[END_REF][START_REF] Bourgade | Convergence of a finite volume scheme for coagulation-fragmentation equations[END_REF]. Still, it is by now known that, for the SCE with locally bounded coagulation kernels growing at most linearly at infinity, the non-conservative approximation also allows one to construct mass-conserving solutions [START_REF] Filbet | Mass-conserving solutions and non-conservative approximation to the Smoluchowski coagulation equation[END_REF][START_REF] Barik | A note on mass-conserving solutions to the coagulation-fragmentation equation by using non-conservative approximation[END_REF]. The last outcome of our analysis is that, in our case, the conservative and non-conservative approximations can be handled simultaneously and both lead to a weak solution to the SCE which might not be the same due to the lack of a general uniqueness result but is mass-conserving. We now outline the results of the paper: In the next section, we state precisely our hypotheses on coagulation kernel and on the initial data together with the definition of solutions and the main result. In Section 3, all weak solutions are shown to be mass-conserving. Finally, in the last section, the existence of a weak solution to the SCE (1.1)-(1.2) is obtained by using a weak L 1 compactness method applied to either the non-conservative or the conservative approximations of the SCE.

Main result

We assume that the coagulation kernel Ψ satisfies the following hypotheses.

Hypotheses 2.1. (H1) Ψ is a non-negative measurable function on (0, ∞) × (0, ∞), (H2) There are β > 0 and k > 0 such that

0 ≤ Ψ(ζ, η) = Ψ(η, ζ) ≤ k(ζη) -β , (ζ, η) ∈ (0, 1) 2 , 0 ≤ Ψ(ζ, η) = Ψ(η, ζ) ≤ kηζ -β , (ζ, η) ∈ (0, 1) × (1, ∞), 0 ≤ Ψ(ζ, η) = Ψ(η, ζ) ≤ k(ζ + η), (ζ, η) ∈ (1, ∞) 2 .
Observe that (H2) implies that

Ψ(ζ, η) ≤ k max ζ -β , ζ max η -β , η , (ζ, η) ∈ (0, ∞) 2 .
Let us now mention the following interesting singular coagulation kernels satisfying hypotheses 2.1.

(a) Smoluchowski's coagulation kernel [START_REF] Smoluchowski | Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen[END_REF] (with

β = 1/3) Ψ(ζ, η) = ζ 1/3 + η 1/3 ζ -1/3 + η -1/3 , (ζ, η) ∈ (0, ∞) 2 . (b) Granulation kernel [16] Ψ(ζ, η) = (ζ + η) θ 1 (ζη) θ 2
, where θ 1 ≤ 1 and θ 2 ≥ 0.

(c) Stochastic stirred froths [START_REF] Clark | Stably coalescent stochastic froths[END_REF] Ψ(ζ, η) = (ζη) -β , where β > 0.

Before providing the statement of Theorem 2.3, we recall the following definition of weak solutions to the SCE (1.1)-(1.2). We set

L 1 -2β,1 (0, ∞) := L 1 ((0, ∞); (ζ -2β + ζ)dζ). Definition 2.2. Let T ∈ (0, ∞] and g in ∈ L 1 -2β,1 (0, ∞), g in ≥ 0 a.e. in (0, ∞). A non- negative real valued function g = g(ζ, t) is a weak solution to equations (1.1)-(1.2) on [0, T ) if g ∈ C ([0, T ); L 1 (0, ∞)) L ∞ (0, T ; L 1 -2β,1 (0, ∞)) and satisfies ∞ 0 [g(ζ, t) -g in (ζ)]ω(ζ)dζ = 1 2 t 0 ∞ 0 ∞ 0 ω(ζ, η)Ψ(ζ, η)g(ζ, s)g(η, s)dηdζds, (2.1
)

for every t ∈ (0, T ) and ω ∈ L ∞ (0, ∞), where ω(ζ, η) := ω(ζ + η) -ω(ζ) -ω(η), (ζ, η) ∈ (0, ∞) 2 .
Now, we are in a position to state the main theorem of this paper.

Theorem 2.3. Assume that the coagulation kernel satisfies hypotheses (H1)-(H2) and consider a non-negative initial condition g in ∈ L 1 -2β,1 (0, ∞). There exists at least one massconserving weak solution g to the SCE (1.1)-(1.2) on [0, ∞), that is, g is a weak solution to (1.1)-(1.2) in the sense of Definition 2.2 satisfying M 1 (g)(t) = M 1 (g in ) for all t ≥ 0, the total mass M 1 (g) being defined in (1.5).

Weak solutions are mass-conserving

In this section, we establish that any weak solution g to (1.1)-(1.2) on [0, T ), T ∈ (0, ∞], in the sense of Definition 2.2 is mass-conserving, that is, satisfies

M 1 (g)(t) = M 1 (g in ), t ≥ 0. (3.1)
To this end, we adapt an argument designed in [2, Section 3] to investigate the same issue for the discrete coagulation-fragmentation equations and show that the behaviour of g for small volumes required in Definition 2.2 allows us to control the possible singularity of Ψ. In order to prove Theorem 3.1, we need the following sequence of lemmas.

Lemma 3.2. Assume that (H1)-(H2) hold. Let g be a weak solution to (1.1)-(1.2) on [0, T ).

Then, for q ∈ (0, ∞) and t ∈ (0, T ),

q 0 ζg(ζ, t)dζ - q 0 ζg in (ζ)dζ = - t 0 q 0 ∞ q-ζ ζΨ(ζ, η)g(ζ, s)g(η, s)dηdζds. (3.2) Proof. Set ω(ζ) = ζχ (0,q) (ζ) for ζ ∈ (0, ∞) and note that ω(ζ, η) =                0, if ζ + η ∈ (0, q), -(ζ + η), if ζ + η ≥ q, (ζ, η) ∈ (0, q) 2 , -ζ, if (ζ, η) ∈ (0, q) × [q, ∞), -η, if (ζ, η) ∈ [q, ∞) × (0, q), 0, if (ζ, η) ∈ [q, ∞) 2 .
Inserting the above values of ω into (2.1) and using the symmetry of Ψ, we have

q 0 [g(ζ, t) -g in (ζ)]ζdζ = 1 2 t 0 ∞ 0 ∞ 0 ω(ζ, η)Ψ(ζ, η)g(ζ, s)g(η, s)dηdζds = - 1 2 t 0 q 0 q q-ζ (ζ + η)Ψ(ζ, η)g(ζ, s)g(η, s)dηdζds - 1 2 t 0 q 0 ∞ q ζΨ(ζ, η)g(ζ, s)g(η, s)dηdζds - 1 2 t 0 ∞ q q 0 ηΨ(ζ, η)g(ζ, s)g(η, s)dηdζds =- t 0 q 0 q q-ζ ζΨ(ζ, η)g(ζ, s)g(η, s)dηdζds - t 0 q 0 ∞ q ζΨ(ζ, η)g(ζ, s)g(η, s)dηdζds,
which completes the proof of Lemma 3.2.

In order to complete the proof of Theorem 3.1, it is sufficient to show that the right-hand side of (3.2) goes to zero as q → ∞. The first step in that direction is the following result.

Lemma 3.3. Assume that (H1)-(H2) hold. Let g be a solution to (1.1)-(1.2) on [0, T ) and consider t ∈ (0, T ). Then

(i) ∞ q [g(ζ, t) -g in (ζ)]dζ = - 1 2 t 0 ∞ q ∞ q Ψ(ζ, η)g(ζ, s)g(η, s)dηdζds + 1 2 t 0 q 0 q q-ζ Ψ(ζ, η)g(ζ, s)g(η, s)dηdζds, (ii) lim q→∞ t 0 q q 0 q q-ζ Ψ(ζ, η)g(ζ, s)g(η, s)dηdζ - ∞ q ∞ q Ψ(ζ, η)g(ζ, s)g(η, s)dηdζ ds = 0. Proof. Set ω(ζ) = χ [q,∞) (ζ) for ζ ∈ (0, ∞) and the corresponding ω is ω(ζ, η) =                0, if ζ + η ∈ (0, q), 1, if ζ + η ∈ [q, ∞), (ζ, η) ∈ (0, q) 2 , 0, if (ζ, η) ∈ (0, q) × [q, ∞), 0, if (ζ, η) ∈ [q, ∞) × (0, q), -1, if (ζ, η) ∈ [q, ∞) 2 .
Inserting the above values of ω into (2.1), we obtain Lemma 3.3 (i).

Next, we readily infer from the integrability of ζ → ζg(ζ, t) and ζ → ζg in (ζ) and Lebesgue's dominated convergence theorem that

lim q→∞ q ∞ q [g(ζ, t) -g in (ζ)]dζ ≤ lim q→∞ ∞ q ζ[g(ζ, t) + g in (ζ)]dζ = 0.
Multiplying the identity stated in Lemma 3.3 (i) by q, we deduce from the previous statement that the left-hand side of the thus obtained identity converges to zero as q → ∞. Then so does its right-hand side, which proves Lemma 3.3 (ii). Then, for t ∈ (0, T ),

(i) lim q→∞ t 0 q 0 ∞ q ζΨ(ζ, η)g(ζ, s)g(η, s)dηdζds = 0, and
(ii) lim q→∞ q t 0 ∞ q ∞ q Ψ(ζ, η)g(ζ, s)g(η, s)dηdζds = 0.
Proof. Let q > 1, t ∈ (0, T ), and s ∈ (0, t). To prove the first part of Lemma 3.4, we split the integral as follows

q 0 ∞ q ζΨ(ζ, η)g(ζ, s)g(η, s)dηdζ = J 1 (q, s) + J 2 (q, s), with J 1 (q, s) := 1 0 ∞ q ζΨ(ζ, η)g(ζ, s)g(η, s)dηdζ, J 2 (q, s) := q 1 ∞ q ζΨ(ζ, η)g(ζ, s)g(η, s)dηdζ.
On the one hand, it follows from (H2) and Young's inequality that

J 1 (q, s) ≤ k 1 0 ∞ q ζ 1-β ηg(ζ, s)g(η, s)dηdζ ≤ k ∞ 0 ζ 1-β g(ζ, s)dζ ∞ q ηg(η, s)dη ≤ k g(s) L 1 -2β,1 (0,∞) ∞ q ηg(η, s)dη
and the integrability properties of g from Definition 2.2 and Lebesgue's dominated convergence theorem entail that lim q→∞ t 0 J 1 (q, s)ds = 0.

(

On the other hand, we infer from (H2) that

J 2 (q, s) ≤ k q 1 ∞ q ζ(ζ + η)g(ζ, s)g(η, s)dηdζ ≤ 2k q 1 ∞ q ζηg(ζ, s)g(η, s)dηdζ ≤ 2kM 1 (g)(s) ∞ q ηg(η, s)dη,
and we argue as above to conclude that lim q→∞ t 0 J 2 (q, s)ds = 0.

Recalling (3.3), we have proved Lemma 3.4 (i).

Similarly, by (H2),

q ∞ q ∞ q Ψ(ζ, η)g(ζ, s)g(η, s)dηdζ ≤ k ∞ q ∞ q (qζ + qη)g(ζ, s)g(η, s)dηdζ ≤ 2k ∞ q ∞ q ζηg(ζ, s)g(η, s)dηdζ ≤ 2kM 1 (g)(s) ∞ q ηg(η, s)dη,
and we use once more the previous argument to obtain Lemma 3.4 (ii). Now, we are in a position to prove Theorem 3.1.

Proof of Theorem 3.1. Let t ∈ (0, T ). From Lemma 3.4 (i), we obtain

lim q→∞ t 0 q 0 ∞ q ζΨ(ζ, η)g(ζ, s)g(η, s)dηdζds = 0, (3.4)
while Lemma 3.3 (ii) and Lemma 3.4 (ii) imply that lim

q→∞ q t 0 q 0 q q-ζ Ψ(ζ, η)g(ζ, s)g(η, s)dηdζds = 0. (3.5) Since t 0 q 0 ∞ q-ζ ζΨ(ζ, η)g(ζ, s)g(η, s)dηdζds ≤ q t 0 q 0 q q-ζ Ψ(ζ, η)g(ζ, s)g(η, s)dηdζds + t 0 q 0 ∞ q ζΨ(ζ, η)g(ζ, s)g(η, s)dηdζds,
it readily follows from (3.4) and (3.5) that the right-hand side of (3.2) converges to zero as q → ∞. Consequently,

M 1 (g)(t) = lim q→∞ q 0 ζg(ζ, s)dζ = lim q→∞ q 0 ζg in (ζ)dζ = M 1 (g in ).
This completes the proof of Theorem 3.1.

Existence of weak solutions

This section is devoted to the construction of weak solutions to the SCE (1.1)-(1.2) with a nonnegative initial condition g in ∈ L 1 -2β,1 (0, ∞). It is achieved by a classical compactness technique, the appropriate functional setting being here the space L 1 (0, ∞) endowed with its weak topology first used in the seminal work [START_REF] Stewart | A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels[END_REF] and subsequently further developed in [START_REF] Barik | A note on mass-conserving solutions to the coagulation-fragmentation equation by using non-conservative approximation[END_REF][START_REF] Camejo | Regular solutions to the coagulation equations with singular kernels[END_REF][START_REF] Camejo | The singular kernel coagulation equation with multifragmentation[END_REF][START_REF] Escobedo | Gelation and mass conservation in coagulation-fragmentation models[END_REF][START_REF] Filbet | Mass-conserving solutions and non-conservative approximation to the Smoluchowski coagulation equation[END_REF][START_REF] Giri | Weak solutions to the continuous coagulation with multiple fragmentation[END_REF][START_REF] Ph | From the discrete to the continuous coagulation-fragmentation equations[END_REF]. Given a non-negative initial condition g in ∈ L 1 -2β,1 (0, ∞), the starting point of this approach is the choice of an approximation of the SCE (1.1)-(1.2), which we set here to be

∂g n (ζ, t) ∂t = B c (g n )(ζ, t) -D θ c,n (g n )(ζ, t), (ζ, t) ∈ (0, n) × (0, ∞), (4.1)
with truncated initial condition

g n (ζ, 0) = g in n (ζ) := g in (ζ)χ (0,n) (ζ), ζ ∈ (0, n), (4.2)
where n ≥ 1 is a positive integer, θ ∈ {0, 1},

Ψ θ n (ζ, η) := Ψ(ζ, η)χ (1/n,n) (ζ)χ (1/n,n) (η) 1 -θ + θχ (0,n) (ζ + η) (4.3) for (ζ, η) ∈ (0, ∞) 2 and D θ c,n (g)(ζ) := n-θζ 0 Ψ θ n (ζ, η)g(ζ)g(η)dη, ζ ∈ (0, n), (4.4)
the gain term B c (g)(ζ) being still defined by (1.3) for ζ ∈ (0, n). The introduction of the additional parameter θ ∈ {0, 1} allows us to handle simultaneously the so-called conservative approximation (θ = 1) and non-conservative approximation (θ = 0) and thereby prove that both approximations allow us to construct weak solutions to the SCE (1.1)-(1.2), a feature which is of interest when no general uniqueness result is available. Note that we also truncate the coagulation for small volumes to guarantee the boundedness of Ψ θ n which is a straightforward consequence of (H2) and (4.3). Thanks to this property, it follows from [START_REF] Stewart | A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels[END_REF] (θ = 1) and [START_REF] Filbet | Mass-conserving solutions and non-conservative approximation to the Smoluchowski coagulation equation[END_REF] (θ = 0) that there is a unique non-negative solution

g n ∈ C 1 ([0, ∞); L 1 (0, n)) to (4.1)-(4.
2) (we do not indicate the dependence upon θ for notational simplicity) which satisfies

n 0 ζg n (ζ, t)dζ = n 0 ζg in n (ζ)dζ -(1 -θ) t 0 n 0 n n-ζ ζΨ θ n (ζ, η)g n (ζ, s)g n (η, s)dηdζds (4.5)
for t ≥ 0. The second term in the right-hand side of (4.5) vanishes for θ = 1 and the total mass of g n remains constant throughout time evolution, which is the reason for this approximation to be called conservative. In contrast, when θ = 0, the total mass of g n decreases as a function of time. In both cases, it readily follows from (4.5) that

n 0 ζg n (ζ, t)dζ ≤ n 0 ζg in n (ζ)dζ ≤ M 1 (g in ), t ≥ 0. (4.6)
For further use, we next state the weak formulation of (4.1)-(4.2): for t > 0 and ω ∈ L ∞ (0, n), there holds

n 0 ω(ζ)[g n (ζ, t) -g in n (ζ)]dζ = 1 2 t 0 n 1/n n 1/n H θ ω,n (ζ, η)Ψ θ n (ζ, η)g n (ζ, s)g n (η, s)dηdζds, (4.7)
where

H θ ω,n (ζ, η) := ω(ζ + η)χ (0,n) (ζ + η) -[ω(ζ) + ω(η)] 1 -θ + θχ (0,n) (ζ + η) for (ζ, η) ∈ (0, n) 2 .
In order to prove Theorem 2.3, we shall show the convergence (with respect to an appropriate topology) of a subsequence of (g n ) n≥1 towards a weak solution to (1.1)-(1.2). For that purpose, we now derive several estimates and first recall that, since g in ∈ L 1 -2β,1 (0, ∞), a refined version of de la Vallée-Poussin theorem, see [START_REF] Châu-Hoàn | Etude de la classe des opérateurs m-accrétifs de L 1 (Ω) et accrétifs dans L ∞ (Ω)[END_REF] or [START_REF] Ph | Weak compactness techniques and coagulation equations[END_REF]Theorem 8], guarantees that there exist two non-negative and convex functions σ 1 and σ 2 in C 2 ([0, ∞)) such that σ ′ 1 and σ ′ 2 are concave,

σ i (0) = σ ′ i (0) = 0, lim x→∞ σ i (x) x = ∞, i = 1, 2, (4.8)
and

I 1 := ∞ 0 σ 1 (ζ)g in (ζ)dζ < ∞, and I 2 := ∞ 0 σ 2 ζ -β g in (ζ) dζ < ∞. (4.9)
Let us state the following properties of the above defined functions σ 1 and σ 2 which are required to prove Theorem 2.3.

Lemma 4.1. For (x, y) ∈ (0, ∞) 2 , there holds

(i) σ 2 (x) ≤ xσ ′ 2 (x) ≤ 2σ 2 (x), (ii) xσ ′ 2 (y) ≤ σ 2 (x) + σ 2 (y),
and

(iii) 0 ≤ σ 1 (x + y) -σ 1 (x) -σ 1 (y) ≤ 2 xσ 1 (y) + yσ 1 (x)
x + y .

Proof. A proof of the statements (i) and (iii) may be found in [START_REF] Ph | Weak compactness techniques and coagulation equations[END_REF]Proposition 14] while (ii) can easily be deduced from (i) and the convexity of σ 2 .

We recall that throughout this section, the coagulation kernel Ψ is assumed to satisfy (H1)-(H2) and g in is a non-negative function in L 1 -2β,1 (0, ∞).

Moment estimates

We begin with a uniform bound in L 1 -2β,1 (0, ∞).

Lemma 4.2.

There exists a positive constant B > 0 depending only on g in such that, for t ≥ 0,

n 0 ζ + ζ -2β g n (ζ, t)dζ ≤ B.
Proof. Let δ ∈ (0, 1) and take

ω(ζ) = (ζ + δ) -2β , ζ ∈ (0, n), in (4.7)
. With this choice of ω,

H θ ω,n (ζ, η) ≤ (ζ + η + δ) -2β -(ζ + δ) -2β -(η + δ) -2β χ (0,n) (ζ + η) ≤ 0
for all (ζ, η) ∈ (0, n) 2 , so that (4.7) entails that, for t ≥ 0,

n 0 (ζ + δ) -2β g n (ζ, t)dζ ≤ n 0 (ζ + δ) -2β g in n (ζ)dζ ≤ ∞ 0 ζ -2β g in (ζ)dζ.
We then let δ → 0 in the previous inequality and deduce from Fatou's lemma that

n 0 ζ -2β g n (ζ, t)dζ ≤ ∞ 0 ζ -2β g in (ζ)dζ, t ≥ 0.
Combining the previous estimate with (4.6) gives Lemma 4.2 with B := g in L 1

-2β,1 (0,∞) . We next turn to the control of the tail behavior of g n for large volumes, a step which is instrumental in the proof of the convergence of each integral on the right-hand side of (4.1) to their respective limits on the right-hand side of (1.1). Lemma 4.3. For T > 0, there is a positive constant Γ(T ) depending on k, σ 1 , g in , and T such that,

(i) sup t∈[0,T ] n 0 σ 1 (ζ)g n (ζ, t)dζ ≤ Γ(T ), and
(ii) (1 -θ) T 0 n 1 n 1 σ 1 (ζ)χ (0,n) (ζ + η)Ψ(ζ, η)g n (ζ, s)g n (η, s)dηdζds ≤ Γ(T ).
Proof. Let T > 0 and t ∈ (0, T ). We set ω

(ζ) = σ 1 (ζ), ζ ∈ (0, n), into (4.7) and obtain n 0 σ 1 (ζ)[g n (ζ, t) -g in n (ζ)]dζ = 1 2 t 0 n 1/n n 1/n σ1 (ζ, η)χ (0,n) (ζ + η)Ψ(ζ, η)g n (ζ, s)g n (η, s)dηdζds - 1 -θ 2 t 0 n 1/n n 1/n [σ 1 (ζ) + σ 1 (η)]χ [n,∞) (ζ + η)Ψ(ζ, η)g n (ζ, s)g n (η, s)dηdζds, recalling that σ1 (ζ, η) = σ 1 (ζ + η) -σ 1 (ζ) -σ 1 (η)
, hence, using (H2) and Lemma 4.1, Owing to the concavity of σ ′ 1 and the property σ 1 (0) = 0, there holds

n 0 σ 1 (ζ)[g n (ζ, t) -g in n (ζ)]dζ ≤ k 2 4 i=1 J i,n (t) -(1 -θ)R n (t), with J 1,n (t) := t 0 1 0 1 0 σ1 (ζ, η)(ζη) -β g n (ζ, s)g n (η, s)dηdζds, J 2,n (t) := t 0 1 0 n 1 σ1 (ζ, η)ζ -β ηg n (ζ, s)g n (η, s)dηdζds,
σ1 (ζ, η) = ζ 0 η 0 σ ′′ 1 (x + y)dydx ≤ σ ′′ 1 (0)ζη , (ζ, η) ∈ (0, ∞) 2 . ( 4.10)
By (4.10), Lemma 4.2, and Young's inequality,

J 1,n (t) ≤ σ ′′ 1 (0) t 0 1 0 1 0 ζ 1-β η -β g n (ζ, s)g n (η, s)dηdζds ≤ σ ′′ 1 (0) t 0 1 0 ζ + ζ -2β g n (ζ, s)dζ 2 ds ≤ σ ′′ 1 (0)B 2 t.
Next, Lemma 4.1 (iii), Lemma 4.2, and Young's inequality give

J 2,n (t) = J 3,n (t) ≤ 2 t 0 1 0 n 1 ζσ 1 (η) + ησ 1 (ζ) ζ + η ζ -β ηg n (ζ, s)g n (η, s)dηdζds ≤ 2 t 0 1 0 n 1 ζ 1-β σ 1 (η) + σ 1 (1)ζ -β η g n (ζ, s)g n (η, s)dηdζds ≤ 2 t 0 1 0 ζ + ζ -2β g n (ζ, s)dζ n 1 σ 1 (η)g n (η, s)dη ds + σ 1 (1) t 0 1 0 ζ + ζ -2β g n (ζ, s)dζ n 1 ηg n (η, s)dη ds ≤ 2σ 1 (1)B 2 t + 2B t 0 n 0 σ 1 (η)g n (η, s)dηds, and
J 4,n (t) ≤ 2 t 0 n 1 n 1 (ησ 1 (ζ) + ζσ 1 (η)) g n (ζ, s)g n (η, s)dηdζds ≤ 4B t 0 n 0 σ 1 (η)g n (η, s)dηds.
Gathering the previous estimates, we end up with

n 0 σ 1 (ζ)[g n (ζ, t) -g in n (ζ)]dζ ≤ k σ ′′ 1 (0) 2 + 2σ 1 (1) B 2 t + 4kB t 0 n 0 σ 1 (η)g n (η, s)dηds -(1 -θ)R n (t),
and we infer from Gronwall's lemma and (4.9) that

n 0 σ 1 (ζ)g n (ζ, t)dζ + (1 -θ)R n (t) ≤ e 4kBt n 0 σ 1 (ζ)g in n (ζ)dζ + σ ′′ 1 (0) 8 + σ 1 (1) 2 Be 4kBt ≤ I + σ ′′ 1 (0) + σ 1 (1) B e 4kBt .
This completes the proof of Lemma 4.3.

Uniform integrability

Next, our aim being to apply Dunford-Pettis' theorem, we have to prevent concentration of the sequence (g n ) n≥1 on sets of arbitrary small measure. For that purpose, we need to show the following result.

Lemma 4.4. For any T > 0 and λ > 0, there is a positive constant L 1 (λ, T ) depending only on k, σ 2 , g in , λ, and T such that

sup t∈[0,T ] λ 0 σ 2 ζ -β g n (ζ, t) dζ ≤ L 1 (λ, T ). Proof. For (ζ, t) ∈ (0, n) × (0, ∞), we set u n (ζ, t) := ζ -β g n (ζ, t). Let λ ∈ (1, n)
, T > 0, and t ∈ (0, T ). Using Leibniz's rule, Fubini's theorem, and (4.1), we obtain

d dt λ 0 σ 2 (u n (ζ, t))dζ ≤ 1 2 λ 0 λ-η 0 σ 2 ′ (u n (ζ + η, t))(ζ + η) -β Ψ θ n (ζ, η)g n (ζ, t)g n (η, t)dζdη. (4.11)
It also follows from (H2) that

Ψ θ n (ζ, η) ≤ Ψ(ζ, η) ≤ 2kλ 1+2β (ζη) -β , (ζ, η) ∈ (0, λ) 2 . (4.12)
We then infer from (4.11), (4.12), Lemma 4.1 (ii) and Lemma 4.2 that

d dt λ 0 σ 2 (u n (ζ, t))dζ ≤kλ 1+2β λ 0 λ-η 0 σ ′ 2 (u n (ζ + η, t))(ζ + η) -β u n (ζ, t)u n (η, t)dζdη ≤kλ 1+2β λ 0 λ-η 0 η -β [σ 2 (u n (ζ + η, t)) + σ 2 (u n (ζ, t))] u n (η, t)dζdη ≤2kλ 1+2β λ 0 η -2β g n (η, t) λ-η 0 σ 2 (u n (ζ + η, t))dζdη ≤2kλ 1+2β B λ 0 σ 2 (u n (ζ, t))dζ.
Then, using Gronwall's lemma, the monotonicity of σ 2 , and (4.9), we obtain

λ 0 σ 2 (ζ -β g n (ζ, t))dζ ≤ L 1 (λ, T ),
where L 1 (λ, T ) := I 2 e 2kλ 1+2β BT , and the proof is complete.

Time equicontinuity

The outcome of the previous sections settles the (weak) compactness issue with respect to the volume variable. We now turn to the time variable.

Lemma 4.5. Let t 2 ≥ t 1 ≥ 0 and λ ∈ (1, n). There is a positive constant L 2 (λ) depending only on k, g in , and λ such that

λ 0 ζ -β |g n (ζ, t 2 ) -g n (ζ, t 1 )|dζ ≤ L 2 (λ)(t 2 -t 1 ).
Proof. Let t > 0. On the one hand, by Fubini's theorem, (4.12), and Lemma 4.2,

λ 0 ζ -β B c (g n )(ζ, t)dζ ≤ 1 2 λ 0 λ-ζ 0 (ζ + η) -β Ψ(ζ, η)g n (ζ, t)g n (η, t)dηdζ ≤ kλ 1+2β λ 0 λ 0 ζ -β η -2β g n (ζ, t)g n (η, t)dηdζ ≤ kλ 1+3β λ 0 ζ -2β g n (ζ, t)dζ 2 ≤ kλ 1+3β B 2 .
On the other hand, since

Ψ θ n (ζ, η) ≤ Ψ(ζ, η) ≤ 2kλ β ηζ -β , 0 < ζ < λ < η < n,
we infer from (4.12) and Lemma 4.2 that

λ 0 ζ -β D θ c,n (g n )(ζ, t)dζ ≤ λ 0 n 0 ζ -β Ψ(ζ, η)g n (ζ, t)g n (η, t)dηdζ ≤ 2kλ 1+2β λ 0 λ 0 ζ -2β η -β g n (ζ, t)g n (η, t)dηdζ + 2kλ β λ 0 n λ ζ -β ηg n (ζ, t)g n (η, t)dηdζ ≤ 2kB 2 (1 + λ 1+β )λ β .
Consequently, by (4.1),

Convergence

We are now in a position to complete the proof of the existence of a weak solution to the SCE (1.1)-(1.2).

Proof of Theorem 2.3. For (ζ, t) ∈ (0, n) × (0, ∞), we set u n (ζ, t) := ζ -β g n (ζ, t). Let T > 0 and λ > 1. Owing to the superlinear growth (4.8) of σ 2 at infinity and Lemma 4.4, we infer from Dunford-Pettis' theorem that there is a weakly compact subset K λ,T of L 1 (0, λ) such that (u n (t)) n≥1 lies in K λ,T for all t ∈ [0, T ]. Moreover, by Lemma 4.5, (u n ) n≥1 is strongly equicontinuous in L 1 (0, λ) at all t ∈ (0, T ) and thus also weakly equicontinuous in L 1 (0, λ) at all t ∈ (0, T ). A variant of Arzelà-Ascoli's theorem [26, Theorem 1.3.2] then guarantees that (u n ) n≥1 is relatively compact in C w ([0, T ]; L 1 (0, λ)). This property being valid for all T > 0 and λ > 1, we use a diagonal process to obtain a subsequence of (g n ) n≥1 (not relabeled) and a non-negative function g such that To complete the proof of Theorem 2.3, it remains to show that g is a weak solution to the SCE (1.1)-(1.2) on [0, ∞) in the sense of Definition 2.2. This step is carried out by the classical approach of [START_REF] Stewart | A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels[END_REF] with some modifications as in [START_REF] Camejo | Regular solutions to the coagulation equations with singular kernels[END_REF][START_REF] Camejo | The singular kernel coagulation equation with multifragmentation[END_REF] and [START_REF] Ph | From the discrete to the continuous coagulation-fragmentation equations[END_REF] to handle the convergence of the integrals for small and large volumes, respectively. In particular, on the one hand, the behavior for large volumes is controlled by the estimates of Lemma 4.3 with the help of the superlinear growth (4.8) of σ 1 at infinity and the linear growth (H2) of Ψ. On the other hand, the behavior for small volumes is handled by (H2), Lemma 4.2, and (4.13).

Finally, g being a weak solution to (1.1)-(1.2) on [0, ∞) in the sense of Definition 2.2, it is mass-conserving according to Theorem 3.1, which completes the proof of Theorem 2.3.

2 ζ 0 Ψ 0 Ψ

 200 by ∂g(ζ, t) ∂t = B c (g)(ζ, t) -D c (g)(ζ, t), (ζ, t) ∈ (0, ∞) 2 , (1.1) with initial condition g(ζ, 0) = g in (ζ) ≥ 0, ζ ∈ (0, ∞),(1.2)where the operator B c and D c are expressed asB c (g)(ζ, t) := 1 (ζ -η, η)g(ζ -η, t)g(η, t)dη (1.3)andD c (g)(ζ, t) := ∞ (ζ, η)g(ζ, t)g(η, t)dη.(1.4)

Theorem 3 . 1 .

 31 Suppose that (H1)-(H2) hold. Let g be a weak solution to (1.1)-(1.2) on [0, T ) for some T ∈ (0, ∞]. Then g satisfies the mass-conserving property (3.1) for all t ∈ (0, T ).

Lemma 3 . 4 .

 34 Assume that (H1)-(H2) hold. Let g be a weak solution to (1.1)-(1.2) on [0, T ).

J 3 σ 1

 31 η)ζη -β g n (ζ, s)g n (η, s)dηdζds, η)(ζ + η)g n (ζ, s)g n (η, s)dηdζds, (ζ)χ [n,∞) (ζ + η)Ψ(ζ, η)g n (ζ, s)g n (η, s)dηdζds.

g

  n -→ g in C w ([0, T ]; L 1 (0, λ))for all T > 0 and λ > 1. Owing to Lemma 4.3 and the superlinear growth (4.8) of σ 1 at infinity, a by-now classical argument allows us to improve the previous convergence tog n -→ g in C w ([0, T ]; L 1 ((0, ∞); (ζ -β + ζ)dζ)).(4.13)

Acknowledgments

This work was supported by University Grant Commission (UGC), India, for providing Ph.D fellowship to PKB. AKG would like to thank Science and Engineering Research Board (SERB), Department of Science and Technology (DST), India for providing funding support through the project Y SS/2015/001306.