Joachim Escher

Philippe Laurencot
email: laurenco@math.univ-toulouse.fr

Bogdan -Vasile

AND Philippe Laurenc ¸ot

Bogdan-Vasile Matioc
email: matioc@ifam.uni-hannover.de

EXISTENCE AND STABILITY OF WEAK SOLUTIONS FOR A DEGENERATE PARABOLIC SYSTEM MODELLING TWO-PHASE FLOWS IN POROUS MEDIA

Keywords: 2010 Mathematics Subject Classification. 35K65, 35K40, 35D30, 35B35, 35Q35 Degenerate parabolic system, weak solutions, exponential stability, thin film, Liapunov functional

published or not. The documents may come

Existence and stability of weak solutions for a degenerate parabolic system modelling two-phase flows in porous media

Introduction

In this paper we consider the following system of degenerate parabolic equations

∂ t f = ∂ x (f ∂ x f ) + R∂ x (f ∂ x h) , ∂ t h = ∂ x (f ∂ x f ) + R µ ∂ x [(h -f )∂ x h] + R∂ x (f ∂ x h) , (t, x) ∈ (0, ∞) × (0, L), (1.1)
which models two-phase flows in porous media under the assumption that the thickness of the two fluid layers is small. Indeed, the system (1.1) has been obtained in [START_REF] Escher | Modelling and analysis of the Muskat problem for thin fluid layers[END_REF] by passing to the limit of small layer thickness in the Muskat problem studied in [START_REF] Escher | A generalised Rayleigh-Taylor condition for the Muskat problem[END_REF] (with homogeneous Neumann boundary condition). Similar methods to those presented in [START_REF] Escher | Modelling and analysis of the Muskat problem for thin fluid layers[END_REF] have been used in [START_REF] Günther | A justification for the thin film approximation of Stokes flow with surface tension[END_REF] and [START_REF] Matioc | Hele-Shaw flow in thin threads: A rigorous limit result[END_REF], where it is rigorously shown that, in the absence of gravity, appropriate scaled classical solutions of the Stokes' and one-phase Hele-Shaw problems with surface tension converge to solutions of thin film equations ∂ t h + ∂ x (h a ∂ 3 x h) = 0, with a = 3 for Stoke's problem and a = 1 for the Hele-Shaw problem.

In our setting f is a nonnegative function expressing the height of the interface between the fluids while h ≥ f is the height of the interface separating the fluid located in the upper part of the porous medium from air, cf. Figure 1. We assume that the bottom of the porous medium, which is located at y = 0, is impermeable and that the air is at constant pressure normalised to be zero. The parameters R and R µ are given by

R := ρ + ρ --ρ + and R µ := µ - µ + R ,
where ρ -, µ -[resp. ρ + , µ + ] denote the density and viscosity of the fluid located below [resp. above] in the porous medium. Of course, we have to supplement system (1.1) with initial conditions

f (0) = f 0 , h(0) = h 0 , x ∈ (0, L), (1.2)
x y

0 L y = f (x) y = h(x) g(x)
.

∂ x f = ∂ x h = 0, x = 0, L. (1.3)
It turns out that the system is parabolic if we assume that h 0 > f 0 > 0 and R > 0, that is, the denser fluid lies beneath. Existence and uniqueness of classical solutions to (1.1) have been established in this parabolic setting in [START_REF] Escher | Modelling and analysis of the Muskat problem for thin fluid layers[END_REF]. Furthermore, it is also shown that the steady states of (1.1) are flat and that they attract at an exponential rate in H 2 solutions which are initially close by.

In this paper we are interested in the degenerate case which appears when we allow f 0 = 0 and h 0 = f 0 on some subset of (0, L). Owing to the loss of uniform parabolicity, existence of classical solutions can no longer be established by using parabolic theory and we have to work within an appropriate weak setting. Furthermore, the system is quasilinear and, as a further difficulty, each equation contains highest order derivatives of both unknowns f and h, i.e. it is strongly coupled. In order to study the problem (1.1) we shall employ some of the methods used in [START_REF] Escher | Global weak solutions for a degenerate parabolic system modeling the spreading of insoluble surfactant[END_REF] to investigate the spreading of insoluble surfactant. However, in our case the situation is more involved since we have two sources of degeneracy, namely when f and g := h -f become zero. It turns out that by choosing (f, g) as unknowns, the system (1.1) is more symmetric:

∂ t f = (1 + R)∂ x (f ∂ x f ) + R∂ x (f ∂ x g) , ∂ t g = R µ ∂ x (g∂ x f ) + R µ ∂ x (g∂ x g) , (t, x) ∈ (0, ∞) × (0, L), (1.4)
since, up to multiplicative constants, the first equation can be obtained from the second by simply interchanging f and g. Corresponding to (1.4) we introduce the following energy functionals:

E 1 (f, g) := L 0 (f ln f -f + 1) + R R µ (g ln g -g + 1) dx and E 2 (f, g) := L 0 f 2 + R(f + g) 2 dx.
It is not difficult to see that both energy functionals E 1 and E 2 dissipate along classical solutions of (1.4). While in the classical setting the functional E 2 plays an important role in the study of the stability properties of equilibria [START_REF] Escher | Modelling and analysis of the Muskat problem for thin fluid layers[END_REF], in the weak setting we strongly rely on the weaker energy E 1 which, nevertheless, provides us with suitable estimates for solutions of a regularised problem and enables us to pass to the limit to obtain weak solutions. Note also that E 1 appears quite natural in the context of (1.4), while, when considering (1.1), one would not expect to have an energy functional of this form.

Our main results read as follows:

Theorem 1.1. Assume that R > 0, R µ > 0. Given f 0 , g 0 ∈ L 2 ((0, L)) with f 0 ≥ 0 and g 0 ≥ 0 there exists a global weak solution (f, g) of (1.4) satisfying

(i) f ≥ 0, g ≥ 0 in (0, T ) × (0, L), (ii) f, g ∈ L ∞ ((0, T ), L 2 ((0, L))) ∩ L 2 ((0, T ), H 1 ((0, L))), for all T > 0 and L 0 f (T )ψ dx - L 0 f 0 ψ dx = - T 0 L 0 ((1 + R)f ∂ x f + Rf ∂ x g) ∂ x ψ dx dt, L 0 g(T )ψ dx - L 0 g 0 ψ dx = -R µ T 0 L 0 (g∂ x f + g∂ x g) ∂ x ψ dx dt for all ψ ∈ W 1 ∞ ((0, L)). Moreover, the weak solutions satisfy (a) f (T ) 1 = f 0 1 , g(T ) 1 = g 0 1 , (b) E 1 (f (T ), g(T )) + T 0 L 0 1 2 |∂ x f | 2 + R 1 + 2R |∂ x g| 2 dx dt ≤ E 1 (f 0 , g 0 ), (c) E 2 (f (T ), g(T )) + T 0 L 0 f ((1 + R)∂ x f + R∂ x g) 2 + RR µ g(∂ x f + ∂ x g) 2 dx dt ≤ E 2 (f 0 , g 0 )
for almost all T ∈ (0, ∞).

Remark 1.2. If f 0 = 0 for instance, a solution to (1.4) is (0, g) where g solves the classical porous medium equation ∂ t g = R µ ∂ x (g∂ x g) in (0, ∞) × (0, L) with homogeneous Neumann boundary conditions and initial condition g 0 .

Additionally to the existence result, we show that the weak solutions constructed in Theorem 1.1 converge at an exponential rate towards the unique flat equilibrium (which is determined by mass conservation) in the L 2 -norm: Theorem 1.3 (Exponential stability). Under the assumptions of Theorem 1.1, there exist positive constants M and ω such that

f (t) - 1 L L 0 f 0 dx 2 2 + g(t) - 1 L L 0 g 0 dx 2 2
≤ M e -ωt for a.e. t ≥ 0.

Remark 1.4. Theorem 1.3 suggests that degenerate solutions become classical after evolving over a certain finite period of time, and therefore would converge in the H 2 -norm towards the corresponding equilibrium, cf. [START_REF] Escher | Modelling and analysis of the Muskat problem for thin fluid layers[END_REF].

The outline of the paper is as follows. In Section 2 we regularise the system (1.4) and prove that the regularised system has global classical solutions, the global existence being a consequence of their boundedness away from zero and in H 1 ((0, L)). The purpose of the regularisation is twofold: on the one hand, the regularised system is expected to be uniformly parabolic and this is achieved by modifying (1.4) and the initial data such that the comparison principle applied to each equation separately guarantees that f ≥ ε and g ≥ ε for some ε > 0. On the other hand, the regularised system is expected to be weakly coupled, a property which is satisfied by a suitable mollification of ∂ x g in the first equation of (1.4) and ∂ x f in the second equation of (1.4). The energy functional E 1 turns out to provide useful estimates for the regularised system as well. In Section 3 we show that the classical global solutions of the regularised problem converge, in appropriate norms, towards a weak solution of (1.4), and that they satisfy similar energy inequalities as the classical solutions of (1.4) for both energy functionals E 1 and E 2 . Finally, we give a detailed proof of Theorem 1.3.

Throughout the paper, we set L p := L p ((0, L)) and

W 1 p := W 1 p ((0, L)) for p ∈ [1, ∞]
, and H 2α := H 2α ((0, L)) for α ∈ [0, 1]. We also denote positive constants that may vary from line to line and depend only on L, R, R µ , and (f 0 , g 0 ) by C or C i , i ≥ 1. The dependence of such constants upon additional parameters will be indicated explicitly.

The regularised system

In this section we introduce a regularised system which possesses global solutions provided they are bounded in H 1 and also bounded away from zero. In Section 3 we show that these solutions converge towards weak solutions of (1.4).

We fix two nonnegative functions f 0 and g 0 in L 2 (the initial data of system (1.4)) and first introduce the space

H 2 B := {f ∈ H 2 ((0, L)) : ∂ x f (0) = ∂ x f (L) = 0}
. We note that for each ε > 0, the elliptic operator (1 -ε 2 ∂ 2

x ) : H 2 B → L 2 is an isomorphism. This property is preserved when considering the restriction

(1 -ε 2 ∂ 2 x ) : {f ∈ C 2+α ([0, L]) : ∂ x f (0) = ∂ x f (L) = 0} → C α ([0, L]), α ∈ (0, 1).
Given f, g ∈ L 2 , we let then

F ε := (1 -ε 2 ∂ 2 x ) -1 f, G ε := (1 -ε 2 ∂ 2 x ) -1 g, (2.1)
and consider the following regularised problem

∂ t f ε = (1 + R)∂ x (f ε ∂ x f ε ) + R∂ x ((f ε -ε)∂ x G ε ) , ∂ t g ε = R µ ∂ x ((g ε -ε)∂ x F ε ) + R µ ∂ x (g ε ∂ x g ε ) , (t, x) ∈ (0, ∞) × (0, L), (2.2a)
supplemented with homogeneous Neumann boundary conditions

∂ x f ε = ∂ x g ε = 0 x = 0, L, (2.2b)
and with regularised initial data

f ε (0) = f 0ε := (1 -ε 2 ∂ 2 x ) -1 f 0 + ε, g ε (0) = g 0ε := (1 -ε 2 ∂ 2 x ) -1 g 0 + ε. (2.2c)
Note that the regularised initial data (f 0ε , g 0ε ) ∈ H 2 B × H 2 B and, invoking the elliptic maximum principle, we have

f 0ε ≥ ε, g 0ε ≥ ε. (2.3) Letting F 0ε := (1 -ε 2 ∂ 2 x ) -1 f 0 and G 0ε := (1 -ε 2 ∂ 2 x
) -1 g 0 , we obtain by multiplying the relation

F 0ε -ε 2 ∂ 2
x F 0ε = f 0 by F 0ε and integrating over (0, L) the following relation

F 0ε 2 2 + ε 2 ∂ x F 0ε 2 2 = L 0 f 0 F 0ε dx≤ f 0 2 F 0ε 2 ,
which gives a uniform L 2 -bound for the regularised initial data:

f 0ε 2 ≤ f 0 2 + ε √ L and g 0ε 2 ≤ g 0 2 + ε √ L. (2.4)
Concerning the solvability of problem (2.2), we use quasilinear parabolic theory, as presented in [START_REF] Amann | Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems[END_REF], to prove the following result:

Theorem 2.1. For each ε ∈ (0, 1) problem (2.2) possesses a unique global nonnegative solution X ε := (f ε , g ε ) with f ε , g ε ∈ C([0, ∞), H 1 ) ∩ C((0, ∞), H 2 B ) ∩ C 1 ((0, ∞), L 2 )
. Moreover, we have

f ε ≥ ε, g ε ≥ ε for all (t, x) ∈ (0, ∞) × (0, L).
In order to prove this global result, we establish the following lemma which gives a criterion for global existence of classical solutions of (2.2): Lemma 2.2. Given ε ∈ (0, 1), the problem (2.2) possesses a unique maximal strong solution

X ε = (f ε , g ε ) on a maximal interval [0, T + (ε)) satisfying f ε , g ε ∈ C([0, T + (ε)), H 1 ) ∩ C((0, T + (ε)), H 2 B ) ∩ C 1 ((0, T + (ε)), L 2 ). Moreover, if for every T < T + (ε) there exists C(ε, T ) > 0 such that f ε ≥ ε/2 + C(ε, T ) -1 , g ε ≥ ε/2 + C(ε, T ) -1 , and max t∈[0,T ] X ε (t) H 1 ≤ C(ε, T ), (2.5)
then the solution is globally defined, i.e. T + (ε) = ∞.

Proof. Let ε ∈ (0, 1) be fixed. To lighten our notation we omit the subscript ε in the remainder of this proof. Note first that problem (2.2) has a quasilinear structure, in the sense that (2.2) is equivalent to the system of equations:

     ∂ t X + A(X)X = F (X) in (0, ∞) × (0, L), BX = 0 on (0, ∞) × {0, L}, X(0) = X 0 on (0, L), (2.6)
where the new variable is X := (f, g) with X 0 = (f 0 , g 0 ), and the operators B and F are respectively given by

BX := ∂ x X, F (X) := ∂ x R(f -ε)∂ x G R µ (g -ε)∂ x F . Letting a(X) := (1 + R)f 0 0 R µ g , the operator A is defined by the relation A(X)Y := -∂ x (a(X)Y ).
We shall prove first that (2.6) has a weak solution defined on a maximal time interval, for which we have a weak criterion for global existence. We then improve in successive steps the regularity of the solution to show that it is actually a classical solution, so that this criterion guarantees also global existence of classical solutions.

Given α ∈ [0, 1], the complex interpolation space

H 2α B := [L 2 , H 2 B ] α is known to satisfy, cf. [1], H 2α B = H 2α , α ≤ 3/4, {f ∈ H 2α : ∂ x f = 0 for x = 0, L} , α > 3/4.
Furthermore, for each β ∈ (1/2, 2] we define the set

V β B := {f ∈ H β B : ε/2 < f }, which is open in H 2 B . Choose now γ := 1/2 -2ξ > 0, where ξ ∈ (0, 1/18). We infer from (2.2c), that X 0 ∈ V 1-ξ B × V 1-ξ B .
In order to obtain existence of a unique weak solution of (2.6) we verify the assumptions of [START_REF] Amann | Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems[END_REF]Theorem 13.1]. With the notation of [START_REF] Amann | Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems[END_REF]Theorem 13.1] we define (σ, s, r, τ

) := (3/2 -3ξ, 1 + ξ, 1 -ξ, -ξ), 2 α := 3/2 -3ξ. Since 1 -ξ -1/2 > γ, we conclude that V 1-ξ B ⊂ C γ ([0, L]), meaning that the elements of a(X) belong to C γ ([0, L]) for X ∈ V 1-ξ B × V 1-ξ B . Since a(X) is positive definite, we conclude in virtue of γ > 2 α -1 that (A, B) ∈ C 1-(V 1-ξ B × V 1-ξ B , E α (0, L)), the notation E α (0, L) being defined in [1, Sections 4 & 8]. Moreover, since (1-ε 2 ∂ 2 x ) -1 ∈ L(L 2 , H 2 B ) we also have F ∈ C 1-(V 1-ξ × V 1-ξ , H -ξ ),
whereby, in the notation of [START_REF] Amann | Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems[END_REF],

H -ξ = H -ξ B because |ξ| < 1/2, cf.
[1, eq. (7.5)]. Thus, we find all the assumptions of [1, Theorem 13.1] fulfilled, and conclude that, for each X 0 ∈ V 2 B , there exists a unique maximal weak

H 3/2-ξ B -solution X = (f, g) of (2.6), that is f, g ∈ C([0, T + ), V 1-ξ B ) ∩ C((0, T + ), H 3/2-ξ B ) ∩ C 1 ((0, T + ), H -1/2-3ξ B ),
and the first equation of (2.6) is satisfied for all t ∈ (0, T + ) when testing with functions belonging to

H 1/2+3ξ B . Moreover, if X [0, T ] is bounded in H 1 B × H 1 B and bounded away from ∂V 1-ξ B
for all T > 0, then T + = ∞, which yields the desired criterion (2.5).

We show now that this weak solution has even more regularity, to conclude in the end that the existence time of the strong solution of (2.6) coincides with that of the weak solution (of course they are identical on each interval where they are defined). Indeed, given δ > 0, it holds that

X ∈ C([δ, T + ), H 3/2-3ξ B ) ∩ C 1 ([δ, T + ), H -1/2-3ξ B
). Hence, if 0 < 2ρ < 2, we conclude from [1, Theorem 7.2] and [7, Proposition 1.1.5] that X is actually Hölder continuous

X ∈ C ρ ([δ, T + ), H 3/2-3ξ-2ρ B
).

Choosing ρ := ξ and µ := 1 -6ξ > 0, we have that H

3/2-3ξ-2ρ B ֒→ C µ ([0, L]), so that the elements of the matrix a(X(t)) belong to C µ ([0, L]) for all t ∈ [δ, T + ). Defining 2 µ := 2 -8ξ > 0, we observe that µ > 2 µ -1 and (A(X), B) ∈ C ρ ([δ, T + ), E µ ((0, L))).
Finally, with 2ν := 3/2 + ξ, our choice for ξ implies

(X(δ), F (X)) ∈ H 2ν-2 B × C ρ ([δ, T + ), H -ξ B ) ⊂ H 2ν-2 B × C ρ ([δ, T + ), H 2ν-2 B ),
and the assertions of [START_REF] Amann | Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems[END_REF]Theorem 11.3] are all fulfilled. Whence, the linear problem

     ∂ t Y + A(X)Y = F (X) in (δ, T + ) × (0, L), BY = 0 on (δ, T + ) × {0, L}, Y (δ) = X(δ) on (0, L), (2.7)
possesses a unique strong H 2ν -solution Y , that is

Y ∈ C([δ, T + ), H 2ν-2 B × H 2ν-2 B ) ∩ C((δ, T + ), H 2ν B × H 2ν B ) ∩ C 1 ((δ, T + ), H 2ν-2 B × H 2ν-2 B
).

In view of [START_REF] Amann | Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems[END_REF]Remark 11.1] we conclude that both X and Y are weak

H 3/2-3ξ B
-solutions of (2.7), whence we infer from [START_REF] Amann | Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems[END_REF]Theorem 11.2] that X = Y, and so

f, g ∈ C([δ, T + ), H -1/2+ξ B ) ∩ C((δ, T + ), H 3/2+ξ B ) ∩ C 1 ((δ, T + ), H -1/2+ξ B
).

Interpolating as we did previously and taking into account that δ was arbitrarily chosen, we have

f, g ∈ C θ ([δ, T + ), C 1+θ ([0, L])) if we set 4θ ≤ ξ. Hence, we find that (X(δ), (A(X), F (X))) ∈ L 2 × C θ ([δ, T + ), H(H 2 B × H 2 B , L 2 × L 2 ) × (L 2 × L 2 )), where H(H 2 B × H 2 B , L 2 × L 2 ) denotes the set of linear operators in L 2 × L 2 with domain H 2 B × H 2 B
which are negative infinitesimal generators of analytic semigroups on L 2 × L 2 , and, in virtue of [1,

Theorem 10.1], X ∈ C((δ, T + ), H 2 B × H 2 B ) ∩ C 1 ((δ, T + ), L 2 × L 2
) for all δ ∈ (0, T + ). Hence, the strong solution of (2.6), which is obtained by applying [1, Theorem 12.1] to that particular system, exists on [0, T + ) and the proof is complete.

The remainder of this section is devoted to prove that T + (ε) = ∞ for the strong solution (f ε , g ε ) of (2.2) constructed in Lemma 2.2. In view of Lemma 2.2, it suffices to prove that (f ε , g ε ) are a priori bounded in H 1 and away from zero. Concerning the latter, we note that we may apply the parabolic maximum principle to each equation of the regularised system (2.2a) separately. Indeed, owing to the boundary conditions (2.2b), the constant function (t, x) → ε solves the first equation of (2.2a) and (2.2b) while we have f 0ε ≥ ε by (2.2c). Consequently,

f ε ≥ ε in [0, T + (ε)) × [0, L]
and, using a similar argument for g ε , we conclude that

f ε ≥ ε, g ε ≥ ε for all (t, x) ∈ [0, T + (ε)) × (0, L). (2.8)
Next, owing to (2.2b) and the nonnegativity of f ε and g ε , it readily follows from (2.2a) that the L 1 -norm of f ε and g ε is conserved in time, that is,

f ε (t) 1 = f 0ε 1 = f 0 1 + εL, g ε (t) 1 = g 0ε 1 = g 0 1 + εL (2.9)
for all t ∈ [0, T + (ε)). The next step is to improve the previous L 1 -bound to an H 1 -bound as required by Lemma 2.2. To this end, we shall use the energy E 1 for the regularised problem, see (2.13) below. As a preliminary step, we collect some properties of the functions (F ε , G ε ) defined in (2.1) in the next lemma.

Lemma 2.3. For all t ∈ (0, T + (ε))

F ε (t) 2 ≤ f ε (t) 2 , G ε (t) 2 ≤ g ε (t) 2 , (2.10) ∂ x F ε (t) 2 ≤ ∂ x f ε (t) 2 , ∂ x G ε (t) 2 ≤ ∂ x g ε (t) 2 , (2.11) ε ∂ 2 x F ε (t) 2 ≤ ∂ x f ε (t) 2 , ε ∂ 2 x G ε (t) 2 ≤ ∂ x g ε (t) 2 .
(2.12)

Proof. The proof of (2.10) is similar to that of (2.4). We next multiply the equation

F ε -ε 2 ∂ 2 x F ε = f ε by -∂ 2
x F ε , integrate over (0, L) and use the Cauchy-Schwarz inequality to estimate the right-hand side and obtain (2.11) and (2.12). Lemma 2.4. Given T ∈ (0, T + (ε)), we have that

E 1 (f ε (T ), g ε (T )) + T 0 L 0 1 2 |∂ x f ε | 2 + R 1 + 2R |∂ x g ε | 2 dx dt ≤ E 1 (f ε (0), g ε (0)). (2.13)
Proof. Using (2.11) and Hölder's inequality, we get

d dt E 1 (f ε , g ε ) = L 0 ∂ t (f ε log(f ε )) + R R µ ∂ t (g ε log(g ε )) dx = - L 0 (1 + R)|∂ x f ε | 2 + R f ε -ε f ε ∂ x f ε ∂ x G ε + R g ε -ε g ε ∂ x g ε ∂ x F ε + R|∂ x g ε | 2 dx ≤-(1 + R) ∂ x f ε 2 2 + R ∂ x f ε 2 ∂ x G ε 2 + R ∂ x g ε 2 ∂ x F ε 2 -R ∂ x g ε 2 2
≤ -

1 2 ∂ x f ε 2 2 - 1 + 2R 2 ∂ x f ε 2 2 -2R ∂ x f ε 2 ∂ x g ε 2 + R ∂ x g ε 2 2
≤ -

1 2 ∂ x f ε 2 2 - R 1 + 2R ∂ x g ε 2 2 .
Integrating with respect to time, we obtain the desired assertion.

Since z ln z -z + 1 ≥ 0 for all z ∈ [0, ∞), relation (2.13) gives a uniform estimate in (ε, t) ∈ (0, 1) × (0, T + (ε)) of (∂ x f ε , ∂ x g ε ) in L 2 ((0, T ), L 2 × L 2 ) in dependence only of the initial condition (f 0 , g 0 ). Indeed, on the one hand, since ln z ≤ z -1 for all z ∈ [0, ∞), we have

z ln z -z + 1 ≤ z(z -1) -(z -1) = (z -1) 2
for all z ≥ 0, (2.14) so that

E 1 (f 0ε , g 0ε ) ≤ L 0 (f 0ε -1) 2 + R R µ (g 0ε -1) 2 dx ≤ C 1 (2.15)
for all ε ∈ (0, 1) by (2.4). On the other hand, owing to the Poincaré-Wirtinger inequality and (2.9), we have

f ε (t) 2 ≤ f ε (t) - 1 L f ε (t) 1 2 + f ε (t) 1 √ L ≤ C ( ∂ x f ε (t) 2 + 1) .
A similar bound being available for g ε , we infer from (2.13), (2.15), and the nonnegativity of E 1 that, for T ∈ (0, T + (ε)),

T 0 f ε (t) 2 H 1 + g ε (t) 2 H 1 dt ≤ C T 0 1 + ∂ x f ε (t) 2 2 + ∂ x g ε (t) 2 2 dt ≤ C [T + E 1 (f 0ε , g 0ε )] ≤ C 2 (T ). ( 2

.16)

We next use this estimate to prove that the solution (f ε , g ε ) of (2.2) is bounded in L ∞ ((0, T ), H 1 × H 1 ) for all T < T + (ε). While the estimates were independent of ε up to now, the next ones have a strong dependence upon ε which explains the need of a regularisation of the original system.

Lemma 2.5. Given (ε, T ) ∈ (0, 1) × (0, T + (ε)), there exists a constant C(ε, T ) > 0 such that the solution (f ε , g ε ) of (2.2) fulfills

f ε (t) H 1 + g ε (t) H 1 ≤ C(ε, T ) for all t ∈ [0, T ].
(2.17)

Proof. We prove first the bound for f ε . Given z ∈ R, let q(z) := z 2 /2. With this notation, the first equation of (2.2a) reads

∂ t f ε -(1 + R)∂ 2 x q(f ε ) = R∂ x ((f ε -ε)∂ x G ε ).
Multiplying this relation by ∂ t q(f ε ) and integrating over (0, L), we get

L 0 ∂ t f ε ∂ t q(f ε ) dx -(1 + R) L 0 ∂ 2 x q(f ε )∂ t q(f ε ) dx = R L 0 ∂ x ((f ε -ε)∂ x G ε )∂ t q(f ε ) dx.
Using an integration by parts and Young's inequality, we come to the following inequality

f ε ∂ t f ε 2 2 + 1 + R 2 d dt ∂ x q(f ε ) 2 2 ≤ 1 2 f ε ∂ t f ε 2 2 + R 2 2 L 0 f ε [∂ x ((f ε -ε)∂ x G ε )] 2 dx.
Whence, we have shown that

f ε ∂ t f ε 2 2 + (1 + R) d dt ∂ x q(f ε ) 2 2 ≤ R 2 L 0 f 3 ε (∂ 2 x G ε ) 2 + f ε (∂ x f ε ) 2 (∂ x G ε ) 2 dx, (2.18)
and the second term on the right-hand side of (2.18) may be estimated, in view of (2.8), by

L 0 f ε (∂ x f ε ) 2 (∂ x G ε ) 2 dx ≤ 1 ε L 0 f 2 ε (∂ x f ε ) 2 (∂ x G ε ) 2 dx = 1 ε L 0 (∂ x q(f ε )) 2 (∂ x G ε ) 2 dx ≤ 1 ε ∂ x G ε 2 ∞ ∂ x q(f ε ) 2 2 . Now, since G ε is the solution of G ε -ε 2 ∂ 2
x G ε = g ε with homogeneous Neumann boundary conditions at x = 0, L, and g ε ≥ ε, the elliptic maximum principle guarantees that G ε ≥ ε. Hence, -ε 2 ∂ 2

x G ε ≤ g ε , and therefore, for x ∈ (0, L),

-ε 2 ∂ x G ε (x) ≤ x 0 g ε dx ≤ g ε 1 and ε 2 ∂ x G ε (x) ≤ L x g ε dx ≤ g ε 1 , so that ε 2 ∂ x G ε ∞ ≤ g ε 1 .
In view of (2.9), we arrive at

L 0 f ε (∂ x f ε ) 2 (∂ x G ε ) 2 dx ≤ 1 ε 5 g ε 2 1 ∂ x q(f ε ) 2 2 ≤ C(ε) ∂ x q(f ε ) 2 2 . (2.19)
Concerning the first term on the right-hand side of (2.18), it follows from (2.8) and (2.12) that

L 0 f 3 ε (∂ 2 x G ε ) 2 dx ≤ 1 ε L 0 f 4 ε (∂ 2 x G ε ) 2 dx ≤ 4 ε q(f ε ) 2 ∞ ∂ 2 x G ε 2 2 ≤ 4 ε 3 q(f ε ) 2 ∞ ∂ x g ε 2 2
.

In order to estimate q(f ε ) ∞ , we choose x ε ∈ (0, L) such that Lf ε (x ε ) = f ε 1 . By the fundamental theorem of calculus, we get

q(f ε )(x) = q(f ε )(x ε ) + x xε ∂ x q(f ε ) dx ≤ 1 2L 2 f ε 2 1 + √ L ∂ x q(f ε ) 2 for all x ∈ (0, L).
Summarising, we obtain in view of (2.18) and (2.19), that

d dt ∂ x q(f ε ) 2 2 ≤ C(ε)(1 + ∂ x g ε 2 2 )(1 + ∂ x q(f ε ) 2 2 ), (2.20)
and from (2.16) and (2.20)

∂ x q(f ε (t)) 2 2 ≤ (1 + ∂ x q(f 0ε ) 2 2 ) exp C(ε) t 0 1 + ∂ x g ε 2 2 ds ≤ C(ε, T ), t ∈ [0, T ].
Since f ε ≥ ε, we then have

∂ x f ε (t) 2 2 ≤ C(ε, T ) for all t ∈ [0, T ].
Using (2.9) and the Poincaré-Wirtinger inequality, we finally obtain that

f ε (t) H 1 ≤ C(ε, T ) for all t ∈ [0, T ].
Moreover, due to the symmetry of (2.2a), g ε satisfies the same estimate as f ε , and this completes our argument.

Proof of Theorem 2.1. The proof is a direct consequence of Lemma 2.2, the lower bounds (2.8), and Lemma 2.5.

We end this section by showing that, though E 2 is not dissipated along the trajectories of the regularised system (2.2), a functional closely related to E 2 is almost dissipated, with non-dissipative terms of order ε. Lemma 2.6. For ε ∈ (0, 1) and T > 0, we have

E 2,ε (f ε (T ), g ε (T )) + T 0 f ε |(1 + R)∂ x f ε + R∂ x G ε | 2 + RR µ g ε |∂ x (F ε + g ε )| 2 dt ≤ E 2,ε (f 0ε , g 0ε ) + εC 2 T 0 ̺ ε (t) dt,
(2.21)

with ̺ ε := ∂ x f ε 2 2 + ∂ x g ε 2 2 and 2E 2,ε (f ε , g ε ) := (1 + R) f ε 2 2 + R g ε 2 2 + R L 0 (F ε g ε + G ε f ε ) dx. (2.22)
Proof. We multiply the first equation of (2.2) by (1+R)f ε +RG ε and integrate over (0, L) to obtain

L 0 ∂ t f ε ((1 + R)f ε + RG ε ) dx = - L 0 f ε ((1 + R)∂ x f ε + R∂ x G ε ) 2 dx + I 1,ε (2.23) with I 1,ε := εR L 0 ∂ x G ε ((1 + R)∂ x f ε + R∂ x G ε ) dx.
Thanks to Hölder's inequality and (2.11), we have

|I 1,ε | ≤ εR(1 + R) ∂ x G ε 2 ∂ x f ε 2 + ∂ x G ε 2 2 ≤ εC ∂ x f ε 2 2 + ∂ x g ε 2 2 .
(2.24)

Similarly, multiplying the second equation of (2.2) by R(F ε + g ε ) and integrating over (0, L) give

R L 0 ∂ t g ε (F ε + g ε ) dx = -RR µ L 0 g ε (∂ x F ε + ∂ x g ε ) 2 dx + I 2,ε (2.25) with I 2,ε := εRR µ L 0 ∂ x F ε ∂ x (F ε + g ε ) dx.
Using again Hölder's inequality and (2.11), we obtain

|I 2,ε | ≤ εRR µ ∂ x F ε 2 2 + ∂ x g ε 2 ∂ x F ε 2 ≤ εC ∂ x f ε 2 2 + ∂ x g ε 2 2 .
(2.26)

Observing that

L 0 ∂ t f ε ((1 + R)f ε + RG ε ) dx + R L 0 ∂ t g ε (F ε + g ε ) dx = 1 + R 2 d dt f ε 2 2 + R L 0 G ε ∂ t F ε -ε 2 ∂ 2 x ∂ t F ε dx + R 2 d dt g ε 2 2 + R L 0 F ε ∂ t G ε -ε 2 ∂ 2 x ∂ t G ε dx = 1 2 d dt (1 + R) f ε 2 2 + R g ε 2 2 + 2R L 0 F ε G ε dx + ε 2 R L 0 (∂ x G ε ∂ t ∂ x F ε + ∂ x F ε ∂ t ∂ x G ε ) dx = 1 2 d dt (1 + R) f ε 2 2 + R g ε 2 2 + 2R L 0 F ε G ε + ε 2 ∂ x F ε ∂ x G ε dx ,
we sum (2.23) and (2.25), use (2.24) and (2.26) to obtain 1 2

d dt (1 + R) f ε 2 2 + R g ε 2 2 + 2R L 0 F ε G ε + ε 2 ∂ x F ε ∂ x G ε dx ≤ - L 0 f ε ((1 + R)∂ x f ε + R∂ x G ε ) 2 dx -RR µ L 0 g ε (∂ x F ε + ∂ x g ε ) 2 dx + εC 2 ̺ ε . Since 2 L 0 F ε G ε + ε 2 ∂ x F ε ∂ x G ε dx = L 0 2F ε G ε -ε 2 G ε ∂ 2 x F ε -ε 2 F ε ∂ 2 x G ε dx = L 0 (G ε f ε + F ε g ε ) dx,
the claimed inequality follows from the above two identities after integration with respect to time.

Weak solutions

Given T ∈ (0, ∞], we let Q T := (0, T ) × (0, L). Furthermore, given ε ∈ (0, 1), we let (f ε , g ε ) be the global strong solution of the regularised problem (2.2) constructed in Theorem 2.1. We shall prove that (f ε , g ε ) converges, in appropriate function spaces over Q T , towards a pair of functions (f, g) which turns out to be a weak solution of (1.4) in the sense of Theorem 1.1.

Recall that, by (2.8), (2.4), (2.9), and (2.13), (f ε , g ε ) satisfies the following estimates

(a) f ε ≥ ε, g ε ≥ ε on Q ∞ , (b)
f 0ε 2 + g 0ε 2 ≤ f 0 2 + g 0 2 + 2 √ L, (c) f ε (t) 1 = f 0 1 + εL, g ε (t) 1 = g 0 1 + εL, (d) E 1 (f ε (t), g ε (t)) + t 0 1 2 ∂ x f ε 2 2 + R 1 + 2R ∂ x g ε 2 2 ds ≤ E 1 (f ε (0), g ε (0)), (3.1)
for t ≥ 0. Using (3.1), we show that:

Lemma 3.1 (Uniform estimates). Let h ∈ {f, g, F, G}. There exists a positive constant C 4 (T ) such that, for all (ε, T ) ∈ (0, 1) × (0, ∞), we have

(i) T 0 h ε (t) 2 H 1 + h ε (t) 3 3 dt ≤ C 4 (T ), (3.2) (ii)
T 0 ∂ t h ε (t) 6/5 (W 1 6 ) ′ dt ≤ C 4 (T ). (3.3)
Proof. The estimate for h ε in L 2 (0, T, H 1 ) is obtained from the energy estimate (2.13), by taking also into account relations (2.9), (

, the nonnegativity of E 1 , and the Poincaré-Wirtinger inequality as in the proof of (2.16). In order to prove the second estimate of (3.2), we note that, since H 1 is continuously embedded in L ∞ and (f ε ) is bounded in L 2 (0, T, H 1 ), (f ε ) is uniformly bounded with respect to ε in L ∞ (0, ∞; L 1 ) ∩ L 2 (0, T ; L ∞ ) for all T > 0. The claimed L 3 -bound then follows from the inequality f ε

3 3 ≤ f ε 2 ∞ f ε 1 . Next, an obvious consequence of the definition of F ε is that F ε p ≤ f ε p for p ∈ [1, ∞]
, from which we deduce the expected bound in L 3 for (F ε ). A similar argument shows that (g ε ) and (G ε ) satisfy the second estimate in (3.2).

In order to prove (ii), consider first h ∈ {f, g}. From (3.2) and Hölder's inequality we obtain that

(f ε ∂ x f ε ), ((f ε -ε)∂ x G ε ), ((g ε -ε)∂ x F ε )
, and (g ε ∂ x g ε ) are uniformly bounded in L 6/5 (Q T ). Therefore, the equations of (2.2a) may be written in the form

∂ t h ε = ∂ x H h ε , for some function H h ε which is uniformly bounded in L 6/5 (Q T ) and satisfies homogeneous Dirichlet conditions H h ε (0) = H h ε (L) = 0. Given φ ∈ W 1 6 , we have | ∂ t h ε |φ L 2 | = L 0 φ∂ x H h ε dx = L 0 H h ε ∂ x φ dx ≤ H h ε 6/5 ∂ x φ 6 .
Consequently,

∂ t h ε (t) (W 1 6 ) ′ ≤ H h ε (t) 6/5 for t ∈ (0, T ).
and the families (f ε ), (g ε ) are both uniformly bounded in L 6/5 (0, T ; W 1 6 ′ ).

Finally, we have (1

-ε 2 ∂ 2 x )p|q L 2 = p|(1 -ε 2 ∂ 2 x )q L 2 for all p, q ∈ H 2 B , and choosing p := (1 -ε 2 ∂ 2 x ) -1 ∂ t f ε , q = (1 -ε 2 ∂ 2 x ) -1 φ with φ ∈ W 1 6 , we have ∂ t F ε |φ L 2 = (1 -ε 2 ∂ 2 x ) -1 ∂ t f ε |φ L 2 = ∂ t f ε |(1 -ε 2 ∂ 2 x ) -1 φ L 2 = ∂ x H f ε |(1 -ε 2 ∂ 2 x ) -1 φ L 2 = -H f ε |(1 -ε 2 ∂ 2 x ) -1 ∂ x φ L 2 .
Since (1 -ε 2 ∂ 2 x ) -1 is a contraction in L 6 , we obtain that

| ∂ t F ε |φ L 2 | ≤ H f ε 6/5 (1 -ε 2 ∂ 2 x ) -1 ∂ x φ 6 ≤ H f ε 6/5 φ W 1 6
, and the assertion (ii), when h = F, follows at once. Invoking a similar argument for (G ε ), we complete the proof.

This lemma enables us to use a result from [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF] and show that (f ε ), (g ε ), (F ε ), and (G ε ) are relatively compact in L 2 (0, T ; C α ([0, L])), provided that α ∈ (0, 1/2). This will allow us to identify a limit point for each of these sequences, and find in this way a candidate for solving (1.4). Indeed, we have: Lemma 3.2. Given h ∈ {f, g, F, G}, and α ∈ (0, 1/2), there exists a subsequence (h ε k ) of (h ε ) which converges strongly in L 2 (0, T ; C α ([0, L])).

Proof. Invoking the Rellich-Kondrachov theorem, we have the following sequence of embeddings

H 1 ֒→ C α ([0, L]) ֒→ W 1 6 ′ , α < 1/2,
with compact embedding H 1 ֒→ C α ([0, L]). Furthermore, in view of Lemma 3.1 (i), the family (h ε ) is uniformly bounded in L 2 (0, T ; H 1 ), while, by Lemma 3.1 (ii), (∂ t h ε ) is uniformly bounded in L 1 (0, T ; W 1 6 ′ ). Whence, the assumptions of [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF]Corollary 4] are all fulfilled, and we conclude that

(h ε ) is relatively compact in L 2 (0, T ; C α ([0, L])).
3.1. Construction of weak solutions. Using the uniform estimates deduced at the beginning of this section, we now establish the existence of a weak solution of (1.4). Owing to Lemma 3.2, there are f, g, F, G ∈ L 2 (0, T ; C α ([0, L])) such that, for α ∈ (0, 1/2),

f ε k → f, g ε k → g, F ε k → F, G ε k → G in L 2 (0, T ; C α ([0, L])). (3.4)
Furthermore, by Lemma 3.1 (i), the subsequences (∂ x f ε k ), (∂ x g ε k ), (∂ x F ε k ), and (∂ x G ε k ) are uniformly bounded in the Hilbert space L 2 (Q T ). Hence, we may extract further subsequences (denoted again by (f ε k ), (g ε k ), (F ε k ), and (G ε k )) which converge weakly:

∂ x f ε k ⇀ ∂ x f, ∂ x g ε k ⇀ ∂ x g, ∂ x F ε k ⇀ ∂ x F, ∂ x G ε k ⇀ ∂ x G in L 2 (Q T ). (3.5)
In fact, we have that

f = F and g = G a.e. in Q T . (3.6)
Indeed, (3.6) follows by multiplying the relation

F ε k -ε 2 k ∂ 2 x F ε k = f ε k by a test function in H 1
, integrating by parts, and letting then k → ∞ with the help of (3.4) and (3.5). In view of (3.4)-(3.6), we then have

f ε k ∂ x f ε k ⇀ f ∂ x f, f ε k ∂ x G ε k ⇀ f ∂ x g in L 1 (Q T ), g ε k ∂ x F ε k ⇀ g∂ x f, g ε k ∂ x g ε k ⇀ g∂ x g in L 1 (Q T ). (3.7)
Using the fact that (f ε , g ε ) are strong solutions of (2.2), we obtain by integration with respect to space and time that

L 0 f ε k (T )ψ dx - L 0 f 0ε k ψ dx =- Q T [(1 + R)f ε k ∂ x f ε k + R(f ε k -ε k )∂ x G ε k ] ∂ x ψ dx dt, L 0 g ε k (T )ψ dx - L 0 g 0ε k ψ dx =-R µ Q T [(g ε k -ε k )∂ x F ε k + g ε k ∂ x g ε k ] ∂ x ψ dx dt, (3.8)
for all T > 0, ε ∈ (0, 1), and

ψ ∈ W 1 ∞ . Since f 0ε k → f 0 and g 0ε k → g 0 in L 2 (3.9)
by classical arguments, we may pass to the limit as k → ∞ in (3.8) and use (3.4), (3.5), (3.7), and (3.9) to conclude that (f, g) is a weak solution of (2.2) in the sense of Theorem 1.1. The fact that (f, g) can be defined globally follows by using a standard Cantor's diagonal argument (using a sequence T n ր ∞).

3.2.

Energy estimates for weak solutions. Letting ε → 0 in the relation (3.1) (c), we find in view of (3.4), that

f (t) 1 = f 0 1 , g(t) 1 = g 0 1 , t ∈ (0, ∞).
We show now that the weak solution found above satisfies the energy estimate

E 1 (f (T ), g(T )) + Q T 1 2 |∂ x f | 2 + R 1 + 2R |∂ x g| 2 dx dt ≤ E 1 (f 0 , g 0 ) (3.10)
for T ∈ (0, ∞). Recall that, by Lemma 2.4, we have

E 1 (f ε k (T ), g ε k (T )) + Q T 1 2 |∂ x f ε k | 2 + R 1 + 2R |∂ x g ε k | 2 dx dt ≤ E 1 (f 0ε k , g 0ε k ) (3.11)
for all k ∈ N. On the one hand, note that (3.4) and Fatou's lemma ensure that

E 1 (f (T ), g(T )) ≤ lim inf k→∞ E 1 (f ε k (T ), g ε k (T )) for T ∈ (0, ∞), (3.12
) while (3.5) implies

Q T |∂ x f | 2 dx dt ≤ lim inf k→∞ Q T |∂ x f ε k | 2 dx dt, Q T |∂ x g| 2 dx dt ≤ lim inf k→∞ Q T |∂ x g ε k | 2 dx dt. (3.13)
We still have to pass to the limit in the right-hand side of (3.11). By (3.9), we may assume that (f 0ε k ) and (g 0ε k ) converge almost everywhere towards f 0 and g 0 , respectively. Furthermore, since 0 ≤ x ln x -x + 1 ≤ 1 + 2x 3/2 for x ≥ 0, we have

E |f 0ε k ln f 0ε k -f 0ε k + 1| dx ≤ |E| + 2 E f 3/2 0ε k dx ≤ |E| + 2|E| 1/4 f 0ε k 3/2 2 ≤ C|E| 1/4
for all k ∈ N and all measurable subsets E of (0, L), meaning that the family (f 0ε k ln f 0ε k -f 0ε k + 1) is uniformly integrable. Clearly, the same is true also for (g 0ε k ln g 0ε k -g 0ε k + 1). We infer then from Vitali's convergence theorem, cf. [5, Theorem 2.24], that the limit of the right-hand side of (3.11) exists and

lim k→∞ E 1 (f 0ε k , g 0ε k ) = E 1 (f 0 , g 0 ).
Whence, passing to the limit in (3.11), we obtain in view of (3.12) and (3.13) the desired estimate (3.10). Finally, we show that weak solutions of (1.4) satisfy

E 2 (f (T ), g(T )) + Q T f ((1 + R)∂ x f + R∂ x g) 2 + RR µ g(∂ x f + ∂ x g) 2 dx dt ≤ E 2 (f 0 , g 0 ) (3.14)
for T ∈ (0, ∞). In virtue of (3.4), (3.5), and (3.6) we have

f ε k → f , ∂ x f ε k ⇀ ∂ x f, and ∂ x G ε k ⇀ ∂ x g in L 2 (Q T ) which implies that f ε k ∂ x f ε k ⇀ √ f ∂ x f and f ε k ∂ x G ε k ⇀ √ f ∂ x g in L 1 (Q T ). Consequently, f ε k ((1 + R)∂ x f ε k + R∂ x G ε k ) ⇀ f ((1 + R)∂ x f + R∂ x g) in L 1 (Q T ).
and, by a similar argument,

√ g ε k (∂ x F ε k + ∂ x g ε k ) ⇀ √ g(∂ x f + ∂ x g) in L 1 (Q T ).
Now, owing to (3.2), the sequence (̺ ε k ) defined in Lemma 2.6 is bounded in L 2 ((0, T )) and we then infer from Lemma 2.6 that both (

f ε k ((1 + R)∂ x f ε k + R∂ x G ε k )) and ( √ g ε k (∂ x F ε k + ∂ x g ε k ))
are bounded in L 2 (Q T ). The previous weak convergences in L 1 (Q T ) may then be improved to weak convergence in L 2 (Q T ) (upon extracting a further subsequence if necessary) and we can then pass to the limit in (2.21) to conclude that (3.14) holds true, using weak lower semicontinuity arguments in the left-hand side and the property ε k ̺ ε k → 0 in L 2 ((0, T )) in the right-hand side.

3.3. Exponential convergence towards equilibria. In this last part of the paper we prove our second main result, Theorem 1.3. The proof is based on the interplay between estimates for the two energy functionals E 1 and E 2 , with the specification that we use E 1 to estimate the time derivative of the stronger energy functional E 2 , and obtain exponential decay of weak solutions in the L 2 -norm.

Recall from (3.1) (c) that

A k := f 0 1 L + ε k = f ε k (t) 1 L and B k := g 0 1 L + ε k = g ε k (t) 1 L
for all k ∈ N and t ∈ [0, ∞). Introducing

F k := L 0 f ε k ln f ε k A k -f ε k + A k + R R µ g ε k ln g ε k B k -g ε k + B k dx + 1 2 L 0 (f ε k -A k ) 2 + R (f ε k -A k ) 2 + (g ε k -B k ) 2 + (g ε k -B k )(F ε k -A k ) +(f ε k -A k )(G ε k -B k )]} dx,
we infer from (3.1) (c) and the proofs of Lemma 2.4 and 2.6 that

dF k dt = L 0 ∂ t f ε k ln f ε k + R R µ ∂ t g ε k ln g ε k dx + 1 2 d dt L 0 (1 + R)f 2 ε k + R g 2 ε k + f ε k G ε k + g ε k F ε k dx ≤ - 1 2 ∂ x f ε k 2 2 - R 1 + 2R ∂ x g ε k 2 2 + ε k C 3 ∂ x f ε k 2 2 + ∂ x g ε k 2 2
≤ -

1 3 ∂ x f ε k 2 2 - R 2 + 2R ∂ x g ε k 2 2
provided k is large enough. Using the Poincaré-Wirtinger inequality, we find a positive constant

C 5 such that dF k dt ≤ -C 5 f ε k -A k 2 2 + g ε k -B k 2 2
(3.15)

for large k and all t ∈ (0, ∞). We show now that the right-hand side of (3.15) can be bounded by -ωF k for some small positive number ω. Indeed, arguing as in Lemma 2.3, we find that

L 0 [(g ε k -B k )(F ε k -A k ) + (f ε k -A k )(G ε k -B k )] dx ≤ g ε k -B k 2 F ε k -A k 2 + f ε k -A k 2 G ε k -B k 2 ≤ 2 g ε k -B k 2 f ε k -A k 2 ≤ f ε k -A k 2 2 + g ε k -B k 2 2 .
(3.16) Recalling (2.14), we end up with

f ε k -A k 2 2 + g ε k -B k 2 2 = L 0 A 2 k f ε k A k -1 2 + B k g ε k B k -1 2 dx ≥ L 0 A 2 k f ε k A k ln f ε k A k - f ε k A k + 1 + B 2 k g ε k B k ln g ε k B k - g ε k B k + 1 dx ≥ min k A k , R µ B k R L 0 f ε k ln f ε k A k -f ε k + A k + R R µ g ε k ln g ε k B k -g ε k + B k dx.
(3.17) Combining (3.15), (3.16), and (3.17), we conclude that if f 0 1 > 0 and g 0 1 > 0, then dF k dt (t) ≤ -ωF k (t)

for some positive constant ω and k sufficiently large. Whence,

f ε k (t) -A k L 2 + g ε k (t) -B k L 2 ≤ Ce -ωt ,
which yields, for k → ∞, the desired estimate by (3.4) and (3.6), as stated in Theorem 1.3. If f 0 = 0 [resp. g 0 = 0], then f = 0 [resp. g = 0], while g [resp. f ] is a weak solution of the onedimensional porous medium equation and converges therefore even in the L ∞ -norm to flat equilibria (if f 0 = 0, then uniqueness of solutions to (2.6) implies that f ε = ε and ∂ t g ε = R µ (g ε ∂ x g ε )), cf. [START_REF] Vázquez | The Porous Media Equation[END_REF]Theorem 20.16]. Convergence in this stronger norm is due to the fact that comparison methods may be used for the one-dimensional porous media equation, while for our system they fail because of the structure of the system.

Figure 1 .

 1 Figure 1. The physical setting

Partially supported by the French-German procope project 20190SE.


Ankik Kumar Giri
email: ankik-kumar.giri@unileoben.ac.at

Philippe Laurençot

Gerald Warnecke

Weak Solutions to the Continuous Coagulation Equation with Multiple Fragmentation

Keywords: Coagulation, Multiple Fragmentation, Unbounded kernels, Existence, Weak compactness

The existence of weak solutions to the continuous coagulation equation with multiple fragmentation is shown for a class of unbounded coagulation and fragmentation kernels, the fragmentation kernel having possibly a singularity at the origin. This result extends previous ones where either boundedness of the coagulation kernel or no singularity at the origin for the fragmentation kernel were assumed.

Introduction

The continuous coagulation and multiple fragmentation equation describes the evolution of the number density f = f (x, t) of particles of volume x ≥ 0 at time t ≥ 0 and reads ∂f (x, t) ∂t = 1 2

x 0 K(x -y, y)f (x -y, t)f (y, t)dy -∞ 0 K(x, y)f (x, t)f (y, t)dy

+ ∞ x b(x, y)S(y)f (y, t)dy -S(x)f (x, t), (1)
with

f (x, 0) = f 0 (x) ≥ 0. ( 2
)
The first two terms on the right-hand side of (1) accounts for the formation and disappearance of particles as a result of coagulation events and the coagulation kernel K(x, y) represents the rate at which particles of volume x coalesce with particles of volume y. The remaining two terms on the right-hand side of (1) describes the variation of the number density resulting from fragmentation events which might produce more than two daughter particles, and the breakage function b(x, y) is the probability density function for the formation of particles of volume x from the particles of volume y. Note that it is non-zero only for x < y.

The breakage function is assumed here to have the following properties y 0 b(x, y)dx = N < ∞, for all y > 0, b(x, y) = 0 for x > y,

and y 0 xb(x, y)dx = y for all y > 0.

(

) 5
The parameter N represents the number of fragments obtained from the breakage of particles of volume y and is assumed herein to be finite and independent of y. This is however inessential for the forthcoming analysis, see Remark 2.3 below. As for the condition [START_REF] Ph | On a class of continuous coagulation-fragmentation equations[END_REF], it states that the total volume of the fragments resulting from the splitting of a particle of volume y equals y and thus guarantees that the total volume of the system remains conserved during fragmentation events.

The existence of solutions to coagulation-fragmentation equations has already been the subject of several papers which however are mostly devoted to the case of binary fragmentation, that is, when the fragmentation kernel Γ satisfies the additional symmetry property Γ(x + y, y) = Γ(x + y, x) for all (x, y) ∈]0, ∞[ 2 , see the survey [START_REF] Ph | On coalescence equations and related models[END_REF] and the references therein. The coagulationfragmentation equation with multiple fragmentation has received much less attention over the years though it is already considered in the pioneering work [START_REF] Melzak | A scalar transport equation[END_REF], where the existence and uniqueness of solutions to (1)-( 2) are established for bounded coagulation and fragmentation kernels K and Γ. A similar result was obtained later on in [START_REF] Mclaughlin | An existence and uniqueness result for a coagulation and multiple-fragmentation equation[END_REF] by a different approach. The boundedness of Γ was subsequently relaxed in [START_REF] Lamb | Existence and uniqueness results for the continuous coagulation and fragmentation equation[END_REF] where it is only assumed that S grows at most linearly, but still for a bounded coagulation kernel. Handling simultaneously unbounded coagulation and fragmentation kernels turns out to be more delicate and, to our knowledge, is only considered in [START_REF] Ph | On a class of continuous coagulation-fragmentation equations[END_REF] for coagulation kernels K of the form K(x, y) = r(x)r(y) with no growth restriction on r and a moderate growth assumption on Γ (depending on r) and in [START_REF] Giri | The continuous coagulation equation with multiple fragmentation[END_REF] for coagulation kernels satisfying K(x, y) ≤ φ(x)φ(y) for some sublinear function φ and a moderate growth assumption on Γ (see also [START_REF] Ph | The discrete coagulation equation with multiple fragmentation[END_REF] for the existence of solutions for the corresponding discrete model). Still, the fragmentation kernel Γ is required to be bounded near the origin in [START_REF] Giri | The continuous coagulation equation with multiple fragmentation[END_REF][START_REF] Ph | On a class of continuous coagulation-fragmentation equations[END_REF] which thus excludes kernels frequently encountered in the literature such as Γ(y, x) = (α + 2) x α y γ-(α+1) with α > -2 and γ ∈ R [START_REF] Mcgrady | Shattering" transition in fragmentation[END_REF]. The purpose of this note is to fill (at least partially) this gap and establish the existence of weak solutions to (1) for simultaneously unbounded coagulation and fragmentation kernels K and Γ, the latter being possibly unbounded for small and large volumes. More precisely, we make the following hypotheses on the coagulation kernel K, multiple-fragmentation kernel Γ, and selection rate S.

Hypotheses 1.1. (H1) K is a non-negative measurable function on [0, ∞[×[0, ∞[ and is symmetric, i.e. K(x, y) = K(y, x) for all x, y ∈]0, ∞[, (H2) K(x, y) ≤ φ(x)φ(y) for all x, y ∈]0, ∞[ where φ(x) ≤ k 1 (1 + x) µ for some 0 ≤ µ < 1 and constant k 1 > 0.

(H3) Γ is a non-negative measurable function on ]0, ∞[×]0, ∞[ such that Γ(x, y) = 0 if 0 < x < y.
Defining S and b by ( 3), we assume that b satisfies [START_REF] Ph | On a class of continuous coagulation-fragmentation equations[END_REF] and there are θ ∈ [0, 1[ and two non-

negative functions k :]0, ∞[→ [0, ∞[ and ω :]0, ∞[ 2 → [0, ∞[ such that, for each R ≥ 1:
(H4) we have Γ(y, x) ≤ k(R) y θ for y > R and x ∈]0, R[, (H5) for y ∈]0, R[ and any measurable subset E of ]0, R[, we have

y 0 1 E (x)Γ(y, x)dx ≤ ω(R, |E|), y ∈]0, R[,
where |E| denotes the Lebesgue measure of E, 1 E is the indicator function of E given by

1 E (x) := 1 if x ∈ E, 0 if x / ∈ E,
and we assume in addition that

lim δ→0 ω(R, δ) = 0, (H6) S ∈ L ∞ ]0, R[.
We next introduce the functional setting which will be used in this paper: define the Banach space X with norm • by

X = {f ∈ L 1 (0, ∞) : f < ∞} where f = ∞ 0 (1 + x)|f (x)|dx,
together with its positive cone

X + = {f ∈ X : f ≥ 0 a.e.}.
For further use, we also define the norms

f x = ∞ 0 x|f (x)|dx and f 1 = ∞ 0 |f (x)|dx, f ∈ X.
The main result of this note is the following existence result:

Theorem 1.2. Suppose that (H1)-(H6) hold and assume that f 0 ∈ X + . Then ( 1)-( 2) has a weak solution f on ]0, ∞[ in the sense of Definition 1.3 below. Furthermore, f (t) x ≤ f 0 x for all t ≥ 0.

Before giving some examples of coagulation and fragmentation kernels satisfying (H1)-(H6), we recall the definition of a weak solution to (1)-( 2) [START_REF] Stewart | A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels[END_REF].

Definition 1.3. Let T ∈]0, ∞]. A solution f of (1)-(2) is a non-negative function f : [0, T [→ X + such that, for a.e. x ∈]0, ∞[ and all t ∈ [0, T [, (i) s → f (x, s) is continuous on [0, T [, (ii) the following integrals are finite t 0 ∞ 0 K(x, y)f (y, s)dyds < ∞ and t 0 ∞ x b(x, y)S(y)f (y, s)dyds < ∞,
(iii) the function f satisfies the following weak formulation of ( 1)-( 2)

f (x, t) = f 0 (x) + t 0 1 2 x 0 K(x -y, y)f (x -y, s)f (y, s)dy - ∞ 0 K(x, y)f (x, s)f (y, s)dy + ∞ x b(x, y)S(y)f (y, s)dy -S(x)f (x, s) ds.
Coming back to (H1)-(H6), it is clear that coagulation kernels satisfying K(x, y) ≤ x µ y ν + x ν y µ for some µ ∈ [0, 1[ and ν ∈ [0, 1[ which are usually used in the mathematical literature satisfy (H1)-(H2), see also [START_REF] Giri | The continuous coagulation equation with multiple fragmentation[END_REF] for more complex choices. Let us now turn to fragmentation kernels which also fit in the classes considered in Hypotheses 1.1.

Clearly, if we assume that

Γ ∈ L ∞ (]0, ∞[×]0, ∞[)
as in [START_REF] Giri | The continuous coagulation equation with multiple fragmentation[END_REF][START_REF] Mclaughlin | An existence and uniqueness result for a coagulation and multiple-fragmentation equation[END_REF], (H4) and (H5) are satisfied with k = Γ L ∞ , θ = 0, and ω(R, δ) = Γ L ∞ δ. Now let us take

S(y) = y γ and b(x, y) = α + 2 y x y α for 0 < x < y, (6)
where γ > 0 and α ≥ 0, see [START_REF] Mcgrady | Shattering" transition in fragmentation[END_REF][START_REF] Peterson | Similarity solutions for the population balance equation describing particle fragmentation[END_REF]. Then

Γ(y, x) = (α + 2)x α y γ-(α+1) for 0 < x < y.
Let us first check (H5). Given R > 0, y ∈]0, R[, and a measurable subset E of ]0, R[, we deduce from Hölder's inequality that

y 0 1 E (x)Γ(y, x)dx = (α + 2)y γ-(α+1) y 0 1 E (x)x α dx ≤ (α + 2)y γ-(α+1) |E| γ γ+1 y 0 x α(γ+1) dx 1 γ+1 ≤ (α + 2)|E| γ γ+1 (1 + α(γ + 1)) -1 γ+1 y α+ 1 γ+1 +γ-(α+1) ≤ C(α, γ)y γ 2 γ+1 |E| γ γ+1 ≤ C(α, γ)R γ 2 γ+1 |E| γ γ+1 . This shows that (H5) is fulfilled with ω(R, δ) = C(α, γ)R γ 2 γ+1 δ γ γ+1 . As for (H4), for 0 < x < R < y, we write Γ(y, x) ≤ (α + 2) R α y γ-(α+1) ≤ (α + 2) R γ-1 if γ ≤ α + 1, (α + 2) R α y γ-(α+1) if γ > α + 1, and (H4) is satisfied provided γ < 2 + α with k(R) = (α + 2) R γ-1 and θ = 0 if γ ∈]0, α + 1] and k(R) = (α+2) R α and θ = γ -(α+1) ∈ [0, 1[ if γ ∈]α+1, α+2[. Therefore, Theorem 1.
2 provides the existence of weak solutions to (1)-( 2) for unbounded coagulation kernels K satisfying (H1)-(H2) and multiple fragmentation kernels Γ given by ( 6) with α ≥ 0 and γ ∈]0, α + 2[. Let us however mention that some fragmentation kernels which are bounded at the origin and considered in [START_REF] Giri | The continuous coagulation equation with multiple fragmentation[END_REF][START_REF] Ph | On a class of continuous coagulation-fragmentation equations[END_REF] need not satisfy (H4)-(H5).

Remark 1.4. While the requirement γ < α + 2 restricting the growth of Γ might be only of a technical nature, the constraint γ > 0 might be more difficult to remove. Indeed, it is well-known that there is an instantaneous loss of matter in the fragmentation equation when S(x) = x γ and γ < 0 produced by the rapid formation of a large amount of particles with volume zero (dust), a phenomenon refered to as disintegration or shattering [START_REF] Mcgrady | Shattering" transition in fragmentation[END_REF]. The case γ = 0 thus appears as a borderline case.

Let us finally outline the proof of Theorem 1.2. Since the pioneering work [START_REF] Stewart | A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels[END_REF], it has been realized that L 1 -weak compactness techniques are a suitable way to tackle the problem of existence for coagulation-fragmentation equations with unbounded kernels. This is thus the approach we use hereafter, the main novelty being the proof of the estimates needed to guarantee the expected weak compactness in L 1 . These estimates are derived in Section 2.2 on a sequence of unique global solutions to truncated versions of (1)-( 2) constructed in Section 2.1. After establishing weak equicontinuity with respect to time in Section 2.3, we extract a weakly convergent subsequence in L 1 and finally show that the limit function obtained from the weakly convergent subsequence is actually a solution to (1)-(2) in Sections 2.4 and 2.5.

Existence

Approximating equations

In order to prove the existence of solutions to (1-2), we take the limit of a sequence of approximating equations obtained by replacing the kernel K and selection rate S by their "cut-off" analogues K n and S n [START_REF] Stewart | A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels[END_REF], where

K n (x, y) := K(x, y) if x + y < n, 0 if x + y ≥ n, S n (x) := S(x) if 0 < x < n, 0 if x ≥ n,
for n ≥ 1. Owing to the boundedness of K n and S n for each n ≥ 1, we may argue as in [START_REF] Stewart | A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels[END_REF]Theorem 3.1] or [START_REF] Walker | Coalescence and breakage processes[END_REF] to show that the approximating equation

∂f n (x, t) ∂t = 1 2 x 0 K n (x -y, y)f n (x -y, t)f n (y, t)dy - n-x 0 K n (x, y)f n (x, t)f n (y, t)dy + n x b(x, y)S n (y)f n (y, t)dy -S n (x)f n (x, t), (7)
with initial condition

f n 0 (x) := f 0 (x) if 0 < x < n, 0 if x ≥ n. ( 8
)
has a unique non-negative solution

f n ∈ C 1 ([0, ∞[; L 1 ]0, n[) such that f n (t) ∈ X + for all t ≥ 0.
In addition, the total volume remains conserved for all t ∈ [0, ∞[, i.e.

n 0 xf n (x, t)dx = n 0 xf n 0 (x)dx. (9)
From now on, we extend f n by zero to ]0, ∞[×[0, ∞[, i.e. we set f n (x, t) = 0 for x > n and t ≥ 0. Observe that we then have the identity

S n f n = Sf n .
Next, we need to establish suitable estimates in order to apply the Dunford-Pettis Theorem [2, Theorem 4.21.2] and then the equicontinuity of the sequence (f n ) n∈N in time to use the Arzelà-Ascoli Theorem [1, Appendix A8.5]. This is the aim of the next two sections.

Weak compactness

Lemma 2.1. Assume that (H1)-(H6) hold and fix T > 0. Then we have: (i) There is L(T ) > 0 (depending on T ) such that

∞ 0 (1 + x)f n (x, t)dx ≤ L(T ) for n ≥ 1 and all t ∈ [0, T ],
(ii) For any ε > 0 there exists

R ε > 0 such that for all t ∈ [0, T ] sup n≥1 ∞ Rε f n (x, t)dx ≤ ε, (iii) given ε > 0 there exists δ ε > 0 such that, for every measurable set E of ]0, ∞[ with |E| ≤ δ ε , n ≥ 1, and t ∈ [0, T ], E f n (x, t)dx < ε.
Proof. (i) Let n ≥ 1 and t ∈ [0, T ]. Integrating [START_REF] Ph | On coalescence equations and related models[END_REF] with respect to x over ]0, 1[ and using Fubini's Theorem, we have

d dt 1 0 f n (x, t)dx = - 1 2 1 0 1-x 0 K n (x, y)f n (x, t)f n (y, t)dydx - 1 0 n-x 1-x K n (x, y)f n (x, t)f n (y, t)dydx + 1 0 n x b(x, y)S(y)f n (y, t)dydx - 1 0 S(x)f n (x, t)dx.
Since K n , f n , and S are non-negative and Γ satisfies (3), we have

d dt 1 0 f n (x, t)dx ≤ 1 0 n x b(x, y)S(y)f n (y, t)dydx = 1 0 1 x Γ(y, x)f n (y, t)dydx + 1 0 n 1 Γ(y, x)f n (y, t)dydx,
Using Fubini's Theorem and (H5) (with R = 1 and E =]0, 1[) in the first term of the right-hand side and (H4) (with R = 1) in the second one, we obtain

d dt 1 0 f n (x, t)dx ≤ 1 0 f n (y, t) y 0 Γ(y, x)dxdy + k(1) 1 0 n 1 yf n (y, t)dydx ≤ω(1, 1) 1 0 f n (x, t)dx + k(1) f n (t) x . (10)
Recalling that f n (t) x = f n (0) x ≤ f 0 for t ≥ 0 by (9), we readily deduce from (10) that

d dt 1 0 f n (x, t)dx ≤ ω(1, 1) 1 0 f n (y, t)dy + k(1) f 0 .
Integrating with respect to time, we end up with

1 0 f n (x, t)dx ≤ f 0 1 + k(1) ω(1, 1) exp(ω(1, 1)t), t ∈ [0, T ].
Using (9) again we may estimate

∞ 0 (1 + x)f n (x, t)dx = 1 0 f n (x, t)dx + n 1 f n (x, t)dx + n 0 xf n (x, t)dx ≤ 1 0 f n (x, t)dx + n 1 xf n (x, t)dx + f 0 ≤ f 0 1 + k(1) ω(1, 1)
exp(ω(1, 1)T ) + 2 =: L(T ).

(ii) For ε > 0, set R ε := f 0 /ε. Then, by [START_REF] Mclaughlin | An existence and uniqueness result for a coagulation and multiple-fragmentation equation[END_REF], for each n ≥ 1 and for all t ∈ [0, T ] we have

∞ Rε f n (x, t)dx ≤ 1 R ε ∞ Rε xf n (x, t)dx ≤ f 0 R ε < ε.
(iii) Fix R > 0. For n ≥ 1, δ ∈ (0, 1), and t ∈ [0, T ], we define

p n (δ, t) = sup R 0 1 E (x)f n (x, t)dx : E ⊂]0, R[ and |E| ≤ δ .
Consider a measurable subset E ⊂]0, R[ with |E| ≤ δ. For n ≥ 1 and t ∈ [0, T ], it follows from the non-negativity of f n , ( 3) and ( 7)-( 8) that

d dt R 0 1 E (x)f n (x, t)dx ≤ 1 2 I n 1 (t) + I n 2 (t) + I n 3 (t), ( 11
)
where

I n 1 (t) := R 0 1 E (x) x 0 K n (x -y, y)f n (x -y, t)f n (y, t)dydx, I n 2 (t) := R 0 1 E (x) R x Γ(y, x)f n (y, t)dydx, I n 3 (t) := R 0 1 E (x) ∞ R Γ(y, x)f n (y, t)dydx.
First, applying Fubini's Theorem to I n 1 (t) gives

I n 1 (t) = R 0 f n (y, t) R y 1 E (x)K n (y, x -y)f n (x -y, t)dxdy = R 0 f n (y, t) R-y 0 1 E (x + y)K n (y, x)f n (x, t)dxdy.
Setting -y + E := {z > 0 : z = -y + x for some x ∈ E}, it follows from (H2) and the above identity that

I n 1 (t) ≤ k 2 1 (1 + R) µ R 0 (1 + y) µ f n (y, t) R 0 f n (x, t)1 -y+E∩]0,R-y[ (x)dxdy. Since -y + E∩]0, R -y[⊂]0, R[ and |-y + E∩]0, R -y[| ≤ | -y + E| = |E| ≤ δ,
we infer from the definition of p n (δ, t) and Lemma 2.1 (i) that

I n 1 (t) ≤ k 2 1 (1 + R) µ R 0 (1 + y) µ f n (y, t)dy p n (δ, t) ≤ k 2 1 L(T )(1 + R) µ p n (δ, t).
Next, applying Fubini's Theorem to I n 2 (t) and using (H5) and Lemma 2.1 (i) give

I n 2 (t) = R 0 f n (y, t) y 0 1 E (x)Γ(y, x)dxdy ≤ ω(R, |E|) R 0 f n (y, t)dy ≤ L(T )ω(R, |E|).
Finally, owing to (H4) and ( 9), we have

I n 3 (t) ≤k(R) R 0 ∞ R 1 E (x)y θ f n (y, t)dydx ≤ k(R)R θ-1 |E| ∞ R yf n (y, t)dy ≤k(R)R θ-1 f 0 |E| ≤ k(R)R θ-1 f 0 δ.
Collecting the estimates on I n j (t), 1 ≤ j ≤ 3, we infer from (11) that there is

C 1 (R, T ) > 0 such that d dt R 0 1 E (x)f n (x, t)dx ≤ C 1 (R, T ) (p n (δ, t) + ω(R, δ) + δ) .
Integrating with respect to time and taking the supremum over all E such that E ⊂]0, R[ with |E| ≤ δ give

p n (δ, t) ≤ p n (δ, 0) + T C 1 (R, T )[ω(R, δ) + δ] + C 1 (R, T ) t 0 p n (δ, s)ds, t ∈ [0, T ].
By Gronwall's inequality (see e.g. [14, p. 310]), we obtain

p n (δ, t) ≤ [p n (δ, 0) + T C 1 (R, T )(ω(R, δ) + δ)] exp {C 1 (R, T )t}, t ∈ [0, T ]. (12)
Now, since f n (x, 0) ≤ f 0 (x) for x > 0, the absolute continuity of the integral guarantees that sup n {p n (δ, 0)} → 0 as δ → 0 which implies, together with (H5) and ( 12) that

lim δ→0 sup n≥1,t∈[0,T ] {p n (δ, t)} = 0.
Lemma 2.1 (iii) is then a straightforward consequence of this property and Lemma 2.1 (i).

Lemma 2.1 and the Dunford-Pettis Theorem imply that, for each t ∈ [0, T ], the sequence of functions (f n (t)) n≥1 lies in a weakly relatively compact set of

L 1 ]0, ∞[ which does not depend on t ∈ [0, T ].

Equicontinuity in time

Now we proceed to show the time equicontinuity of the sequence (f n ) n∈N . Though the coagulation terms can be handled as in [START_REF] Giri | The continuous coagulation equation with multiple fragmentation[END_REF][START_REF] Ph | On a class of continuous coagulation-fragmentation equations[END_REF][START_REF] Stewart | A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels[END_REF], we sketch the proof below for the sake of completeness. Let T > 0, ε > 0, and

φ ∈ L ∞ ]0, ∞[ and consider s, t ∈ [0, T ] with t ≥ s. Fix R > 1 such that 2L(T ) R < ε 2 , (13)
the constant L(T ) being defined in Lemma 2.1 (i). For each n, by Lemma 2.

1 (i), ∞ R |f n (x, t) -f n (x, s)|dx ≤ 1 R ∞ R x{f n (x, t) + f n (x, s)}dx ≤ 2L(T ) R . (14)
By ( 7), [START_REF] Walker | Coalescence and breakage processes[END_REF], and ( 14), we get

∞ 0 φ(x){f n (x, t) -f n (x, s)}dx ≤ R 0 φ(x){f n (x, t) -f n (x, s)}dx + ∞ R |φ(x)||f n (x, t) -f n (x, s)|dx ≤ φ L ∞ t s 1 2 R 0 x 0 K n (x -y, y)f n (x -y, τ )f n (y, τ )dydx + R 0 n-x 0 K n (x, y)f n (x, τ )f n (y, τ )dydx + R 0 n x b(x, y)S(y)f n (y, τ )dydx + R 0 S(x)f n (x, τ )dx dτ + φ L ∞ ε 2 . ( 15
)
By Fubini's Theorem, (H2), and Lemma 2.1 (i), the first term of the right-hand side of (15) may be estimated as follows:

1 2 R 0 x 0 K n (x -y, y)f n (x -y, τ )f n (y, τ )dydx = 1 2 R 0 R y K n (x -y, y)f n (x -y, τ )f n (y, τ )dxdy = 1 2 R 0 R-y 0 K n (x, y)f n (x, τ )f n (y, τ )dxdy ≤ k 2 1 2 R 0 R-y 0 (1 + x) µ (1 + y) µ f n (x, τ )f n (y, τ )dydx ≤ k 2 1 L(T ) 2 2 .
Similarly, for the second term of the right-hand side of (15), it follows from (H2) that

R 0 n-x 0 K n (x, y)f n (x, τ )f n (y, τ )dydx ≤ k 2 1 R 0 n-x 0 (1 + x) µ (1 + y) µ f n (x, τ )f n (y, τ )dydx ≤ k 2 1 L(T ) 2 .
For the third term of the right-hand side of (15), we use Fubini's Theorem, (H4), (H5), and Lemma 2.

≤ R 0 f n (y, τ ) y 0 1 ]0,R[ (x)Γ(y, x)dxdy + k(R) R 0 ∞ R y θ f n (y, τ )dydx ≤ω(R, R) R 0 f n (y, τ )dy + k(R) R 0 ∞ R yf n (y, τ )dydx ≤[ω(R, R) + Rk(R)] L(T ).
Finally, the fourth term of the right-hand side of (15) is estimated with the help of (H6) and Lemma 2.1 (i) and we get

R 0 S(x)f n (x, t)dx ≤ S L ∞ ]0,R[ L(T ).
Collecting the above estimates and setting

C 2 (R, T ) = 3k 2 1 L(T ) 2 2 + ω(R, R) + Rk(R) + S L ∞ ]0,R[ L(T ) the inequality (15) reduces to ∞ 0 φ(x){f n (x, t) -f n (x, s)}dx ≤ C 2 (R, T ) φ L ∞ (t -s) + φ L ∞ ε 2 < φ L ∞ ε, (16)
whenever t -s < δ for some suitably small δ > 0. The estimate (16) implies the time equicontinuity of the family {f n (t), t ∈ [0, T ]} in L 1 ]0, ∞[. Thus, according to a refined version of the Arzelà-Ascoli Theorem, see [12, Theorem 2.1], we conclude that there exist a subsequence (f n k ) and a non-negative function f

∈ L ∞ (]0, T [; L 1 ]0, ∞[) such that lim n k →∞ sup t∈[0,T ] ∞ 0 {f n k (x, t) -f (x, t)} φ(x) dx = 0, (17)
for all T > 0 and φ ∈ L ∞ ]0, ∞[. In particular, it follows from the non-negativity of f n and f , (9), and (17) that, for t ≥ 0 and R > 0,

R 0 xf (x, t)dx = lim n k →∞ R 0 xf n k (x, t)dx ≤ f 0 x < ∞.
Letting R → ∞ implies that f (t) x ≤ f 0 x and thus f (t) ∈ X + .

Passing to the limit

Now we have to show that the limit function f obtained in ( 17) is actually a weak solution to (1)-(2). To this end, we shall use weak continuity and convergence properties of some operators which define now: for g ∈ X + , n ≥ 1, and x ∈]0, ∞[, we put

Q n 1 (g)(x) = 1 2 x 0 K n (x -y, y)g(x -y)g(y)dy, Q n 2 (g)(x) = n-x 0 K n (x, y)g(x)g(y)dy, Q 1 (g)(x) = 1 2 x 0 K(x -y, y)g(x -y)g(y)dy, Q 2 (g)(x) = ∞ 0 K(x, y)g(x)g(y)dy, Q 3 (g)(x) = S(x)g(x), Q 4 (g)(x) = ∞ x
b(x, y)S(y)g(y)dy,

and

Q n = Q n 1 -Q n 2 -Q 3 + Q 4 , Q = Q 1 -Q 2 -Q 3 + Q 4 .
We then have the following result: Lemma 2.2. Let (g n ) n∈N be a bounded sequence in X + , ||g n || ≤ L, and g ∈ X + such that g n ⇀ g in L 1 ]0, ∞[ as n → ∞. Then, for each R > 0 and i ∈ {1, . . . , 4}, we have

Q n i (g n ) ⇀ Q i (g) in L 1 ]0, R[ as n → ∞. (18)
Proof. The proof of (18) for i = 1, 2 is the same as that in [START_REF] Giri | The continuous coagulation equation with multiple fragmentation[END_REF][START_REF] Stewart | A global existence theorem for the general coagulation-fragmentation equation with unbounded kernels[END_REF] to which we refer. The case i = 3 is obvious since φS belongs to L ∞ ]0, R[ by (H6) and (18) follows at once from the weak convergence of (g n ) in

Existence

Now we are in a position to prove the main result.

Proof of Theorem 1.2. Fix R > 0, T > 0, and consider t ∈ [0, T ] and φ ∈ L ∞ ]0, R[. Owing to Lemma 2.2, we have for each s ∈ [0, t],

R 0 φ(x){Q n k (f n k (s))(x) -Q(f (s))(x)}dx → 0 as n k → ∞. (24)
Arguing as in Section 2.3, it follows from (H2), (H4)-(H6), and Lemma 2.1 (i) that there is C 3 (R, T ) > 0 such that, for n ≥ 1, and s ∈ [0, t], we have

R 0 φ(x)Q n (f n (s))(x)dx ≤ C 3 (R, T ) φ L ∞ ]0,R[ . (25)
Since the right-hand side of (25) is in L 1 ]0, t[, it follows from (24), (25) and the dominated convergence theorem that

t 0 R 0 φ(x){Q n k (f n k (s))(x) -Q(f (s))(x)}dxds → 0 as n k → ∞. ( 26
)
Since φ is arbitrary in L ∞ ]0, R[, Fubini's Theorem and (26) give

t 0 Q n k (f n k (s))ds ⇀ t 0 Q(f (s))ds in L 1 ]0, R[ as n k → ∞. (27)
It is then straightforward to pass to the limit as n k → ∞ in ( 7)-( 8) and conclude that f is a solution to (1)-( 2) on [0, ∞[ (since T is arbitrary). This completes the proof of Theorem 1.2.

  The selection function S(x) describes the rate at which particles of volume x are selected to fragment. The selection function S and breakage function b are defined in terms of the multiple-fragmentation kernel Γ by the identities y)dy, b(x, y) = Γ(y, x)/S(y).

  x)f n (y, τ )dxdy + x)f n (y, τ )dydx

  L 1 ]0, ∞[. For i = 4, we consider φ ∈ L ∞ ]0, R[ and use (3) and Fubini's Theorem to compute, for r > R, R 0 φ(x){Q 4 (g n )(x) -Q 4 (g)(x)}dx = )S(y)b(x, y){g n (y) -g(y)}dydx ≤ )S(y)b(x, y){g n (y) -g(y)}dxdy + ∞ R R 0 φ(x)Γ(y, x){g n (y) -g(y)}dxdy .

Remark 2 . 3 .

 23 It is worth pointing out that the assumption (4) y 0 b(x, y)dx = N is only used to prove (20) and it is clear from that proof that the assumption sup y∈]0,R[ y 0 b(x, y)dx < ∞ for all R > 0 is sufficient. Thus, Theorem 1.2 is actually valid under this weaker assumption.

Acknowledgments

A.K. Giri would like to thank International Max-Planck Research School, Magdeburg, Germany and FWF Austrian Science Fund for their support. Part of this work was done while Ph. Laurençot enjoys the hospitality and support of the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK.

This can be further written as

with

We use (H6) and (4) to observe that, for y ∈]0, R[,

This shows that the function y →

We next infer from (H4) that, for

On the one hand, (21) guarantees that the function y → R 0 φ(x)Γ(y, x)dx belongs to L ∞ ]R, r[ and the weak convergence of (g n ) to g in

On the other hand, we deduce from (21) and the boundedness of (g n ) and g in

which is asymptotically small (as r → ∞) uniformly with respect to n. We thus conclude that

Substituting (20) and ( 22) into (19), we obtain

for all r > R. Owing to (23), we may let r → ∞ and conclude that (18) holds true for i = 4 thanks to the arbitrariness of φ and the proof of Lemma 2.2 is complete.