Exponential Decay Towards Equilibrium for the Inhomogeneous Aizenman-Bak Model

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Exponential Decay Towards Equilibrium for the Inhomogeneous Aizenman-Bak Model

J. A. Carrillo¹, L. Desvillettes², K. Fellner³

1 ICREA (Instituci´o Catalana de Recerca i Estudis Avan¸cats) and Departament de Matem`a- tiques, Universitat Aut`onoma de Barcelona, E-08193 Bellaterra, Spain.

E-mail:carrillo@mat.uab.es

2 CMLA - ENS de Cachan, 61 Av. du Pdt. Wilson, 94235 Cachan Cedex, France.

E-mail:desville@cmla.ens-cachan.fr

3 Faculty of Mathematics, University of Vienna, Nordbergstr. 15, 1090 Wien, Austria.

E-mail:klemens.fellner@univie.ac.at, Tel:+43-1-4277/50720, Fax:+43-1-4277/50650

Abstract: The Aizenman-Bak model for reacting polymers is considered for spatially inhomogeneous situations in which they diffuse in space with a non- degenerate size-dependent coefficient. Both the break-up and the coalescence of polymers are taken into account with fragmentation and coagulation constant kernels. We demonstrate that the entropy-entropy dissipation method applies directly in this inhomogeneous setting giving not only the necessary basic a priori estimates to start the smoothness and size decay analysis in one dimension, but also the exponential convergence towards global equilibria for constant diffusion coefficient in any spatial dimension or for non-degenerate diffusion in dimension one. We finally conclude by showing that solutions in the one dimensional case are immediately smooth in time and space while in size distribution solutions are decaying faster than any polynomial. Up to our knowledge, this is the first result of explicit equilibration rates for spatially inhomogeneous coagulation- fragmentation models.

Key words:Coagulation-fragmentation continuous in size models with spatial diffusion, exponential equilibration rate, entropy-entropy dissipation method, reaction-diffusion systems.

AMS subject classification:35B40, 35B45, 82D60

1. Introduction

We analyze the spatial inhomogeneous version of a size-continuous model for reacting polymers or clusters of aggregates:

∂tf−a(y)4xf =Q(f, f). (1.1) Here, f = f(t, x, y) is the concentration of polymers/clusters with length/size y ≥ 0 at time t ≥ 0 and point x∈ Ω ⊂ R^d, d ≥ 1. These polymers/clusters diffuse in the environment Ω. This set is assumed to be a smooth bounded domain with normalized volume, i.e., |Ω| = 1. In the one dimensional case, we will set Ω = (0,1). Equation (1.1) is to be considered with homogeneous Neumann boundary condition

∇xf(t, x, y)·ν(x) = 0 on∂Ω (1.2) withνthe outward unit normal toΩ, so that there is no polymer flux through the physical boundary. We assume the diffusion coefficienta(y) to be non-degenerate in the sense that there exista_∗, a^∗∈R⁺ such that

0< a_∗≤a(y)≤a^∗. (1.3) On the other hand, the reaction termQ(f, f) of (1.1) models chemical degrada- tion -break-up or fragmentation- and polymerization -coalescence or coagulation- of polymers/clusters. More precisely, the full collision operator reads as

Q(f, f) =Qc(f, f) +Qb(f, f) =Q⁺(f, f)−Q⁻(f, f)

=Q⁺_c(f, f)−Q⁻_c(f, f) +Q⁺_b(f, f)−Q⁻_b(f, f) (1.4) with obvious definitions of the coagulation Qc(f, f), fragmentation or break-up Qb(f, f), loss Q⁻(f, f) and gainQ⁺(f, f) operators which are determined from the four basic terms in (1.4):

1. Coalescence of clusters of sizey⁰≤y andy−y⁰ results into clusters of sizey:

Q⁺_c(f, f) :=

Z y

0

f(t, x, y−y⁰)f(t, x, y⁰)dy⁰. (1.5) 2. Polymerization of clusters of size y with other clusters of sizey⁰ produces a

loss in its concentration:

Q⁻_c(f, f) := 2f(t, x, y) Z ∞

0

f(t, x, y⁰)dy⁰. (1.6) 3. Break-up of clusters of sizey⁰ larger thany contributes to create clusters of

sizey:

Q⁺_b(f, f) := 2 Z ∞

y

f(t, x, y⁰)dy⁰. (1.7) 4. Break-up of polymers of sizeyreduces its concentration:

Q⁻_b(f, f) :=y f(t, x, y). (1.8)

This kind of models finds its application not only in polymers and cluster aggre- gation in aerosols [S16, S17, AB, Al, Dr] but also in cell physiology [PS], popula- tion dynamics [Ok] and astrophysics [Sa]. Here, fragmentation and coagulation kernels are all set up to constants as in the original Aizenman-Bak model [AB].

This will be of paramount importance in the basic a-priori estimates. The con- servation of the total number of monomers at timet≥0 quantified by

Z

Ω

N(t, x)dx, where N(t, x) :=

Z ∞

0

y f(t, x, y)dy

is the basic conservation law satisfied by equation (1.1) since the reaction term (1.4) satisfies

Z

Ω

Z ∞

0

y Q(f, f)dy dx= 0,

and thus, assuming initially a positive total number of monomers, we formally conclude

Z

Ω

Z ∞

0

y f(t, x, y)dy dx= Z

Ω

N(t, x)dx= Z

Ω

N0(x)dx:=N∞>0. (1.9) Another macroscopic quantity of interest is the number density of polymers,

M(t, x) :=

Z ∞

0

f(t, x, y)dy, (1.10)

that together with the total number of monomersN(t, x) satisfies the reaction- diffusion system

∂tN− 4x

Z ∞

0

ya(y)f(t, x, y)dy

= 0, (1.11)

∂tM− 4x

Z ∞

0

a(y)f(t, x, y)dy

=N−M², (1.12) becoming a closed decoupled system in the constant diffusion case (a(y) :=a) :

∂tN−a4xN = 0, (1.13)

∂tM−a4xM =N−M². (1.14) The definition of the full collision operator has to be understood in the weak sense as

< Q(f, f), ϕ >=

Z ∞

0

Z ∞

0

(f(y⁰⁰)−f(y)f(y⁰)) (ϕ(y) +ϕ(y⁰)−ϕ(y⁰⁰))dy dy⁰ (1.15) for any smooth functionϕ(y), wherey⁰⁰=y+y⁰and the dependence on (t, x) of the density function has been dropped for notational convenience. An alternative weak formulation that can be useful in several arguments below is obtained integrating by parts in theQ⁺_b part giving

< Q(f, f), ϕ >=−2 Z ∞

0

ϕ(y)f(y)dy Z ∞

0

f(y⁰)dy⁰+ Z ∞

0

Z ∞

0

f(y)f(y⁰)ϕ(y⁰⁰)dy dy⁰ + 2

Z ∞

0

f(y)Φ(y)dy− Z ∞

0

y f(y)ϕ(y)dy (1.16)

for any smooth function ϕ, the functionΦbeing the primitive of ϕ(∂_yΦ=ϕ) such thatΦ(0) = 0.

Let us consider the (free-energy) entropy functional associated to any positive densityf as

H(f)(t, x) = Z ∞

0

(flnf−f)dy , (1.17) and the relative entropy H(f|g) = H(f)−H(g) of two states f and g not necessarily with the sameL¹_y-norm. Then, the entropy formally dissipates as

d dt

Z

Ω

H(f)dx= − Z

Ω

Z ∞

0

a(y)|∇xf|²

f dy dx (1.18)

− Z

Ω

Z ∞

0

Z ∞

0

(f⁰⁰−f f⁰) ln f⁰⁰

f f⁰

dy dy⁰dx :=−DH(f) with obvious notations.

Global existence and uniqueness of classical solutions has been studied in [Am, AW] for some particular cases, namely, for constant diffusion coefficient or dimension one with additional restrictions for the coagulation and fragmentation kernel not including the AB model. The initial boundary-value problem to (1.1)- (1.2) was then analyzed in [LM02-1], for much more general coagulation and fragmentation kernels including the AB model (1.5) - (1.8), proving the global existence of weak solutions satisfying the entropy dissipation inequality

Z

Ω

H(f(t))dx+ Z t

0

D_H(f(s))ds≤ Z

Ω

H(f₀)dx for allt≥0.

The equilibrium states for which the entropy dissipation vanishes are better understood after applying a remarkable inequality proven in [AB, Propositions 4.2 and 4.3]. A modified version of this inequality (reviewed in section 2) reads as:

Z ∞

0

Z ∞

0

(f⁰⁰−f f⁰) ln f⁰⁰

f f⁰

dy dy⁰≥M H(f|f^√_{N ,N}) + 2(M−√

N)². (1.19) Herein,f^√_{N ,N}denotes a distinguished, exponential-in-size distribution with the very momentsM =√

N andN :

f^√_{N ,N}(t, x, y) =e⁻

√y N .

These distributionsf^√_{N ,N}appear as analog to the so-called intermediate or local equilibria in the study of inhomogeneous kinetic equation (e.g. [DV01, CCG, FNS, DV05, FMS, NS]). Finally, the conservation of mass (1.9) identifies (at least formally) the global equilibriumf_∞ with constant momentsM_∞² =N =N_∞ :

f∞=e⁻

√y

N∞ . (1.20)

The analogy to intermediate equilibria carries over to the following additivity of relative entropies

H(f|f_∞) =H(f|f^√_{N ,N}) +H(f^√_{N ,N}|f_∞). (1.21)

It is worthy to point out that even if f^√_{N ,N} and f_∞ do not have the same L¹_y−norm, its global relative entropy

Z

Ω

H(f^√_{N ,N}|f_∞)dx= 2 sZ

Ω

N dx− Z

Ω

√ N dx

!

≥0

is a nonnegative quantity, as easily checked via Jensens’s inequality. In [LM02-1], it is proved thatf_∞attracts all global weak solutions inL¹(Ω×(0,∞)) of (1.1)- (1.2) but no time decay rate is obtained. This result is the analog to convergence results along subsequences for the classical Boltzmann equation in [De].

Other existence and uniqueness results for inhomogeneous coagulation-frag- mentation models were given in [CD] and the references therein. Finally, let us mention that the conservation law (1.9) is known not to hold for certain coagulation-fragmentation kernels, phenomena known as gelation [ELMP], and the convergence or not towards typical self-similar profiles for the pure coagu- lation models is a related issue; we refer to [Le, LM05, MP]. We refer finally to [LM02-1, LM03, LM04] for an extensive list of related literature.

Let us now discuss some works on the study of the long time asymptotics for related models. Qualitative results concerning a discrete version of a coagulation- fragmentation system, the Becker-D¨oring system, have been obtained in [CP, LW, LM02-2] and the references therein. We emphasize that global explicit de- cay estimates towards equilibrium were obtained for the Becker-D¨oring system without diffusion in [JN] by entropy-entropy dissipation methods. Other tech- niques have recently been developed for inhomogeneous kinetic equations. We refer to [MN] for a spectral approach and to [V06] for a general description of the concept of hypocoercivity. Note that the presence of diffusion instead of advection makes possible in our present context, not to use the concept of hypocoercivity.

In this work we prove exponential decay towards equilibrium with explicit rates and constants. Our key result, Lemma 2 in section 2, establishes a func- tional inequality between entropy and entropy dissipation provided lower and upper bounds on the momentM and (1.3). We are able to apply this functional inequality to solutions of (1.1)-(1.2) in the next two situations.

In the special case of size-independent diffusion coefficientsa(y) =a, we show, as first application of Lemma 2, the exponential decay towards equilibrium in all space dimensions d≥1 by exploiting the closed system (1.13)–(1.14) forN andM in section 2.

In the case of general diffusion coefficients a(y) satisfying (1.3) we prove in section 3 a-priori estimates in the one-dimensional case d= 1, which entail an entropy-entropy dissipation estimate with a constant sufficient to conclude expo- nential decay via a suitable Gronwall argument. These two cases are summarized in the following Theorem :

Theorem 1.Let Ω be a smooth bounded connected open set ofR^d,d ≥1 and assume a constant diffusion coefficienta(y) =a >0or letΩbe the interval(0,1) and consider a diffusion coefficient satisfying (1.3). Let us also assume thatf₀6=

0is a nonnegative initial datum such that (1 +y+ lnf₀)f₀∈L¹((0,1)×(0,∞)).

In the casea(y) =a >0assume further thatM0(x)is anL^∞(Ω)-function.

Then, the global weak solutions f(t, x, y) of (1.1)–(1.2) decay exponentially to the global equilibrium state (1.20)with explicitly computable constantsC₁,C₂ and rateα, both in global relative entropy :

Z

Ω

H(f(t)|f∞)dx≤C1

Z

Ω

H(f0|f∞)dx

e^{−α t}, (1.22) and inL¹_x,y sense :

kf(t,·,·)−f_∞k_L1 x,y ≤C₂

Z

Ω

H(f₀|f_∞)dx

e^−α2t (1.23) for all t≥0, where f∞ is defined by (1.20) andN∞ >0 is determined by the conservation of mass (1.9).

In the one dimensional case, it is further possible to interpolate the expo- nential decay in a ”weak” norm like L¹ with polynomially growing bounds in

”strong” norms like (weighted)L¹_y(H_x¹) in order to get an exponential decay in a ”medium” norm likeL¹_y(L^∞_x). Thus, the decay toward equilibrium can be ex- tended to these stronger norms. The following proposition is proved at the end of section 4:

Proposition 1.Under the assumptions of Theorem1for the case d= 1, for all t∗ >0 and q≥0, there are explicitly computable constants C3, α >0 such that whenevert≥t∗,

Z ∞

0

(1 +y)^qkf(t,·, y)−f∞(y)kL^∞_x dy≤C3e^{−α t}. (1.24) A bootstrap argument in the spirit of the proof of Proposition 1 allows to replace theL^∞_x norm by any Sobolev norms in (1.24).

2. Entropy entropy dissipation estimate

Please note that in this section we will systematically use the shortcuts : M =

Z

Ω

Z ∞

0

f(x, y)dydx , N = Z

Ω

Z ∞

0

yf(x, y)dydx . We start by reminding the reader the following functional inequality:

Lemma 1.[AB, Proposition 4.3]Letg:=g(y)be a function ofL¹₊((0,∞))with finite entropyglng∈L¹((0,∞)), then

Z ∞

0

Z ∞

0

g(y)g(y⁰) lng(y+y⁰)dy dy⁰≤ Z ∞

0

g(y)dy

Z ∞

0

g(y⁰) lng(y⁰)dy⁰

− Z ∞

0

g(y)dy ²

. (2.1)

This inequality allows to show the dissipation inequality (1.19). Following the original paper [AB] or the survey [LM04], one finds that

Z ∞

0

Z ∞

0

(f⁰⁰−f f⁰) ln f⁰⁰

f f⁰

dy dy⁰ ≥M H(f|f^√_{N ,N}) + (M −√ N)² +M²

N M²ln N

M² + 1− N M²

. (2.2) In fact, after expanding the left hand side of (2.2), one applies Lemma 1 to the term

Z ∞

0

Z ∞

0

−f f⁰lnf⁰⁰dy dy⁰, while for the term

Z ∞

0

Z ∞

0

f f⁰ f⁰⁰ f f⁰ln

f⁰⁰ f f⁰

dy dy⁰

one uses Jensen’s inequality for the convex functionxlnxand further that Z ∞

0

Z ∞

0

f⁰⁰dy dy⁰=N.

Then, after directly calculating the remaining terms one obtains (2.2), as in [LM04], and moreover the inequality (1.19) when applying the elementary in- equalityxln(x) + 1−x≥(1−√

x)²forx≥0 to the last term on the right hand side of (2.2).

For the subsequent large-time analysis, we will rather study the relative en- tropy with respect to the global equilibrium, which dissipates according to (1.18) and (1.19) as

d dt

Z

Ω

H(f|f∞)dx≤ − Z

Ω

Z ∞

0

a(y)|∇_xf|²

f dy dx (2.3)

− Z

Ω

h

M H(f|f^√_{N ,N}) + 2(M−√ N)²i

dx :=−D(f).

We introduce a Lemma enabling to estimate the entropy off by means of its entropy dissipation. This is a functional estimate, that is, the functionf in this Lemma does not depend ont and has not necessarily something to do with the solution of our equation.

Lemma 2.Assume (1.3). Let f :=f(x, y)≥ 0 be a measurable function with moments satisfying 0<M_∗ ≤M(x)≤ kMkL^∞_x and 0 < N_∞ =N. Then, the following entropy entropy dissipation estimate holds

D(f)≥ C(M∗, N∞, a∗, P(Ω)) kMkL^∞_x

Z

Ω

H(f|f∞)dx , (2.4) with a constant C(M_∗, N_∞, a_∗, P(Ω))depending only onM_∗,N_∞,a_∗ and the Poincar´e constant P(Ω).

Proof.- Step 1.-We start with the right-hand side of (2.4) by using the addi- tivity (1.21) and calculating

Z

Ω

H(f|f∞)dx= Z

Ω

H(f|f^√_{N ,N})dx+ 2p N−√

N

. (2.5)

Step 2.-The second term of (2.5) is bounded as : pN−√

N ≤ 2

√N_∞

hkM −√ Nk²_L2

x+kM−Mk²_L2 x

i

. (2.6)

Indeed, since√ N−√

N is orthogonal to√

N−M in L²_x, we have pN−√

N ≤N−√ N

2

√ N

= 1

√N_∞k√ N−√

Nk²_L2

x ≤ 1

√N_∞k√

N−Mk²_L2 x, and further, we obtain (2.6) by expandingk√

N−Mk²_L2

x and Young’s inequality 1

2k√

N−Mk²_L2

x− kM −Mk²_L2 x ≤ k√

N−M−M+Mk²_L2 x. Thus, we obtain (using 0<M∗< M)

Z

Ω

H(f|f_∞)dx≤max{M⁻¹_∗ ,2N∞⁻¹²} Z

Ω

M H(f|f^√_{N ,N})dx+ 2kM−√ Nk²_L2

x

+ 4

√N_∞kM−Mk²_L2 x

≤ max{M⁻¹_∗ ,2N⁻

1

∞2} Z

Ω

Z ∞

0

Z ∞

0

(f⁰⁰−f f⁰) ln f⁰⁰

f f⁰

dy dy⁰dx

+ 4

√N∞

kM−Mk²_L2

x, (2.7)

by the inequality (1.19)

Step 3.-Next, the variance ofM, i.e. the last term on the right hand side of (2.7) is controlled by the first, ”Fisher”-type term of (2.3). Denoting withP(Ω) the constant of Poincar´e’s inequality, we estimate using Cauchy-Schwartz

Z

Ω

Z ∞

0

a(y)|∇xf|²

f dydx≥ a∗

kMkL^∞_x

Z

Ω

Z ∞

0

|∇xf|² f dy

Z ∞

0

f dy

dx

≥ a_∗ kMk_L^∞

x

Z

Ω

Z ∞

0

∇xf dy

2

dx= a_∗ kMk_L^∞

x

Z

Ω

|∇xM|²dx

≥ a_∗

P(Ω)kM(t,·)k_L^∞

x

kM−Mk²_L2

x. (2.8)

We remark that the seemingly more natural estimate Z

Ω

Z ∞

0

a(y)|∇_xf|²

f dydx≥ 4 P(Ω)k√

M−√ Mk²_L2

x,

provides a bound which does not seem sufficient to conclude like in step 2.

Step 4.-Finally, combining (2.7) and (2.8), we have Z

Ω

H(f|f∞)dx≤max{M⁻¹_∗ ,2N_∞⁻¹²,4P(Ω)kM(t,·)k_L^∞

x

a_∗√

N_∞ }D(f), which yields the proof of lemma 2.

Now, let us directly apply this entropy-entropy dissipation estimate to prove the constant diffusion part of Theorem 1. In the constant diffusion case, the equations for the first two momentsM(t, x) andN(t, x) become the closed sys- tem (1.13)-(1.14). The existence and uniqueness of global, classical solutions with globalL^∞ bounds from below and above are standard thanks to the max- imum principle applied to the equations forN and further for M. We refer, for instance, to [Ro, Ki] for details, to conclude with:

Lemma 3.Let Ωbe a smooth bounded connected open set ofR^d,d≥1 and let us assume that the initial data M0(x) andN0(x)6= 0 are nonnegativeL^∞(Ω)- functions. Then, there exist increasing functionst7→M∗(t),N∗(t)and decreas- ing functionst7→M^∗(t),N^∗(t)such that the unique global bounded solutions of the system (1.13)-(1.14) satisfy

0< M∗(t)≤M(t, x)≤M^∗(t)<∞, (2.9) 0< N_∗(t)≤N(t, x)≤N^∗(t)<∞, (2.10) for allt >0.

Proof of Theorem 1: Case aconstant, d≥1.- Let us fixt∗ >0. From (2.9)- (2.10), we have 0<M∗≤M(t, x)≤ M^∗<∞and 0<N∗≤N(t, x)≤ N^∗<∞ for allt≥t_∗, and thus

D(f)≥ C(M∗, N∞, a∗, P(Ω)) M^∗

Z

Ω

H(f|f∞)dx

for all t ≥ t∗ due to (2.4) with the constant C(M∗, N∞, a∗, P(Ω)) given in Lemma 2. As a direct consequence, we get

d dt

Z

Ω

H(f|f∞)dx≤ −C(M∗, N∞, a∗, P(Ω)) M^∗

Z

Ω

H(f|f∞)dx for allt≥t∗. Gronwall’s lemma implies estimate (1.22).

Next, convergence inL¹ as stated in Theorem 1 follows from the functional inequality of Csiszar-Kullback type [Cs, Ku] :

kf(t,·,·)−f∞k²_L1 x,y ≤2

Z

Ω

M(t, x)dx+p N∞

Z

Ω

H(f(t)|f∞)dx . (2.11) The proof is standard, see [CCD] for related inequalities, and it is shown via a Taylor expansion of the functionϕ(f) =fln(f)−f up to second order around f_∞. Indeed, for a function ζ(x, y) ∈ (inf{f(x, y), f_∞(y)},sup{f(x, y), f_∞(y)}), we get

Z

Ω

H(f|f_∞)dx= Z

Ω

Z ∞

0

− y

√N_∞(f−f_∞) + 1

2ζ(f −f_∞)²

dy dx

and the first term vanishes due to the conservation law (1.9). For the second term, we apply H¨older’s inequality

kf−f_∞k²_L1

x,y ≤ kζk_L1 x,y

Z

Ω

Z ∞

0

1

ζ(f−f_∞)²dy dx withkζk_L1 x,y≤

Z

Ω

M dx+p N_∞.

Noticing that t ∈ [0, t∗] 7→ f(t,·,·) ∈ L¹_x,y is bounded, we finally get (1.23), which concludes the proof of Theorem 1.

Remark 1.As a consequence of the previous result, we also showed that the unique global bounded solutions of the system (1.13)-(1.14) satisfy M(t, x) → M_∞ = √

N_∞ and N(t, x) → N_∞ as t → ∞ in L¹(Ω) exponentially fast with explicit constants. In fact, we first remark that H(f|f_∞) =H(f|fM,N) + H(fM,N|f∞) with

ES:=

Z

Ω

H(f_M,N|f∞)dx= Z

Ω

2√

N(ξlnξ−ξ+ 1)+N_∞^−1/2√ N−p

N_∞² dx,

whereξ=^√M

N and

fM,N(t, x, y) =M² N e^−MNy.

It is obvious thatESis positive since the minimum ofξlnξ−ξ+ 1 is zero, and it can be written as

Z

Ω

H(f_M,N|f_∞)dx= Z

Ω

MlnM²

N −2(M−√

N) + 2p

N_∞−√ N

dx ,

by using the conservation of mass (1.9). SinceH(f|fM,N)≥0, then (1.22) implies the exponential convergence to zero of ES by the above additivity property.

Finally, a simple Taylor expansion shows that, for allt≥t_∗ kp

N(t)−M(t)k²_L2 x+kp

N(t)−p N_∞k²_L2

x≤L Z

Ω

H(fM,N(t)|f_∞)dx , with

L= max

1, M^∗

√N∗

,√ N^∗

,

that implies by trivial arguments the exponential convergence inL¹(Ω) towards equilibrium for M and N. In fact, the system (1.13)-(1.14) might have been studied by a direct application of the techniques in [DF05, DF06].

3. A-priori Estimates

In the sequel, we shall discuss the general diffusion coefficient, i.e., size dependent verifying (1.3) but we restrict to the one dimensional case, d= 1 (we shall not recall this fact in the various Lemmas). We begin the proof of Theorem 1.

Lemma 4.Assume thatf₀6= 0 is a non-negative initial datum such that (1 + y)f₀ ∈ L¹((0,1)×(0,∞)). Then, there exists M^∗₀ > 0 such that solutions of (1.1)–(1.8)satisfy

sup

t≥0

Z

Ω

Z ∞

0

f(t, x, y)dy dx ≤ Z

Ω

M(t, x)dx ≤ M^∗₀ (3.1) Proof.-We estimate theL¹(Ω)-norm ofM(t, x) by integrating equality (1.14), obtaining

d dt

Z

Ω

M(t, x)dx= Z

Ω

N(t, x)dx− Z

Ω

M(t, x)²dx

≤ Z

Ω

N0(x)dx− Z

Ω

M(t, x)dx 2

by H¨older’s inequality and the conservation of mass (1.9). Therefore, for allt≥0 Z

Ω

M(t, x)dx≤max (Z

Ω

M₀(x)dx, Z

Ω

N₀(x)dx 1/2)

:=M^∗₀ showing (3.1).

We now turn to a control of theL¹_y(L^∞_x )-norm off:

Lemma 5.Assuming that the nonnegative initial datum f0 6= 0 satisfies (1 + y+ lnf0)f0 ∈L¹((0,1)×(0,∞)). Then, the number density of polymers M ∈ L¹+L^∞(0,∞;L^∞(0,1)). More precisely, there existm_∞>0 and anL¹₊(0,∞)- function m₁(t)such that the solution of (1.1)–(1.8)satisfies

Z ∞

0

sup

0<x<1

f(t, x, y)dy ≤ m_∞+m₁(t) (3.2) and as a consequence,

kM(t,·)kL^∞_x ≤m_∞+m1(t) a.e.t≥0.

Proof.-In order to estimate theL^∞_x -norm ofM(t, x), we first use the entropy dissipation (2.3) ofH(f|f∞) to deduce that

2 Z ∞

0

Z 1

0

Z ∞

0

∂x

pf²

dy dx dt≤ Z ∞

0

Z 1

0

Z ∞

0

a(y) 2a_∗

(∂xf)²

f dy dx dt

≤ H(f0|f∞)

2a_∗ :=µ1. (3.3)

Now, we integrate

pf(t, x, y)−p

f(t,x, y) =˜ Z x

˜ x

∂x

pf(t, ξ, y)dξ

with respect to ˜x∈(0,1) and estimate sup

0<x<1

pf(t, x, y)− Z 1

0

pf(t,x, y)˜ d˜x ²

≤ Z 1

0

∂x

pf(t, ξ, y)2

dξ.

Hence, after further integration with respect to y ∈ (0,∞), we apply Young’s and H¨older’s inequalities to show

Z ∞

0

sup

0<x<1

f(t, x, y)dy≤2 Z ∞

0

Z 1

0

∂_xp

f(t, ξ, y)² dξ dy

| {z }

+ 2 Z ∞

0

Z 1

0

f(t,x, y)˜ d˜x dy

| {z }

.

:=m1(t) ≤m_∞

In particular, we have, due to (3.1) and (3.3), that Z ∞

0

m₁(t)dt≤µ₁= H(f0|f_∞)

2a_∗ , m_∞= 2M^∗₀. Finally,

kM(t,·)kL^∞_x ≤ Z ∞

0

sup

0<x<1

f(t, x, y)dy , which completes the proof of Lemma 5.

Note that the estimates (3.1) and (3.2) are somehow in duality, a fact that will become essential below. We now prove a Lemma showing that the total number of clustersR1

0M(t, x)dxis bounded below by a strictly positive constant : Lemma 6.Assume that f0 6= 0 is a nonnegative initial datum such that (1 + y+ lnf0)f0 ∈ L¹((0,1)×(0,∞)). Then, there exists a constant M0∗ >0 such that for all timest≥0, one has

Z

Ω

M(t, x)dx≥ M_0∗, (3.4)

wheref is a solution of (1.1)–(1.8).

Proof.-We remind that d

dt Z 1

0

M(t, x)dx= Z 1

0

N0(x)dx− Z 1

0

M(t, x)²dx, so that

d dt

Z 1

0

M(t, x)dx≥ Z 1

0

N₀(x)dx− kM(t,·)k_L^∞

x

Z 1

0

M(t, x)dx

≥ Z 1

0

N₀(x)dx−(m_∞+m₁(t)) Z 1

0

M(t, x)dx.

Then, d dt

e^{R0t(m∞+m1(s))ds} Z 1

0

M(t, x)dx

≥ Z 1

0

N₀(x)dx e^{R0t(m∞+m1(s))ds},

and, recallingµ₁≥R+∞

0 m₁(s)ds), we deduce Z 1

0

M(t, x)dx≥ Z 1

0

M0(x)dx e^{−R0t(m∞+m1(σ))dσ} +

Z 1

0

N0(x)dx Z t

0

e^{−Rst(m∞+m1(σ))dσ}ds

≥e^{−µ1−m∞t} Z 1

0

M0(x)dx+ Z t

0

e^{−(t−s)m∞−µ1}ds Z 1

0

N0(x)dx

≥e^−µ1

e^−m∞t Z 1

0

M0(x)dx+1−e^−m∞t m_∞

Z 1

0

N0(x)dx

.

Distinguishing here betweent <1 andt≥1, for instance, we obtain M_0∗:=e^−µ1 inf

e^−m∞

Z 1

0

M0(x)dx,1−e^−m∞ m_∞

Z 1

0

N0(x)dx

,

which concludes the proof of lemma 6.

Next, we show the uniform control in time of all moments with respect to sizey of the solutions. Let us define the moment of orderp >1 by

Mp(f)(t) :=

Z 1

0

Z ∞

0

y^pf(t, x, y)dy dx for allt≥0.

Lemma 7.We assume that f0 6= 0 is a nonnegative initial datum such that (1 +y+ lnf₀)f₀ ∈L¹((0,1)×(0,∞)). Then, the solution f of (1.1)–(1.8)has moments M_p(f)(t) uniformly bounded in time t > t_∗ > 0 and for any p > 1, i.e., there exist explicit constantsM^∗_p(f0, m_∞, m1, p)such that

Mp(f)(t)≤ M^∗_p, for a.e.t > t∗>0. (3.5) Proof.-We proceed in two steps:

Step 1.- We first assume that Mp(f)(t∗)<∞for certain p >1 and t∗ >0.

Using the weak formulation (1.16), it is easy to check that

< Q(f, f), y^p>=−2 Z ∞

0

y^pf(y)dy M(t, x) + Z ∞

0

Z ∞

0

f(y)f(z)(y+z)^pdy dz

−p−1 p+ 1

Z ∞

0

f(y)y^p+1dy.

Taking into account Lemma 5 and (y+z)^p≤C_p⁰(y^p+z^p), we deduce

< Q(f, f), y^p>≤ 2(C_p⁰ −1) Z ∞

0

y^pf(y)dy[m_∞+m1(t)]−p−1 p+ 1

Z ∞

0

f(y)y^p+1dy for allp >1. Integrating in space, we find that the evolution of the moment of orderp >1 is given by

d

dtM_p(f)(t)≤2(C_p⁰ −1)M_p(f)(t) [m_∞+m₁(t)]−p−1

p+ 1M_p+1(f)(t). (3.6)

Trivial interpolation of thep+ 1-order moment with the moment of order one implies

Mp(f)(t)≤ 1 ^p−1

Z

Ω

N0(t, x)dx+ Mp+1(f)(t) for all >0, and thus

d

dtM_p(f)(t)≤2(C_p⁰ −1)M_p(f)(t) [m_∞+m₁(t)]−p−1 p+ 1 1

M_p(f)(t) +D for certain constantD. Choosing >0 such that

2(C_p⁰ −1)m_∞−p−1 p+ 1 1 ≤ −1

2 we obtain

d

dtM_p(f)(t)≤ −1

2M_p(f)(t) + 2(C_p⁰ −1)m₁(t)M_p(f)(t) +D for a.e.t > t_∗. According to Duhamel’s formula,

Mp(f)(t)≤Mp(f)(t_∗) exp

2(C_p⁰ −1) Z t

t∗

m1(s)ds−t−t_∗ 2

+D Z t

t∗

exp

2(C_p⁰ −1) Z t

s

m₁(τ)dτ −t−s 2

ds (3.7)

which shows that the moment M_p(f)(t) is bounded by a constantM^∗_p for a.e.

t > t∗ sincem1(t)∈L¹((0,∞)) by Lemma 5.

Moreover, it follows from (3.6) that the boundedness ofM_p(t_∗) immediately implies that

Z T

t∗

Mp+1(f)(t)dt <∞

for allT >0, and thus the finiteness ofM_p+1(f)(t) for a.e. t > t_∗ and a simple induction argument enables then to conclude the bounds on all higher moments.

Step 2.- It remains to show that for given nontrivial initial datay f0 ∈L¹_x,y and for a p > 1 and a time t_∗ >0 we have that M_p(t_∗) < ∞. We start with the following observation [MW, Appendix A]: For a nonnegative integrable func- tiong(y)6= 0 on (0,∞), there exists a concave functionΦ(y), depending on g, smoothly increasing fromΦ(0)>0 toΦ(∞) =∞such that

Z ∞

0

Φ(y)g(y)dy <∞.

Moreover, the functionΦcan be constructed to satisfy Φ(y)−Φ(y⁰)≥C y−y⁰

yln²(e+y) (3.8)

for 0< y⁰ < y withC not depending ong. We refer to [MW, Appendix A] for all the details of this ”by-now standard” construction.

To show now that M_p(t_∗) < ∞ for a p > 1 and a time t_∗ > 0, we take functions Φ(x, y) constructed for nontrivialy f₀(x, y)∈L¹_y(0,∞) a.e.x∈(0,1) and calculate - similar to Step 1 - the moment

M_1,Φ(f)(t) = Z

Ω

Z ∞

0

y Φ(x, y)f(x, y)dy dx.

For the fragmentation part, we use (3.8) for 0< y⁰< y and estimate y Φ(y)Qf(f) = 2

Z y

0

y⁰(Φ(y⁰)−Φ(y))dy⁰f(y)

≤ −C ln⁻²(e+y)y⁻¹ Z y

0

y⁰(y−y⁰)dy⁰f(y)

=−C ln⁻²(e+y)y²

6 f(y)≤ −Cδy^2−δf(y),

for all δ >0 and a positive constantCδ, where the (t, x)-dependence has been dropped for notational convenience. Hence, by estimating the coagulation part similar to Step 1 making use of the concavity ofΦ, we obtain that

d

dtM1,Φ(f)(t)≤3(m_∞+m1(t))M1,Φ(f)(t)−CδM_2−δ(f)(t),

and boundedness of the moment M_1,Φ follows by interpolation as well as the finiteness ofM_2−δ(f)(t_∗) analogously to Step 1.

Next, we show thatM andN are bounded below uniformly (with respect to tandx) for allt≥t_∗>0.

Proposition 2.Under the assumptions of theorem1, lett∗>0be given. Then, there are strictly positive constantsM∗ andN∗ such that for all t≥t_∗>0,

M(t, x)≥ M_∗ and N(t, x)≥ N_∗. Proof.-We write the equation satisfied byf in this way :

∂tf−a(y)∂xxf =g1−y f− kM(t,·)kL^∞_x f, whereg1 is nonnegative. Then

(∂t+a(y)∂xx)

f e^{ty+R0tkM(s,·)kL∞x ds}

=g2, whereg2 is nonnegative.

Now, we recall that the solutionh:=h(t, x) of the heat equation

∂th−a ∂xxh=G,

with homogeneous Neumann boundary condition on the interval (0,1), where a >0 is a constant andG:=G(t, x)∈L¹, is given by the formula

h(t, x) = 1 2√ π

Z 1

−1

˜h(0, z)

∞

X

k=−∞

√1

a te^{−(2k+x−z)24a t} dz + 1

2√ π

Z t

0

Z 1

−1

G(s, z)˜

∞

X

k=−∞

1

pa(t−s)e^{−(2k+x−z)24a(t−s)} dzds, with ˜hand ˜Gdenoting the “evenly mirrored around 0 in thexvariable” functions handG.

Therefore, for allt₁, t≥0, andx∈(0,1), y∈R+, f(t1+t, x, y)e^{(t1+t)y+R0t1 +tkM(s,·)kL∞x ds}

≥ 1 2√ π

Z 1

−1

f˜(t1, z, y) 1 pa(y)te⁻

(x−z)2

4a(y)t e^{t1y+R0t1kM(s,·)kL∞x ds}dz,

so that whent∈[t_∗,2t_∗] (and since|x−z|<2) : f(t1+t, x, y)≥ 1

√2π a^∗t_∗ Z 1

0

f(t1, z, y)e^−a∗1t∗ e^{−2t∗y−2t∗m∞−µ1}dz

≥C Z 1

0

f(t1, z, y)e^−2t∗ydz,

whereC >0 depends on the constantsa_∗, a^∗,m_∞, µ₁andt_∗>0.

We recall that for allt≥t_∗, Z 1

0

Z ∞

0

y²f(t, x, y)dy dx≤ M^∗₂, and thus, for anyA >0, we deduce

N(t1+t, x)≥C e^−2t∗A Z 1

0

Z A

0

f(t1, z, y)y dydz

≥C e^−2t∗A Z 1

0

N(t1, x)dx− M^∗₂/A

=C e^−2t∗A Z 1

0

N₀(x)dx− M^∗₂/A

,

due to the conservation law (1.9) and Z 1

0

Z ∞

A

y f(t, x, y)dy dx≤ M^∗₂/A.

Choosing nowA, we get thatN(t₁+t, x)≥ N_∗for someN_∗>0 which does not depend ont₁. Using Lemma 6,

M(t1+t, x)≥C Z 1

0

Z ∞

0

f(t1, z, y)e^−2ydydz

≥C e^−2t∗A Z 1

0

Z A

0

f(t1, z, y)dydz

≥C e^−2t∗A

M_0∗− M^∗₂/A²

.

Once again choosing A, we get that M(t1+t, x) ≥ M_∗. Since M_∗ does not depend ont1, we get Proposition 2.

4. Proofs of Theorem 1 and Proposition 1 if Ω= (0,1).

With Proposition 2 and Lemma 5 providing the moment bounds required by the entropy-entropy dissipation Lemma 2 in the one dimensional caseΩ= (0,1), we turn now to the

Proof of Theorem 1: Case Ω= (0,1).-According to Lemma 2, d

dt Z 1

0

H(f|f_∞)dx≤ −D(f)≤ − C kMk_L^∞

x

Z 1

0

H(f|f_∞)dx ,

wherekMkL^∞_x (t)≤m_∞+m1(t) is inL¹_t+L^∞_t by Lemma 5. Hence, fort_∗>0, Z 1

0

H(f(t)|f∞)dx≤ Z 1

0

H(f(t∗)|f∞)dxexp Z t

t∗

− C kMkL^∞_x

ds

.

Knowing that m1(t) ∈ L¹_t with R∞

0 m1(t)dt ≤ µ1, we consider the sets A :=

{s >0 :m1(s)≥1} andBt:={s∈[0, t] :m1(s)<1}. We readily find that

|A|= Z

A

ds≤ Z ∞

0

m1(t)dt≤µ1 and |Bt|=t− Z

A∩[0,t]

ds≥t−µ1.

Moreover, Z t

t_∗

− C kMkL^∞_x

ds≤ Z

B_t

− C kMkL^∞_x

ds≤ − C

(1 +m_∞)(t−µ1)

finishing the proof of (1.22). The proof of the L¹-decay estimate (1.23) follows the same arguments as in the case of constant diffusion done in Section 2 using Csiszar-Kullback type inequalities.

Finally, we show Proposition 1. Let us denote byC_T any constant of the form C(t) (1 +T)^s, where s∈Rand C(t) is bounded on any interval [t_∗,+∞) with t_∗>0.

Proof of Proposition1.-We observe using the bounds (3.5) and (3.1) that for allq≥0,

Z T

0

Z 1

0

Z ∞

0

(1 +y)^qQ⁺(f, f)dy dx dt≤ Z T

0

Z 1

0

Z ∞

0

(1 +y)^q+1

q+ 1 f(t, x, y)dy dx dt +

Z T

0

Z 1

0

Z ∞

0

Z ∞

0

(1 +y+z)^qf(t, x, y)f(t, x, z)dz dy dx dt

≤2^q+1(M^∗₀+M^∗_q+1)T+ 2^q Z T

0

kM(t,·)k_L^∞

x (M^∗₀+M^∗_q)dt≤C_T. According to the properties of the heat kernel (Cf. [DF06] for example), we know that for anyε >0 andt∗>0,

kf(·,·, y)k_L3−ε([t∗,T]×Ω)≤CT

kf(0,·, y)k_L1

x+kQ⁺(f, f)(·,·, y)k_L1([0,T]×Ω)

.

As a consequence, Z ∞

0

(1 +y)^qkf(·,·, y)k_L^3−ε_{([t∗,T]×Ω)}dy≤CT. Then, for allr∈[2,3[

Z ∞

0

(1 +y)^qkQ⁺(f, f)(·,·, y)k_Lr/2([t_∗,T]×Ω)dy

≤ Z ∞

0

(1 +y)^q+1

q+ 1 kf(·,·, y)k_Lr([t∗,T]×Ω)dy +

Z ∞

0

(1 +y)^qk Z ∞

0

f(·,·, y⁰)f(·,·, y−y⁰)dy⁰k_Lr/2([t∗,T]×Ω)dy

≤C_T + Z ∞

0

Z ∞

0

(1 +y+z)^qkf(·,·, y)f(·,·, z)k_Lr/2([t∗,T]×Ω)dydz

≤CT +

2^q−1 Z ∞

0

(1 +y)^qkf(·,·, y)kL^r([t∗,T]×Ω)dy 2

≤CT. Using again the properties of the heat kernel (still described in [DF06]), we see that for anys∈[1,∞) andt_∗>0,

Z ∞

0

(1 +y)^qkf(·,·, y)k_Ls([t∗,T]×Ω)dy≤CT. The above argument can now be used withr= 4 and shows that

Z ∞

0

(1 +y)^qkQ⁺(f, f)(·,·, y)kL²([t∗,T]×Ω)dy≤C_T.

As a consequence, the standard energy estimate on the heat kernel implies that Z ∞

0

(1 +y)^qkf(T,·, y)k_H1

xdy≤CT.

Then, using a Gagliardo-Niremberg type interpolation and Theorem 1, we obtain Z ∞

0

(1 +y)^qkf(T,·, y)−f_∞(y)k_L^∞

x dy≤ Z ∞

0

(1 +y)^qkf(T,·, y)−f_∞(y)k^3/4_H1 x

×

kf(T,·, y)−f∞(y)k^1/4_L1 x

dy

≤ Z ∞

0

(1 +y)^4q/3kf(T,·, y)−f_∞(y)k_H1 xdy

^3/4 Z ∞

0

kf(T,·, y)−f_∞(y)k_L1 xdy

^1/4

≤C_T^3/4 exp(−Cst T)≤Cstexp(−Cst T), which concludes the proof of proposition 1.

Acknowledgements.-JAC acknowledges the support from DGI-MEC (Spain) project MTM2005-08024. KF is partially supported by the WWTF (Vienna) project ”How do cells move?” and the Wittgenstein Award 2000 of Peter A.

Markowich. JAC and KF appreciate the kind hospitality of the ENS de Cachan.

The authors want to express their gratitude to the reviewer who helped us to improve this work.

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