Large time asymptotics for a
continuous coagulation-fragmentation model with degenerate size-dependent diffusion
Laurent Desvillettes,
1Klemens Fellner,
23Abstract
We study a continuous coagulation-fragmentation model with constant kernels for reacting polymers (see [AB]). The polymers are set to diffuse within a smooth bounded one-dimensional domain with no-flux boundary conditions. In particular, we consider size-dependent diffusion coefficients, which may degenerate for small and large cluster-sizes. We prove that the entropy-entropy dissipation method applies directly in this inhomogeneous setting. We first show the necessary basic a priori estimates in dimension one, and secondly we show faster-than-polynomial convergence towards global equilibria for diffusion coefficients which vanish not faster than linearly for large sizes. This extends the previous results of [CDF], which assumes that the diffusion coefficients are bounded below.
Key words: Continuous coagulation-fragmentation with spatial diffusion de- generate in size, faster than polynomial equilibration rates, entropy-entropy dissipation method,
AMS subject classification: 35B40, 35B45, 82D60
1 Introduction
We analyse the spatially inhomogeneous version of a size-continuous model for reacting polymers or clusters of aggregates:
∂tf−a(y)∂xxf =Q(f, f). (1.1) Here,f :=f(t, x, y) is the concentration of polymers/clusters with length/size y≥0 at timet≥0 and pointx∈[0,1]. These polymers/clusters diffuse in the environment. Equation (1.1) is to be considered with homogeneous Neumann boundary condition
∂xf(t,0, y) =∂xf(t,1, y) = 0, (1.2)
1CMLA, ENS Cachan, CNRS & IUF, PRES UniverSud, 61 Av. du Pdt. Wilson, 94235 Cachan Cedex, France. E-mail:[email protected]
2DAMTP, Centre of Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK. E-mail:[email protected], Tel:+44-1223-7/60454
3on leave from the Faculty of Mathematics, University of Vienna, Nordbergstr. 15, 1090 Wien, Austria. E-mail:[email protected], Tel:+43-1-4277/50720
so that there is no polymer flux through the physical boundary.
The diffusion coefficient a := a(y) is supposed to be bounded above and below on compact intervals y ∈ [δ, δ−1] for all 0 < δ < 1, but can possibly degenerate for small and large sizes (that is, tend to ∞ at y = 0 and tend to 0 at y → ∞). More precisely, we shall assume that there exist a∗ > 0, δ7→a∗(δ)>0 such that :
y7→a(y) measurable satisfies
(a(y)≤a∗(δ), ∀y∈[δ, δ−1]
0<1+ya∗ ≤a(y), ∀y∈[0,∞). (1.3) The reaction term Q(f, f) of (1.1) models chemical degradation (break- up or fragmentation) and polymerisation (coalescence or coagulation) of poly- mers/clusters. The full collision operator sums a gain- and a loss-term from both coagulation and fragmentation :
Q(f, f) =Q+(f, f)−Q−(f, f)
= Z y
0
f(t, x, y−y′)f(t, x, y′)dy′+ 2 Z ∞
y
f(t, x, y′)dy′
−2f(t, x, y) Z ∞
0
f(t, x, y′)dy′−y f(t, x, y). (1.4) These four terms model the following phenomena: Coagulation of clusters of sizey′≤y andy−y′ results into clusters of sizey, break-up of clusters of size y′ larger thany creates clusters of sizey, coagulation of clusters of sizey with clusters of sizey′ produces a loss, as does, finally, break-up of clusters of sizey.
This kind of models finds its application not only in polymers and cluster aggregation in aerosols [S16, S17, AB, Al, Dr] but also in cell physiology [PS], population dynamics [Ok] and astrophysics [Sa]. Here, fragmentation and coag- ulation kernels are all set to constants as in the original Aizenman-Bak model [AB]. This will be of paramount importance in the basic a-priori estimates as well as in the use of the entropy-entropy dissipation method.
A fundamental conservation-of-mass law follows from the collision invariance R∞
0 y Q(f, f)dy= 0, entailing that the total number of monomers (or mass of polymers)
N(t, x) :=
Z ∞ 0
y f(t, x, y)dy
(assumed initially to be positive) is formally conserved for timest≥0 : Z 1
0
N(t, x)dx= Z 1
0
Z ∞
0
y f(t, x, y)dy dx= Z 1
0
N(0, x)dx:=N∞>0. (1.5) Another macroscopic quantity of interest is the number density of polymers,
M(t, x) :=
Z ∞
0
f(t, x, y)dy,
that together with the total number of monomers N(t, x) (formally) satisfies the (non-closed) reaction-diffusion system
∂tN−∂xx
Z ∞
0
ya(y)f(t, x, y)dy
= 0, (1.6)
∂tM−∂xx
Z ∞
0
a(y)f(t, x, y)dy
= N−M2. (1.7) The definition of the full collision operator has to be understood in the weak sense. Integrating by parts the gain term of the fragmentation operator, we obtain for any smooth functionϕ:=ϕ(y) and functionf :=f(y) such that the integrals exist:
Z ∞ 0
Q(f, f)(y)ϕ(y)dy= Z ∞
0
Z ∞ 0
[ϕ(y′′)−ϕ(y)−ϕ(y′)]f(y)f(y′)dy dy′ + 2
Z ∞ 0
Φ(y)f(y)dy− Z ∞
0
y ϕ(y)f(y)dy , (1.8) where the functionΦ denotes the primitive ofϕ(∂yΦ=ϕ) withΦ(0) = 0 and y′′=y+y′.
Let us consider the (free-energy) entropy functional associated to any posi- tive densityf :=f(y) as
H(f) = Z ∞
0
(flnf−f)dy ,
and the relative entropy H(f|g) = H(f)−H(g) of two states f and g (not necessarily with the same L1y-norm). Then, the entropy (integrated w.r.t. x) formally dissipates for solutions of eq. (1.1) as
d dt
Z 1
0
H(f)dx=− Z 1
0
Z ∞ 0
a(y)|∂xf|2
f dy dx (1.9)
− Z 1
0
Z ∞ 0
Z ∞ 0
(f′′−f f′) ln f′′
f f′
dy dy′dx :=−D(f) withf′:=f(t, x, y′) andf′′:=f(t, x, y′′).
In the present paper, we shall rather use a weaker dissipation inequality (see (3.1) below), which is obtained by using a remarkable inequality proven in [AB, Propositions 4.2 and 4.3]. It reads (for functions ofyonly) as (Cf. [CDF]):
Z ∞
0
Z ∞
0
(f′′−f f′) ln f′′
f f′
dy dy′≥M H(f|fN) + 2(M −√
N)2. (1.10) Herein, fN := fN(y) denotes a distinguished, exponential-in-size distribution with the very momentsM =√
N andN : fN(y) =e−√yN.
These distributionsfN 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 (1.5) identifies (at least formally) the global equilibriumf∞ with constant momentsM∞2 =N = N∞:
f∞=e−√Ny∞. (1.11)
The analogy to intermediate equilibria carries over to the following additivity of relative entropies
H(f|f∞) =H(f|fN) +H(fN|f∞). (1.12) It is worthy to point out that even iffN andf∞do not have the sameL1y−norm, its global relative entropyR1
0 H(fN|f∞)dx= 2(
qR1
0 N dx−R1 0
√N dx)≥0 is a nonnegative quantity, as easily checked via Jensens’s inequality.
Global existence and uniqueness of classical solutions to equations of the form (1.1)-(1.4) has been studied in [Am, AW] but with restrictions for the coagulation and fragmentation kernel which do not enable to cope with the Aizenman-Bak model (1.4). For the initial boundary-value problem to (1.1)- (1.2), global existence of weak solutions was then proven in [LM02-1], assuming only the first condition of (1.3), and for much more general coagulation and fragmentation kernels including the Aizenman-Bak model (1.4).
In [LM02-1], it is also proven that f∞ attracts all global weak solutions in L1([0,1]×(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].
In the present paper, we are able to obtain explicit rates and constants for the decay to equilibrium for degenerate diffusion coefficients as stated in as- sumption (1.3). As a trade-off we are not able to recover exponential decay.
Nevertheless we shall show decay faster than any polynomial, and in fact of the type exp(−(lnt)β) for allβ <2. Up to our knowledge, this is the first result of explicit convergence for inhomogeneous coagulation-fragmentation models with degenerate diffusion, and the first example in which the entropy/entropy dissi- pation method leads to a convergence with such a strange rate.
Our key Lemma 3.1 in section 3 establishes a functional inequality between entropy and entropy dissipation provided we have lower and upper bounds on the moment M, an upper bound on higher moments, and assumption (1.3).
While Lemma 3.1 holds in all space dimensions, it is in the one-dimensional case that we are able to apply this functional inequality to solutions of (1.1)-(1.4), in which the entropy dissipation entails sufficiently strong a-priori estimates.
The decay to equilibrium with a rate given above follows finally via a suitable Gronwall argument (see section 4). Our main Theorem reads as :
Theorem 1.1 Consider a diffusion coefficient satisfying (1.3). Let us also as- sume that f0 6= 0 (i.e. f0 is not constant zero, for which the theorem holds
trivially by settingf∞= 0forN∞= 0) is a nonnegative initial datum such that (1 +y+ lnf0)f0∈L1((0,1)×(0,∞)).
Then, the global weak solutions f(t, x, y)∈L∞loc(R+;L1((0,1)×(0,∞))) of (1.1)–(1.4)satisfying the entropy/entropy dissipation estimate
Z 1
0
H(f(t, x,·))dx+ Z t
0
D(f(s,·,·))ds≤ Z 1
0
H(f0(x,·))dx
decay to the global equilibrium state(1.11)with the following rate: For allβ <2, there existsCβ>0(which can be explicitly bounded above w.r.t. f0,a∗ anda∗) such that (for all t >0)
Z 1
0
H(f(t,·)|f∞)dx≤Cβe−(lnt)β, (1.13) and:
kf(t,·,·)−f∞kL1x,y ≤Cβe−(lnt)β, (1.14) wheref∞ is defined by (1.11) [and N∞>0 is determined by the conservation of mass (1.5)].
Remark 1.1 The second part of (1.3) [that is, the at most linear degeneracy of the diffusion coefficient for large sizes] is unavoidable in our method. To illustrate why this is so, let us calculate the evolution of second order moments M2(f)(t) :=R1
0
R∞
0 y2f(t, x, y)dydx using the weak formulation (1.8). We find (droppingt andxfor notational convenience)
d
dtM2(f) = Z 1
0
Z ∞
0
Z ∞
0
2y y′f(y)f(y′)dydy′dx− Z 1
0
Z ∞
0
y3
3f(y)dydx , and, using Young’s inequality a(y)y a(yy′′) ≤ 12
y2
a2(y)+a(y2(y′)2′)
, d
dtM2(f)≤ Z 1
0
Z ∞
0
a(y′)f(y′)dy′ Z ∞
0
2y2 a(y)f(y)dy
dx−
Z 1
0
Z ∞ 0
y3
3 f(y)dydx.
Then, we notice that the Fisher information being bounded as in (1.9)“almost”
implies that R∞
0 a(y′)f(y′)dy′ ∈L∞t,x (this is not quite true, Cf. lemma 2.2 for a more precise statement). Interpolating2y2/a(y)betweeny3 andy, and using a Gronwall argument leads to a global bound on M2(f), and the appearance of a third order moment. It is such bounds on moments (Cf. Lemma 3.1) which yield an explicit decay towards equilibrium.
But of course, such an interpolation holds only when a(y) ≥ a∗(1 +y)−δ withδ < 1, which is a slightly degraded version of assumption (1.3). We think therefore that it is not possible to significantly relax the condition on the diffusion coefficient for largey with our method.
Remark 1.2 On the opposite, it is possible to relax the condition on the dif- fusion coefficient a := a(y) for small y to allow an unbounded yet integrable inverse, i.e. R1
0 a−1(y)dy <∞, if we assume initial data f0:=f0(x, y)∈L∞x,y. For such initial data a multiplier technique (see [CP][CDF1, Lemma 3.2]) shows the propagation of the L∞ bound for all times and Lemma 2.2 can be suitably modified.
Remark 1.3 We cannot expect the explicit decay rate given in eq. (1.13), (1.14) to be optimal, since it is a consequence of estimates on moments (more precisely, the dependence w.r.t. pof the bounds on the moment of orderp) which are probably not optimal themselves, as well as a consequence of the entropy- entropy dissipation estimate, which is certainly not optimal in the steps 2 and 3.
We nevertheless suspect that the convergence towards equilibrium might not be exponential, the degeneracy when y→ ∞ of the diffusion rate playing here the same role as the degeneracy whenv → ∞ in soft potentials for the Boltzmann equation (Cf. [TV]).
It is further possible to interpolate the faster-than-polynomial decay in a
“weak” norm likeL1 with polynomially growing bounds in “strong” norms like (weighted)L1y(Hx1) in order to get faster-than-polynomial decay in a “medium”
norm like L1y(L∞x ). Thus, the decay toward equilibrium can be extended to these stronger norms. This idea is used in the proof of the following proposition (Cf. the end of section 4):
Proposition 1.1 Under the assumptions of Theorem 1.1, for all q ≥ 0 and β <2, there are (explicitly computable) constants Cβ,q such that
Z ∞ 0
(1 +y)qkf(t,·, y)−f∞(y)kL∞x dy≤Cβ,qe−(lnt)β, (1.15) for allt≥t∗>0.
A bootstrap argument in the spirit of the proof of Proposition 1.1 can even allow to replace theL∞x norm by any Sobolev norm in (1.15).
Note that on one hand no extra assumption on the initial datum is needed for this Proposition, since the regularity w.r.t. xis created thanks to the diffusive character of eq. (1.1). On the other hand, one definitely needs extra assumptions on the initial datum (typicallyR∞
0 (1 +y)kf0(t,·, y)kL∞x dy <+∞) if one wishes to use Proposition 1.1 (or more simply estimate (4.6)) in order to prove a result of uniqueness for weak solutions of eq. (1.1). We shall come back to the issue of uniqueness with general initial data before stating lemma 2.2.
Explicit rates of decay for coagulation-fragmentation models without diffu- sion have been obtained in [AB] and, for the Becker-D¨oring model, in [JN].
Explicit rates of decay for reversible reaction-diffusion models (correspond- ing to a finite number of possible size for polymers) have been obtained in
[DF05, DF06, DF]. Non-constructive exponential rates via a contradiction ar- gument was shown for general drift-diffusion-reaction systems in [Gr¨o, GGH]. In [DF07], the case of a degenerate diffusion in reaction-diffusion models is studied.
The first result of explicit rate of decay for inhomogeneous coagulation- fragmentation models was proven in [CDF], under the physically unrealistic assumption that the diffusion is bounded below and above. The present paper is devoted to the removal of this assumption.
The method of proof makes use of the entropy–entropy dissipation method (Cf. [Des] for a general introduction to this method in the context of kinetic equations). It is in particular reminiscent of works in which “slowly growing a priori bounds” appear, such as [TV] and [DF06].
Moreover, one uses here bounds on moments in which one keeps track of the constants (w.r.t. the order of the moment) so that some summability of those bounds can be recovered in the end. This idea is already present in papers such as [BGP]. The strange functions of time recovered in (1.13) is directly related to the summability mentioned above. (Cf. the end of the proof of theorem 1.1).
2 A-priori Estimates
We begin with a-priori estimates used in the proof of Theorem 1.1. In the lemmas and propositions of this section, f, M or N always refer to a global weak solution of (1.1)–(1.2) satisfying the assumptions of Theorem 1.1.
Lemma 2.1 There exists M∗0>0 (depending only on the initial datum) such that
sup
t≥0
Z 1
0
M(t, x)dx≤ M∗0. (2.1)
Proof.-The proof is a consequence of integrating (1.7) and Jensen’s inequal- ity:
d dt
Z 1
0
M(t, x)dx≤N∞− Z 1
0
M(t, x)dx 2
. Cf. [CDF, Lemma 4] for more details.
We now turn to a control ofM(t,·) inL∞x . Note that the estimate obtained in Lemma 2.2 below is a significant step towards a proof of uniqueness of weak solutions to eq. (1.1) with general initial datum of finite mass and entropy. One would in fact need the same kind of estimate forN in order to get such a result (see e.g. [LM04]).
Lemma 2.2 The number density of polymers M := M(t, x) lies in [L2 + L∞](0,∞;L∞x(0,1)). More precisely, there existm∞>0and an L1∩L2(0,∞)- functionm2(t)such that (for a.e. t≥0)
kM(t,·)kL∞x ≤m∞+m2(t).
Proof.- We integrate
f(t, x, y)−f(t,x, y) = 2˜ Z x
˜ x
pf(t, ξ, y)∂x
pf(t, ξ, y)dξ with respect to ˜x∈(0,1) andy∈R+, and get
Z ∞ 0
f(t, x, y)− Z 1
0
f(t,x, y)˜ d˜x dy≤2
Z ∞ 0
Z 1
0 |p
f(t, ξ, y)| |∂x
pf(t, ξ, y)|dξdy
≤2 Z ∞
0
Z 1
0
a(y)−1f(t, x, y)dxdy
12 Z ∞
0
Z 1
0
a(y)|∂x
pf(t, x, y)|2dxdy 12
≤
a−∗1 Z 1
0
(M(t, x) +N(t, x))dx
1 2
(D(f(t)))1/2, using assumption (1.3). Then,
M(t, x)≤ Z 1
0
M(t,x)˜ d˜x+a−∗1/2(M∗0+N∞)1/2D(f(t))1/2.
The first term in this estimate belongs to L∞t thanks to Lemma 2.1 and the second one toL2t. Note that theL2function can be split as a sum of anL1and anL∞function. This completes the proof of Lemma 2.2.
Lemma 2.3 There exists a constant M0∗ > 0 such that (for all t ≥ 0), one has
Z 1
0
M(t, x)dx≥ M0∗. (2.2)
Proof.- A Gronwall type proof exploiting the estimate d
dt Z 1
0
M(t, x)dx≥N∞−(m∞+m2(t)) Z 1
0
M(t, x)dx , can be found in [CDF, Lemma 6].
Next, we show the (uniform for timet≥t∗>0) control 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
ypf(t, x, y)dy dx
for allt≥0. The proof uses the fact that the fragmentation term (in terms of creation of moments) somehow dominates the non-linear coagulation term. This property was already used for homogeneous coagulation-fragmentation models (with so-called strong fragmentation) in the works of [BC, Cos], see also [Des, MW] for related kinetic models. Here, we used crucially Lemma 2.2 in order to
adapt this idea to the spatially inhomogeneous case. The proof keeps record of the dependence uponp of the uniform bound of the moment of order p: this will enable us in the proof of Theorem 1.1 to conclude to the “exponential of the square of the logarithm” convergence rate via a summation argument.
Note also that the bounds which are obtained in Lemma 2.4 below ensure that the mass is conserved along the solutions of eq. (1.1) [that is, no gela- tion occurs], provided that some moments initially exist. We refer to [CnDF1], [CnDF2] and the references therein for a much broader approach to this problem (though in the “discrete iny” setting).
Lemma 2.4 For any p >1 andt∗>0, one has
Mp(f)(t)≤(22pC)p=:M∗p, for t≥t∗>0, (2.3) for a constantC=C(t∗, f0)depending only on the initial datum andt∗.
Proof.-We prove Lemma 2.4 in three steps . We denoteµ2=km2kL2(where m2 is defined in Lemma 2.2)
Step 1.-As in [CDF, Lemma 7], the evolution of the moment of orderp >1 is governed by
d
dtMp(f)(t)≤(2p−2)Mp(f)(t) [m∞+m2(t)]−p−1
p+ 1Mp+1(f)(t). (2.4) Trivial interpolation of the (p+ 1)-order moment with the moment of order one implies thanks to Young’s inequality and the conservation of mass (1.5) that
−p−1
p+ 1Mp+1(f)(t)≤ ǫ−p
p+ 1N∞− p
p+ 1ǫ−1Mp(f)(t) for allǫ >0. Thus
d
dtMp(f)(t)≤
(2p−2)(m∞+m2(t))− p p+ 1ǫ−1
Mp(f)(t) + ǫ−p p+ 1N∞. Moreover, according to Duhamel’s formula and usingRt
t∗m2ds≤µ2√
t−t∗, we estimate for allp >1
Mp(f)(t)≤Mp(f)(t∗)eθ(t−t∗)+ ǫ−p p+ 1N∞
Z t
t∗
eθ(t−s)ds , (2.5) whereθis the function defined by
θ(r) =
(2p−2)m∞−1 2ǫ−1
r+ (2p−2)µ2√ r .
Choosing thenǫ−1= 22pm∞, it is easy to verify thatθis bounded above. More precisely,
θ(r) = (2p−2)µ2
√ r− r
√rp
, where√rp:= 2p−2 22p−1−2p+ 2
µ2
m∞ >0,
and 0 < θ(r) ≤ θ(rp/4) < 2mµ22
∞ for r ∈ (0, rp), θ(rp) = 0, θ(r) < 0 for r ∈ (rp,∞). Then, we obtain the estimate
θ(r)≤I{r≤2r
p}
µ22
2m∞ −I{r>2r
p}(2p−2)µ2(1− 1
√2) r
√rp
, (2.6)
since√r−√rrp ≤ −(1−√12)√rrp forr >2rp. Finally, we end up with Mp(f)(t)≤Mp(f)(t∗)
e
µ2 2
2m∞ +e−(2p−2)µ2(1−√12)2√rp
+(22pm∞)pN∞
e
µ2 2 2m∞2rp+
Z t
0
e−(2p−2)µ2(1−√12)
t−s
√rp ds
≤Mp(f)(t∗)
e
µ2 2 2m∞ + 1
+ (22pm∞)pN∞
e
µ2 2
2m∞2rp+ 6rp
m∞ µ22
(2.7)
≤ C Mp(f)(t∗) + (22pm∞)p . for a constantC=C(N∞, m∞, µ2) and for allt≥t∗>0.
Step 2.-Next, we construct a sequence{tk}fork= 2,3,4, . . . as follows : Let us fixt∗>0, sett2= t2∗, and assume thatM2(f)(t∗/2)<∞. Let 0< λ <1 be some parameter (in fact, we shall takeλ:= inf{1,(mµ∞
2 )2t2∗}). By (2.7), we have for allt∈[tk, tk+λ2rk] that
Mk(f)(t)≤ Mk(f)(tk) + (22km∞)k2rkN∞
e µ
22
2m∞ + 1 + 3m∞ µ22
. Reinserting into (2.4) and integrating over [tk, tk+λ2rk] yields
k−1 k+ 1
Z tk+λ2rk
tk
Mk+1(f)(t)dt≤Mk(f)(tk) + (2k−2)
λ2rkm∞+µ2
pλ2rk
× Mk(f)(tk) + (22km∞)k2rkN∞
e
µ2 2
2m∞ + 1 + 3m∞ µ22
. Then, dividing by the interval lengthλ2rk and recalling that√rk <21−k µm2 we obtain ∞
Z tk+λ2rk
tk
Mk+1(f)(t) λ2rk
dt≤C22kMk(f)(tk) + (22kC)k,
where C = C(N∞, m∞, µ2, λ) depends only on N∞, m∞, µ2, and λ. Thus, there exists a timetk+1∈[tk, tk+λ2rk] such that
Mk+1(f)(tk+1)≤C22kMk(f)(tk) + (22kC)k. Moreover, by iteration ink (and with a larger constantC)
Mk(f)(tk)<(22(k−1)C)(k−1) ⇒ Mk+1(f)(tk+1)≤(22kC)k.
Finally, considering that t∞= lim
k→∞tk≤t2+
∞
X
k=2
λ2rk≤t2+ 2λ( µ2
m∞)2
∞
X
k=2
22−2k=t2+λ( µ2
m∞)22 3, we chooseλ <(t∗−t2)32(mµ∞2 )2 to ensuret∞≤t∗ and hence
Mk(f)(t∗)≤(22kC)k for allk= 2,3,4, . . . , so that
∀t≥t∗, Mk(f)(t)≤(22kC)k for allk= 2,3,4, . . . , whereC=C(N∞, m∞, µ2, t∗) depends only onN∞,m∞,µ2, andt∗.
Step 3.- It remains to show that for given nontrivial initial data such that y f0∈L1x,y, there exists a timet0≤t4∗ such thatMp(f)(t0)<∞for somep >1 and, further, a timet1≤ t2∗ such thatM2(f)(t1)<∞.
We start with the following observation [MW, Appendix A]: For a nonneg- ative integrable function g(y) 6= 0 on (0,∞), there exists a concave function Φ(y), depending on g, smoothly increasing from Φ(0)>0 to Φ(∞) =∞such
that Z ∞
0
Φ(y)g(y)dy <∞. Moreover, the function Φ can be constructed to satisfy
Φ(y)−Φ(y′)≥C y−y′
yln2(e+y) (2.8)
for 0< y′< y withC not depending ong. We refer to [MW, Appendix A] for all the details of this “by-now standard” construction.
To show now thatMp(f)(t0)<∞ for ap >1 and a time t0≤ t4∗, we take functions Φ(x, y) constructed for nontrivialy f0(x, y)∈L1y(0,∞) a.e. x∈(0,1) [herexis only a parameter] and calculate - similar to Step 1 - the moment
M1,Φ(f)(t) = Z 1
0
Z ∞
0
yΦ(x, y)f(x, y)dy dx.
For the fragmentation part, we use (2.8) for 0< y′< y and estimate 2
Z y
0
y′(Φ(y′)−Φ(y))dy′f(y)≤ − C ln2(e+y)y
Z y
0
y′(y−y′)dy′f(y)
= C
ln2(e+y) y2
6 f(y)≤ −Cδy2−δf(y), for allδ >0 and a positive constantCδ [the (t, x)-dependence has been dropped for notational convenience]. Hence, by estimating the coagulation part similar to [CDF, Lemma 7], making use of the concavity of Φ, we obtain
d
dtM1,Φ(f)(t)≤3(m∞+m2(t))M1,Φ(f)(t)−CδM2−δ(f)(t),
and boundedness of the moment M1,Φ follows by interpolation as well as the finiteness ofM2−δ(f)(t0) (for somet0∈(0, t∗/4)) analogously to Step 2. Writing a differential inequality for M3/2(f) yields then the existence of some t1 ∈ (t0, t∗/2) such thatM2(f)(t1)<∞. This yields the assumption at the beginning of step 2 and concludes the proof.
Next, we show that M is bounded below uniformly (with respect to t and x) for allt≥t∗>0.
Proposition 2.1 Lett∗>0be given. Then, there is a strictly positive constant M∗ (depending ont∗, a∗ and a∗(δ) as in assumption (1.3),m∞, µ1 :=km2kL1
and the initial datum) such that for allt≥t∗>0, M(t, x)≥ M∗.
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 ety+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 and G:=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+x4a t−z)2 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 functionshandG“evenly mirrored around 0 in the xvariable”, that is
˜h(t, x) =
h(t, x) x∈[0,1], h(t,−x) x∈[−1,0[, 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−(x4a(y)−z)2t et1y+R0t1kM(s,·)kL∞x dsdz ,
where we have neglected g2 (and all the terms corresponding to k 6= 0 in the sum) as nonnegative. Consideringt ∈ [t∗,2t∗] for some t∗ >0, recalling that kM(s,·)kL∞x ≤m∞+m2(s) withR∞
0 m2(s)ds≤µ1<∞ by Lemma 2.2, and since|x−z|<2 we have for allδ≤y≤1/δ:
f(t1+t, x, y)≥ 1 2√ π
Z 1
−1
f˜(t1, z, y) 1
pa(y)te−a(y)1 te−t y−Rtt11+tkM(s,·)kL∞x dsdz
≥ 1
p2π a∗(δ)t∗ Z 1
0
f(t1, z, y)e−a1+y∗t∗ e−2t∗y−2t∗m∞−µ1dz
≥C Z 1
0
f(t1, z, y)e−(2t∗+a∗1t∗)ydz,
where C > 0 depends on the constants a∗, a∗(δ), m∞, µ1 and t∗ > 0. Using Lemma 2.3, for allK large enough, and 0< δ <1,
M(t1+t, x)≥C e−(2t∗+a∗1t∗)1δ Z 1
0
Z 1/δ
δ
f(t1, z, y)dydz
≥C e−(2t∗+a∗1t∗)1δ
M0∗−δ N∞−K δ− Z
H(f)dx/lnK
, where we have used thatR∞
1
δ f(y)dy≤δN andRδ
0 f(y)dy≤K δ+Rδ
0 flnlnKf dx.
Choosing δ and K, we get that M(t1+t, x) ≥ M∗. Moreover, since M∗ = M∗(a∗, a∗, m∞, µ1, H(f0), t∗) does not depend ont1, we get Proposition 2.1.
3 Entropy-entropy dissipation estimate
For the subsequent large-time analysis, we shall study the relative entropy with respect to the global equilibrium, which dissipates according to (1.9) and (1.10) as
d dt
Z
Ω
H(f|f∞)dx≤ −D1(f), (3.1) D1(f) =
Z
Ω
Z ∞
0
a(y)|∇xf|2
f dy dx+ Z
Ω
hM H(f|fN) + 2(M−√ N)2i
dx (3.2) We introduce a lemma enabling to estimate the entropy of f by means of its entropy dissipation. This is a functional estimate, that is, the functionf in this lemma does not depend ont and does not necessarily have something to do with the solution of our equation. Moreover, since this lemma is true in any dimension of space, we replace the interval [0,1] by any bounded measurable subset Ω of RN of measure 1. Then, for all quantity S, we denote by ¯S its average w.r.t. x∈Ω, that is ¯S=R
ΩS(x)dx.
Lemma 3.1 Assume (1.3), in particular 0 < 1+ya∗ ≤ a(y) for all y ∈ [0,∞).
Let f := f(x, y) ≥ 0 be a measurable function from Ω×[0,+∞[ to R with moments satisfying 0 < M∗ ≤ M(x) := R∞
0 f(x, y)dy ≤ kMkL∞x and 0 <
N∞ :=R
Ω
R∞
0 y f(x, y)dydx= N. Let p >1 and assume that the moment of order2pis finite, i.e. R
Ω
R∞
0 y2pf(x, y)dxdy=M2p<+∞. Then, the following entropy-entropy dissipation estimate holds for allA≥1:
D1(f)≥ C AkMkL∞x
Z
Ω
H(f|f∞)dx−C M2p
A2p+1, (3.3)
with a constantC=C(M∗, N∞, a∗, P(Ω))depending as specified only on M∗, N∞,a∗, and the Poincar´e constantP(Ω).
Proof.- Step 1.- We start with the right-hand side of (3.3) by using the additivity (1.12) and calculating
Z
Ω
H(f|f∞)dx= Z
Ω
H(f|fN)dx+ 2p N−√
N
, (3.4)
where we recall that ¯S:=R
ΩS(x)dx(with|Ω|= 1).
Step 2.-The second term of (3.4) – which measures how farN is from being constant – is bounded as :
pN−√
N ≤ 2
√N∞
hkM−√
Nk2L2x+kM−Mk2L2x
i. (3.5)
Indeed, since √ N−√
N is orthogonal to √
N−M in L2x, and, thus, k√ N − M+M −√
Nk2L2x =k√
N−Mk2L2x+kM −√
Nk2L2x, we have pN−√
N ≤N −√ N2
√N ≤ 1
√N∞k√ N−√
Nk2L2x ≤ 1
√N∞k√
N−Mk2L2x, and further, we obtain (3.5) by expandingk√
N−Mk2L2x and by using Young’s inequality
1 2k√
N−Mk2L2x− kM −Mk2L2x≤ k√
N−M−M+Mk2L2x. Thus, we obtain (using 0<M∗< M)
Z
Ω
H(f|f∞)dx≤maxn
M−∗1,2N∞−12oZ
Ω
M H(f|fN)dx+ 2kM−√ Nk2L2x
+ 4
√N∞kM−Mk2L2x. (3.6)
Step 3.-For a cut-off sizeA >0, we introduce the finite size density integral MA(t, x) := RA
0 f(t, x, y)dy and its complement MAc(t, x) := R∞
A f(t, x, y)dy and proceed to estimate the last term in (3.6) in the following way :
kM−Mk2L2x= Z
Ω
MA−MA+MAc −MAc2 dx
≤2kMA−MAk2L2x+ 4 A2p
Z
Ω
Z ∞
0
ypf(t, x, y)dy 2
dx
≤2kMA−MAk2L2x+ 4
A2pkMkL∞xM2p, (3.7) for anyp >1, where the last term has been estimated thanks to Cauchy-Schwarz inequality.
Step 4.-Next, the variance ofMA, i.e. the first term on the right-hand side of (3.7) is controlled by the first, “Fisher”-type term of (3.1). Denoting by P(Ω) the constant of Poincar´e’s inequality, we estimate using Cauchy-Schwartz inequality and assumption (1.3), forA≥1:
kMA−MAk2L2x ≤P(Ω) Z
Ω
∇x Z A
0
f dy
2
dx
≤P(Ω) Z
Ω
Z A
0
a(y)|∇xf|2 f dy
Z A
0
f a(y)dy
dx
≤P(Ω)1 +A
a∗ kMkL∞x
Z
Ω
Z ∞ 0
a(y)|∇xf|2 f dydx
≤P(Ω)2A
a∗ kMkL∞x
Z
Ω
Z ∞ 0
a(y)|∇xf|2
f dydx , (3.8) since we have a(y)1 ≤1+ya∗ ≤1+Aa∗ fory∈[0, A].
Step 5.-Finally, combining (3.6) with (3.7) and (3.8), and further with (3.2), we have (still forA≥1)
Z
Ω
H(f|f∞)dx≤C(M∗, N∞, P(Ω), a∗)kMkL∞x A D1(f) +C(M∗, N∞)D1(f) +C(N∞)M2pkMkL∞x A−2p,
which yields the proof of Lemma 3.1, sincekMkL∞x A≥ M∗.
4 Proof of Theorem 1.1
With Proposition 2.1 and Lemmas 2.2 and 2.4 providing the moment bounds required by the entropy-entropy dissipation Lemma 3.1 in the one dimensional case Ω = (0,1), we turn now to the
Proof.- [Theorem 1.1] We divide the proof into two steps: First, we show polynomial convergence rates, and, secondly, we prove faster than polynomial rates via a summation argument.
Step 1.-We denote byC1,C2, etc., various constants which only depend on f0,a∗ anda∗. According to Lemmas 2.4 and 3.1, for anyA >1
d dt
Z 1
0
H(f|f∞)dx≤ −D1(f)
≤ − C1
kMkL∞x
1 A
Z 1
0
H(f|f∞)dx+C2C3p 28p2
A2p+1, (4.1) where kMkL∞x (t) ≤ m∞+m2(t) by Lemma 2.2. By choosing A = A(t) = max{2, A∗(t)} whereA∗(t) is defined by
C2C3p 28p2 (A∗)2p+1 =1
2 C1
kMkL∞x
1 A∗
Z 1
0
H(f|f∞)dx , it follows that
C2C3p 28p2 A2p+1 ≤1
2 C1
kMkL∞x
1 A
Z 1
0
H(f|f∞)dx . Hence, denotingC4=C1/(2C2), we see that
1
A ≤C3−1/2 C4R1
0 H(f|f∞)dx kMkL∞x 28p2
!2p1 , and we obtain
d dt
Z 1
0
H(f|f∞)dx≤ −1 2
C1C3−1/2 kMkL∞x
C4R1
0 H(f|f∞)dx kMkL∞x 28p2
!2p1 Z 1
0
H(f|f∞)dx, which integrates (fort > t∗>0) like
Z 1
0
H(f(t)|f∞)dx −2p1
− Z 1
0
H(f(t∗)|f∞)dx −2p1
≥ C5
p24p Z t
t∗
ds kMk1+
1 2p
L∞x
≥ C5
p24p
1 (1 +m∞)1+2p1
Z t
t∗
I{m2(s)<1}ds≥ C5
p24p
t−t∗−µ1
(1 +m∞)1+2p1 (since |{m2 ≥1}| ≤µ1 whereµ1 =km2kL1 as defined in Lemma 2.2). Using the fact thatR1
0H(f(t∗)|f∞)dx≥0 and denoting C8= ((1 +m∞)/C5)2,C7= t∗+µ1, we get
Z 1
0
H(f(t)|f∞)dx≤C8pp2p28p2(t−C7)−2p.
Step 2.-We may now sum over all 2≤p∈N
X
p≥2
(t−C7)2p (2C8)pp2p28p2
Z 1
0
H(f(t)|f∞)dx≤1, i.e.
Z 1
0
H(f(t)|f∞)dx≤L(t−C7), where (for all 1< α <2)
L−1(t) = X
q≥1,even
tq
(C10q)q22q2 = X
q≥1,even
tqe−2q2ln 2−qln(q C10)
≥C(α) X
q≥1,even
tqe−α(αq)αα−1+(q+2α )2, since
e2q2ln 2+qln(q C10)+(q+2α )2 =O(eα(αq)
α α−1
).
Note that our assumption onαimplies that αα−1 >2.
Finally, choosingp(even) such thatt∈[exp((p/α)α−11),exp(((p+2)/α)α−11)]
and applying to the following elementary computation on these intervals : e(lnt)[2(α−1)]
tp ≤e(p+2α )2−p(αp)α−11 =e(p+2α )2−α(pα)α−1α , we have
L−1(t)≥C(α)e[ln2(α−1)(t)]
for alltlarge enough. and 1< α <2. This ends the proof of (1.13). The proof of theL1-decay estimate (1.14) follows from the Csiszar-Kullback inequality.
In the proof of Proposition 1.1, we use explicit Lr bounds (r ≥1) for the 1D heat equation. Similar bounds were already established in [DF06]. Here we prove an improved version allowing, in particular, unbounded diffusion co- efficients. As these bounds will be used pointwise in y, we will suppress for notational convenience the dependence ony.
Lemma 4.1 Letudenote the solution of the 1D heat equation (t >0, x∈[0,1], and (constant) diffusivity a) with homogeneous Neumann boundary condition, i.e.
∂tu−a ∂xxu=g , ∂xu(t,0) =∂xu(t,1) = 0, (4.2) and assume for the initial data u(0, x) =u0(x)and the source termg(t, x)that
u0∈Lp([0,1]), g∈Lp([0,+∞)×[0,1]).
