PARABOLIC-ELLIPTIC KELLER-SEGEL EQUATION
G. EGA ˜NA, S. MISCHLER
Abstract. The present paper deals with the parabolic-elliptic Keller-Segel equation in the plane in the general framework of weak (or “free energy”) solutions associated to initial datum with finite massM, finite second moment and finite entropy. The aim of the paper is threefold:
(1) We prove the uniqueness of the “free energy” solution on the maximal interval of existence [0, T∗) withT∗=∞in the case whenM ≤8π andT∗ <∞in the case whenM >8π. The proof uses a DiPerna-Lions renormalizing argument which makes possible to get the “optimal regularity” as well as an estimate of the difference of two possible solutions in the criticalL4/3 Lebesgue norm similarly as for the 2dvorticity Navier-Stokes equation.
(2) We prove immediate smoothing effect and, in the caseM <8π, we prove Sobolev norm bound uniformly in time for the rescaled solution (corresponding to the self-similar variables).
(3) In the caseM <8π, we also prove weightedL4/3 linearized stability of the self-similar profile and then universal optimal rate of convergence of the solution to the self-similar profile.
The proof is mainly based on an argument of enlargement of the functional space for semigroup spectral gap.
Keywords: Keller-Segel model; chemotaxis; weak solutions; free energy; entropy method; log- arithmic Hardy-Littlewood-Sobolev inequality; Hardy-Littlewood-Sobolev inequality; subcritical mass; uniqueness; large time behavior; self-similar variables.
AMS Subject Classification: 35B45, 35B30, 35D05, 35K15, 35B40, 35D10, 35K60
Contents
1. Introduction 1
2. A posteriori estimates - Proof of Theorem 1.4 6
3. Uniqueness - Proof of Theorem 1.3 15
4. Self-similar behaviour - Proof of Theorem 1.5 17
References 23
1. Introduction
The aim of the paper is to prove uniqueness of “free energy” weak solutions to the the so-called parabolic-elliptic Keller-Segel equation in the plane associated to initial datum with finite mass M ≥0, finite polynomial moment and finite entropy, and in the subcritical caseM <8πto prove optimal rate of convergence to self-similarity of these solutions. In [19] our analysis will be extended to the parabolic-parabolic Keller-Segel equation in a similar context.
The Keller-Segel (KS) system for chemotaxis describes the collective motion of cells that are attracted by a chemical substance that they are able to emit ([34, 27]). We refer to [8] and the references quoted therein for biological motivation and mathematical introduction. In this paper we are concerned with the parabolic-elliptic KS model in the plane which takes the form
∂tf = ∆f− ∇(f∇c) in (0,∞)×R2, (1.1)
c := −¯κ = −κ∗f in (0,∞)×R2,
1
with κ:= 2π1 log|z|, so that in particular
−∇c= ¯K:=K ∗f, K:=∇κ= 1 2π
z
|z|2.
Here t ≥ 0 is the time variable, x ∈ R2 is the space variable, f = f(t, x) ≥ 0 stands for the mass density of cellswhile c=c(t, x)∈Ris thechemo-attractant concentration which solves the (elliptic) Poisson equation−∆c=f in (0,∞)×R2.
The evolution equation (1.1) is complemented with an initial condition
(1.2) f(0, .) =f0 in R2,
where throughout this paper, we shall assume that
(1.3) 0≤f0∈L12(R2), f0logf0∈L1(R2).
Here and below we define the weight functionhxi:= (1 +|x|2)1/2and the weighted Lebesgue space Lpk(R2) for 1≤p≤ ∞,k≥0, by
Lpk(R2) :={f ∈L1loc(R2); kfkLp
k:=kfhxikkLp <∞}, as well asL1+(R2) the cone of nonnegative functions ofL1(R2).
The fundamental identities are that any solution to the Keller-Segel equation (1.1) satisfies at least formally the conservation of mass
(1.4) M(t) :=
Z
R2
f(t, x)dx= Z
R2
f0(x)dx=:M, the second moment equation
(1.5) M2(t) :=
Z
R2
f(t, x)|x|2dx=C1(M)t+M2,0, M2,0:=
Z
R2
f0(x)|x|2dx, C1(M) := 4M 1−8πM
, and the free energy-dissipation of the free energy identity
(1.6) F(t) +
Z t
0
DF(s)ds=F0, where the free energy F(t) =F(f(t)),F0=F(f0) is defined by
F =F(f) :=
Z
R2
flogf dx+1 2
Z
R2
f¯κ dx, and the dissipation of free energy is defined by
DF=DF(f) :=
Z
R2
f|∇(logf) +∇¯κ|2dx.
It is worth emphasizing that the critical mass M∗ := 8π is a threshold because one sees from (1.5) that there does not exist nonnegative and mass preserving solution when M > 8π (the identity (1.5) would imply that the second moment becomes negative in a finite time shorter than T∗∗:= 2πM2,0/[M(8π−M)]).
On the one hand, in the subcritical caseM <8π, thanks to the logarithmic Hardy-Littlewood Sobolev inequality (see e.g. [3, 18])
(1.7) ∀f ≥0, Z
R2
f(x) logf(x)dx+ 2 M
Z Z
R2×R2
f(x)f(y) log|x−y|dxdy≥C2(M), withC2(M) :=M(1 + logπ−logM), one can easily check (see [8, Lemma 7]) that for subcritical massM <8π, there holds
(1.8) H:=H(f) =
Z
flogf ≤C3(M)F+C4(M),
with C3(M) := 1/ 1− M8π
, C4(M) := C3(M)C2(M)M/(8π). Then from (1.8) and the very classical functional inequality (see for instance [8, Lemma 8])
(1.9) H+ :=H+(f) =
Z
f(logf)+≤ H+1
4M2+C5(M),
with C5(M) := 2Mlog(2π) + 2/e, one concludes that (1.4), (1.5) and (1.6) provide a convenient family of a priori estimates in order to define weak solutions, namely
H+(f(t)) +M2(f(t)) +C3(M) Z t
0
DF(f(s))ds≤ (1.10)
≤C3(M)F0+5
4M2,0+ 2C1(M)t+C4(M) +C5(M), where the RHS term is finite under assumption (1.3) on f0, since
F0 ≤ H0+ 1 4π
Z Z
f0(x)f0(y) (log|x−y|)+dxdy (1.11)
≤ H0+ 1 4π
Z Z
f0(x)f0(y)|x−y|2dxdy≤ H0+1
πM M2,0, with H0:=H(f0). In other words, we have
(1.12) AT(f) := sup
t∈[0,T]
H+(f(t)) +M2(f(t))}+ Z T
0
DF(f(s))ds≤C(T) ∀T ∈(0, T∗) with T∗= +∞and a constantC(T) which depends onM,M2,0,H0and the final timeT.
On the other hand, in the critical case M = 8π and the supercritical case M > 8π, the above argument using the logarithmic Hardy-Littlewood Sobolev inequality (1.7) fails, but one can however prove that (1.12) holds with T∗ = +∞ when M = 8πand that (1.12) holds with some T∗∈(0, T∗∗] whenM >8π(see [6] for details as well as Remark 2.3 below).
Definition 1.1. For any initial datumf0 satisfying (1.3)and any final time T∗>0, we say that (1.13) 0≤f ∈L∞(0, T;L1(R2))∩C([0, T);D0(R2)), ∀T ∈(0, T∗),
is a weak solution to the Keller-Segel equation in the time interval(0, T∗)associated to the initial condition f0 wheneverf satisfies (1.4),(1.5)and
(1.14) F(t) +
Z t
0
DF(s)ds≤ F0 ∀t∈(0, T∗),
as well as the Keller-Segel equation (1.1)-(1.2)in the distributional sense, namely (1.15)
Z
R2
f0(x)ϕ(0, x)dx= Z T∗
0
Z
R2
f(t, x)n
(∇x(logf) + ¯K)· ∇xϕ−∂tϕo dxdt for any ϕ∈Cc2([0, T)×R2).
It is worth emphasizing that thanks to the Cauchy-Schwarz inequality, we have Z
R2
f|∇x(logf) + ¯K|dx≤M1/2D1/2F , and the RHS of (1.15) is then well defined thanks to (1.10).
This framework is well adapted for the existence theory.
Theorem 1.2. For any initial datumf0 satisfying (1.3)there exists at least one weak solution on the time interval (0, T∗)in the sense of Definition 1.1 to the Keller-Segel equation (1.1)-(1.2)with T∗= +∞whenM ≤8πandT∗<+∞ whenM >8π.
We refer to [8, Theorem 1] for the subcritical case M ∈(0,8π) and to [6] for the critical and supercritical casesM ≥8π.
Our first main result establishes that this framework is also well adapted for the well-posedness issue.
Theorem 1.3. For any initial datumf0 satisfying (1.3)there exists at most one weak solution in the sense of Definition 1.1 to the Keller-Segel equation (1.1)-(1.2).
Theorem 1.3 improves the uniqueness result proved in [20] in the class of solutionsf ∈C([0, T];
L12(R2))∩L∞((0, T)×R2) which can be built under the additional assumptionf0∈L∞(R2) (see also [24] where a uniqueness result is established for a related model). Our proof follows a strategy introduced in [23] for the 2D viscous vortex model. It is based on a DiPerna-Lions renormalization trick (see [21]) which makes possible to get the optimal regularity of solutions for small time and then to follow the uniqueness argument introduced by Ben-Artzi for the 2D viscous vortex model (see [4, 10]).
Next we consider the smoothness issue and the long time behaviour of solution for subcritical mass issue. For that last purpose it is convenient to work with self-similar variables. We introduce the rescaled functionsg andudefined by
(1.16) g(t, x) :=R(t)−2f(logR(t), R(t)−1x), u(t, x) :=c(logR(t), R(t)−1x), with R(t) := (1 + 2t)1/2. The rescaled parabolic-elliptic KS system reads
∂tg = ∆g+∇(gx−g∇u) in (0,∞)×R2, (1.17)
u = −κ∗g in (0,∞)×R2. Our second main result concerns the regularity of the solutions.
Theorem 1.4. For any initial datum f0 satisfying (1.3)the associated solution f is smooth for positive time, namely f ∈C∞((0, T∗)×R2), and satisfies the identity (1.6)on(0, T∗). Moreover, when M <8π, the rescaled solution g defined by (1.16)satisfies
(1.18) sup
t≥0
Mk(g(t))≤max((k−1)k/2M, Mk(f0)) ∀k≥2, as well as
(1.19) sup
t≥ε
kg(t, .)kW2,∞ ≤ C ∀ε >0, for some explicit constant C which depends onε,M,F0 andM2,0.
It is worth mentioning thatLpbound ongfor positive time and forp∈[1,∞) was known but non uniformly in time and as an a priori bound, while (1.19) is proved as an a posteriori estimate. Our proof is merely based on the same estimates as those established in [8], on a bootstrap argument and on the remark that the rescaled free energy provides uniform in time estimates.
From now on in this introduction, we definitively restrict ourself to the subcritical caseM <8π and we focus on the long time asymptotic of the solutions. It has been proved in [8, Theorem 1.2]
that the solution given by Theorem 1.2 satisfies
(1.20) g(t, .)→G in L1(R2) as t→ ∞, where Gis a solution to the rescaled stationary problem
∆G+∇(Gx−G∇U) = 0 in R2, (1.21)
0≤G, Z
R2
G dx=M, U =−K ∗G.
Moreover, the uniqueness of the solutionGto (1.21) has been proved in [8, 5], see also [15, 16, 17], so thatG=GM stands for the unique self-similar profile with same mass M asf0 and it is given
in implicit form by
(1.22) G=M e−G∗κ−|x|2/2
R
R2e−G∗κ−|x|2/2dx and that U =−G∗κsatisfies
(1.23) ∆U + M
R
R2eU−|x|2/2dxeU−|x|2/2= 0.
Our third main result is about the convergence to self-similarity.
Theorem 1.5. For any M ∈(0,8π), and any finite real numbers F0, M4,0, there exists a (non explicit) constant C such that for any initial datum f0 satisfying (1.3)with
M0(f0) =M, M4(f0)≤M4,0, F(f0)≤ F0,
the associated solution in self-similar variables g defined by (1.16)-(1.1)satisfies the optimal rate convergence
kg(t, .)−GkL4/3 ≤C e−t ∀t≥1, where Gstands for the self-similar profile with same massM asf0.
Let us emphasize that the strong assumption M4(f0) < ∞ can be weaken. For instance, assuming only Mk0(f0) < ∞ for some k0 > 3, the same proof leads to the same optimal rate, and with the sole assumption M2(f0)<∞, one can get a not optimal rate of convergence to the self-similar profile in the sense thatkg(t, .)−GkL4/3 ≤Cηe−ηtfor allt≥1 and for someη∈(0,1), Cη ∈(0,∞).
For some particular class of initial data (essentially for an initial datumf0which is close enough to the self-similar profile G in the sense of the strongly confining norm L2(eν|x|2dx), ν > 0) it has been proved that the associated solution converges with exponential rate when M is small enough in [7] and for any M ∈(0,8π) in [16, 17]. In these last works, the linear stability of the linearized rescaled equation around the self-similar profile is established and that is the cornerstone of these nonlinear stability results. Our proof follows a strategy of “enlarging the functional space of semigroup spectral gap” initiated in [32] for studying long time convergence to the equilibrium for the homogeneous Boltzmann equation, and then developed in [30, 25, 12, 11, 29] (see also [31]) in the framework of kinetic equations and growth-fragmentation equations. More precisely, taking advantage of the linear stability of the linearized rescaled equation established in [17] in the small space L2(eν|x|2dx) we prove that the same result holds in the more larger spaceL4/3k , k > 3/2.
Then gathering the long time convergence (without rate) to self-similarity (1.20) with the estimates of Theorem 1.4, we get that any solution reaches a small L4/3k -neighborhood of G in finite time and we conclude to Theorem 1.5 by nonlinear stability inL4/3k ∩L14.
Let us end the introduction by describing the plan of the paper. In Section 2 we present some functional inequalities which will be useful in the sequel of the paper, we establish several a posteriori bounds satisfied by any weak solution, and we prove Theorem 1.4. Section 3 is dedicated to the proof of the uniqueness result stated in Theorem 1.3. In Section 4 we prove the long time behaviour result as stated in Theorem 1.5.
Acknowledgments.The authors gratefully acknowledge the support of the MADCOF ANR project (ANR-08-BLAN-0220). E.G. would like to thank the CEREMADE at Universit´e Paris- Dauphine for its kind hospitality in 2012 and 2013 where the work has been initiated and mostly written. He is also grateful to the MADCOF ANR project for the two several months grants that it provided to him. S.M. would like to thank the mathematics department of the Universidad de La Habana for its hospitality in summer 2013 where the work has been concluded.
2. A posteriori estimates - Proof of Theorem 1.4
We start by presenting some elementary functional inequalities which will be of main importance in the sequel. The two first estimates are picked up from [23, Lemma 3.2] but are probably classical and the third one is a variant of the Gagliardo-Niremberg-Sobolev inequality.
Lemma 2.1. For any0≤f ∈L1(R2)with finite mass M and finite Fisher information I=I(f) :=
Z
R2
|∇f|2 f , there holds
∀p∈[1,∞), kfkLp(R2)≤CpM1/pI(f)1−1/p, (2.1)
∀q∈[1,2), k∇fkLq(R2)≤CqM1/q−1/2I(f)3/2−1/q. (2.2)
For any 0≤f ∈L1(R2)with finite massM, there holds
∀p∈[2,∞) kfkLp+1(R2)≤CpM1/(p+1)k∇(fp/2)k2/(p+1)L2 . (2.3)
For the sake of completeness we give the proof below.
Proof of Lemma 2.1. We start with (2.2). Letq∈[1,2) and use the H¨older inequality:
k∇fkqLq = Z
∇f√ f
q
fq/2≤
Z |∇f|2 f
q/2Z
fq/(2−q)
(2−q)/2
=I(f)q/2kfkq/2Lq/(2−q). Denoting by q∗ = 2q/(2−q) ∈ [2,∞) the Sobolev exponent associated to q in dimension 2, we have, thanks to a standard interpolation inequality and to the Sobolev inequality,
kfkLq/(2−q) = kfkLq∗/2≤ kfk1/(qL1 ∗−1)kfk(qLq∗∗−2)/(q∗−1)
(2.4)
≤ Cqkfk1/(qL1 ∗−1)k∇fk(qLq∗−2)/(q∗−1). Gathering these two inequalities, it comes
k∇fkLq ≤CqI(f)1/2kfk1/(2(qL1 ∗−1))k∇fk(qLq∗−2)/(2(q∗−1)), from which we deduce (2.2).
We now establish (2.1). Forp∈(1,∞), writep=q∗/2 =q/(2−q) withq:= 2p/(1 +p)∈[1,2) and use (2.4) and (2.2):
kfkLp ≤ Cpkfk
q∗ −11 +q∗ −2q∗ −1(1q−12)
L1 I(f)q∗ −2q∗ −1(32−1q), from which one easily concludes.
We verify (2.3). From the Sobolev inequality and the Cauchy-Schwarz inequality, we have kw2(1+1/p)kL1(R2) = kw1+1/pk2L2(R2)≤ k∇(w1+1/p)k2L1(R2)
≤ (1 + 1/p)2kw1/pk2L2k∇wk2L2(R2)
(2.5)
and we conclude to (2.3) by taking w:=fp/2.
The proof of (1.19) in Theorem 1.4 is split in several steps that we present as some intermediate autonomous a posteriori bounds.
Lemma 2.2. For any weak solution f and any finale time T ∈ (0, T∗) there exists a constant C:=C(M,AT(f))such that
(2.6) 1
2 Z T
0
I(f(t))dt≤C.
In particular, in the subcritical case M < 8π the constant C only depends on M, H0, M2,0 and T ∈(0,∞).
Proof of Lemma 2.2. We write DF(f) =
Z
f|∇(logf + ¯κ)|2
≥ Z
f|∇logf|2+ 2 Z
∇f· ∇¯κ=I(f)−2 Z
f2.
On the other hand, for any A >1, using the Cauchy-Schwarz inequality and the inequality (2.1) forp= 3, we have
Z
f21f≥A ≤ Z
f1f≥A
1/2Z
f31/2
≤ Z
f (logf)+
logA 1/2
C33M I(f)21/2
,
from what we deduce for A=A(M,H+(f)) large enough, and more precisely takingAsuch that logA= 16H+(f)C33M,
(2.7)
Z
f21f≥A≤C33/2M1/2H+(f)1/2
(logA)1/2I(f)≤1 4I(f).
Together with the first estimate, we find 1
2I(f) ≤ DF(f) + 2 Z
f21f≤A
≤ DF(f) + 2M exp(16H+(f)C33M),
and we conclude thanks to (1.12) in the general case and thanks to (1.4)–(1.11) in the subcritical
caseM <8π.
Remark 2.3. As we have already mentioned we are not able to use the logarithmic Hardy- Littlewood Sobolev inequality (1.7) in the critical and supercritical cases. However, introducing the Maxwell function M :=M(2π)−1exp(−|x|2/2)of massM and the relative entropy
H(h|M) :=
Z
R2
(hlog(h/M)−h+M)dx
one classically shows that any solution f to the Keller-Segel equation (1.1)formally satisfies d
dtH(f(t)|M) = −I(f(t)) + Z
f(t)2+ 2M
≤ −I(f(t)) +M A+C33/2M1/2H+(f(t))1/2
(logA)1/2 I(f(t)) + 2M (∀A >0)
= −I(f(t)) +M exp 4C33MH+(f(t))) + 2M
= −I(f(t)) +M exp
C6H(f(t)|M) + 2M,
for a constant C6 = C6(M), where we have used (2.7), we have made the choice logA :=
4C33MH+(f(t)) and we have used a variant of inequality (1.9). This differential inequality pro- vides a local a priori estimate on the relative entropy which is the key estimate in order to prove local existence result for supercritical mass as well as global existence result for critical mass in[6].
As an immediate consequence of Lemmas 2.1 and 2.2, we have Lemma 2.4. For anyT ∈(0, T∗), any weak solutionf satisfies
f ∈Lp/(p−1)(0, T;Lp(R2)), ∀p∈(1,∞), (2.8)
K ∈¯ Lp/(p−1)(0, T;L2p/(2−p)(R2)), ∀p∈(1,2), (2.9)
∇xK ∈¯ Lp/(p−1)(0, T;Lp(R2)), ∀p∈(2,∞).
(2.10)
Proof of Lemma 2.4. The bound (2.8) is a direct consequence of (2.6) and (2.1). The bound (2.9) then follows from the definition ofK, the Hardy-Littlewood-Sobolev inequality (see e.g. [28, Theorem 4.3])
(2.11)
1
|z|∗f
L2r/(2−r)(R2)
≤CrkfkLr(R2), ∀r∈(1,2), with r=pand (2.8). Finally, from (2.6) and (2.2) we have
∇f ∈L3q−22q (0, T;Lq(R2)), ∀q∈(1,2).
Applying the above Hardy-Littlewood-Sobolev inequality to∇xK¯ =K ∗(∇xf) withr=q, we get
∇xK ∈¯ L3q−22q (0, T;L2−q2q (R2)), ∀q∈(1,2),
which is nothing but (2.10).
Lemma 2.5. Any weak solutionf satisfies Z
R2
β(ft1)dx+ Z t1
t0
Z
R2
β00(fs)|∇fs|2dxds (2.12)
≤ Z
R2
β(ft0)dx+ Z t1
t0
Z
R2
(β0(fs)fs2−β(fs)fs)+dxds,
for any times 0 ≤ t0 ≤ t1 < T∗ and any renormalizing function β : R → R which is convex, piecewise of class C1 and such that
|β(u)| ≤C(1 +u(logu)+), (β0(u)u2−β(u)u)+≤C(1 +u2) ∀u∈R. Proof of Lemma 2.5. We write
∂tf −∆xf = ¯K · ∇xf+f2, and we split the proof into three steps.
Step 1. Continuity. Consider a mollifier sequence (ρn) onR2, that isρn(x) :=n2ρ(nx), 0≤ρ∈ D(R2),R
ρ= 1, and introduce the mollified functionftn :=ft∗xρn. Clearly,fn∈C([0, T), L1(R2)).
Using (2.8) and (2.10), a variant of the commutation Lemma [21, Lemma II.1 and Remark 4] tells us that
(2.13) ∂tfn−K · ∇¯ xfn−∆xfn=rn, with
rn:= (f2)∗ρn+ ( ¯K · ∇xf)∗ρn−K · ∇¯ xfn→f2 in L1(0, T;L1loc(R2)).
The important point here is that f2,|∇xK|¯ f ∈L1((0, T)×R2), thanks to (2.10) and (2.8).
As a consequence, the chain rule applied to the smooth functionfn reads (2.14) ∂tβ(fn) = ¯K · ∇xβ(fn) + ∆xβ(fn)−β00(fn)|∇xfn|2+β0(fn)rn,
for any β ∈ C1(R)∩Wloc2,∞(R) such that β00 is piecewise continuous and vanishes outside of a compact set. Because the equation (2.13) with ¯K fixed is linear, the difference fn,k :=fn−fk satisfies (2.13) with rn replaced by rn,k :=rn−rk →0 in L1(0, T;L1loc(R2) and then also (2.14) (with again fn and rn changed infn,k andrn,k). In that last equation, we choose β(s) =β1(s) whereβA(s) =s2/2 for|s| ≤A,βA(s) =A|s|−A2/2 for|s| ≥Aand we obtain for any non-negative functionχ∈Cc2(Rd),
Z
R2
β1(fn,k(t, x))χ(x)dx≤
≤ Z
R2
β1(fn,k(0, x))χ(x)dx+ Z t
0
Z
R2
|rn,k(s, x)|χ(x)dxds +
Z t
0
Z
R2
β1(fn,k(s, x))
−f χ+ ∆χ(x)−K(s, x)¯ · ∇χ(x) dxds,
where we have used that divxK¯ =f, that |β01| ≤ 1 and that β100 ≥0. Because f0 ∈L1, we have fn,k(0) → 0 in L1(R2), and we deduce from the previous inequality, the convergence rn,k → 0 in L1(0, T;L1loc(R2)), the convergence β1(fn,k) ¯K → 0 in L1(0, T;L1loc(R2)) (because β1(s)≤ |s|, because fn,k→0 inL3(0, T, L3/2(R2)) by (2.8) withp= 3/2 and because ¯K ∈L6(0, T;L3(R2))⊂ L3/2(0, T;L3(R2)) by (2.9) withp= 6/5) and the convergence β1(fn,k)f →0 inL1(0, T;L1(R2)) that
sup
t∈[0,T]
Z
R2
β1(fn,k(t, x))χ(x)dx −→
n,k→∞0.
Since χ is arbitrary, we deduce that there exists ¯f ∈ C([0,∞);L1loc(R2)) so that fn → f¯ in C([0, T];L1loc(R2)),∀T >0. Together with the convergencefn→f in C([0,∞);D0(R2)) and the a priori bound (1.10), we deduce thatf = ¯f and
(2.15) fn→f in C([0, T];L1(R2)), ∀T >0.
Step 2. Linear estimates. We come back to (2.14), which implies, for all 0 ≤ t0 < t1, all χ∈Cc2(R2),
Z
R2
β(ftn
1)χ dx+ Z t1
t0
Z
R2
β00(fsn)|∇xfsn|2χ dxds= Z
R2
β(ftn
0)χ dx (2.16)
+ Z t1
t0
Z
R2
nβ0(fsn)rnχ+β(fsn) ∆χ−β(fsn) divx( ¯Kχ)o dxds.
Choosing 0 ≤χ ∈Cc2(R2) and β ∈C1(R)∩Wloc2,∞(R) such thatβ00 is non-negative and vanishes outside of a compact set, and passing to the limit asn→ ∞, we get
Z
R2
β(ft1)χ dx+ Z t1
t0
Z
R2
β00(fs)|∇xfs|2χ dxds≤ Z
R2
β(ft0)χ dx (2.17)
+ Z t1
t0
Z
R2
nh
β0(f)f2−β(f)fi
χ+β(f)h
∆χ−K · ∇χ¯ io dxds.
By approximating χ ≡ 1 by the sequence (χR) with χR(x) = χ(x/R), 0 ≤ χ ∈ D(R2), we see that the last term in (2.17) vanishes and we get (2.12) in the limitR→ ∞ for any renormalizing functionβ with linear growth at infinity.
Step 3. superlinear estimates. Finally, for anyβsatisfying the growth condition as in the statement of the Lemma, we just approximateβby an increasing sequence of smooth renormalizing functions βR with linear growth at infinity, and pass to the limit in (2.12) in order to conclude.
Lemma 2.6. For any weak solutionf, any timeT ∈(0, T∗)and anyp≥2, there exists a constant C:=C(M,AT, T, p)such that for any0≤t0< t2≤T
(2.18) kf(t1)kpLp+1 2
Z t1
t0
k∇x(fp/2)k2L2dt≤ kf(t0)kpLp+C.
Proof of Lemma 2.6. We define the renormalizing functionβK :R+→R+, K≥2, by βK(u) :=up if u≤K, βK(u) := Kp−1
logKulogu if u≥K, so thatβK is convex and piecewise of classC1, and moreover there holds
βK0 (u)u2−βK(u)u= (p−1)up+11u<K+Kp−1
logKu21u>K, and
β00K(u) =p(p−1)up−21u<K+Kp−1 logK
1 u1u>K.
Thanks to Lemma 2.5, we may write Z
R2
βK(ft1)dx+ 4 p0
Z t1
t0
Z
R2
|∇x(fp/2)|21f≤Kdxds+Kp−1 logK
Z t1
t0
Z
R2
|∇xf|2
f 1f≥Kdxds
≤ Z
R2
βK(ft0)dx+ (p−1) Z t1
t0
Z
R2
fp+11f≤Kdxds+Kp−1 logK
Z t1
t0
Z
R2
f21f≥Kdxds,
where p0 ∈ [1,∞] stands for the conjugate exponent associated to p ∈ [1,∞] and defined by 1/p+ 1/p0= 1.
On the one hand, using the splitting f = (f∧A) + (f−A)+, we have T1 := (p−1)
Z t1
t0
Z
R2
fp+11f≤Kdxds
≤ (p−1) 2pApM T + (p−1) 2p Z t1
t0
Z
R2
fA,Kp+1dxds,
where we have definedfA,K:= min((f−A)+, K−A),K > A >0. Moreover, thanks to inequality (2.3) and the same trick as in the proof of Lemma 2.2, we have
Z
R2
fA,Kp+1dx ≤ Cp
Z
R2
fA,Kdx Z
R2
|∇(fA,Kp/2)|2dx
≤ Cp
H+(f) logA
Z
R2
|∇x(fp/2)|21f≤Kdx.
As a consequence, we obtain
T1≤(p−1) 2pApM T + 1 p0
Z t1
t0
Z
R2
|∇x(fp/2)|21f≤Kdxds, forA=A(p,AT)>1 large enough.
On the other hand, thanks to the Sobolev inequality (line 2) and the Cauchy-Schwarz inequality (line 3), we have
T2 := Kp−1 logK
Z
R2
f21f≥Kdx≤4Kp−1 logK
Z
R2
(f−K/2)2+dx
≤ 4Kp−1 logK
Z
R2
|∇(f−K/2)+|dx2
= 4Kp−1 logK
Z
R2
|∇f|1f≥K/2dx2
≤ 4Kp−1 logK
Z
R2
|∇f|2
f 1f≥K/2dx Z
f1f≥K/2dx
≤ 4Kp−1 logK
n4 p2
Z
R2
|∇(fp/2)|2 2 K
p−1
1f≤K+ Z
R2
|∇f|2 f 1f≥K
o H+(f) log(K/2)
≤ 1 p0
Z
R2
|∇(fp/2)|21f≤Kdx+1 2
Kp−1 logK
Z
R2
|∇f|2
f 1f≥Kdx, for any K≥K∗=K∗(p,AT)>max(A,2) large enough.
All together, we have proved that for some constantAandK∗ only depending onp,T and the initial datumf0, and for anyK≥K∗ there holds
Z
R2
βK(ft1)dx+ 2 p0
Z t1
t0
Z
R2
|∇x(fp/2)|21f≤Kdxds≤ Z
R2
βK(ft0)dx+ (p−1) 2pApM T.
We conclude to (2.18) by passing to the limitK→ ∞.
Lemma 2.7. Any weak solutionf is smooth, that is
f ∈Cb∞((ε, T)×R2), ∀ε, T, 0< ε < T < T∗, so that in particular it is a “classical solution” for positive time.
Proof of Lemma 2.7. For any timet0∈(0, T) and any exponentp∈(1,∞), there existst00∈(0, t0) such that f(t00)∈Lp(R2) thanks to (2.8), from what we deduce using (2.18) on the time interval (t00, T) that
(2.19) f ∈L∞(t0, T;Lp(R2)) and ∇xf ∈L2((t0, T)×R2).
Next, by writingK=K1|z|≤1+K1|z|≥1∈L3/2+L∞, it is easily checkedkK ∗fkL∞ ≤C(kfkL3+ kfkL1), and then ¯K ∈L∞(t0, T;L∞(R2)) because of (2.19) and (1.13). We thus have
(2.20) ∂tf+ ∆xf =f2+ ¯K · ∇xf ∈L2((t0, T)×R2), ∀t0>0,
so that the maximal regularity of the heat equation in L2-spaces (see Theorem X.11 stated in [9]
and the quoted reference) provides the bound
(2.21) f ∈L2(t0, T;H2(R2))∩L∞(t0, T;H1(R2)), ∀t0>0.
Thanks to (2.21), an interpolation inequality and the Sobolev inequality, we deduce that∇xf ∈ Lp((t0, T)×R2) for any 1 < p <∞, whence ¯K · ∇xf ∈ Lp((t0, T)×R2), for all t0 > 0. Then the maximal regularity of the heat equation inLp-spaces (see Theorem X.12 stated in [9] and the quoted references) provides the bound
(2.22) ∂tf,∇xf ∈Lp((t0, T)×R2), ∀t0>0
and then the Morrey inequality implies the Holderian regularity f ∈ C0,α((t0, T)×R2) for any 0 < α <1, and any t0 >0. Observing that the RHS term in (2.20) has then also an Holderian regularity, we deduce that
∂tf, ∂xf, ∂x2ixjf ∈Cb0,α((t0, T)×R2), ∀T, t0; 0< t0< T < T∗,
thanks to the classical Holderian regularity result for the heat equation (see Theorem X.13 stated in [9] and the quoted references). We conclude by (weakly) differentiating in time and space the equation (2.20), observing that the resulting RHS term is still a function with Holderian regularity,
applying again [9, Theorem X.13] and iterating the argument.
Proof of Theorem 1.4. We split the proof into seven steps.
Step 1. The regularity off has been yet established in Lemma 2.7.
Step 2. First, we claim that the free energy functionalF is lsc in the sense that for any bounded sequence (fn) of nonnegative functions ofL12(R2) with same massM <8πand such thatF(fn)≤A and fn* f in D0(R2), there holds
(2.23) 0≤f ∈L12(R2) and F(f)≤lim infF(fn).
The proof of (2.23) is classical (see [13, 14, 8]) and we just sketch it for the sake of completeness.
Because of (1.8) and (1.9), we have H+(fn) +M2(fn)≤AT for anyn≥1, and we may apply the Dunford-Pettis lemma which implies that fn* f in L1(R2) weak. Now, introducing the splitting F =Fε+Rε,Fε=H+Vε, with
Vε(g) := 1 2
Z Z
R2×R2
g(x)g(y)κ(x−y)1|x−y|≥ε, Rε(g) := 1
2 Z Z
R2×R2
g(x)g(y)κ(x−y)1|x−y|≤ε,
we clearly have that Fε(f)≤lim infFε(fn) becauseHis lsc andVεis continuous for theL1 weak convergence. On the other hand, using the convexity inequality uv≤ulogu+ev ∀u >0,v ∈R
and the elementary inequality (logu)−≤u−1/2 ∀u∈(0,1), we have forε∈(0,1) andλ >1
|Rε(g)| = 1 4π
Z Z
R2×R2
g(x)1g(x)≤λg(y) (log|x−y|)−1|x−y|≤ε + 1
4π Z Z
R2×R2
g(x)1g(x)≥λg(y) log(|x−y|−1)1|x−y|≤ε
≤ λ 4π
Z
R2
g(y)dy Z
|z|≤ε
(log|z|)−dz + 1
4π Z
R2
g(x)1g(x)≥λ Z
g(y) logg(y) +|x−y|−1 dy
≤ λ
3M ε3/2+ 1 4π
H+(g) logλ
H+(g) + 2πε ,
and we get that supn|Rε(fn)| →0 asε→0 from which we conclude thatF is lsc. Now, we easily deduce that the free energy identity (1.6) holds. Indeed, since f is smooth for positive time, for any fixedt∈(0, T∗) and any given sequence (tn) of positive real numbers which decreases to 0, we clearly have
F(f(tn)) =F(t) + Z t
tn
DF(f(s))ds.
Then, thanks to the Lebesgue convergence theorem, the lsc property of F and the fact that f(tn)* f0weakly inD0(R2), we deduce from the above free energy identity for positive time that
F(f0)≤lim inf
n→∞ F(f(tn))≤ lim
n→∞
F(t) + Z t
tn
DF(f(s))ds =F(t) + Z t
0
DF(f(s))ds.
Together with the reverse inequality (1.14) we conclude to (1.6).
Step 3. From now on, we assume thatM <8π is subcritical and we prove the uniform in time estimates (1.18) and (1.19). We start with the a priori additional moment estimate (1.18). Because we will show the uniqueness of solution without using that additional moment estimates, these ones are rigorously justified thanks to a standard approximation argument, see [8] for details. Denoting g the rescaled solution (1.16) and
Mk:=
Z
R2
g(x)|x|kdx
we compute with Φ(x) = |x|k, k ≥2, thanks to the antisymmetry of the kernel and the Holder inequality
d
dtMk = k2Mk−2−k Mk− 1 2π
Z
R2
Φ0(x)g(t, x) Z
R2
g(t, y) x−y
|x−y|2dydx
= k2Mk−2−k Mk
− 1 4π
Z
R2
Z
R2
g(t, y)g(t, x) (Φ0(x)−Φ0(y)) x−y
|x−y|2dydx
≤ k2M2/kMk1−2/k−k Mk, from which we easily conclude that (1.18) holds.
Step 4. Defining the rescaled free energy E(g) and the associated dissipativity of rescaled free energyD(g) by
E(g) :=
Z
g(1 + logg) +1 2 Z
g|x|2+ 1 4π
Z Z
g(x)g(y) log|x−y|dxdy DE(g) :=
Z
g|∇(logg+|x|2
2 +κ∗g)|2,
we have that any solutiong to the rescaled equation (1.17) satisfies
(2.24) d
dtE(g) +DE(g) = 0 on [0,∞).
On the one hand, as for (1.8), the following functional inequality (2.25)
Z
glogg+1 2 Z
g|x|2≤C3(M)E(g) +C4(M) ∀g∈L1+(R2) holds, and together with (1.9), we find
(2.26)
Z
g(logg)++1 4
Z
g|x|2≤C3(M)E(g) +C7 ∀g∈L1+(R2),
where C7 :=C4+C5. As a consequence of (2.24) and (2.26), we get the uniform in time upper bound on the rescaled free energy for the solutiong of (1.17)
(2.27) sup
t≥0
Z
gt(loggt)++1 4
Z
gt|x|2≤C3(M)E(f0) +C7(M).
Step 5. As in the proof of Lemma 2.6, we easily get that the rescalled solution gof the rescaled equation (1.17) satisfies for any p∈[2,∞)
d
dtkgkpLp+ 4
p0k∇(gp/2)k2L2 = 2 (p−1)kgkpLp+ (p−1)kgkp+1Lp+1
≤ 2 (p−1)M + 3(p−1)kgkp+1Lp+1.
Writing s = s∧A+ (s−A)+, so that sp+1 ≤2p+1(s∧A)p+1+ 2p+1(s−A)p+1+ , and using the Gagliardo-Nirenberg-Sobolev type inequality (2.5) in order to get
Z
(g−A)p+1+ ≤ Cp Z
|∇(g−A)p/2+ |2 Z
(g−A)+
≤ Cp Z
|∇(gp/2)|2H+(g) logA for any A >1, we deduce
d
dtkgkpLp+k∇(gp/2)k2L2 ≤ 2pM+ 3p2p+1ApM + 3p2p+1 Z
(g−A)p+1+
≤ C8(M, p, A) +CpH+(g)
logA k∇(gp/2)k2L2. TakingAlarge enough, we obtain
(2.28) d
dtkgkpLp+1
2k∇(gp/2)k2L2 ≤C9(M, p,E0).
Using the Nash inequality
kwk2L2(R2)≤CNkwkL1(R2)k∇wkL2(R2)
with w:=gp/2, we conclude with d
dtkgkpLp+ 1
CN2 kgk−pLp/2kgk2pLp ≤C9(M, p,E0).
Defining u(t) := kg(t)kpLp first with p = 2, so that kg(t)kp/2Lp/2 = M, we recognize the classical nonlinear ordinary differential inequality
u0+c u2≤C on (0,∞),
for some constantscandC (which only depend onM andE0) from which we deduce the bound (2.29) ∀ε >0∃ C=C(ε, c, C) sup
t≥ε
kg(t)kpLp≤ C,
with p = 2. In order to get the same uniform estimate (2.29) in all the Lebesgue spaces Lp, p ∈ (2,∞), we may proceed by iterating the same argument as above with the choice p = 2k, k ∈ N∗. Coming back to (2.28) with p = 2, we also deduce that for any ε, T > 0 there exists C=C(ε, T,E0) so that
sup
t0≥ε
Z t0+T
t0
k∇g(s)k2L2(R2)ds≤ C.
Step 6. The function gi:=∂xig satisfies
∂tgi−∆gi− ∇(xgi) =gi+ 2g gi−∂xi(∇u· ∇g), from which we deduce that
d dt
Z
|gi|p+p(p−1) Z
|∇gi|2|gi|p−2≤ (2.30)
≤(3p−2) Z
|gi|p+ 2p Z
g|gi|p+p Z
∂xi(∇u· ∇g)gi|gi|p−2. Forp= 2, we have for anyt≥ε
T(t) := 4 Z
g|gi|2+ 2 Z
∂xi(∇u· ∇g)gi
≤ 4kgkL3kgik2L3+ 2k∇u· ∇gk2L2+1
2k∂igik2L2
thanks to the Holder inequality, an integration by part and the Young inequality. Next, we have for any t≥ε
T(t) ≤ C1kgik4/3L2 k∇gik2/3L2 +C2k∇gk2L2+1
2k∇gik2L2
where we have used the classical Gagliardo-Niremberg inequality (see (85) in [9, Chapter IX] and the quoted references)
(2.31) kwkLr(R2)≤CGNkwk1−aLq(R2)k∇wkaL2(R2), a= 1−q
r, 1≤q≤r <∞,
with w := gi, r = 3, q = 2, the uniform bound established in step 5 and the fact that ∇u =
−K ∗g ∈L∞((ε,∞)×R2) thanks to the same argument as in the proof of Lemma 2.7. Last, by the Young inequality we get for anyt≥ε
T(t) ≤ 2
3C13/2kgik2L2+1
3k∇gik2L2+C2k∇gk2L2+1
2k∇gik2L2, from which we deduce from (2.30)
d dt
Z
|gi|2+ Z
|∇gi|2≤C3k∇gk2L2 on (ε,∞),
with C3 := 4 +23C13/2+C2. Remarking that for any fixed ε ∈ (0,1) and any t1 ≥2ε, we may define t0∈(t1−ε, t1) so that
k∇g(t0)k2L2 = inf
(t1−ε,t1)
k∇gk2L2 ≤2 ε
Z t1
t1−ε
k∇g(s)k2L2ds≤C4
thanks to the bound established at the end of step 5, we deduce from the above differential inequality that
kgi(t1)k2L2 ≤ kgi(t0)k2L2+C3
Z t1
t0
k∇g(s)k2L2ds≤C5,
where again C5 := C4+C3C4ε/2 only depends on ε, M and E0. Coming back to the above differential inequality again, we easily conclude that for any ε > 0, there exists a constant Cε = C(ε, M,E0) so that
(2.32) sup
t≥ε
nk∇g(t)k2L2+ Z t+1
t
kD2g(s)k2L2
o≤ Cε.
