A Note onL

Keller-Segel Model

Jian-Guo Liu¹·Jinhuan Wang^1,2

Received: 21 August 2014 / Accepted: 1 June 2015 / Published online: 10 June 2015

© Springer Science+Business Media Dordrecht 2015

Abstract In this note we establish the uniformL^∞-bound for the weak solutions to a de- generate Keller-Segel equation with the diffusion exponent _n+2²ⁿ < m <2−_n²under a sharp condition on the initial data for the global existence. As a consequence, the uniqueness of the weak solutions is also proved.

Keywords Displacement convexity·Log-Lipschitz·Yudovich’s type theorem·Stability in Wasserstein metric

1 Introduction

In this note, we consider the following degenerate Keller-Segel equations

⎧⎪

⎨

⎪⎩

ρt=ρ^m−div(ρ∇c), x∈Rⁿ, t≥0,

−c=ρ , x∈Rⁿ, t≥0, ρ(x,0)=ρ0(x), x∈Rⁿ,

(1.1)

with the diffusion exponentm∈(_n²ⁿ₊₂,2−2/n),n≥3, and the initial dataρ0∈L¹₊(Rⁿ)∩ L^∞(Rⁿ). Hereρ(x, t ) represents the bacteria density andc(x, t ) represents the chemical

The work of J.-G. Liu was partially supported by KI-Net NSF RNMS grant No. 1107291 and NSF grant DMS 1514826. Jinhuan Wang is partially supported by National Natural Science Foundation of China (Grant number: 11301243).

B

jhwang@math.tsinghua.edu.cn J.-G. Liu

jliu@phy.duke.edu

1 School of Mathematics, Liaoning University, Shenyang 110036, P.R. China

2 Department of Physics and Department of Mathematics, Duke University, Durham, NC 27708, USA

substance concentration and it is given by

c= 1

n(n−2)α(n)

Rⁿ

1

|x−y|ⁿ⁻²ρ(y)dy, (1.2)

whereα(n)is the volume of then-dimension unit ball.

Ifρ∈L¹(Rⁿ)∩L^m(Rⁿ), then the associated free energy of the model (1.1) can be written as

F(ρ)= 1 m−1

Rⁿρ^m(x, t )dx−1 2

Rⁿρ(x, t )c(x, t )dx. (1.3) In fact, the Hardy-Letterwood-Sobolev inequality and (1.2) imply that the second term on the right side of (1.3) is bounded. And formally the following entropy-dissipation equality holds

d dt^F

ρ(·, t ) +

Rⁿρ ∇

m

m−1ρ^m−1−c

²dx=0. (1.4)

The model (1.1) can be written in the gradient flow form:

ρt+div(ρv)=0, (1.5)

where the drift velocityv= −∇μ,μis the chemical potential giving by μ=δF

δρ = m

m−1ρ^m−1−c.

The entropy-dissipation equality (1.4) implies that the velocity fieldvsatisfies the estimate T

0

v²_L2(Rⁿ,ρdx;Rⁿ)dt= T

0

Rⁿ

∇ m

m−1ρ^m−1−c

²ρdxdt <+∞, (1.6) if the initial free energyF(ρ0)is finite.

There are many results on the existence and the blow-up for the degenerate Keller-Segel models in literatures, c.f. [3,4,8,20,21]. For the degenerate system, there exist two im- portant diffusion exponents. One ism=m^∗=2−2/nand it produces an exact balance between the diffusion and the nonlocal aggregation in the mass invariant scaling. This expo- nent is usually referred to as the critical exponent and it is connected to the Fujita exponent.

The global existence for small initial data, and blow-up behavior for large initial data to the parabolic-elliptic model (1.1) withm >1 were established by Sugiyama and her collabora- tors in [20,21]. The global existence to the parabolic-parabolic system withm=m^∗ was given by [5] under a sharp condition on the initial data. Another important diffusion expo- nent ism=mc=_n+²ⁿ₂and it was referred to as the energy critical in [4] and it is the critical exponent for the existence of positive stationary solution to the associated Lane-Emden equation. At this exponent the free energy is conformal invariant and the steady solution is unstable [8]. Some results on the global existence and the blow-up at this exponent under a sharp initial condition was established in [8].

For the model (1.1) withmc< m < m^∗, the following exact criteria of the initial data on the global existence and the blow-up in a finite time was established in [9].

Assumption 1

ρ0∈L¹₊ Rⁿ

∩L^m Rⁿ

, F(ρ0) <F^∗, (1.7) whereF^∗is given by

F^∗= 2−²_n−m (m−1)(1−²_n)

2n²α(n) C(n)

^n(m−1)

2n−2−mn

M

2n−m(n+2) 2n−2−mn

0 >0, (1.8)

M0 is the initial massρ0L¹(Rⁿ)and C(n)is the best constant of the Hardy-Littlewood- Sobolev inequality, see [15, pp. 106].

Now we recall the results of the paper [9], which is helpful for proving our main results.

Proposition 1.1 Assume the initial densityρ0satisfies Assumption1. Let

s^∗=

2n²α(n)M

2n−m(n+2) n−2 0

C(n)

^n(m−1)

2n−2−mn

>0. (1.9)

Then the following statements are true (1) (global existence) if

ρ0

Lⁿ²ⁿ+2(Rⁿ)<

s^{∗ n−2}

2n(m−1), (1.10)

then there exists a global weak solution to Eqs. (1.1) and it satisfies (i) there exists a constantμ1<1 such that

ρ(·, t )²ⁿ

n+2 <

μ1s^∗_2n(m−1)ⁿ⁻²

, for allt >0. (1.11) (ii) Time and space derivative regularities, for anyT >0 and for any 1< p <∞,

∇ρ^m+p2−1∈L²

0, T;L² Rⁿ

, (1.12)

∂tρ∈L²

0, T;W_loc^−1,s Rⁿ

, (1.13)

wheres=min{_m^2m₊₁,_nm+^nm(m+1)_(n−m)(m+1)}>1.

(2) (finite time blow-up) if

ρ0

L 2n n+2(Rⁿ)>

s^∗_2n(m−1)ⁿ⁻²

(1.14) andm2(0) <∞, then any weak solution must blow up in a finite time, i.e.∃T^∗>0 such thatρ(·, t )

L 2n

n+2 → ∞ast→T^∗.

Remark 1.1 One can see clearly from Proposition1.1that the exact criterion (s^∗)^{2n(mn−2−1)} given above is actually a relation betweenLⁿ²ⁿ⁺²norm and the total mass of the weak solution.

Moreover, the conditionF(ρ0) <F^∗exclude the caseρ0

Lⁿ²ⁿ+2(Rⁿ)=(s^∗)^{2n(mn−2−1)}.

A more general model is given by

⎧⎪

⎨

⎪⎩

ρ_t=ρ^m−div(ρ∇c), x∈Rⁿ, t≥0, αct−c+βc=ρ , x∈Rⁿ, t≥0, ρ(x,0)=ρ0(x), c(x,0)=c0(x) x∈Rⁿ,

(1.15)

whereα, β≥0, m >0.

Remark 1.2 Proposition 1.1shows that there exists a large mass global weak solution to Eqs. (1.1) withmc< m < m^∗. Recently, Bedrossian [2] proved that some large mass global weak solutions exist for the Keller-Segel equations (1.15) withm=m^∗,α=0 andβ >0.

The uniformL^∞-bounds for weak solutions were obtained by using a bootstrap iterative technique in [6]. This note will give a uniformL^∞-bound under Assumption1and the sharp condition (1.10). As a consequence, the uniqueness will be proved in Theorem1.1.

The uniqueness of the Keller-Segel model (1.15) has been concerned by many scholars recently. The uniqueness of weak solutions to the model (1.15) withm=1 can be obtained easily from an energy method for the caseα=0, β >0. While for the high dimensional degenerate Keller-Segel model (1.15) with m=1, it seems that the uniqueness can not be obtained directly from standard energy estimates or comparison methods in the solution spaceL^∞([0, T );L¹∪L^∞(Rⁿ)). However, with some additional regularities, there are some results on the uniqueness given by using the classical PDE theory. For example, Sugiyama [19] proved the uniqueness for 1-D Keller-Segel model (1.15) withm >1, α=0, β >0, and additional assumption on the regularities∂tρ∈L¹_loc(R×(0, T )), ∂xρ∈L¹(R×(0, T )).

Miura-Sugiyama in [17] and Kagei-Kawakami-Sugiyama in [18] respectively proved the uniqueness of weak solutions to the model (1.15) withα=1 or 0, with additional Hölder regularity by adapting the duality method coupled with the vanishing viscosity duality method. However, it is still an open problem for the Hölder regularity for the degenerate Keller-Segel equations.

Notice that there exists a weak comparison principle for the radial symmetric solution to the Keller-Segel system with dimensionn≥2 [8,13]. However, these systems don’t have the comparison principle in the classical sense due to the non-local aggregation with a log- Lip singular potential. This kind of log-Lip singularity also appeared in the two dimensional incompressible Euler equation and the uniqueness was proved by Yudovich in 1963 [22] in the class of the boundedL^∞solution.

Three methods were recently developed to adapt the Yudovich’s method to the Keller- Segel models. The first method is the optimal transport method, see Carrillo, Lisini and Mainini in [7], they proved the uniqueness in the class of bounded solution and bounded Fisher information. This method may also be suitable for other systems with the gradi- ent flow structure. The second method is the DiPerna-Lions’ renormalizing argument [10], which can give a uniqueness of the solution in theL^pspace. The method can be used to equa- tions with linear diffusion or the model with the semigroup structure. Recently, Egana and Mischler proved the uniqueness of the entropy weak solution for the two dimensional Keller- Segel equation [11] inL^∞([0, T );L¹(R²))∩C([0, T );D(R²)). However, this method can not be used for the degenerate Keller-Segel system. The final method is the Brownian mo- tion method. The uniqueness for Eqs. (1.15) withm=1, α=β=0 was also proved [16] by

using this method. All these three methods are based on Lagrangian coordinates. This note is based on the first method mentioned above and the main result is given below.

Define the space X2(M0):=

ρ∈L¹

Rⁿ :ρ≥0,

Rⁿρ(x)dx=M0,

Rⁿ|x|²ρ(x)dx <∞

with Wasserstein metricW2.

Theorem 1.1 (Space-Time UniformL^∞Estimate and Uniqueness) Assume that the initial density satisfies Assumption1and the sharp condition (1.10), then we have

(i) If the initial dataρ0∈L¹∩L^∞(Rⁿ), then the model (1.1) has a global weak solution with regularity,

ρ∈L^∞ R+;L¹₊

Rⁿ

∩L^∞ Rⁿ

.

Furthermore, ifρ0∈X2(M0), then for anyt >0,ρ∈X2(M0).

(ii) Ifρ0, u0∈X2(M0)∩L^∞(Rⁿ) andρ , uare two solutions as already obtained in (i) with initial dataρ0, u0, then for any fixedT >0, there exist positive constantsCandC¯ depending only onT,M0andL^∞(0, T;L¹∩L^∞(Rⁿ))norms ofρandusuch that for any 0< t < T the following inequality holds

W2

ρ(·, t ), u(·, t )

≤ ¯Cmax

W2(ρ0, u0),

W2(ρ0, u0)e^−Ct ,

which implies that the weak solutions to the model (1.1) are unique and stable with respective to the initial data.

This note is organized as follows. Section2gives the uniformL^∞-estimate of the weak solutions to the model (1.1) under Assumption1and the sharp condition (1.10). In Sec- tion3, some basic properties to the model (1.1) are given. Using these properties and the L^∞estimate for the weak solutions, we prove the uniqueness for the weak solutions to the degenerate Keller-Segel equations.

2 UniformL^∞Estimate for the Weak Solutions

In this section we will extend the result of Proposition1.1to a uniformL^∞bound by utiliz- ing a bootstrap iterative technique [6] under Assumption1, the sharp condition (1.10), and ρ0∈L^∞.

Lemma 2.1 (TheL^pk Estimate) Assume the initial densityρ0∈L^∞(Rⁿ), and satisfies As- sumption1and the sharp condition (1.10). Let(ρ , c)be a weak solution with initial data ρ0. Letpk=2^k+_n+²ⁿ₂+1 fork∈N. Then we have

d

dtρ^p_L^kpk≤ −ρ^p_L^kpk+Cp_k^q2

ρ^p_L^k−1_pk−1η₁

+

ρ^p_L^k−1_pk−1η₂

, k=1,2,· · ·. (2.1) whereCis a fixed constant independent ofpk, andη1, η2are constants satisfyingη1, η2≤2.

Proof Takingpkρ^pk−1as a test function in the first equation of (1.1), we have d

dt

Rⁿρ^pkdx= −4pkm(pk−1) (m+pk−1)²

Rⁿ|∇ρ^m+pk−12 |²dx+(pk−1)

Rⁿρ^pk+1dx

≤ −2C₁

Rⁿ|∇ρ^m+pk−12 |²dx+pk

Rⁿρ^pk+1dx,

where 0< C1≤^2p_(m+p^km(pk−1)

k−1)² is a fixed constant.

Now we will focus on estimating the last term

Rⁿρ^pk+1dx,

Rⁿρ^pk+1dx=ρ

m+pk−1 2 ^{m2(pk+1)+pk−1}

L 2(pk+1) m+pk−1

≤G^α

2(pk+1) m+pk−1

1 ∇ρ^{m+pk2−1αm2(pk+1)+pk−1}

L² ·ρ^m+pk2−1(1_Lr^{−α)m2(pk+1)+pk−1}, (2.2) whereG1=S⁻

α

n 2,Snis the best constant for the Sobolev inequality, and m+pk−1

2 r=p_k−1, α=

1

r−^m+p_2(pk+1)^k−1

1

r−ⁿ⁻²_2n , 1−α=

m+pk−1 2(pk+1) −ⁿ⁻²_2n

1

r−ⁿ⁻²_2n . (2.3) The Young’s inequality implies that

d dt

Rⁿ

ρ^pkdx≤ −2C1

Rⁿ|∇ρ^m+pk−12 |²dx+σ1∇ρ^{m+pk−12 q1αm+2(pk+1)pk−1}

L²

+C(σ1)(pk)^q2G^αq2

2(pk+1) m+pk−1

1 ρ

m+pk−1

2 ^q_L²r^{(1−α)m2(pk+1)+pk−1}, (2.4) whereC(σ1)=(σ1q1)^−q2/q1q₂⁻¹,q1=^m+p_α(pk^k₊⁻¹₁₎, i.e.,q1α_m^2(p_+p^k+1)

k−1=2, and q2= m+pk−1

(m+pk−1)−α(pk+1)≤1+n. (2.5) Thus, takingσ1=C1in (2.4), we get

d dt

Rⁿρ^pkdx≤ −C1

Rⁿ|∇ρ^m+pk2−1|²dx+C(σ1)(pk)^q2G^αq2

2(pk+1) m+pk−1 1

ρ^p_L^kpk⁻¹−1

η₁

, (2.6) where

η1:=2(pk+1)q2(1−α)

r(m+pk−1) =m−2+²_n(pk+1)

m−2+²_np_k−1 ≤2. (2.7) On the other hand,

ρ^p_L^kpk =ρ

m+pk−1 2 ^m+pk−12pk

L 2pk m+pk−1

≤G

2pk m+pk−1

2 ∇ρ^{m+pk2−1θm+pk−12pk}

L² ρ^m+pk2−1(1_Lr^{−θ )m+pk−12pk} , (2.8)

whereG2=S⁻

θ

n 2,ris the same constant as one in (2.2), and θ=

1

r−^m+_2p^pk_k⁻¹

1

r −ⁿ_2n⁻² , 1−θ=

m+pk−1 2pk −ⁿ_2n⁻²

1

r−ⁿ_2n⁻² . Similar to (2.4), we have

ρ^p_L^kpk≤G^{θ 2}

2pk m+pk−1

2 C(σ¯ 1)ρ^p_L^k_pk−1^{((1−θ ))2}+σ1∇ρ^{m+pk−12 2}

L², (2.9)

whereC(σ¯ 1)=(σ11)^−2/1−₂¹,1=^m+p_θp^k_k⁻¹, and2=_(m+^m+p_pk−^k₁₎⁻₋¹_θpk. Hence from (2.6) and (2.9), we deduce

d dt

Rⁿρ^pkdx≤ −

Rⁿρ^pkdx+G^{θ 2}

2pk m+pk−1

2 C(σ¯ 1)ρ^p_L^kpk⁻¹−1^η2

+C(σ1)p_k^q2G^αq2

2(pk+1) m+pk−1

1 ρ^p_L^kpk⁻¹−1^η1. whereη2:=_[(m+p^(m+p_k^k₋^−1)p_1)−θp^{k(1−θ )}_k]pk−1=_m^m₋⁻₁₊¹⁺2^2npk

npk−1 ≤2. Define C1(pk):=C(σ1)G^αq2

2(pk+1) m+pk−1

1 , C2(pk):=G^{θ 2}

2pk m+pk−1 2 C(σ¯ 1).

A simple computation gives

C1(pk)→C(m, n, M0), aspk→ ∞. (2.10) In fact,

C1(p_k)=(m+pk−1)−α(pk+1) m+pk−1

C1

m+pk−1 α(pk+1)

_−(m+ ^α(pk+1)

pk−1)−α(pk+1)

S⁻

α(pk+1) (m+pk−1)−α(pk+1)

n .

Hereα∼O(1), 1−α∼O(1)aspk→ ∞, andSnis independent ofpk. So, we can get the limit relation (2.10).

HenceC1(pk)is uniformly bounded for anyk≥1. A similar discussion gives thatC2(pk) is also uniformly bounded. So, letC(n, m) >1 be a common upper bound ofC1(pk)and C2(pk), we obtain the following differential inequality

d

dtρ^p_L^kpk ≤ −ρ^p_L^kpk+C(n, m)p_k^q2

ρ^p_L^k_pk−1⁻¹ η1+

ρ^p_L^k_pk−1⁻¹ η2

.

Proof of (i) of Theorem1.1 First, we need to estimate y0(t ):=

Rⁿρ^2+n2n+2(x, t )dx=

Rⁿρ^p0(x, t )dx.

Similar to the proof obtaining (2.1), we have d

dtρ^p_L^0p0≤ −ρ^p_L^0p0+Cp ^q₀² ρⁿ⁺²²ⁿ

L 2n n+2

η₁

+ ρⁿ⁺²²ⁿ

L 2n n+2

η₂ ,

whereC,q2andη1,η2are constants independent ofk. So, by using the uniform upper bound (1.11) ofρ

Lⁿ²ⁿ+2, in Proposition1.1, we deduce

y0(t )= ρ^p_L^0p0≤C(n, m, M0). (2.11) Letyk(t ):= ρ(·, t )^p_L^kpk, solving the differential inequality (2.1), we obtain

e^tyk(t )

≤C(m, n)p^q_k²

y_k^η₋²₁+y_k^η₋¹₁ e^t

≤2C(m, n)4¹⁺ⁿ2^(n+1)kmax

1,sup

t≥0y_k−1² (t )

e^t, (2.12)

where the last inequality used 1< q2≤n+1, see (2.5). Letak:=2C(m, n)4¹⁺ⁿ2^(n+1)k>1, K=max{yk(0),1}for allk≥1. Integrating (2.12), it holds that

yk(t )≤akmax

1,sup

t≥0

y_k−1² (t ) 1−e^−t

+yk(0)e^−t

≤akmax

1,sup

t≥0

y_k−1² (t ), yk(0)

≤akmax

supt≥0y_k−² ₁(t ), K

, (2.13)

From (2.13) after some iterative steps, we have yk(t )≤ak(a_k−1)²(a_k−2)²²· · ·(a1)^2k−1max

supt≥0y₀^2k(t ), K^2k−1

≤

2C(m, n)2^k−1

4ⁿ⁺¹2^k−1

2ⁿ⁺¹_k+2(k−1)+2²(k−2)+···+2^k−1(k−(k−1))

×max

sup

t≥0y₀^2k(t ), K^2k−1

=

2C(m, n)4ⁿ⁺¹2^k−1

2ⁿ⁺¹2·2^k−k−2

max

sup

t≥0y₀^2k(t ), K^2k−1

,

Taking the power 1/p_kto the above inequality, then ρL^pk ≤2C(m, n)4ⁿ⁺¹2²⁽ⁿ⁺¹⁾max

sup

t≥0y0(t ), K

. (2.14)

Thus, (2.11) and (2.14) imply thatρL^∞is uniformly bounded.

Now we will prove ifρ0∈X2(M0), then for anyt >0,ρ∈X2(M0).

In fact, from [9], there exists a global weak solution to the model (1.1) under Assump- tion1and the sharp condition (1.10). Using a similar proof to the step 7 and the step 11 of [4, Theorem 2.11], we can get the weak solutionρsatisfies mass conservation, and it’s second moment satisfies the following equation:

dm2(t )

dt =

2n−2(n−2)

m−1 _Rⁿρ^mdx+2(n−2)F(ρ).

Noticing that the conditionm <2−2/nimplies the coefficient 2n−²⁽ⁿ⁻²⁾_m−1 <0, we can get that the second moment is bounded for any fixedT. So, the weak solutionρbelongs to the

spaceX2(M0).

3 Stability and Uniqueness of the Weak Solutions

In this section, we use the result on L^∞-bound of the weak solutions ρ obtained in the previous section to prove the uniqueness of the weak solution to the model (1.1) under Assumption1 and the sharp condition (1.10). Moreover, the stability in the Wasserstein metric is obtained by using the method given in [7]. In fact, all the conditions proposed in [7, Definition 1.1] are true for our model (1.1). For completeness, we will provide the details of the stability proof.

3.1 Preliminaries

Letρ1, ρ2∈X2(M0). Define theW2-metric betweenρ1andρ2as W2(ρ1, ρ2)=

Rⁿ|x−τ (x)|²ρ1(x)dx 1/2

.

Hereτ is the unique optimal transport map betweenρ1 andρ2,τρ1=ρ2, i.e., for every continuous and bounded functionφ:Rⁿ→R, it holds

Rⁿφ (x)ρ2(x)dx=

Rⁿφ τ (x)

ρ1(x)dx.

Now we recall the following two important results for proving the uniqueness. One is the Log-Lipschitz property of∇c. It was proved by Kato [12] for the 2D case. Carrillo et al. in [7] stated that the generalization to high dimension is almost identical. We recall it in the following lemma.

Lemma 3.1 Assume thatρ∈L¹(Rⁿ)∩L^∞(Rⁿ)andcsatisfies (1.2), then we have

|∇c(x)− ∇c(y)| ≤C|x−y|

1+log⁻|x−y|

ρL¹∩L^∞, (3.1) and

|∇c(x)− ∇c(y)|²≤C²φ

|x−y|²

ρ²_L1∩L^∞, (3.2)

where

φ (r):=

rlog²r, r≤e^−1−√2, r+2(1+√

2)e^−1−√2, r > e^−1−√2 (3.3) is a continuous and differentiable concave function on(0,∞).

Following the paper [14, Theorem 2.9], we can get the second key result, which is the estimate on∇ ¯c− ∇cL²in terms of the Wasserstein metric.

Lemma 3.2 Letρ, ρ¯ ∈X2(M0), and(ρ,¯ c)¯ and(ρ , c)satisfy Eq. (1.2). Then

∇ ¯c− ∇cL²≤

max{ ¯ρL^∞,ρL^∞}1/2

W2(ρ, ρ).¯ (3.4) For completeness to the reader, we will outline proofs of Lemmas 3.1and 3.2in the Appendix.

3.2 Stability and Uniqueness

Proposition 3.1 (Stability in Wasserstein Metric) Letρbe a weak solution to the problem (1.1) with initial dataρ0∈X2(M0)∩L^∞(Rⁿ). Then the following properties hold

(i) For anyρ¯∈X2(M0)∩L^∞(Rⁿ), the mapt→W2(ρ(·, t ),ρ(·, t ))¯ is absolutely contin- uous and there exists a positive constantCdepending only onM0,ρ_∞, and ¯ρ_∞ such that for a.e.t∈(0,+∞), it holds

1 2

d

dtW₂²(ρ ,ρ)¯ ≤F(ρ)¯ −F(ρ)+Cω

W₂²(ρ ,ρ)¯

, (3.5)

whereφ (r)is defined in (3.3), and

ω(r)= M0rφ

M₀⁻¹r

, forr >0. (3.6)

(ii) Let u be another bounded weak solution with the initial condition u0∈X2(M0)∩ L^∞(Rⁿ). Then for any fixedT >0, there exists aε0depending only onT,M0,ρ_∞ and ¯ρ_∞such that

(a) ifW2(ρ0, u0) < ε0, we have W2

ρ(·, t ), u(·, t )

≤

max{M0,1}

W2(ρ0, u0)e^−Ct

for anyt < T; (3.7) (b) ifW2(ρ0, u0)≥ε0, then for anyt < T, the following relation holds

W2

ρ(·, t ), u(·, t )

≤ ¯Cmax

W2(ρ0, u0),

W2(ρ0, u0)e^−Ct

, (3.8)

whereCis the same with the case (i),C¯is a constant depending only onT,M0,ρ∞

and ¯ρ∞.

Remark 3.1 Notice that the results of Proposition 3.1were proved in [7] for the Keller- Segel models in any dimension both parabolic-elliptic and fully parabolic. Moreover, their techniques applied equally well for degenerate cases once theL^∞bounds are known. Here, we include a proof for the sake of completeness.

Proof Step 1. Absolutely continuous.

Sinceρ(x, t )is a weak solution to Eq. (1.5) and it has the time regularity (1.13), and the velocity fieldvsatisfies (1.6), then [1, Theorem 8.3.1] implies thatρ(·, t )is an absolutely continuous curve inX2(M0). So, from [1, Theorem 8.4.7], we have that for any fixedρ¯∈ X2(M0), the derivative ofW₂²(ρ(·, t ),ρ)¯ respect to time t exists almost everywhere and satisfies the following equation

1 2

d dtW₂²

ρ(·, t ),ρ¯

=

Rⁿv(x, t )·

x−τ1(x)

ρ(x, t )dx, for a.e.t∈ [0, T ), (3.9) whereτ1is the optimal map betweenρ(x, t )andρ(x), and¯ v= −∇(_m−1^m ρ^m−1−c). More- over, the Hölder inequality gives

_T

0

Rⁿv(x, t )·

x−τ1(x)

ρ(x, t )dxdt

≤ _T

0

Rⁿ|v(x, t )|²ρdx 1/2

Rⁿ|x−τ1(x)|²ρdx 1/2

dt.

From (1.6) with finite second moments ofρandρ, we know¯ _dt^dW₂²(ρ(·, t ),ρ)¯ ∈L¹([0, T )).

Then the fundamental theorem of Calculus for Lebesgue integrals implies thatW₂²(ρ(·, t ),ρ)¯ is absolutely continuous on[0, T ).

Step 2. Displacement convexity.

According to [1, Subsection.9.3], we know that the power functional 1

m−1

Rⁿρ^mdx, m≥1−1 n

has the displacement convexity property, and it is a lower semi-continuous functional. Thus, from [1, pp. 231, (10.1.7)], it holds

Rⁿρ¯^mdx−

Rⁿρ^mdx≥

Rⁿξ(x)·

τ1(x)−x

ρdx, (3.10)

whereξ(x)=m∇ρ^m−1∈L²(μ;Rⁿ) (it was given by (1.12) withp=m) belongs to the Fréchet sub-differential of

Rⁿρ^mdxatμ:=ρdx.

Hence, (3.9) and (3.10) imply that 1

2 d dtW₂²

ρ(·, t ),ρ¯

=

Rⁿ

m

m−1∇ρ^m−1− ∇c

·

τ1(x)−x

ρ(x, t )dx

≤ 1 m−1

Rⁿ

ρ¯^m−ρ^m dx−

Rⁿ∇c(x, t )·

τ1(x)−x

ρ(x, t )dx

=F(ρ)¯ −F(ρ)+1 2

Rⁿ

|∇ ¯c|²− |∇c|² dx

−

Rⁿ∇c(x, t )·

τ1(x)−x

ρ(x, t )dx. (3.11)

Notice that the following two equalities hold true (i) Identity

|∇ ¯c|²− |∇c|²= |∇ ¯c− ∇c|²+2∇c·(∇ ¯c− ∇c). (3.12) (ii) Sinceτ1is the optimal transport plan betweenρandρ, we can get¯

Rⁿ∇c·(∇ ¯c− ∇c)dx=

Rⁿ(cρ¯−cρ)dx

=

Rⁿ

c τ1(x), t

−c(x, t ) ρdx

=

Rⁿ

₁

0

d dsc

τ_s(x), t dsρdx

=

Rⁿ

1 0

∇c τs(x), t

·

τ1(x)−x

dsρdx. (3.13) So, (3.11), (3.12) and (3.13) give that

1 2

d dtW₂²

ρ(·, t ),ρ¯

≤F(ρ)¯ −F(ρ)+1 2

Rⁿ

|∇ ¯c− ∇c|² dx

+

Rⁿ

1 0

∇c τs(x), t

− ∇c(x, t )

·

τ1(x)−x

ρ(x, t )dsdx.

(3.14)

Step 3. Log-Lipschitz estimate.

Denote

I:=

Rⁿ

₁

0

∇c τs(x), t

− ∇c(x, t )

·

τ1(x)−x dsρdx.

Using Lemma3.1with the concavity and the increase property ofφgiven by (3.3), we have

|I| ≤W2(ρ ,ρ)¯ 1

0

Rⁿ|∇c τs(x), t

− ∇c(x, t )|²ρdx 1/2

ds

≤W2(ρ ,ρ)¯ ₁

0

RⁿC²φ

|τ_s(x)−x|² ρdx

1/2

ds

≤

M0CW2(ρ ,ρ)¯ ₁

0

φ

M₀⁻¹W₂²(ρ , ρs) ds

≤

M0CW2(ρ ,ρ)¯ 1

0

φ

M₀⁻¹W₂²(ρ ,ρ)¯ ds.

The last inequality is valid since for anys∈ [0,1], it holds W₂²(ρ , ρs)=s²W₂²(ρ ,ρ).¯ Thus by the definition of the functionω(x)in (3.6), we know that

|I| ≤Cω

W₂²(ρ ,ρ)¯

. (3.15)

So, from (3.14) together with (3.4) and (3.15), we obtain 1

2 d

dtW₂²(ρ ,ρ)¯ ≤F(ρ)¯ −F(ρ)+1

2max{ρL^∞, ¯ρL^∞}W₂²(ρ ,ρ)¯ +Cω

W₂²(ρ ,ρ)¯

≤F(ρ)¯ −F(ρ)+Cω

W₂²(ρ ,ρ)¯ ,

where the last inequality used the fact thatω(r)≥rfor anyr >0. Hence we obtain (3.5).

Step 4. Stability.

Using (3.5) and [1, Lemma 4.3.4], it holds for a.e.t∈(0, T ) 1

2 d dsW₂²

ρ(·, s), u(·, s)

s=t≤1 2

d dsW₂²

ρ(·, s), u(·, t )

s=t+1 2

d dsW₂²

ρ(·, t ), u(·, s)

s=t

≤2Cω W₂²

ρ(·, t ), u(·, t ) ,

i.e., d dtW₂²

ρ(·, t ), u(·, t )

≤4Cω W₂²

ρ(·, t ), u(·, t )

, for a.e.,t∈(0, T ), (3.16) whereConly depends onρ∞,u∞, and the initial mass.

First, we will prove the argument (a). Denote y(t ):=W₂²

ρ(·, t ), u(·, t ) ,

and let

M(s)=

M₀e^−1−√2 s

1 ω(r)dr,

whereω(r)= −rln_M^r

0. For 0≤s≤M0e^−1−√2, it holds M(s)=ln

ln s M0

−ln(1+√

2). (3.17)

Moreover, a simple computation with (3.16) gives d

dt^M y(t )

= − 1 ω(y(t ))

d

dty(t )≥ −4C, i.e.,

−M y(t )

+M y(0)

≤4Ct, for anyt∈ [0, T ). (3.18) On the other hand, choose 0< ε0<1 depending only onT andM0such that it satisfies

0< ε0<

M0e^−1−√2, (3.19)

and

0< ε0<

M0e^−1−√2 max{M0,1}

¹

2e^4CT

. (3.20)

We first consider the casey(0) < ε²₀. Sincey(t )is absolutely continuous int, andy(0) <

ε²₀, (3.19) implies thaty(t ) < M0e^−1−√2 fort1. AssumeT1 is the first time such that y(T1)=M0e^−1−√2, then for anyt∈ [0, T1], (3.17) and (3.18) give thaty(t )satisfies

y(t )≤M₀^1−e−4Cty(0)^e−4Ct ≤max{M0,1}y(0)^e−4Ct, for anyt∈ [0, T1]. (3.21) From (3.21), we know that

M0e⁻¹⁻

√2=y(T1)≤max{M0,1}y(0)^e−4CT1. That is

T1≥ − 1 4Cln

ln M0e^−1−√2 max{M0,1}/ln

y(0)

> T ,

where the last inequality used the condition (3.20) andy(0) < ε₀². Thus for any fixedT >0, we have

y(t )≤max{M0,1} y(0)e^−Ct

, for anyt∈ [0, T ), (3.22) This gives (3.7).

Now we prove the case y(0)≥ε₀². Ify(0)≥ε²₀, there exist two subcases: (i) for any t∈ [0, T ),y(t )≥ε²₀; (ii) there exists at0such thaty(t0) < ε₀².

For the subcase (i), by the definition ofω(r), we know that there exists aC(ε0)such that ω(r)≤C(ε0)rforr≥ε²₀. Hence using (3.16), we get

d

dty(t )≤4Cω y(t )

≤C(ε0)y(t ),