Uniqueness for Keller-Segel-type chemotaxis models

We prove uniqueness in the class of integrable and bounded nonnegative solutions in the energy sense to the Keller-Segel (KS) chemotaxis system. Our proof works for the fully parabolic KS model, it includes the classical parabolic-elliptic KS equation as a particular case, and it can be generalized to nonlinear diffusions in the particle density equation as long as the diffusion satisfies the classical McCann displacement convexity condition. The strategy uses Quasi-Lipschitz estimates for the chemoattractant equation and the above-the-tangent characterizations of displacement convexity. As a consequence, the displacement convexity of the free energy functional associated to the KS system is obtained from its evolution for bounded integrable initial data.

Here, n is the number/mass density of a bacteria/cell population and c represents the concentration of a chemical attractant that can suffer chemical degradation and that it is produced by the cells themselves due to chemotactic interaction. The parameters κ, χ, η, θ, γ might be suitable functions, assumed to be constant in this simplified model. We can perform a time scaling and a suitable change of variables, that is τ = κt, ρ(x, τ ) = θχ ηκ n(x, τ /κ), v(x, τ ) = χ κ c(x, τ /κ). The system is therefore reduced to ∂ t ρ = ∆ρ − div (ρ∇v), where α ≥ 0 and ε ≥ 0 are constants (α = γ/η, ε = κ/η). In case ε = 0, it restricts to the classical parabolic-elliptic Patlak-KS model For ε > 0, the natural free energy functional associated to the dynamics of the system (1.1) is In the case ε = 0, corresponding to (1.2), this Liapunov functional is at least formally equivalent to  Let us emphasize that the main properties we need to get uniqueness of solution are the boundedness of the densities and the Fisher information (1.5). They together imply that the velocity field of the continuity equation for the density ρ is a well defined object belonging to the right functional space, see section 2 for details. Moreover, the boundedness of the density implies that we have a uniform in bounded time intervals estimate on the quasi-Lipschitz constant of part of the velocity field. These are the basic properties that imply the uniqueness for bounded solutions. Let us finally mention that part of the strategy is related to the uniqueness of solutions to fluid and aggregation equations developed in [34,27,5,28,19,7,29]. The main novelty here is the interplay between the diffusive and the aggregation parts. The main results of this work are: Let ρ 1 , ρ 2 be two bounded solutions on [0, T ] × R d to the Cauchy problem associated to (1.2), with initial datum ρ 0 . Then ρ 1 = ρ 2 .
The proof of uniqueness as stated in Theorems 1.3 and 1.4 will be a consequence of a more general property: we will show that bounded solutions satisfies a strong gradient flow formulation by means of a family of evolution variational inequalities. This formulation is similar to the one for semi-convex functionals and implies a non-expansivity property of the distance between two solutions. This non-expansivity property yields uniqueness. All these results will be stated in Theorem 3.1. Moreover this formulation lead also to a relaxed convexity property of the energy functional as stated in Theorem 4.1.
There is a huge literature about the KS system and their variations, so we just restrict here to discuss the main results concerning about bounded solutions. In the classical parabolic-elliptic KS equation ε = α = 0 and d = 2, global in time bounded solutions in the subcritical case m < 8π have been obtained joining the results in [12,24,14]. Actually, the global existence of weak solutions satisfying all properties in Definition 1.1 except the L ∞ bound was obtained in [12] while L ∞ -bounds in bounded time intervals can be obtained from the results in [24,14]. The same techniques could eventually be used to get local in time bounded solutions for all masses, although such a result is not present in the literature. Let us also mention the recent preprint [17] in which the authors actually show that the L ∞ -norm of the solution decays in time like for the heat equation in the subcritical case m < 8π for more restricted initial data. L ∞ -apriori estimates were obtained in the classical parabolic-elliptic KS equation ε = 0 with d ≥ 2 and α ≥ 0 for small L d/2 initial data in [20,21]. These results together with similar arguments as in [12] to get the free energy dissipation property and thus the Fisher information bounds, could lead to the existence of bounded solutions in these cases. We emphasize that these L ∞ estimates show that the solution in bounded time intervals is bounded by a constant that depends only on the L ∞ -norm of the initial data, the initial free energy, and the final time. In particular, existence of bounded solutions is expected if ρ 0 ∈ L ∞ (R d ), and this explains the presence of such an assumption in the previous definitions.
Concerning the fully parabolic KS system, we find global in time solutions satisfying all properties stated in Definition 1.2 except the L ∞ bounds in [15] for d = 2 and the subcritical mass case m < 8π. L ∞ -apriori estimates were obtained in [25] for the fully parabolic case but in bounded domains. It is reasonable to expect that this strategy should work for the whole space case, although it is not written as such in the literature. Results in higher dimensions concerning solutions with L ∞ estimates for small initial data can be found in [8] but estimates on the free energy dissipation are missing there. We finally refer to [11,13,14,22] for different results concerning the existence of solutions satisfying the boundedness of the Fisher information and/or the uniform bounds of the solutions for particular choices of ε ≥ 0, α ≥ 0, and nonlinear diffusions.
As mentioned before, Theorems 1.3 and 1.4 are based on the derivation of quasi-Lipschitz estimates for the chemoattractant v. This is the reason behind the additional assumption on the initial datum v 0 in Theorem 1.4. We will clarify the use of quasi-Lipschitz estimates of the chemoattractant in the next section together with a quick summary of the main properties of optimal transport that we need in this work. Section 3 is devoted to show the main uniqueness results, derived from a more general property of bounded solutions for the Keller-Segel model. In fact, we will show that for bounded solutions we can obtain evolution variational inequalities. In Section 4 we show that these evolution variational inequalities lead to certain convexity of the associated free energy functional. In order not to break the flow of the argument, we postpone to Section 5 the rigorous derivation of the quasi-Lipschitz estimates of the elliptic and parabolic equations for v. In Section 5, we will also prove a strengthening of Theorem 1.4, with more general initial data. Finally, Section 6 is devoted to show how to adapt these arguments to Keller-Segel models with nonlinear diffusion. 3

Preliminary notions
2.1. Some elliptic and parabolic regularity estimates. The proofs of our results are based on the technique used by Yudovich [34] for treating uniqueness in the case of incompressible Euler equations for fluidodynamics. In particular, we exploit a quasi-Lipschitz property for the velocity field of the continuity equation for ρ in (1.1) and (1.2). This property comes from the regularity that v gains being solution to the second equation in (1.1) and (1.2).
, by exploiting some estimates of the Newtonian potential, ∇v satisfies the following log-Lipschitz property (see [6] and [30,Chapter 8], [32] and also [34] , where C is a suitable positive constant, depending only on ρ L 1 and ρ L ∞ and log − denotes the negative part of the natural logarithm function. As a consequence, we get the estimate Indeed, for large values of |x − y| the estimate (2.1) is quite obvious, since it is immediate to show that ∇B 0,d * ρ is a bounded function in the whole space with a direct estimate using the fact that ρ ∈ L 1 ∩ L ∞ (R d ). The log-Lipschitz property itself can be justified through standard elliptic regularity, as we will do in Section 5. Analogous facts hold if we consider the equation −∆v + αv = ρ, appearing in (1.2), or more general uniformly elliptic operators, so that we have About the parabolic equation for v in (1.1), the quasi-Lipschitz property also carries over, since formally inequality (2.1) translates in terms of the parabolic metric to If v is the unique solution to the Cauchy problem for the parabolic equation . For a more complete insight into these properties, it will be convenient to recall some facts about the Zygmund class and its role in elliptic and parabolic regularity. However, in order not to introduce some not really necessary notation before the proof of our main results, we prefer to postpone the proof of Proposition 2.1 and Proposition 2.2 to Section 5. Indeed, in Section 5 we will develop a more rigorous discussion about the log-Lipschitz estimates, and thanks to some refined parabolic regularity we will also prove a slight strengthening of Theorem 1.4.
We recall a formula for the differentiation of the squared Wasserstein distance along solutions of the continuity equation Then the curve is absolutely continuous with respect to the Wasserstein distance, [3,Theorem 8 where T t is the optimal map between ρ t andρ (see [ (2.5) Here we are lettingḢ 1 (R d ) be the space of Lebesgue measurable functions v : is defined by duality with functions having finite L 2 (R d ) norm of the gradient only. By the way, we can also consider the space H 1 (R d ) = W 1,2 (R d ). In fact, from the proof in [27, Proposition 2.8] it is not difficult to see that the same estimate holds considering the H −1 (R d ) space given by duality with the full norm ( ∇v 2

Bounded solutions as gradient flows: EVI and uniqueness
The uniqueness Theorems 1.3 and 1.4 are consequences of a general result interpreting bounded solutions to (1.1) (resp. (1.2)) as the trajectory of the gradient flow of the functional (1.3) (resp. (1.4)) in the appropriate metric setting. We prove that bounded solutions satisfy a family of evolution variational inequalities (EVI). Among different notions of gradient flow in metric sense, the EVI formulation is stronger than other formulations and typically corresponding to a convex structure, as in [3, Theorem 11.2.1] for the theory in the Wasserstein setting.
Notation for the energy functional. Before giving the proof, we introduce some uniform notation for working with the full functional (1.3) even in the parabolic-elliptic case. Let ρ ∈ We are considering the free energy functional On the other hand, if ε = 0 it is understood that v is given by B α,d * ρ. Therefore the parameter ε only indicates if we are considering problem (1.1) or (1.2). In particular, this writing of the functional as in (1.3) is valid in general, even for ε = 0, except for two particular cases: ε = α = 0 and d = 1, 2, as discussed in the introduction. In these two cases, we need to renormalize the free energy functional. Given ρ * ∈ M 2 (R d ; m) a smooth and compactly supported density and v * = B 0,d * ρ * , we redefine (1.3) for ε = α = 0 and d = 1, 2 as , as ρ − ρ * has zero mean, see [4,32] for more details.
In the rest of this work, when referring to the free energy functional F ε,α (ρ, v), we will be using (1.3) for any ε ≥ 0, α ≥ 0, except for ε = α = 0 and d = 1, 2 where the free energy functional is given by (3.1).
Let us observe that now all the integrals involved in the definition of F ε,α are well defined and finite for ε ≥ 0, α ≥ 0 and ρ, v as above. The negative part of the entropy term can be classically treated by the Carleman inequality, see for instance [9,Lemma 2.2] where the second moment bound on the density is used. The boundedness of the density controls the positive contribution of the entropy term together with the integrability of vρ in case ε > 0 since v ∈ W 1,2 (R d ). For ε = 0 the integrability of vρ in case α > 0 is implied by the Newtonian potential case α = 0 since the singularity of the Bessel potential at the origin is the same. The integrability for α = ε = 0 and d ≥ 3 results directly from the Hardy-Littlewood-Sobolev inequality for the Newtonian potential. For α = ε = 0 and d = 1, 2 we use the behavior at infinity of the density ρ. Actually, α = ε = 0 and d = 1 is a trivial case since the Newtonian potential is given by B 0,1 (x) = |x|. For α = ε = 0 and d = 2 since log(e + |x| 2 )ρ ∈ L 1 (R d ) then vρ ∈ L 1 (R d ) using the logarithmic HLS inequality, see for instance [10].
Notation for the ambient metric space. We let with the convention that X 0 = M 2 (R d ; m) and D 0 (z 1 , z 2 ) = W 2 (ρ 1 , ρ 2 ). Moreover, for z = ρ ∈ X 0 × L ∞ (R d ), F 0,α (z) will be understood to be F 0,α (ρ, v) with v = B α,d * ρ, as usual when ε = 0. In the space X ε the metric derivative of an absolutely continuous curve t → z t is denoted and defined by and it exists for L 1 -a.e. t > 0. The local metric slope of the functional F ε,α is defined by These two abstractly defined objects are used to give the notion of curves of maximal slope in general metric setting, see [2, §3], , according to Definition 1.2. If ε = 0, let z t = ρ t be a bounded solution to problem (1.2), starting from z 0 = ρ 0 ∈ X 0 ∩ L ∞ (R d ), according to Definition 1.1. Then the three following properties hold: i) The evolution variational inequality (EVI) formulation: ii) The energy dissipation equality (EDE) in metric sense: the map t → F ε,α (z t ) is locally Lipschitz continuous and iii) The following expansion control property: given another bounded solution t → ζ t , with initial datum ζ 0 in the same space of z 0 above, there exists a constant C, depending on ρ L ∞ ((0,T )×R d ) and v 0 W 2,∞ (R d ) (and the same quantities associated to ζ), such that there holds Proof. We first introduce the auxiliary functional for ρ and v being as in the definition of F ε,α at the beginning of this section, so that The proof is organized in four steps.

Step1. Quasi-Lipschitz Estimate implies control of the evolution of the Wasserstein distance.-Thanks to the assumption (1.5), we learn that the Fisher information
Exploiting the differentiability properties of the entropy functional, we can use the above-the-tangent formulation of displacement convexity to get for L 1 -a.e. t ∈ (0, T ) where T t denotes the optimal transport map between ρ t andρ. We refer to [2, §3.3.1] for an intuitive proof of this fact, and to [3, Chapter 10] for the theory in full generality. In particular, the finiteness of the Fisher information of ρ t implies that the second term is finite, so that this differentiation formula is meaningful. If ε > 0 (resp. ε = 0), letv ∈ W 1,2 (R d ) (resp.v = B α,d * ρ). Take Using the notation x s t := (1 − s)x + sT t (x), s ∈ [0, 1], and taking into account that and (3.7), we obtain for L 1 -a.e. t ∈ (0, T ) Let us denote by II t the last term in the right hand side above. The crucial point is to treat such term using the log-Lipschitz property of ∇v. Notice that, if ε = 0, we are in the assumptions of Proposition 2.1 and we apply (2.1), where the constant C depends in principle only on (m, α, d and) the L ∞ norm of ρ t , which we are assuming to be uniformly bounded on (0, T ). In the case ε > 0, still by the uniform space-time L ∞ assumption on ρ t and the W 2,∞ assumption on v 0 , we are in the framework of Proposition 2.2, so that we can apply the estimate (2.3). In this case the constant will depend also on (ε and) v 0 W 2,∞ (R d ) . Since ϕ is concave, we can also use the Jensen inequality, and letting ρ s t = x s t # ρ t be the Wasserstein geodesic connecting ρ t andρ we have (3.8) The last inequality holds since geodesic interpolation ensures for all s ∈ [0, 1] and since ϕ is non decreasing. We recall that the constant C in (3.8) depends only on (ε, α, d, the mass m and) the L ∞ ((0, T ) × R d ) norm of ρ and, in the case ε > 0, the W 2,∞ (R d ) norm of v 0 . Inserting this in the estimate for I t , we have for L 1 -a.e. t ∈ (0, T ) where ω is the function defined in (3.2). Since ρ t satisfies the continuity equation and (1.5), the uniform L ∞ bound of ρ t implies that T 0 ξ t 2 L 2 (R d ,ρt;R d ) dt < +∞. Therefore t → ρ t is absolutely continuous with respect to W 2 and by (2.4) Inserting this into (3.9), and recalling the definition of I t , we finally obtain for L 1 -a.e. t ∈ (0, T ).
Step 2: EVI for the parabolic-parabolic case.-Recalling thatv ∈ W 1, 2 (R d ), observing that ∆v t ∈ L 2 (R d ) for a.e.-t ∈ (0, T ) and using the elementary identity |a| 2 − |b| 2 = |a − b| 2 + 2 b, a − b for every a, b ∈ R k , the variation of the second part of the functional (1.3) (that is, F ε,α − Φ ε,α ) can be written as (3.11) Therefore, we deduce (3.12) Now, we use again (3.10), leading to (3.13) By using the duality betweenḢ 1 andḢ −1 , the Young inequality, and (2.5) we have 14) where Q is the largest of the L ∞ norms ofρ and ρ t over the time interval (0, T ). Taking into account that ω is given by (3.2) and that mϕ(m −1 x 2 ) ≥ x for every x > 0, combining (3.13) and 9 (3.14) we get, up to introducing a new constant C, ∈ (0, T ). The new constant C depends as usual on (ε α, d, m and) Step 3: EVI for the parabolic-elliptic case.-When either d ≥ 3 or α > 0, we can repeat the proof of the parabolic-parabolic case, letting ε = 0 therein and recalling thatv is no more an arbitrary W 1,2 (R d ) function but is given by convolution withρ. In particular we arrive to the corresponding of (3.13), and the second line therein can now be estimated as follows. Using the Moreover, recalling the estimate (2.5) (which works both inḢ −1 and H −1 where Q is the largest of the L ∞ norms ofρ and ρ t over the time interval [0, T ]. Inserting these estimates in (3.13) we obtain for L 1 -a.e. t ∈ (0, T ), where the constant C depends only on ε, α, d, m, ρ L ∞ ((0,T )×R d ) , ρ L ∞ (R d ) .
Step 4: Conclusion.-We are ready to prove the three points in the statement of the theorem. The proof of i) is a consequence of (3.15) for the case ε > 0, and (3.16) for the case ε = 0, taking into account that α ≥ 0 and that ω(D 2 (z t ,z)) ≥ ω(W 2 2 (ρ t ,ρ)) being ω increasing. It is a standard fact that the gradient flow formulation in EVI sense implies the one in EDE sense in (3.4). Indeed, the proof of ii) follows from (3.3) and (3.5) and can be exactly carried out as in [2,Proposition 3.6].
Proof of Theorems 1.3 and 1.4. The main theorems in the introduction are now a straightforward consequence of the expansion control iii) in Theorem 3.1. Both Theorems follow from the inequality (3.5) observing that G −1 (G(0) + 4Ct) = G −1 (−∞) = 0.

ω-convexity of the functional
In this section we show another consequence of the EVI formulation of bounded solutions. For the functional F ε,α the following relaxed ω-convexity along geodesics holds, see [18] for ω-convexity of functionals on measures. We assume that bounded solutions to (1.1) (resp. (1.2) for ε = 0) verify that for some T > 0 This assumption has been proved in several cases, see the introduction for more details.
Using the fact that s → z s is a geodesic, the right hand side is nonnegative, thus The lower semi continuity of t → F ε,α (z s t ) and the continuity of r → D 2 (z s r , z i ), i = 0, 1 yield Since s → z s is a geodesic we have D 2 (z s , z 0 ) = s 2 D 2 (z 1 , z 0 ) and D 2 (z s , z 1 ) = (1 − s) 2 D 2 (z 0 , z 1 ) and we conclude.

A refined result in Zygmund spaces
This section is devoted to give a rigorous justification of the estimates stated in subsection 2.1. We will also give a slight improvement of Theorem 3.1 and Theorem 1.4 by guaranteing a suitable quasi-Lipschitz estimate under a more general condition on the initial datum v 0 . The right framework is that of Zygmund spaces. These classes of functions were introduced in [35], and they belong to the more general framework of Besov spaces.
Zygmund estimates and log-Lipschitz regularity in the elliptic case. We first introduce the basic Zygmund class Λ 1 (R d ) as the set of continuous bounded functions f over R d such that It is well known that functions in the Zygmund class are in general not Lipschitz, possibly nowhere differentiable. Indeed, functions in Λ 1 (R d ) enjoy a log-Lipschitz modulus of continuity. Therefore, for any f ∈ Λ 1 (R d ) there exists a positive constant C such that we refer for instance to [36,Chapter 2,§3]. More generally, following for instance [33,Chapter 5] we may define the class Λ n (R d ) for any n ∈ N as follows. We let Λ 0 = L ∞ (R d ) and we say that all the derivatives of f of order n − 1 belong to Λ 1 (R d ). In the usual notation of Besov spaces, Λ n corresponds to B n ∞, ∞ . In this framework we have Proof of Proposition 2.1. If α > 0, from the general theory on Bessel potentials (see for instance [33,Chapter 5,[3][4][5][6]) we learn that by convolution with the Bessel kernel B α,d we indeed get two indices of regularity in Λ n spaces. Therefore, if ρ ∈ L ∞ (R d ), we indeed get that v = B α,d * ρ belongs to Λ 2 (R d ), and thus ∇v ∈ Λ 1 (R d ) and (2.1) follows. For the case α = 0 we address to the references mentioned in Section 2 (it is also possible to directly check that ∇v ∈ L ∞ (R d ), and then the Newtonian potential behaves like the Bessel potential near the origin so that ∇v is also log-Lipschitz).
With respect to the parabolic metric, the definition of Zygmund spaces adapts as follows. We have Λ 0 (Q T ) := L ∞ (Q T ), and Λ 1 (Q T ) is the space of continuous bounded functions f overQ T such that there hold In particular, we see that f ∈ Λ 2 (Q t ) implies f ∈ L ∞ ((0, T ); W 1,∞ (R d )), with ∇f satisfying (5.1), so that finally f satisfies also (2.3). When dealing with parabolic equations, it is suitable to consider spaces of functions defined with respect to the parabolic metric, since it is natural to deal with functions which have derivative up to order k with respect to time and 2k with respect to space. For classic results, we refer for instance to [16] or to the monograph [26], where estimates are derived in Sobolev and Hölder spaces of this kind, see Chapter 4 therein.
In [16] we find that if the forcing term of the heat equation has bounded mean oscillation (BMO), still with respect to the parabolic metric, than the same holds true for second order space derivatives and first order time derivatives of the solution. This would be enough for deducing that first derivatives in space are in the Zygmund class with respect to the parabolic metric and that therefore they satisfy a log-Lipschitz estimate. The results in [16] deal only with null initial datum, but they can be generalized to more general data with suitable regularity requirements. Some extensions involving initial data in Zygmund classes are found in [1,23], based on direct estimates on fundamental solutions. Summing up, we have Proof of Proposition 2.2. Suppose that v is the solution (convolution with fundamental operator) of the forced heat equation ∂ t v = ∆v + ρ. Suppose ρ ∈ Λ 0 (Q T ) and v 0 ∈ Λ 2 (R d ). Then we have v ∈ Λ 2 (Q T ). See [16] for the case v 0 = 0, see [23,Theorem 4] for a general result. If we consider the second equation of (1.1) with α > 0, the fundamental solution is just multiplied by a decaying exponential at infinity and the same result carries over. Therefore, a sufficient condition in order This gives a rigorous justification of the assumptions on the initial datum of Theorem 1.4. However a refined analysis shows that this assumption can be weakened, as we do next.
Initial datum in Λ 1 (R d ). We have to consider the weighted Zygmund space Λ −1 2 (Q T ), defined as the corresponding space Λ 2 (Q T ), with the addition of a time weight which is divergent as t → 0. In particular, locally in Q T functions in Λ −1 2 (Q T ) have the same smoothness as the ones in Λ 2 (Q T ), but this regularity does no more extend to the closure of Q T . More precisely, by definition f ∈ Λ −1 2 (Q T ) 13 means that f ∈ Λ 1 (Q T ), sup x,y∈R d t∈[0,T ] √ t |∇f (x, t) − 2∇f ((x + y)/2, t) + ∇f (y, t)| |x − y| < +∞ (5.2) and the second finite differences of f and ∇f with respect to time verify the corresponding estimates, as in the definition of Λ 2 (Q T ), still with the addition of the weight t 1/2 . for a constant C depending on ρ L ∞ ((0,T )×R d ) , v 0 Λ 1 (R d ) , ρ L ∞ (R d ) . Moreover the EDE (3.4) holds, and the expansion control property holds in this form: given another bounded solution t → ζ t as above with initial datum ζ 0 ∈ (M 2 (R d ; m)∩L ∞ (R d ))×(Λ 1 (R d )∩ W 1,2 (R d )) there is
Thus we deduce the weighted analogous of (2.3), that is where C is a new suitable positive constant depending on the data and ϕ is defined in (2.2). Following the line of the proof Theorem 3.1 we reach the estimate (3.8) for II t , which now has to be changed because we have to use (5.7), obtaining |II t | ≤ Ct −1/2 W 2 (ρ t ,ρ) mϕ(m −1 W 2 2 (ρ t ,ρ)) = Ct −1/2 ω(W 2 2 (ρ t ,ρ)). We can repeat all the other steps which lead to (3.14), obtaining the corresponding EVI with the additional weight t −1/2 , which directly lead to (5.3). We conclude as in Step 4 of the proof of