On a Class of Diffusion-Aggregation Equations

We investigate the diffusion-aggregation equations with degenerate diffusion $\Delta u^m$ and singular interaction kernel $\mathcal{K}_s = (-\Delta)^{-s}$ with $s\in(0,\frac{d}{2})$. We analyze the regime %($m>2-2s/d$, $d$ is the dimension) where the diffusive forces are stronger than the aggregation forces. In such regime, we show existence, uniform boundedness and H\"{o}lder regularity of solutions in the case that either $s>\frac{1}{2}$ or $m<2$. Uniqueness is proved for kernels with $s>1$.


Introduction
We consider the following degenerate diffusion equations with drifts: with nonnegative initial data u(x, 0) = u 0 ∈ L 1 (R d ) ∩ L ∞ (R d ), where the degeneracy arises due to the range of m, m > 1. The nonlocal drift is of the form When d ≥ 3, we can write K s u = (−∆) −s u for some c = c(d, s) > 0 which is a typical representation of the aggregating effect between density particles, with smaller s representing stronger aggregation at near-distances and therefore more singular. For larger s, we consider stronger force at long-distances. In dimension two, the kernel of (−∆) −s is of a different form, for simplicity we restrict to d ≥ 3.
The model arises from the macroscopic description of cell motility due to cell adhesion and chemotaxis phenomena, see [6,10]. The degenerate diffusion models the repulsion between cells to take over-crowding effects into consideration [26] and it is also widely seen in many physical applications, including fluids in porous medium. The homogeneous singular kernel models the attractive interactions between cells. The competition between non-local aggregation and diffusion is one of the core of subject of diffusionaggregation equations.
To find the balance of diffusion and concentration effects, we use a scaling argument, also see [5,11]. Define u r (x, t) := r d u(rx, r d(m−1)+2 t), then formally (−∆) −s u r = r d−2s (−∆) −s u. It is straightforward to check ∂ t u r = ∆u m r − r 2d−dm−2s ∇ · (u r ∇ · K s u r ). So m = 2 − 2s/d leads to a compensation effect between diffusion and aggregation. We call the range m > 2 − 2s/d subcritical where the diffusion dominates over the aggregation.
When s = 1, K 1 represents the Newtonian potential and the equation (1.1) is the degenerate Patlak-Keller-Segel equation. In the range m > 2 − 2/d, the well-posedness, boundedness and continuity regularity properties of solutions have been established, see [2,12]. When m = 2 − 2/d, it has been shown in [5,9,15] that there is a critical value of the mass and the behaviour of the solutions is determined by the initial mass. If the initial mass is large, solutions may blow up in finite time. If m < 2 − 2/d, the aggregation dominates and the problem is called supercritical, see [2,4,5]. Again in this regime, we have finite time blow-up of solutions.
In this paper, we consider the natural extension of the Newtonian potential with more near-range singularity, i.e. K s = (−∆) −s if 0 < s < 1 and with more long-range singularity if s > 1 (see (2.1)). For this kernel, to the best of our knowledge, only stationary solutions have been analyzed before in [10]. It has been shown there that stationary solutions are radially symmetric decreasing with compact supports and enjoy certain regularity properties in most of the subcritical regime. Our goal here is to initiate investigating the dynamic equation (1.1), starting with its well-posedness and regularity properties.
Many questions stay open as we discuss below.
Summary of our result. We approach the problem by two approximations: regularization of the gradient of the kernel and elimination of the degeneracy, see (3.2). The key as well as the hard step is to show a prior boundedness estimates of solutions. We will firstly prove uniform L p -regularity properties (for some p → ∞) of solutions to the approximate problems in the subcritical regime. This can be seen as an variate result as compared to [2,18,25] where Keller-Segel systems or equations are considered, see Theorems 3.1-3.4. We are going to use Sobolev inequalities, properties of fractional Laplacian and the equation to show some differential iterative inequalities which will eventually give us a uniform in time L ∞ bound. The idea of the proof is to control the aggregation term by the degenerate diffusion. Very importantly in each estimate, we should not break the scaling (1.3) and this turns out to be a useful hint for us, for example the choice of exponents in inequalities, see (3.16). And the condition m > 2 − 2s d is essential in the proof.
In the subcritical range with 1/2 < s ≤ 1, the uniform bounds are obtained separately when 2−2s/d < m < 2 and m ≥ 2, and only for the former range of m when s ≤ 1/2. The proofs for the three cases are different. s = 1 2 is critical, because |∇K s | is only locally integrable when s > 1 2 . Boundedness of solutions in the case {m ≥ 2, s ≤ 1/2} is unknown, though we believe it is true. While likely a technical challenge, extending the results seems to require some different ideas.
When s > 1, again the regimes 2 − 2s/d < m < 2 and m ≥ 2 are treated separately. However the proofs are even more different. In this regime the tool is limited, for example we can not use the fractional differentiation, instead we use Young's convolution inequality to treat the singular convolution integral. Technically we are required to use three arguments for different parts of the iterative steps, see Theorem 3.4. Let us mention that the difficulty for m > 2 arises as well in [10] where stationary states of (1.1) are studied. More precisely in Theorem 1 [10], stationary solutions are shown to be in W 1,∞ (R d ) only when 2 − 2s/d < m ≤ 2.
With aforementioned a priori bounds, we obtain existence and bounds for the solution to the original problem (1.1) by compactness, see Theorem 4.1, 4.3. The hard part is to justify u∇(−∆) −s u = u∇K s * u when s ≤ 1/2, where u is the weak solution of (1.1), because in such cases ∇K s * u is not well-defined for u ∈ L 1 ∩ L ∞ . To overcome this difficulty, the following estimate can be proved under the condition m < 2, ∇u ∈ L 2 loc ([0, ∞), L 2 (R d )). Using this, we will show in Lemma 4.2 that Next let us state the uniqueness result.
Theorem 1.2 (Uniqueness). Suppose m, s are in the subcritical range and s > 1, Then the weak solution to (1.1) with initial data u 0 is unique.
Uniqueness result is rather limited, it is only shown here for s > 1 in the frame ofḢ −1 (R d ), following the approach of [2,3] where they consider the case when s = 1.
Now we look at the regularity of solutions to (1.1). Theorem 1.3 (Hölder Regularity). Suppose m, s are in the subcritical range and s ∈ ( 1 2 , d 2 ). Let u(·, t) be a weak solution to (1.1) with non-negative initial data u 0 ∈ L 1 (R d ) ∩ L ∞ (R d ). Then the following holds (a) For any τ > 0, u is Hölder continuous in R d × (τ, ∞).
When 1/2 < s < d 2 , ∇K s u is a well-defined and bounded vector field in R d for all u ∈ L 1 (R d )∩L ∞ (R d ) and then the interior Hölder estimate is a consequence of [19] where the porous medium equation with locally integrable drift terms is considered. We also study the regularity of solutions on the boundary t = 0 if the initial data is Hölder continuous which is given in Theorem 6.1.
The regularity result is left open in the regime s ≤ 1/2 and the main difficulty comes from ∇(−∆) −s u. As mentioned before, we can have boundedness of ∇(−∆) −s u in L 2 loc ([0, ∞), L 2 (R d )) when m < 2. But this bound is not strong enough to obtain uniform Hölder estimates, according to [13,19]. We need some more careful analysis which could be employed in future research.
Let us comment that our results and proofs adapt to more general kernels given by K s u = K s * u where |K s (x, y, t)|, |∇ x K s (x, y, t)|, |D 2 x K s (x, y, t)| share the same singularity as |x − y| −d+2s , |x − y| −d−1+2s , |x − y| −d−2+2s respectively near x = y. Some modifications are needed if we only assume |K s (x, y, t)|, |∇ x K s (x, y, t)|, |D 2 x K s (x, y, t)| to be bounded away from x = y. Lastly let us mention that a lot of open questions remain to be investigated even in the subcritical regime, existence result for s < 1/2 (and m > 2), uniqueness result for s < 1. And there are even more questions in the supercritical regime. Some of these open questions closely related to us will be stated in the outline.
Outline of the paper.
We assume the space dimension d ≥ 3 for the simplicity of computation, and also assume that m, s are in the subcritical range in the whole paper. Section 2 contains preliminary definitions and notations. Section 3 deals with a priori estimates of solutions and the proof is given separately for Acknowledgements. The author would like to thank his advisor Inwon Kim for her guidance and stimulating discussions. The author would also like to thank Franca Hoffmann, Kyungkeun Kang and Monica Visan for helpful discussions and suggestions.

Preliminaries and Notations
• Let us start with discussing the fractional potential operator K s = (−∆) −s .
We use the notation −(−∆) r with r ∈ (0, 1] for fractional Laplace operator which is defined on the Schwartz class of functions on R d by Fourier multiplier with symbol −|ξ| 2r , see chapter V [24]. Alternatively, −(−∆) r can also be realized as the following singular integral in the sense of Cauchy principal value, see [21].
We denote the constant before the above integral as c d,r . The domain of the operator can be extended naturally to the Sobolev space W 2r,2 (R d ). We will write The bilinear form associated to the space W r,2 (R d ) is define to be the following with reference to [21] and Section 3 [7]. For v, w ∈ W r,2 (R d ) Using Parsevals identity and definitions, we have for 0 < r 1 < r The inverse operator is denoted by −(−∆) −s which can be realized as the convolution of a function with the Riesz potential (2.1) and K s (x, y) : Here s can be any number in (0, d 2 ) and u is a function integrable enough for (2.1) to make sense. We refer readers to [8,21,23] for more details.
When s > 1 2 , ∇K s u is well defined for u ∈ L 1 (R d ) ∩ L ∞ (R d ). When s < 1 2 , if we further assume that u is γ-Hölder continuous with γ ≥ 1 − 2s, then ∇K s u is defined via a Cauchy principal value • We now give the following notion of weak solutions to (1.1). The notion is similar to the one in [3,8].
and for all test • Next we collect some known results which will be used later.
For any function u : Then there exists a constant C depending only on α, r, q, s such that Condition (2.4) can be replaced by If s = 0, the inequality is classical and (2.4) can be replaced by This lemma about Gagliardo-Nirenberg Inequality is not given in the most general form, which is unnecessary for our purpose. We refer readers to [22] for the classical Gagliardo-Nirenberg inequality.
To the best of our knowledge, the validity of the inequality with fractional derivatives is proved in Corollary 1.5 [17]. But they did not cover the case when q = 1 and (2.6) holds. We postpone the completion of the proof to the appendix.
The following lemma is useful which can be proved by using Calderón-Zygmund inequality. We refer readers to Theorem 4.3.3 [16] for the details.
Lemma 2.4. There exists a constant C > 0 such that for all 1 < p < ∞ and u ∈ W 1,p Recall the homogeneous Sobolev spaceḢ s (R d ): The homogeneous Sobolev space is the space of tempered distributions f over R d , the Fourier transform of which belongs to L 1 loc (R d ) and satisfies Proposition 2.5. If |s| < d 2 ,Ḣ s can be considered as the dual space ofḢ −s through the bilinear functional: for any H 1 is the subset of tempered distributions with locally integrable Fourier transforms and such that |∇f | ∈ L 2 (R d ).
For details and more properties, we refer readers to the book [1].

Notations.
We write N as all natural numbers and N + as all positive natural numbers. For p ≥ 1, γ ∈ (0, 1), for simplicity, we denote Given two points (x, t), (y, s) ∈ R d+1 , we define the distance between them to be We write B R (x) as a ball in R d centered at x with radius R. We denote B R := B R (0). The scaled parabolic cylinders are written as We denote the scaled parabolic cylinders near t = 0 by The standard parabolic cylinders are denoted by Q r := Q(r, 1) and Q 0 r := Q 0 (r, 1). Throughout this paper, the constants {C} represent universal constants, by which we mean various constants that only depends on m, d, s, γ and L 1 , L ∞ or C γ norms of the initial data u 0 . We may write C(A) or C A to emphasize the dependence of C on A.
We write A B if A ≤ CB for some universal constant C. When writing A D B, we mean A ≤ CB where C depends on universal constants and D (with particular emphasis on the dependence of D). By A ∼ B, we mean both A B and B A are satisfied.
Let S be a measurable set in R d . The indicator function χ S (x) equals 1 if x ∈ S and it equals 0 otherwise.
Then there is 1. V s,ǫ is a smooth vector field and V s, holds for all x for some C > 0 only depending on d, s. For small ǫ > 0, we consider u ǫ which solves the following problem: V s,ǫ is smooth and compactly supported. The convolution integral V s,ǫ * u is well-defined since V s,ǫ bounded. Equation (3.2) is uniformly parabolic. The existence and uniqueness of a solution u ǫ is proved in Theorem 4.2 [3] and the solution is smooth.
In the following theorems, we are going to prove that u ǫ are uniformly bounded independent of ǫ. As mentioned before, we will treat the following cases separately: We use a refined iteration method and this approach can be found in Lemma 5.1 [20].
is non-negative. Let u := u ǫ be the solution to (3.2). Then there exists a constant C such that for all n ≥ 1 and t ∈ R + there is The constant C only depends on d, s, m and the L 1 , L ∞ norms of u 0 .
Proof. Without loss of generality, let us suppose that the total mass of u 0 is 1 and so is the total mass of u(·, t) by the equation. Since u is smooth, for n ≥ 3 − m we multiply u n−1 on both sides of (3.2) and find By property 2. of V s,ǫ , we obtain Then By Gagliardo-Nirenberg interpolation inequality It can be checked that α > 2 − 2s if and only if s > 1 2 . The conditions of Lemma 2.3 are satisfied. Let θ(n) := α + n+1−l l β. Then the above two equalities give Since m < 2, {θ(n)} is decreasing as n → ∞ and the limit equals 2−2s d + 1 1 2 + 1 d which is less than 2. Very importantly when n = 3 − m, θ(3 − m) < 2 is equivalent to So for all n ≥ 3 − m, θ = θ(n) ∈ (τ, 2 − τ ) for some τ (m, s) > 0. Then By Hölder's inequality, for any small δ > 0 Since θ(n) < 2 uniformly, {c n } are uniformly bounded in n for all n ≥ 3 − m. By Gagliardo-Nirenberg inequality By direct calculations, γn l < 2. Then by Young's inequality where c, C are independent of n. Then n k = 2 k (2 − m) − 1 + m. Since m < 2, we have n k → ∞ as k → ∞.
Notice for all k ≥ 1, n k ≥ n 1 = 3 − m. Thus we can take n = n k+1 in the above for all k ≥ 1. Then l = n k and n k ∼ 2 k . If writing A k = R d u n k dx, we have proved for all k ≥ 1 To conclude the proof we need the following lemma.
(t) and suppose {B 0 (t), B k (0)} are uniformly bounded with respect to k ∈ N, t > 0. Then {B k (t)} are uniformly bounded for all t > 0 and k ∈ N + .
We refer readers to Lemma 3.1 [19] for the proof. Now we consider the case when m ≥ 2 and s > 1 2 .
Theorem 3.2. Theorem 3.1 holds in the regime: m ≥ 2, s ∈ ( 1 2 , 1]. Proof. Denote u j = max{u − j, 0},ũ j = min{u, j} and so u = u j +ũ j . For some n ≥ 2, let us multiply u n−1 1 on both sides of (3.2). We have for some C m > 0 bounded from below for all m ≥ 2, we obtain Let us now estimate X: We will first consider X 1 . By the fact that Then for any small δ > 0 In the last inequality (3.9), we applied where we picked α = 1 2 So by (3.9), for some universal small δ > 0 For Y l with l = n − 1, n, as proved before (in Theorem 3.1) By Fourier transformation and Hölder's inequality, where we pick Thus ǫ p ∼ δ n d+1 and since −q It is not hard to check that for all n ≥ 2, β 1 (n) ≤ β 1 (2) < 1. And so c n is uniformly bounded for all n ≥ 2. So we proved that for any small δ > 0 Y n ≤ C δ n cn u Recall here γ n ≤ 2 + C n . Now we let n = 2 k for k = 1, 2.. and A k = R d u n k 1 dx. By Lemma 3.1, u 1 (x, t) is uniformly bounded for all t ≥ 0 and so is u(x, t).
For n ≥ 3 − m, denote l = n+m−1 2 ≥ 1. We multiply u n−1 on both sides of (3.2) and obtain (3.24) Let χ(x) = χ |x|≤1 (x) be the indicator function. Let A 1 := χ∇ · V s,ǫ and A 2 := (1 − χ)∇ · V s,ǫ . It is not hard to see A 2 is bounded and 1. A 1 is compactly supported and |A 1 |(z) ≤ |z| −d−2+2s . 2. |A 1 | bounded in L d d+2−2s ′ (R d ) for all 1 < s ′ < s. We fix one s ′ such that We have By Young's convolution inequality X 1 ≤ u n p u q with p, q ≥ 1, By Gagliardo-Nirenberg and Young's inequalities, where α, β are given by l np If d 2s ′ −2 ≥ n, take np l = q l =: r. Then α = β. According to (3.25), r can be computed by 1 Then and so p ≥ 1, q ≥ 1 is satisfied due to the assumption d 2s ′ −2 ≥ n. Then by (3.25), (3.26) (3.28) We claim It is only nontrivial to verify the second formula of the claim. Let us compute with equality holds when n = 3 − m. We get thatα n+1 l < 2 is exactly equivalent to m > 2 − 2s ′ d . Also since the sum of the exponents in (3.28) equals n+1 l ∼ 2 + c n for some universal c > 0, we obtain for some universal ǫ > 0 independent of n.
As done several times before, by Young's inequality we derive that for some universal constants C, c. If d 2s ′ −2 ∈ (l, n), take q = d 2s ′ −2 , p = 1 and α, β satisfying (3.27). Since it is immediately to check that α, β ∈ (0, 1). Then by (3.25), (3.26) We claim that Actually 2 − α n l + β 1 l is away from 0 independent of n. We omit the proof which is a direct computation. Also since for some universal c > 0, by Hölder's inequality in this case, again we have Secondly suppose d 2s ′ −2 ≤ l. We take p = 1, q = d 2s ′ −2 ≤ l in (3.25). Then since u 1 is bounded, the set {u > 1} is of finite measure. Thus by Jensen's inequality Thus where α is given by l n + ( 1 2 + 1 d )α = 1. From this we get In the last inequality we used that And we need 2 − n l α > 0 to be bounded away from 0 uniformly in n. Actually As before by Hölder's inequality As for X 2 , note X 2 ≤ C u n 1 , therefore it can be handled similarly as we bound (3.32). In all by (3.24)(3.29)(3.31)(3.33) and taking δ to be small, we proved for all n ≥ 3 − m ∂ t u n 1 + c ∇u l 2 2 ≤ C δ n c + C δ n c u l 2+ c n 1 .
for some universal constants C, c > 0. As in (3.5), we can bound ∇u l 2 2 from below. Then as before, taking n k = 2 k (2 − m) − 1 + m and A k = u n k 1 , we end the proof by applying Lemma 3.1.
For n > 1, we multiply u n−1 1 on both sides of (3.2) where u 1 = (u − 1) + . We have Since m ≥ 2, we have For Y using the notation u = u 1 +ũ, we have By Young's convolution inequality, the above

Existence of Solutions
In this section, we show existence of weak solutions to (1.1). We are going to take ǫ → 0 in equation (3.2 Then |V s,ǫ | is locally integrable near the origin and so independent of ǫ. The situation is in some sense better.
We have the following theorem.
Then there exists a weak solution u to (1.1) with initial data u 0 and u preserves the mass.
Using the estimate given in section 3 and the fact that V s,ǫ * u ǫ ∞ , ∇K s * u ǫ ∞ are uniformly bounded independent of ǫ, t, the proof is almost the same as the proofs in Theorem 1,2,7 in [2]. We omit the proof. The proof of conservation of mass is similar to those in the proof of Theorem 4.3 given below.
Let us focus on the situation when s ≤ 1 2 . We need the following a prior estimates.
We used the fact that u ǫ , u are uniformly bounded in height. Keep in mind that u ǫ → u in L 1 (R d ×[0, T ]). Then to show the integral converges to 0 as ǫ → 0, we only need to estimate the first term of (4.8) which is denoted as X. Suppose max t∈[0,T ] ξ(·, t) = 0| B c R ξ for some R ξ > 0 and then by (3.1), Again by (4.2), we have Notice (4.7) and the equation deduce the mass preservation of u: for all t > 0, R d udx = R d u 0 dx. Finally let us mention that the property u ∈ C([0, T ], L 1 (R d )) follows from [2,3].

Uniqueness
In this section, we consider the uniqueness of weak solutions to (1.1) in the regime s > 1. In general, the problem is open. Proof. (of Theorem 5.1) We will follow the approach of [2,3] and estimate the difference of weak solutions inḢ −1 . Suppose u 1 , u 2 are two weak solutions to (1.1) with the same initial data u 0 . For each t > 0, define φ(·, t) through ∆φ(x, t) = u 1 (x, t) − u 2 (x, t) and lim |x|→∞ φ(x, t) = 0.

Hölder Regularity
In this section we look at the case when s > 1/2 and prove Theorem 1.3. Let u be a solution to (1.1) and denote V (x, t) = ∇K s u(x, t). Then we can rewrite the equation as u t = ∆u m + ∇ · (V u).  Therefore V (x, t) is uniformly bounded. Let us consider (6.1) and the notion of solutions is the same as Definition 2.1 after replacing K s u by V . We give both the interior regularity and the regularity up to t = 0 results of solutions to (6.1). Here we only need m > 1. Fix any point x ∈ B 1 2 and without loss of generality, we can assume x = 0. The first goal is to obtain η k M ≥ osc Q 0 (a k r,b 2k ) v for all integers k, (6.3)