THE AGGREGATION EQUATION WITH NEWTONIAN POTENTIAL: THE VANISHING VISCOSITY LIMIT

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1 THE AGGREGATIO EQUATIO WITH EWTOIA POTETIAL: THE VAISHIG VISCOSITY LIMIT ELAIE COZZI 1, GUG-MI GIE 2, JAMES P. KELLIHER 3 Abstract. The viscous and inviscid aggregation equation with ewtonian potential models a number of different physical systems and has close analogs in 2D incompressible fluid mechanics. We consider a slight generalization of these equations in the whole space establishing well-posedness of the viscous and inviscid equations, spatial decay of the viscous solutions, and the convergence of viscous solutions to the inviscid solution as the viscosity goes to zero. Compiled on Saturday 22 nd April, 217 at 11:42 Contents 1. Introduction 1 2. Measuring persistence of spatial decay 4 3. The linear viscous problem 8 4. The nonlinear viscous problem The inviscid problem 2 6. Total mass and infinite energy The vanishing viscosity limit for GAG ν ) for velocities in L The vanishing viscosity limit for GAG ν ) for velocities not in L The vanishing viscosity limit in the L -norm Concluding Remarks 47 Acknowledgements 47 References Introduction In this work we study on R d, d 2, the viscous or inviscid) aggregation equation with ewtonian potential, t ρ ν + divρ ν v ν ) = ν ρ ν, AG ν ) v ν = Φ ρ ν, ρ ν ) = ρ. Here, ν is the viscosity and Φ is the fundamental solution of the Laplacian, or ewtonian potential so Φ = δ and div v ν = ρ ν ). The density is ρ ν, the velocity is v ν, and ρ is the initial density. Many variations on these equations are considered in the literature, primarily by using potentials other than the ewtonian or by using more general diffusive terms. We restrict our attention to the ewtonian potential with linear diffusion, for we will be concerned with 1

2 2 ELAIE COZZI, GUG-MI GIE, JAMES P. KELLIHER analyzing the viscous ν > ) and inviscid ν = ) aggregation equation using techniques adapted from the study of 2D fluid mechanics. The aggregation equation models numerous physical problems. For the ewtonian potential, as in AG ν ), this includes chemotaxis, where AG ν ) for ν > is a limiting case of the Keller-Segel system see Section 5.2 of [23]) and has been extensively studied. In this context, ρ ν measures the density of cells bacteria or cancer cells, for instance) and v ν is the gradient of the concentration of a chemoattractant. References most closely related to the approach to the aggregation equation taken in this paper include [2, 3, 4, 15, 16, 22, 23]. We will, in fact, consider a slightly more general set of equations of the form GAG ν ) t ρ ν + v ν ρ ν = σ 2 ρ ν ) 2 + ν ρ ν, v ν = σ 1 Φ ρ ν, ρ ν ) = ρ, where σ 1, σ 2 are constants with σ 1. When σ 1 = 1, σ 2 = 1, GAG ν ) reduces to AG ν ) since then divρ ν v ν ) = v ν ρ ν + div v ν ρ ν = v ν ρ ν ρ ν ) 2. The special case σ 1 = 1, σ 2 = 1 has been used to model type-ii superconductivity when ν = see [22] and the references therein). In such applications, ρ is not assumed to have a distinguished sign, so there is no real difference between studying σ 1 = 1, σ 2 = 1 and σ 1 = 1, σ 2 = 1. At least one other special case of GAG ν ) has been studied in the literature: GAG ) with σ 1 = 1, σ 2 = are derived from AG ) by making a transformation of variables in 1.6) of [2]. This transformation applies only in the special case of aggregation patch initial data analogous to a vortex patch for fluids) for AG ). Although this transformation only works for aggregation patch initial data the authors of [2] go on to use this special case of GAG ) throughout their analysis of aggregation patches. A general well-posedness result is not needed in [2] and hence not established there, but such a result was one of our motivations for studying GAG ν ), the parameters σ 1, σ 2 merely interpolating between AG ) and the equations studied in [2] when ν = ). We will study GAG ν ) in all of R d, but we note that much of what we find extends naturally to a bounded domain if, as is typically done, one uses no-flux boundary conditions, ρ ν n =. This is because such boundary conditions eliminate all troublesome boundary integrals. The situation is the same for 2D incompressible fluids using no-flux conditions on the vorticity, though for fluids such conditions have no real physical meaning. We will find establishing the existence of weak viscous solutions to GAG ν ) no more difficult than doing the same for AG ν ) except for keeping track of the constants σ 1 and σ 2. In most applications compact support of the initial density would suffice, but such compact support is not conserved for viscous solutions. We will find that density having spatial decay of a specific type is conserved, however, and we will find it convenient to work in a space having such decay. Roughly speaking, this space, which we call L 2 see Definition 2.1), consists of L 2 densities having sufficiently rapid algebraic decay at infinity. In outline, our proof of existence of solutions to GAG ν ) proceeds as follows. We first define in Section 2 the spaces L 2 and H1 in analogy with L2 R d ) and H 1 R d ) and prove that H 1 is compactly embedded in L2 M for all M <. We show existence of solutions to a linearized version of GAG ν ) in Section 3 using an abstract functional analytic approach due to J.-L. Lions. We expend most of our effort showing that such solutions lie in L 2 L for all time and then obtaining bounds over time on all L p norms of the density that are uniform in viscosity. In Section 4 we use a sequence of these linearized solutions to approximate a solution to the viscous aggregation equation taking advantage of the compact embedding of

3 THE AGGREGATIO EQUATIO VAISHIG VISCOSITY LIMIT 3 H 1 in L2 M to obtain a solution via a compactness argument. We address uniqueness of solutions only later in Theorem 7.5.) Our viscous existence results suit our needs in later sections when we examine the vanishing viscosity limit; in particular, we need uniform-in-viscosity bounds on L p -norms of the density and on the time of existence of solutions to obtain the limit. Much more is known, however, about the existence time of solutions and how it relates to the initial mass of the density, at least for AG ν ) for nonnegative ρ as summarized in Sections 5.2, 5.3 of [23]). See also the proof of existence of global-in-time renormalized solutions to GAG ν ) for the special case σ 1 = 1, σ 2 = 1 in [22] assuming only that ρ L 1. We return in Section 5 to the well-posedness of weak solutions to the generalized inviscid aggregation equation. We adapt the approach of Marchioro and Pulvirenti in [21] which originates in their earlier text [2]) to prove existence and uniqueness of Lagrangian solutions to the 2D Euler equations, combining it with some ideas from Chapter 8 of Majda and Bertozzi s [19]. Marchioro and Pulvirenti s argument is both economical and elegant, but both of these virtues are impacted by the need to handle non-divergence free vector fields. Fundamentally, this is because the Jacobian of the transformation induced by the flow map is no longer 1 but involves ρ. This introduces into the argument new terms that require us to assume some regularity of the initial density. Only after proving existence with such regularity can a limiting argument be made to treat initial densities lying in L 2 L. Our proofs of existence and uniqueness are simplified by adding the assumption that ρ is compactly supported.) We will also adapt Marchioro and Pulvirenti s approach to establishing higher regularity of inviscid solutions, as their approach translates with only minor difficulties to non-divergence free vector fields. The varying effects of σ 1 and σ 2 begin to become apparent in Section 6 when we examine the behavior of the total mass of the density, mρ ν ) := R ρ ν. We will find that mρ ν ) is d conserved only when σ 1 +σ 2 =. The mass is particularly important in 2D where the energy of the solutions is infinite. The lack of finite energy was no real obstacle for studying weak solutions because only the L norm of the velocity played a role in the estimates. But when treating uniqueness of regular viscous solutions and proving that the vanishing viscosity limit holds, the L 2 norm of the velocity play an important role. As is the case with the 3D avier-stokes equations, the uniqueness of weak solutions is an open problem: we will content ourselves with proving uniqueness of solutions having sufficient regularity. Even for regular solutions uniqueness is not a simple matter, for one needs to control both the density and velocity of the difference between two solutions. Uniqueness will follow, however, from the vanishing viscosity arguments we make beginning in Section 7, where we show that V V ) : v ν v in L, T ; H 1 ), ρ ν ρ in L, T ; L 2 ) as ν for a T > as stated in Theorem 4.5. When d = 3, v ν and v both lie in L 2 R d ). This is no longer in general) the case when d = 2. When σ 1 + σ 2 =, however, because the total mass of the densities ρ ν and ρ are conserved over time, the infinite parts of the energies cancel, giving v ν v L 2 R 2 ). In both of the cases d 3 or d = 2 with σ 1 + σ 2 = V V ) holds, as we show in Section 7. In Section 8 we consider the remaining case where d = 2 but σ 1 + σ 2. In this case the total mass of the densities are not conserved over time and the infinite parts of the energies do not cancel. We will nonetheless be able to isolate the infinite parts of the energy and use them to define a spatially smooth corrector θ ν that lies in weak-l 2 and all higher L p spaces

4 4 ELAIE COZZI, GUG-MI GIE, JAMES P. KELLIHER and show that in place of V V ) we have V V ) : v ν v θ ν in L, T ; H 1 ), ρ ν ρ in L, T ; L 2 ) as ν, θ ν in L, T ; C k ) for all k. As can be seen from V V ), V V ) both the velocity and density converge strongly in the vanishing viscosity limit. Indeed, the arguments in Sections 7 and 8 involve showing the simultaneous convergence of both the velocities and the densities. In Section 9 we use the results from Sections 7 and 8 along with uniform bounds in viscosity on Hölder norms of solutions to GAG ν ) to prove that the vanishing viscosity limit holds in the L -norm of the density for more regular initial data. In Section 1 we make some concluding remarks. We follow the convention that = L 2 R d ). We write, for the L2 -inner product and, ) for the pairing in the duality between H 1 R d ) and H 1 R d ). We make the convention that C stands for an unspecified positive constant that is independent of any significant parameters. Its value may vary from expression to expression. If its explicit dependence upon certain parameters is significant we write Ca 1,..., a n ). In particular, C t) is a positive, continuous, nondecreasing function of t [, ). We will find various uses for the following cutoff function: Definition 1.1. Let a be a radially symmetric function in C R d ) taking values in [, 1], supported in B 2 ) with a 1 on B 1 ) and with ax) nonincreasing in x. For any R 1 define a R ) = a /R). ote that a R x) is nondecreasing in R for any fixed x R d. We also define b R x) = a 2R x) a R x), noting that b R is supported on the annulus of inner, outer radii R, 4R. For any p 1, p 2 [1, ], p 1 p 2, we define f L p 1 L p 2 = f L p 1 + f L p 2 for the Banach space L p 1 L p 2. If a displayed equation with equation number m.n) consists of multiple equalities then m.n) k refers to the k-th equality in the equations. For example, 4.3) 2 refers to the equation v n = σ 1 Φ ρ n 1. We will use many times the following simple form of a classical result: Lemma 1.2 Grönwall s inequality). Fix T >. Let L, α, and β be nonnegative continuous functions on the interval [, T ] with α non-decreasing and assume that M L 1, T )). If then Lt) + Lt) + Ms) ds αt) + Ms) ds αt) exp βs)ls) ds for all t [, T ] βs) ds for all t [, T ]. 2. Measuring persistence of spatial decay The main issue we will face in obtaining weak solutions to GAG ν ) is not regularity, which can be dealt with in a very classical manner, but rather spatial decay. The difficulty is that even if we assume compact support of the initial density as we will in Theorem 5.7 for the inviscid solutions diffusion ensures that compact support is lost for all positive time. We will

5 THE AGGREGATIO EQUATIO VAISHIG VISCOSITY LIMIT 5 find, however, that algebraic spatial decay as characterized by the space L 2 of Definition 2.1 will persist. We explore in this section key properties of this space that we will use in the following sections. Definition 2.1. Fix a real number, let p [1, ], and let a, b R be as in Definition 1.1. For any integer k, define the function space W k,p Rd ) to be the subspace of W k,p R d ) with the norm, f W k,p Let H k Rd ) := W k,2 Rd ), L p Rd ) := W,p Rd ). Rd ) := af W k,p R d ) + sup R b R f W k,p R d ). R 1 Lemma 2.2. Each of the following holds for all d 2: 1) Fix q [1, 2] and let n q = d 2 q with f L q C f L 2, f L q n q R d ) with f L q nq 2q. If f L2 Rd ) for some > n q then f L q R d ) C f L 2, and 1 a R )f L q CR nq) f L 2 for all R ) 2) We have, Φ f LL C f L 1 L C f L 2, 2.2) L the first inequality holding for all f L 1 L, the second holding for all f L 2 L for some > d/2. In 2.2), LL = LLR d ) is the space of bounded log-lipschitz vector fields with { } gx) gy) g LL := g L + sup x y log x y : x, y Rd, < x y e 1. 3) For any p d/d 1), ] there exists Cp) > such that for all f L 1 L p, Φ f L p C f L 1 L p. 2.3) 4) For any p d, ] there exists Cp) > such that for all f L 1 L, p 2 Φ f L Cp) f 2 p p f L + C f L ) 5) If f L 2 L for some > 1 + d/2 then for all R 1, ) d 1 2 d+1 d+1 b R Φ f) L C f L 2 + f f R 1 C f L 2 L R ) Hence, Φ f L 1 with Φ f L 1 C f L 2 L.) L Proof. 1) Fix q [1, 2]. Then for any R 1, b R f L q 1 L 2 q 2q b R f CR d 2 q 2q R f L 2 = CR nq) f supp b R ) L ) This along with af L q C af shows that f L q n q Observe that L 2 with f L q nq C f L 2. n 1 a 2 n = a 2 n a 2 n 1) + a 2 n 1 a 2 n 2) + + a 2 a 1 ) + a 1 = a + b 2 k. k=

6 6 ELAIE COZZI, GUG-MI GIE, JAMES P. KELLIHER Then, using 2.6), n 1 a 2 nf L q af L q + b 2 kf L q C f L 2 + C k= n+1 2 k nq) f L 2 C f L 2, since > n q. But a 2 n increases monotonically to 1. Thus f L q with f L q C f L 2 by the monotone convergence theorem. Similarly, 1 a 2 n)f L q C 2 k nq) f L 2 C2 n nq) f L 2. k=n Given R >, choose n such that 2 n R < 2 n+1. Then 1 a R )f L q 1 a 2 n)f L q C2 n nq) f L 2 CR nq) f L 2. So we see that more generally 2.1) holds. 2) In 2D, the first inequality in 2.2) is Lemma 8.1 of [19]. It can be proved in all dimensions in a manner very similar to that of Theorem 3.1 of [25], so we suppress the proof. The second inequality in 2.2) follows from 1). 3) We have k= Φ ρ L p a Φ L 1 ρ L p + 1 a) Φ L p ρ L 1 < for all p > d/d 1), giving 2.3). 4) Observe that for any p d, ], Φ f L a Φ) f L + 1 a) Φ) f L C a Φ L p f L p + C 1 a) Φ L f L 1, where p = p/p 1) [1, d/d 1)). But by Lebesgue interpolation, f L p f 2 p 2 p p f L, and 2.4) follows. 5) Assume that x 1, and let R = x /8 so that x / supp a 2R. We write Φ fx) Φ a R f)x) + Φ 1 a R )f)x) and bound the two terms separately. For the first term, Φ a R f)x) C x y 1 d a R y) fy) dy C x 1 d a R f L 1 C f L 2 x 1 d, supp a R since x y 3 x /4 for y supp a R. For the other term, we use 2.1) and 2.4) to obtain Φ 1 a R )f)x) C 1 a R )f 2 p 2 p p f L + C 1 a R )f L 1 C R ) 2 2 p p f L 2 p 2 p f L + CR d 2 ) f L 2. But d/2 > 1 and 2/p > 21+d/2)/p = 2+d)/p 1 as long as p d, 2+d]. Choosing p = d + 1 yields 2.5). Remark 2.3. An implication of 2.1) is that we could replace b R with 1 a R in Definition 2.1. The compact support of b R, however, makes it more convenient in most applications. Lemma 2.4. Let >. Then H 1 Rd ) is continuously and compactly embedded in L 2 R d ) and in L 2 M Rd ) for any M <. Moreover, if f n ) is a bounded sequence in H 1 Rd ) then there exists f L 2 Rd ) such that a subsequence of f n ) converges to f in L 2 M Rd ) for all M <.

7 THE AGGREGATIO EQUATIO VAISHIG VISCOSITY LIMIT 7 Proof. Let f n ) be a bounded sequence in H 1. We will construct a subsequence f n k ) and a function f L 2 Rd ) for which f nk f in L 2 M Rd ) for any M <, giving the compact embedding of H 1 Rd ) in L 2 M Rd ) and hence also in L 2 R d ). Because supp a k, k = 1, 2,... is bounded, H 1 supp a k ) is compactly embedded in L 2 supp a k ). We can thus extract a subsequence of f n ), which we relabel as f n 1) ) n=1, and a g 1 L 2 supp a 1 ) such that f n 1) supp a1 g 1 in L 2 supp a 1 ) as n. We continue this process inductively, constructing sequences f n k) ) n=1 and g k) with g k L 2 supp a k ) and f n k) supp ak g k in L 2 supp a k ) as n. At each step, we choose the subsequence f k) n f n k 1) ) n=1. We have g j supp ak = g k for j > k since g j g k L 2 supp a k ) lim n [ g k f j) n L 2 supp a k ) + g j f j) n L 2 supp a k ) ] =. ) n=1 from Hence, we can define a function f pointwise on R d by fx) = lim k g k x). Observe that 1 a k )f n < Ck for all n follows from 2.1). Hence, also 1 a k )f < Ck, so f L 2. We now construct a subsequence of f n ) as follows. We set the first term of the subsequence equal to f 1) j 1, where j 1 is chosen sufficiently large to ensure that f 1) j 1 f L 2 supp a 1 ) < 1. Proceeding inductively, we set the k-th term of the subsequence equal to f k) j k, where j k satisfies j k 1 < j k and f k) j k f L 2 supp a k ) < 1/k. Relabeling this sequence as f n ), we have that f n f L 2 supp a n) < 1/n for all n. Although we have established that the limiting function f is in L 2, we cannot conclude that f n f in L 2. We can, however, show that f n f in L 2 M for any M <, as follows. Fix M <, and observe that f n f L 2 M = af f n ) + sup R M b R f f n ) af f n ) + C sup k M b k f f n ). R 1 k ow let ε >. For any fixed k we can choose > sufficiently large that f f n L 2 supp a 2k ) < k M ε for any n and any k k. Then for all n, f n f L 2 M C sup k k k M f f n L 2 supp a 2k ) + C sup k>k k M k 1 a k )f f n ) Ck M k M ε + Ck M k k Cε + k M ). We used here that 1 a k )f f n ) 1 a k )f + 1 a k )f n Ck. Choosing k large enough that k M < ε we have f n f L 2 < Cε for all n. Since this holds for M all ε >, it follows that f n f L 2 as n. M Corollary 2.5. Let α and assume that f L 2 Rd ) for some integer > α + d/2. Then x α fx) L 1 R d ) with x α f L 1 C f L 2. Proof. For any R 1 we have R α b R x α f 4R) α R α b R f 4 α R b R f CR R f L 2 = C f L 2. This along with ax) x α fx) C af shows that x α fx) L 2 α with x α fx) L 2 α C f L 2. That x α fx) L 1 R d ) with x α f L 1 C f L 2 then follows from 1) of Lemma 2.2.

8 8 ELAIE COZZI, GUG-MI GIE, JAMES P. KELLIHER Lemma 2.6 shows that time-continuity in L 2 with boundedness in L 2 implies time-continuity in all lower Lebesgue norms. We note that this result follows for all L q -norms with q 1, 2] easily by 1) of Lemma 2.2 and Lebesgue space interpolation, so it is the L 1 case that is most important to us. Lemma 2.6. If f C[, T ]; L 2 ) L, T ; L 2 ) for > d/2 then f C[, T ]; Lq ) for all q [1, 2]. Proof. Let ε >. For any t [, T ] let δ > be small enough that fs) ft) L 2 < ε for all s [, T ] t δ, t + δ). Fix q [1, 2] and let n q = d 2 q 2q. Then by 1) of Lemma 2.2, for any R 1, fs) ft) L q a R fs) ft)) L q + 1 a R )fs) ft)) L q fs) ft) L q B 2R )) + CR nq) f L 2 CR nq fs) ft) + CR nq) f L 2 CR nq ε + CR nq). Set R = ε 1. Then nq 1 fs) ft) L q Cε, from which f C[, T ]; L q ) follows, since n q n 1 = d/2 <. 3. The linear viscous problem In this section, we investigate solutions to the linear parabolic problem, { t ξ + v f ξ = σ 2 fξ + ν ξ + g, ξ) = f), 3.1) where f, g are given functions of space and time and v f := σ 1 Φ f. In Section 4, we will use a sequence of solutions to 3.1) to obtain existence of a solution to the nonlinear problem in GAG ν ). In the limit we will have f = ξ = ρ ν, so that we will want f and ξ to exist in the same function spaces. Definition 3.1. Fix > 1 + d/2 and define the solution space Y := {h C[, T ]; L 2 ) L 2, T ; H 1 ) L, T ; L 2 ): t h L 2, T ; H 1 )}. We place no norm on Y, however.) Assume that f Y L [, T ] R d ), and let v f = σ 1 Φ f. Assume that g L 2, T ; L 2 ). We say that ξ Y is a weak solution to the linear problem 3.1) on the interval [, T ] if ξ) = f) and t ξt), ϕ)+ν ξt), ϕ σ 1 + σ 2 ) ft)ξt), ϕ v f t)ξt), ϕ = g, ϕ for a.e. t [, T ] for all ϕ H 1. Equality in 3.2) is to hold in the sense of distributions on, T ). Remark 3.2. By t ξ in Definition 3.1 we mean the weak time derivative of f see and Appendix E.5 of [11]). Thus, t ξ L 2, T ; H 1 ) means that T tξt)vt) dt = T ξt)v t) dt for all real-valued v Cc, T )), the integrations being over Banach space H 1 ) valued functions. We will show in Theorem 3.3 that we can also treat t ξ as a distributional derivative. Also, by the initial condition ξ) = f) we mean that ξt) f) in L 2 as t +, which makes sense because ξ C[, T ]; L 2 ). 3.2)

9 THE AGGREGATIO EQUATIO VAISHIG VISCOSITY LIMIT 9 In many of the proofs that follow we will want to make an energy argument by applying 3.2) with a ϕ that lies in the solution space. In other cases we wish to apply 3.2) with a test function in Cc, T ) R d ) to prove the existence of a solution. This can be justified in an entirely standard way. For completeness, we give the proof below. Theorem 3.3. Let Y := {h C[, T ]; L 2 ) L 2, T ; H 1 ): t h L 2, T ; H 1 )} and assume that f, g, and v f are as in Definition 3.1. The function ξ Y with ξ) = f) is a weak solution as in Definition 3.1 if and only if any of the following hold: 1) 3.2) holds for all ϕ Cc R d ). 2) For all ϕ Cc, T ) R d ) we have T T T t ξ, ϕ) + ξv f ϕ + σ 1 + σ 2 )fξϕ ν ξ ϕ) = gϕ or R d R d T T 3.3) ξ t ϕ + ξv f ϕ + σ 1 + σ 2 )fξϕ ν ξ ϕ) = gϕ. R d R d That is, 3.1) holds on, T ) R d in the sense of distributions. 3) For all ϕ C[, T ]; L 2 ) L 2, T ; H 1 ) T T t ξ, ϕ) + ξv f ϕ + σ 1 + σ 2 )fξϕ ν ξ ϕ) = R d 4) For all ϕ Y we have T T ξ, t ϕ)+ ξv f ϕ + σ 1 + σ 2 )fξϕ ν ξ ϕ) R d T = ξt )ϕt ) f)ϕ)) gϕ. R d R d T R d gϕ. Finally, if ξ is a weak solution as in Definition 3.1 then 1)-4) hold with any t [, T ] in place of T. Proof. We give the proof for g as forcing plays no significant role in the proofs. Assume first that ξ is a solution as given in Definition 3.1. Then 1) follows since Cc R d ) H 1. ext we prove 2). Let ϕt, x) = ϕ 1 t)ϕ 2 x), where ϕ 1 D, T ) and ϕ 2 DR d ). In light of Remark 3.2 we see that integrating 3.2) in time gives either equality in 3.3). Because D, T ) R d ) = D, T )) D R d ) by the Schwartz kernel theorem, it follows that 3.3) holds for all ϕ D, T ) R d ). This establishes 2), since Cc, T ) R d ) = D, T ) R d ) as sets. To prove 3), we first show that 3.3) 1 holds for any ϕ in C [, T ]; Cc R d )), which is clearly dense in C[, T ]; L 2 ) L 2, T ; H 1 ) and in Y ). So let ϕ C [, T ]; Cc R d )). Letting h ε be as in Lemma 3.8, define ϕ ε t, x) = h ε t)ϕt, x). Since ϕ ε Cc, T ) R d ), 3.3) 1 holds for ϕ ε by 2). But T T 3ε/2 ) T t ξ, ϕ) t ξ, ϕ ε ) + t ξt) H 1 ϕ ϕ ε )t) H 1 dt T 3ε/2 by the continuity of the Lebesgue integral. The same kind of bound holds for the other terms in 3.3) 1. Thus, 3.3) 1 holds for ϕ.

10 1 ELAIE COZZI, GUG-MI GIE, JAMES P. KELLIHER ow let ϕ n ) be a sequence in C [, T ]; Cc R d )) converging to ϕ in C[, T ]; L 2 ) L 2, T ; H 1 ). Then 3.3) 1 holds for each ϕ n, and taking advantage of f and, by 2) of Lemma 2.2, v f both lying in L, T ) R d ), it is easy to see that 3) holds in the limit as n. Then 4) follows the same way, except that we use Lemma 3.9 for the time integral. Also note that 1)-4) clearly hold with any t [, T ] in place of T. We now prove the reverse implications. That 1) implies that ξ is a weak solution follows from the density of Cc R d ) in H 1. That 2) implies 1) follows from applying either form of 3.3) with a test function of the form ϕt, x) = ϕ 1 t)ϕ 2 x), since 3.2) is to hold in the sense of distributions in time. Finally, that 3) implies 2) and that 4) implies 3) follows by handling the time integral using Lemma 3.9 as we did in the proof of 4). To obtain the existence of solutions, we use the following extremely general result due to Lions and Magenes [17]. We quote the result as it appears in Theorem 1.9 of [5]. Theorem 3.4. [J.-L. Lions] Let H be a Hilbert space with the subspace V continuously and densely embedded in H. Fix T > and suppose that at; u, v): V V R is a bilinear form satisfying for some constants M, α, C > : 1) for every u, v V the function t at; u, v) is measurable; 2) at; u, v) M u V v V for a.e. t [, T ] for all u, v V ; 3) at; v, v) α v 2 V C v 2 H for a.e. t [, T ] for all v V. Given g L 2, T ; V ) and u H, there exists a unique u with such that u) = u and u L 2, T ; V ) C[, T ]; H) and t u L 2, T ; V ) t ut), v) + at; ut), v) = gt), v) for a.e. t [, T ] for all v V. We will apply Theorem 3.4 with H = L 2 R d ), V = H 1 R d ), and noting that formally, at; u, v) = ν u, v σ 1 + σ 2 ) fu, v v f, u v, 3.4) t ξt), ϕ + at; ξt), ϕ) = t ξt) ν ξt) σ 2 fξt) + v f ξt), ϕ, in accordance with 3.1). Theorem 3.5. Let ν > and T > and let > 1 + d/2. There exists a unique weak solution to 3.1) as in Definition 3.1. Moreover, the norms on ξ in L 2, T ; H 1 ) and in L, T ; L 2 ) can be bounded strictly in terms of f) L 2, f L 2,T ;L 2 L ), ξ L 2,T ;L 2 ), g L 2,T ;L 2 ), and ν 1. Proof. Let H = L 2 R d ), V = H 1 R d ), and define the bilinear form on V V as in 3.4). Then the existence and uniqueness of a weak solution ξ as in Definition 3.1 but with ξ L 2, T ; H 1 ) C[, T ]; L 2 ) and test functions in H 1 follows from Theorem 3.4 once we verify the three required properties of a as follows: 1) This follows from f Y L, t) R d ) and 2.2). 2) We have at; u, v) ν u v + σ 1 + σ 2 f L u v + v f L u v M u V v V.

11 THE AGGREGATIO EQUATIO VAISHIG VISCOSITY LIMIT 11 3) We have, But at; v, v) = ν v 2 σ 1 + σ 2 ) fv, v v f, v v. v f, v v = 1 2 v f, v 2 = 1 2 div v f, v 2 = σ 1 2 f, v 2. A density argument is used to integrate by parts here. We simply note that we have sufficient regularity and decay of f and v so that the first and last expressions make sense.) Thus at; v, v) = ν v 2 σ 1 + 2σ 2 fv, v 2 ν v 2 σ 1 + 2σ 2 f 2 L v 2 = ν v 2 C v 2 = ν v 2 + v 2 ) C + ν) v 2 = ν v 2 V C + ν) v 2 H. We now show that, in fact, ξ Y we note that it is the inability to verify 3) for H = L 2, V = H 1 that requires us to explicitly show this). By 4) of Theorem 3.3 applied with the test function ϕ = b 2 Rξ Y, we have By Lemma 3.9, we have t ξ, b 2 Rξ + ξv f, b 2 Rξ) + σ 1 + σ 2 ) f, b 2 Rξ 2 ν ξ, b 2 Rξ) ) = b R ξt) 2 b R f) 2 t ξ, b 2 Rξ = Thus 3.5) can be rewritten as g, b 2 Rξ. t ξ, b 2 Rξ + b R ξt) 2 b R f) 2. t ξ, b 2 Rξ ξv f, b 2 Rξ) σ 1 + σ 2 ) f, b 2 Rξ 2 + ν ξ, b 2 Rξ) ) = g, b 2 Rξ. ow observe that b 2 Rξ t ξ = 1 b R ξt) 2 b R f) 2), R d 2 ξv f, b 2 Rξ) = v f, 2b R ξ 2 b R + b 2 Rξ ξ 2 b R v f L b R L ξ 2 + b R v f L ξ b R ξ Cν 1 R 2 ξ 2 + ν 4 b R ξ 2, f, b 2 Rξ 2 f L b R ξ 2, ν ξ, b 2 Rξ) = ν ξ, 2b R ξ b R + b 2 R ξ = ν b R ξ 2 + 2ν b R ξ ξ, b R, 2ν b R ξ ξ, b R 2ν b R L ξ b R ξ CνR 1 ξ b R ξ CνR 2 ξ 2 + ν 4 b R ξ 2, g, b 2 Rξ b R g b R ξ 1/2) b R g 2 + 1/2) b R ξ )

12 12 ELAIE COZZI, GUG-MI GIE, JAMES P. KELLIHER We used 5) of Lemma 2.2 to bound b R v f in L, and we used Young s inequality in the second inequality above and in the estimates for the last two terms. In the one term involving a time derivative we applied Lemma 3.9, using t ξ, b 2 R ξ) = tb 2 Rξ), ξ). Thus, b R ξt) ν b R ξ 2 b R f) 2 + b R ξ 2 + Applying Lemma 1.2 Grönwall s lemma) gives b R ξt) 2 + ν b R g 2 ) C νr 2 ξ 2 + C b R ξ 2. b R ξ 2 CR 2 + C t)ν 1 tr 2) e C t)t, 3.6) where we used f) L 2, g L2, T ; L 2 ) for some > 1 + d/2 with 2 an integer. It follows from 3.6) that b R ξt) 2 + ν b R ξ 2 C t)r 2k ν k e C t)t 3.7) for k = 1 and all ν ν for any fixed ν >. We now proceed by induction. Let Sk) be the statement that 3.7) holds. We have shown that Sk) is true for k = 1; now suppose that it holds up to some k 1 <. We refine the estimates, ξv f, b 2 Rξ) Cν 1 R 2 ξ 2 L 2 supp b R ) + ν 4 b R ξ 2, 2ν b R ξ ξ, b R CνR 2 ξ 2 L 2 supp b R ) + ν 4 b R ξ 2. But since b 2R x) + b R x) + b R/2 x) = 1 on the support of b R, we can write ξ 2 L 2 supp b R ) b 2R + b R + b R/2 )ξ 2 ν k 1) C t)r/2) 2k 1) e C t)t + C t)r 2k 1) e C t)t + C t)2r) 2k 1) e C t)t ) C t)r 2k 1) ν k 1) e C t)t. This follows from 3.7) applied with R as well as with R replaced by both R/2 and 2R. With these refinements the argument that led to 3.6) now gives b R ξt) 2 + ν b R ξ 2 CR 2 + C t)ν k tr 2k) e Ct)t. So 3.7) holds for k as well by induction. We must stop at k =, however, the bounds b R f) 2, b R g 2 L 2,T ;L 2 ) CR 2 being the limiting factors. This shows that ξ L, T ; L 2 ) L2, T ; H 1 ). Finally, the dependence of the various constants C t) on the data occurs only through f) L 2 and f L 2,T ;L 2 L ). Hence, the norms on ξ in L2, T ; H 1 ) and in L, T ; L 2 ) can be bounded strictly in terms of f) L 2, f L 2,T ;L 2 L ), ξ L 2,T ;L 2 ), and g L 2,T ;L 2 ). Theorem 3.6. Fix an integer k and define the space Y k := {h C[, T ]; H k ) L 2, T ; H k+1 ) L, T ; H k ): t h L 2, T ; H k 1 )}

13 THE AGGREGATIO EQUATIO VAISHIG VISCOSITY LIMIT 13 but place no norm on it). Assume that f Y k, g L2, T ; H k ). Let ξ be the unique weak solution to 3.1) given by Theorem 3.5. Then ξ Y k. Proof. This regularity result for L 2 -based spaces rather than L 2 -based spaces is classical, based on a sequence of smooth Galerkin approximations to the solution. We give only a formal bootstrapping argument to explain how to obtain the L 2 -based result from the L2 - based result. Taking i of 3.1) for any i = 1,..., d gives formally with i ξ) = i f), where t i ξ + v f i ξ = σ 2 f i ξ + ν i ξ + G, 3.8) G := σ 2 i fξ i v f ξ + i g. That is, i ξ also satisfies 3.1) with different forcing and initial data. But G L 2, T ; L 2 ), so by Theorem 3.5 there exists a unique weak solution, which we will call γ i, to 3.1) with forcing function G and initial data i f). ow, γ i is a weak solution to 3.1) and i ξ is formally a solution to 3.1) with the same initial data and forcing as γ i. In fact, γ i = i ξ follows from the Galerkin-approximation argument referred to earlier. Hence, i ξ Y. Because this holds for all i we have ξ Y and thus ξ C[, T ]; H 1 ) L 2, T ; H 2 ) L, T ; H 1 ). Since 3.1) holds in the sense of distributions by 2) of Theorem 3.3, it follows that t ξ L 2, T ; H k 1 ) so that ξ Y k. This gives the result for k = 1. The result follows for any k by repeating this same process k 1 more times. In Theorem 3.7 we obtain uniform-in-viscosity bounds on the norms of ξ in L [, T ]; L q ) over time for sufficiently regular solutions. These bounds can be obtained for weak solutions as well but only with considerable additional technical difficulties due to the lack of a priori knowledge that the solution is continuous over time in the L q -norm. Because our use of Theorem 3.7 is only to show the analogous result for nonlinear solutions in the next section, we limit ourselves to regular solutions, which is all we will need. Theorem 3.7. Let ν, T > and assume that f Y k for some > 1 + d/2, k > d/2. Let ξ be the regular solution to 3.1) given by Theorem 3.6 without forcing g ). For any q [1, ] we have σ ) 1 ξt) L q f) L q exp q + σ 2 fs) L ds. 3.9) Further, for q = 2 we have ξt) 2 + 2ν σ1 ξ 2 f) 2 exp + 2σ 2 fs) L ) ds. 3.1) Proof. By Theorem 3.6 and Sobolev embedding ξ C[, T ]; L q ) for all q [2, ), while by Lemma 2.6 ξ C[, T ]; L q ) for all q [1, 2). Assume that q is a rational number in 1, ) with q = m 1 /m 2 in lowest terms for m 1 even. This insures that ξ q. The conclusions we reach for such rational q s will hold for all q [1, ) by the continuity of Lebesgue norms. If done formally, the argument we will make is very simple: multiply 3.1) by ϕ = ξ q 1, integrate over space and time, perform several integrations by parts, and in the end obtain a bound on ξt) L q. In fact, for q 2 there is little more to the argument since we have restricted ourselves to sufficiently regular solutions. One of these integrations by parts, however, introduces a factor of ξ q 2 which is singular when q < 2. We will remove this

14 14 ELAIE COZZI, GUG-MI GIE, JAMES P. KELLIHER singularity by multiplying ξ q 1 by a factor that vanishes when ξ is near zero. This factor will be derived from a function λ ε C R) parameterized by ε, 1/2) and defined so that { 3/2)ε, x < ε, λ ε x) = x, x > 2ε and so that λ ε, λ ε with λ ε C where C is independent of ε. So instead of simply using ϕ = ξ q 1 we use ϕ := λ εξ q )ξ q 1. ϕ = f ε ξ)ξ q, where f ε x) := λ εx q )/x. Then f ε C R) with f ε L 3/2)ε) 1 λ L ε Cε 1, f ε L = xλ εx q )qx q 1 λ εx q ) x 2 L x Cε 2 Cq2ε) q + C) Cε 2 + ε q 2 ). We can write this as ) 2 2 qλ 3ε εx q )x q L + λ x εx q ) ) L x It follows immediately from this that ϕ L, T ; L 1 L ) since ξ q belongs in this same space. For time continuity, we have ϕt) ϕs) L r f ε ξs))ξ q t) ξ q s)) L r + ξ q t)f ε ξt)) f ε ξs))) L r f ε L ξ q t) ξ q s) L r + f ε L ξ q t) L ξt) ξs) L r. We conclude that ϕ C[, T ]; L r ) for all r [1, ). Then ϕ = q 1)λ εξ q )ξ q 2 ξ+qλ εξ q )ξ 2q 1) ξ L 2, T ; L 2 ) since ξ L2, T ; L 2 ) and the singularity in ξ q 2 is removed because λ εx) = for x < ε. Hence, ϕ L 2, T ; H 1 ) and thus has sufficient regularity to apply in 3) of Theorem 3.3. This gives t ξλ εξ q )ξ q 1 = ξv f λ εξ q )ξ q 1 ) R d R d t + σ1 + σ 2 )fξλ εξ q )ξ q 1 ν ξ λ εξ q )ξ q 1 ) ) 3.11). R d We were able to replace the pairing in the first integral by a spatial integral because t ξ L 2, T ; H k 1 ) by Theorem 3.6 and ϕ = λ εξ q )ξ q 1 L, T ; L 2 ). Because of the regularity and decay given by Theorem 3.6, we can easily integrate all but the time integral by parts, as follows: ξv f λ εξ q )ξ q 1 ) = divξv f )λ εξ q )ξ q 1 R d R d = σ 1 fλ R εξ q )ξ q v f ξ)λ εξ q )ξ q 1, d R d v f ξ)λ εξ q )ξ q 1 = 1 v f λ ε ξ q )) = 1 divv f )λ ε ξ q ) R d q R d q R d = σ 1 fλ ε ξ q ) q R d fξλ εξ q )ξ q 1 = fλ εξ q )ξ q, R d R d ν ξ λ εξ q )ξ q 1 ) = q 1)ν λ εξ q )ξ q 2 ξ 2 R d R d qν λ εξ q )ξ 2q 1) ξ 2. R d

15 THE AGGREGATIO EQUATIO VAISHIG VISCOSITY LIMIT 15 ote that fλ ε ξ q ) L, T ; L 1 L ) since f lies in this same space and λ ε ξ q ) L, T ; L ). In the last inequality we used λ ε, λ ε to conclude that the two integrals were. For q = 2, though, these terms would simplify to ν λ εξ 2 ) ξ 2 2ν λ εξ 2 )ξ 2 ξ ) R d R d We will return to this issue at the end of the proof. From 3.11) we now have t ξλ εξ q )ξ q 1 σ 2 fλ εξ q )ξ q + σ ) 1 R d R d q fλ εξ q ). 3.13) We will now take lim ε of each term in 3.13). The two terms on the right-hand side are easy: Since λ ε ξ q ) ξ q only on the set E ε t) := {x R d : ξ q t, x) < 2ε}, we have fλ ε ξ q ) fξ q fλ ε ξ q ) ξ q ) R d R d E εt) 4ε f 4ε f L 1,T ) R d ) Cε, R d fλ εξ q )ξ q fξ q fξ q λ εξ q ) 1) R d R d R d t fξ q λ εξ q ) 1 Cε f, E εt) R d since λ ε C, ξ q 2ε on E ε t), and λ εξ q ) = 1 on R d \ E ε t). Hence as ε, fλ ε ξ q ) fξ q, fλ εξ q )ξ q fξ q. R d R d R d R d This leaves the time integral. We have, t ξλ εξ q )ξ q 1 = lim a R t ξλ εξ q )ξ q 1 = 1 R d R R d q lim t a R λ ε ξ q )) R R d = 1 q lim a Rλ ε ξt) q ) L 1 a R λ ε f) q ) L 1) = 1 R q lim a R λ ε ξt) q ) λ ε f) q )) R R d = 1 λ ε ξt) q ) λ ε f) q )). q R d The first equality holds by the dominated convergence theorem, and the third follows by integrating by parts in time. For the final equality, we used that λ ε ξt, x) q ) λ ε f, x) q ) = λ εξt, x) q ) λ ε f, x) q ) ξt, x) q f, x) q ξt, x) q f, x) q λ ε L ξt, x) q f, x) q = C ξt, x) q f, x) q to conclude that λ ε ξt) q ) λ ε f) q ) L 1 R d ) even though neither term alone lies in L 1 R d )). This allowed us to apply the dominated convergence theorem to take R. This same bound then also allows us to apply the dominated convergence theorem to take ε, so that lim ε R d t ξλ εξ q )ξ q 1 = 1 q R d ξt) q f) q ) = 1 q ξt) q L q f) q L q ).

16 16 ELAIE COZZI, GUG-MI GIE, JAMES P. KELLIHER Therefore, in the limit as ε, 3.13) becomes 1 ξt) q L f) q ) t q q L q R d ) σ1 q + σ 2 fξ q. Rearranging and applying Lemma 1.2 Grönwall s lemma) gives ξt) q L f) q q L exp σ q 1 + qσ 2 fs) L ) ds, 3.14) from which 3.9) follows for all q [1, ]. The result for q = 2 in 3.1) can be obtained by keeping the terms in 3.12). It can also be obtained directly by a simplification of the argument we just gave, since there is no need to introduce either a R or λ ε when q = 2. We used the following lemmas in the proofs above. Lemma 3.8. Let T > and fix t [, T ]. We can define functions h ε D, T )) parameterized by ε, t /2) such that h ε = 1 on [3ε/2, t 3ε/2], h ε = on [, ε/2] [t ε/2, T ], h ε, and h ε ) δ ) δ t ) as ε, the convergence being as Radon measures on [, T ]. Proof. Let η Cc R) be supported on 1/2, 1/2), nonnegative with R η = 1, and define η ε ) = ε d η /ε). Set h ε t) := η ε s ε) η ε s t + ε)) ds. It is easy to see that h ε has all the stated properties. Lemma 3.9. For all ξ, ϕ Y, T t ξ, ϕ) = T ξ, t ϕ) + ξt ), ϕt )) ξ), ϕ)). Proof. Fix ξ Y and assume at first that ϕ C [, T ]; Cc R d )). Let h ε be as in Lemma 3.8. Then h ε ϕ D, T ) R d ) so T lim ε t ξ, h ε ϕ) = lim = lim = T ε T ε T ξ, t h ε ϕ)) T h ε t)ξt), t ϕt)) dt lim h ε εt)ξt), ϕt)) dt ξt), t ϕt)) dt + ξt ), ϕt )) ξ), ϕ)). In the last step we used the dominated convergence theorem for the first integral, since h ε t)ξt), t ϕt)) ξt) H 1 t ϕt) H 1 L 1 [, T ]) and in the second integral we used the continuity of ξt), ϕt)) in time. The result then follows from the density of C [, T ]; C c R d )) in Y.

17 THE AGGREGATIO EQUATIO VAISHIG VISCOSITY LIMIT The nonlinear viscous problem Definition 4.1 gives our definition of a weak solution to the generalized aggregation equation. This definition applies for both viscous and inviscid solutions. In this section we treat viscous solutions, leaving inviscid solutions to Section 5. Definition 4.1. Fix > 1 + d/2 and let Y be as in Definition 3.1. Let ν and ρ L 2 L. We say that ρ ν Y is a weak solution to the generalized aggregation equations GAG ν ) on the interval [, T ] with initial density ρ if ρ ν ) = ρ with t ρ ν t), ϕ)+ν ρ ν t), ϕ σ 1 + σ 2 ) ρ ν t)) 2, ϕ ρ ν t)v ν t), ϕ = for a.e. t [, T ] for all ϕ H 1. Equality in 4.1) is to hold in the sense of distributions on, T ). 4.1) Theorem 4.2. The function ρ ν Y L [, T ] R d ) with ρ ν ) = ρ is a weak solution as in Definition 4.1 if and only if any of the following hold: 1) 4.1) holds for all ϕ Cc R d ). 2) For all ϕ Cc, T ) R d ) we have T T t ρ ν, ϕ) + ρ ν v ν ϕ + σ 1 + σ 2 )ρ ν ) 2 ϕ ν ρ ν ϕ ) = or R d T ρ ν t ϕ + ρ ν v ν ϕ + σ 1 + σ 2 )ρ ν ) 2 ϕ ν ρ ν ϕ ) =. R d That is, GAG ν ) holds on, T ) R d in the sense of distributions. 3) For all ϕ C[, T ]; L 2 ) L 2, T ; H 1 ) T T t ρ ν, ϕ) + ρ ν v ν ϕ + σ 1 + σ 2 )ρ ν ) 2 ϕ ν ρ ν ϕ ) =. R d 4) For all ϕ Y we have T T ρ ν, t ϕ) + ρ ν v ν ϕ + σ 1 + σ 2 )ρ ν ) 2 ϕ ν ρ ν ϕ ) R d 4.2) = ρ ν T )ϕt ) ρ ϕ)). R d Finally, if ρ ν is a weak solution as in Definition 4.1 then 1)-4) hold with any t [, T ] in place of T. Proof. This follows from Theorem 3.3, since ρ ν is a weak solution to 3.1) as in Definition 3.1 with f = ρ ν and ρ = f). Remark 4.3. In Theorem 4.2 we used the assumption that ρ ν lies not just in Y, but also in L [, T ] R d ), so that it can serve as a valid f in Definition 3.1. It is only in showing that 4) follows from ρ ν being a weak solution to the nonlinear problem that this additional assumption is required, however. To establish the existence of solutions as in Definition 4.1 we will construct a sequence of approximations as follows: ρ t, x) = ρ x), v n = σ 1 Φ ρ n 1, t ρ n + v n ρ n = σ 2 ρ n 1 ρ n + ν ρ n, ρ n ) = ρ 4.3)

18 18 ELAIE COZZI, GUG-MI GIE, JAMES P. KELLIHER for n = 1, 2,.... In 4.3) 2,3,4 ρ n is a regular solution to the linear problem as given by Theorem 3.6 with f = ρ n 1. ote that div v n = σ 1 ρ n 1. Proposition 4.4. Fix T > with T < σ 2 ρ L ) 1 or T < if σ 2 =. Assume that ρ H k for some > 1 + d/2, k > d/2. Let ν >, n, t [, T ]. We have When σ 2 we have ρ n t) 2 + 2ν When σ 2 = we have ρ n t) L ρ L 1 σ 2 ρ L t. 4.4) ρ n t) L q ρ L q 1 σ 2 ρ L t) σ 1 +1 qσ 2 q [1, ), ρ n 2 = ρ 2 1 σ 2 ρ L t) σ 1 4.5) +2 σ 2. ρ n t) 2 + 2ν ρ n t) L q ρ L q exp σ 1 q 1 ρ L t ) q [1, ), ρ n 2 = ρ 2 exp σ 1 ρ L t). 4.6) Proof. We proceed by induction. Let Sk) be the statement that 4.4) holds for n = k. Certainly, S) holds trivially. Assume that Sn 1) holds. We will use this to show that Sn) holds. From 3.9) with ξ = ρ n, f = ρ n 1 ) we have ρ n t) L q ρ L q exp ow, by the induction hypothesis, ρ n 1 s) L q 1 σ 1 + σ 2 ρ n 1 s) L ρ ds L 1 σ 2 ρ L s ds { σ2 1 log 1 σ = 2 ρ L t), σ 2, ρ L t, σ 2 =. ) ds. 4.7) Taking the limit as q of both sides of 4.7), it follows by the continuity of Lebesgue norms that for σ 2, ρ n t) L ρ L exp log 1 σ 2 ρ L t)) = ρ L 1 σ 2 ρ L t) 1, and ρ n t) L ρ L if σ 2 =. Thus, Sn 1) = Sn), and we see by induction that in fact Sn) holds for all n. Returning to 4.7), we see that 4.5) 1 and 4.6) 1 hold. The bounds in 4.5) 2, 4.6) 2 follow by the same argument specifically for q = 2 and using the energy bound in 3.1). Theorem 4.5. Fix T > with T < σ 2 ρ L ) 1 or T < if σ 2 =. ote that [, T ] is within the time of existence for the inviscid problem see Theorem 5.7). Let ν > and assume that ρ L L 2 for some > 1 + d/2. Then there exists a weak solution to GAG ν ) as in Definition 4.1 on the time interval [, T ] with ρ ν t) L ρ L 1 σ 2 ρ L t. 4.8)

19 When σ 2, we have ρ ν t) 2 + 2ν THE AGGREGATIO EQUATIO VAISHIG VISCOSITY LIMIT 19 ρ ν t) L q ρ L q 1 σ 2 ρ L t) σ 1 +1 qσ 2 q [1, ), ρ ν 2 ρ 2 1 σ 2 ρ L t) σ 1 4.9) +2 σ 2. When σ 2 =, we have ρ ν t) 2 + 2ν ρ ν t) L q ρ L q exp σ 1 q 1 ρ L t ) q [1, ), ρ ν 2 ρ 2 exp σ 1 ρ L t). 4.1) Further, if ρ H k for a positive integer k then ρν Y k. The space Y k is defined in Theorem 3.6.) Proof. Because of the bounds in Proposition 4.4, we can make a standard argument to prove the existence of solutions along the same lines as that for the existence of solutions to the avier-stokes equations for instance, see pages of [9]). For completeness, we give a full argument here. Let ρ,j t, x) = η 1/j ρ x), j = 1, 2,..., where η is a Friedrich s mollifier and note that ρ,j Y m for all m. Let ρj) n ) be the sequence of linear solutions defined in 4.3), but with the starting solution being ρ,j instead of ρ. ote that ρ j) n Y m for all m as well by Theorem 3.6. Then define ρ n := ρ n) n. By Proposition 4.4 and Theorem 3.5 we see that ρ n ) is bounded in L, T ; L 2 ) L 2, T ; H 1 ) L2, T ; H 1 ). But L 2, T ; H 1 ) is weakly compact, so ρ n ) converges weakly to some ρ in L 2, T ; H 1 ). The compact embedding given by Lemma 2.4 implies that some subsequence of ρ n ), which we relabel as ρ n ), converges strongly to ρ in L 2, T ; L 2 M ) for all M < ; hence also ρ n t) ρt) in L 2 M for all M < for almost all t [, T ]. Moreover, by the uniqueness of limits, Lemma 2.4 also gives that ρt) L 2 for almost all t in [, T ]. Let ϕ Cc, T ) R d ). Then using 2) of Theorem 3.3, since each ρ n is a weak solution as in Definition 3.1 with f = ρ n 1, we have T = lim ρn t ϕ + ρ n v ρn 1 ϕ + σ 1 + σ 2 )ρ n 1 ρ n ϕ ν ρ n ϕ ). n R d ow, ρ n and ρ n 1 are each bounded in L 1 L by Proposition 4.4, and v ρn 1 is bounded in L as well by 2) of Lemma 2.2. This allows us to apply the dominated convergence theorem to the first three terms. We can then take the limit as n approaches infinity, using ρ n t), ρ n 1 t) ρt) in L 2 M and thus in L2 ) and for almost all t [, T ]. By 1) and 4) of Lemma 2.2, v ρn 1 t) vt) in L for almost all t [, T ], and we conclude that T = lim ρn t ϕ + ρ n v ρn 1 ϕ + σ 1 + σ 2 )ρ n 1 ρ n ϕ ν ρ n ϕ ) n R d T = ρ t ϕ + ρv ϕ + σ 1 + σ 2 )ρ 2 ϕ ν ρ ϕ ). R d Here, we also used that ρ n ρ weakly in L 2, T ; H 1 ). By Proposition 4.4, ρ n ) is uniformly bounded in L [, T ] R d ). Hence ρ L [, T ] R d ). Since ρ n ) ρ in L 2, it follows from Theorem 4.2 that ρ ν = ρ is a weak solution to

20 2 ELAIE COZZI, GUG-MI GIE, JAMES P. KELLIHER GAG ν ) as in Definition 4.1. Then for all q 2, ρt) L q ρt) ρ n t) L q + ρ n t) L q ρt) ρ n t) 2 q ρt) L + ρ n t) L ) q 2 q + ρ n t) L q. Taking the limit as n and using the bounds in 4.5) and 4.6) gives 4.9) and 4.1) for q 2, ) and then, by the continuity of Lebesgue norm, we obtain 4.8). For q = 2, 4.9) and 4.1) follow directly from 4.5) and 4.6), using also that the convergence of ρ n to ρ weakly in L 2, T ; H 1 ) implies that ρ n L 2,T ;H 1 ) ρ L 2,T ;H 1 ). For q [1, 2) we apply 1) of Lemma 2.2 to show that ρt) ρ n t) L q and so obtain 4.9) and 4.1). That ρ Y k implies ρν Y k follows similarly using Theorem 3.6. Uniqueness of solutions to GAG ν ), even for higher regularity solutions, is not an entirely simple matter. It will follow as a consequence of our proofs of the vanishing limit in Sections 7 and 8 see Theorem 7.5). 5. The inviscid problem Establishing existence and uniqueness of weak solutions in Eulerian variables as formulated in Definition 4.1 is quite difficult. This is in contrast to the 2D Euler equations, for which existence of solutions for bounded initial vorticity can be established quite easily using a sequence of solutions to the avier-stokes equations. This argument for the 2D Euler equations is possible because the L p -norms of the avier-stokes vorticity ω can be bounded uniformly over viscosity in the whole plane or in a bounded domain if ω = u n = is used as a boundary condition; see, for instance, Section 4.1 of [18] for the bounded domain argument). Then a weak solution to the Euler equations can be obtained using the velocity formulation. A velocity formulation of GAG ν ) is possible as we briefly describe in Section 1, but a long detour into developing properties of the pressure is required in order to properly develop a weak formulation. Instead, we will work with Lagrangian solutions, adapting the economical and elegant proofs for 2D solutions to the Euler equations given by Marchioro and Pulvirenti in [21], which originates in their earlier text [2]. We will also use some of the ideas from Chapter 8 of [19]. In [3], the authors obtain well-posedness in the special case of AG ). Their approach could in principle be extended to the more general equations in GAG ). In brief, the authors of [3] first construct smooth solutions then use an approximating sequence of such solutions to obtain a Lagrangian solution by demonstrating convergence of the flow maps as in [19]). This approach is reversed in [21], where Lagrangian solutions are first constructed by obtaining the convergence of a sequence of flow maps for approximating linearizations of the 2D Euler equations. A very simple argument then shows that regularity of the initial data is propagated over time. Considerable complications arise when adapting Marchioro and Pulvirenti s arguments for GAG ) because the underlying velocity field is not divergence-free analogous complications are dealt with in [3]). This requires the assumption of some regularity on the initial data to obtain weak solutions, removing this assumption a posteriori via a separate argument similar to the proof of existence in Chapter 8 of [19]. Before giving the proof of well-posedness of inviscid solutions, let us motiviate our definition of a Lagrangian solution by observing formally that if ρ = ρ solves GAG ) and X is the

21 flow map for v = v then THE AGGREGATIO EQUATIO VAISHIG VISCOSITY LIMIT 21 d dt ρt, Xt, x)) = σ 2ρt, Xt, x)) 2. Integrating along flow lines gives ρt, Xt, x)) = ρ x) 1 σ 2 tρ x). 5.1) This motivates the following definition of a Lagrangian solution to GAG ): Definition 5.1. Fix T >. Let X : [, T ] R d R d with Xt, ) a homeomorphism for all t [, T ] and let ρ L R d ). Define ρ: [, T ] R d R by ρt, x) = ρ X 1 t, x)) 1 σ 2 tρ X 1 t, x)) 5.2) and let v := σ 1 Φ ρ. Here, X 1 is defined by X 1 t, Xt, x)) = x for all t, x) [, T ] R d. Then X or more fully X, ρ, v) is a Lagrangian solution to the inviscid generalized aggregation equations GAG ) with initial density ρ if X is the flow map for v; that is, if for all t, x R d. Xt, x) = x + vs, Xs, x)) ds The form of ρ in 5.1) or 5.2) also yields a sharp time of existence for our Lagrangian solutions. If we do not consider the sign of ρ we obtain an upper limit on the existence time that is the same as that for viscous solutions in Theorem 4.5. Hence we should expect that if say, σ 2 < and ρ >, so that the inviscid solution exists for all time, then the existence time for viscous solutions might be considerably longer than the bound given in Theorem 4.5. An open question is whether, for all sufficiently small viscosity, viscous solutions to GAG ν ) exist for as long as the inviscid solution exists, as was established for the 3D avier-stokes and Euler equations in [8]. Because we do not assume that the initial data has a distinguished sign that is compatible with the signs of σ 1, σ 2, we do not have a maximum or comparison principle. This is what makes this a difficult problem. Issues of existence times of viscous solutions in relation to the total mass of ρ have been well-studied for AG ν ): see Sections 5.2 and 5.3 of [23].) We will need the following simple proposition that shows that the Lebesgue norms of the density of Lagrangian solutions depend only upon the initial density and time. Proposition 5.2. Let X, ρ, v) be a Lagrangian solution as in Definition 5.1 with initial density ρ L 1 L. If T < σ 2 ρ L ) 1 or T < if σ 2 =, then for all t [, T ], we have ρ ρt) L L 1 σ 2 ρ L t, 5.3) and for all q [1, ), ρt) L q { ρ L q 1 σ 2 ρ L t) σ 1 qσ 2 1, σ2, ρ L q exp σ 1 q 1 ρ L t ), σ 2 =. 5.4) Moreover, vt) L C T ) for all t [, T ], and if ρ is compactly supported, then there exists R = RT ) such that ρt) remains supported in B RT ) ) for all t [, T ].

22 22 ELAIE COZZI, GUG-MI GIE, JAMES P. KELLIHER Proof. The bound in 5.3) follows immediately from 5.2). For q [1, ), we have ρ y) ) ρt) L q = q 1 q det Xt, y) dy, 1 σ 2 tρ y) where we made the change of variables x = Xt, y). But R d s det Xs, y) = div vs, Xs, y)) det Xs, y) = σ 1 ρs, Xs, y)) det Xs, y) = σ 1ρ y) 1 σ 2 sρ y) det Xs, y) 5.5) see, for instance, page 3 of [7]) so that Jt, y) := det Xt, y) = { 1 σ 2 tρ y)) σ 1 σ 2, σ 2, e σ 1ρ y)t, σ 2 =. Hence when σ 2, ) 1 ρt) L q = ρ y) q 1 σ 2 tρ y)) σ 1 q q σ 1 dy ρ L q 1 σ 2 ρ L t) σ 1 R d qσ ) This, with the analogous bound for σ 2 =, gives 5.4). Then vt) L C T ) for all t [, T ] follows from 5.3), 5.4) for q = 1, and 2.2). The bound on the velocity then immediately yields the bound on the compact support of ρt). In what follows, we make use of three lemmas that appear at the end of this section as well as Lemma 9.5, which we defer to Section 9 because we use Littlewood-Paley theory in its proof. We start by showing that a Lagrangian solution if it exists maintains the Hölder regularity of the initial density. Theorem 5.3. Let ρ C k,α R d ) L 1 R d ) for some integer k and α, 1). Fix T > with T < σ 2 ρ L ) 1 or T < if σ 2 =. Assume that ρ is a Lagrangian solution to GAG ) on the interval [, T ]. Then ρ L, T ; C k,α ). Moreover, ρt) C k,α Ct, σ 1, ρ L 1, ρ C k,α), 5.7) and when k 1, ρ is a classical solution with ρ C k, T ; C k,α ). Proof. It follows as in Theorem of [7]) that x Xt, x) x has norm 1 in C θt), where θt) = e ct, c = C T ) ρ L 1 L, and this same bound holds for the inverse flow map. Thus, by Lemma 5.9, with ρt) C αθt), ρt) C αθt) = ρt) L + ρt) Ċαθt) C T ) ρ L + C T ) ρ Ċα C T ) ρ L + C T ) ρ Ċα = C T ) ρ C α. Hence by Lemma 9.5, vt) = σ 1 Φ ρt) C 1,αθt) with X 1 t) α Ċ θt) vt) C αθt) C ρt) L 1 + C ρt) C αθt) C T ) ρ L 1 + ρ C α), 5.8) where we also used Proposition 5.2.