HYDRODYNAMIC LIMIT OF THE KINETIC CUCKER-SMALE FLOCKING MODEL TRYGVE K. KARPER, ANTOINE MELLET, AND KONSTANTINA TRIVISA

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1 HYDODYNAMIC LIMIT OF THE KINETIC CUCKE-SMALE FLOCKING MODEL TYGVE K. KAPE, ANTOINE MELLET, AND KONSTANTINA TIVISA Abstract. The hyroynamic limit of the kinetic Cucker-Smale flocking moel is investigate. The starting point is the moel consiere in [19], which in aition to the free-transport of iniviuals an the Cucker-Smale alignment operator, inclues a strong local alignment term. This term was erive in [2] as the singular limit of an alignment operator ue to Motsch an Tamor [25]. The moel is enhance with the aition of noise an a confinement potential. The objective of this work is the rigorous investigation of the singular limit corresponing to strong noise an strong local alignment. The proof relies on the relative entropy metho an entropy inequalities which yiel the appropriate convergence results. The resulting limiting system is an Euler-type flocking system. Contents 1. Introuction 1 2. Existence theory 4 3. Main result 6 4. Proof of Theorem Appenix A. Local well-poseness of the Euler-flocking system 19 eferences Introuction Mathematical moels aime at capturing parts of the flocking behavior exhibite by animals such as birs, fish, or insects, are currently receiving wiesprea attention in the mathematical community. Many of these moels have sprung out in the wake of the seminal paper by Cucker an Smale [9]. The typical approach is base on particle moels where each iniviual follows a simple set of rules. To ate, the majority of stuies on flocking moels have been on the behavior of the particle moel or the corresponing kinetic equation. For practical purposes, if the number of iniviuals in the flock is very high, it might be esirable to ientify regimes where the complexity of the moel may be reuce. This paper is a moest contribution to this complex task. The starting Date: May 3, Mathematics Subject Classification. Primary:35Q84; Seconary:35D3. Key wors an phrases. flocking, kinetic equations, hyroynamic limit, relative entropy, Cucker-Smale, self-organize ynamics. The work of T.K was supporte by the esearch Council of Norway through the project The work of A.M was supporte by the National Science Founation uner the Grant DMS The work of K.T. was supporte by the National Science Founation uner the Grants DMS an DMS

2 2 KAPE, MELLET, AND TIVISA point for our stuy is the following kinetic Cucker-Smale equation on (, T f t + v x f + iv v (fl[f] x Φ v f = σ v f + β iv v (f(v u. (1.1 Here, f := f(t, x, v is the scalar ensity of iniviuals, 1 is the spatial imension, β, σ are some constants an Φ is a given confinement potential. The alignment operator L is the usual Cucker-Smale (CS operator, which has the form L[f] = K(x, yf(y, w(w v w y, (1.2 with K being a smooth symmetric kernel. The last term in (1.1 escribes strong local alignment interactions, where u enotes the average local velocity, efine by fv v u(t, x = f v. This strong alignment term was introuce in [2] as the following singular limit of the Motsch-Tamor (MT alignment operator φ(x yf(y, w(w v w y lim L[f] = lim = (u v. (1.3 φ δ φ δ φ(x yf(y, w w y Since the MT term is relatively new in the literature, a remark on its purpose seems appropriate. The MT operator was introuce in [25] to correct a eficiency in the stanar Cucker-Smale moel. Specifically, since the CS operator L[ ] is weighte by the ensity, the effect of the term is almost zero in sparsely populate regions. The MT operator instea weights by a local average ensity. To arrive at (1.1, the rationale is to let the MT operator govern alignment at small scales an the CS operator the large scales. Our equation (1.1 is then obtaine in the local limit (1.3 which seems appropriate at the mesoscopic level. Since the kinetic equation (1.1 is pose in 2+1 imensions, obtaining a numerical solution of (1.1 is very costly. In fact, the most feasible approach seems to be Monte-Carlo methos using solutions of the unerlying particle moel with a large number of particles an realizations. Consequently, it is of great interest to etermine parameter regimes where the moel may be reuce in complexity. The goal of this paper is to stuy the singular limit of (1.1 corresponing to strong noise an strong local alignment, that is σ, β. More precisely, we are concerne with the limit ɛ in the following equation: f ɛ t + v x f ɛ + iv v (f ɛ L[f ɛ ] x Φ v f ɛ = 1 ɛ vf ɛ + 1 ɛ iv v(f ɛ (v u ɛ. (1.4 This scaling can alternatively be obtaine from (1.1 by the change of variables x = ɛ x, t = ɛ t, an assuming that K(x, y = ɛ K(x y. In the remaining parts of this paper, we shall establish with rigorous arguments that f ɛ (t, xe v u(t,x 2 2, where an u are the ɛ limits of ɛ = f ɛ v, ɛ u ɛ = f ɛ v v. As a consequence, we will conclue that the ynamics of f is totally escribe by the following Euler-Flocking system t + iv x (u =, (1.5 (u t + iv x (u u + x = K(x, y(x(y[u(y u(x] y x Φ. (1.6

3 HYDODYNAMIC LIMIT OF THE KINETIC CUCKE-SMALE FLOCKING MODEL 3 The result will be precisely state in Theorem 3.1 with the proof coming up in Section 4. The proof is establishe via a relative entropy argument proviing in aition a rate of convergence in ɛ. The relative entropy metho relies on the weakstrong uniqueness principle establishe by Dafermos for systems of conservation laws amitting convex entropy functional [11] (see also [12]. It has been successfully use to stuy hyroynamic limits of particle systems [16, 21, 24, 27]. In our case, the result will be somewhat restricte as we nee the existence of smooth solutions to (1.5 which we only know locally in time. emark 1.1. An important issue in the stuy of Cucker-Smale type equations is whether a given moel leas to flocking behavior. By flocking, it is usually meant that the velocity u(x, t converges, for large t, to a constant velocity ū. Here, we note that flocking can only occur in (1.5-(1.6 if the confinement potential an the pressure are in balance. Inee, if (, ū (with ū constant solves (1.6, then M = Φ an hence = e Φ x e Φ. We thus see that flocking only occurs when the ensity is very ilute an, in particular, oes not have compact support Formal erivation of (1.5 - (1.6. For the convenience of the reaer, let us now give the formal arguments for why (1.5 - (1.6 can be expecte in the limit. First, we note that to obtain an interesting limit as ɛ in (1.4, the right-han sie shoul converge to zero v f ɛ + iv v (f ɛ (v u ɛ. If this is the case, the limit f can only have the following form f ɛ f(t, x, v = (t, xe v u(t,x 2 2. Hence, it seems plausible that the evolution of f (in the limit can be governe by equations for the macroscopic quantities an u alone. To erive equations for an u, let us first integrate (1.4 with respect to v ɛ t + iv( ɛ u ɛ =. (1.7 Hence, by assuming the appropriate converge properties an passing to the limit, we obtain the continuity equation (1.5. To formally erive (1.6, let us multiply (1.4 by v an integrating with respect to v to obtain ( ɛ u ɛ t + iv x (v vf ( ɛ v (1.8 K(x, y ɛ (x ɛ (y(u ɛ (x u ɛ (y y + ɛ x Φ =. Passing to the limit in (1.7 an (1.8 (assuming that f ɛ e u v 2 2, ɛ an u ɛ u, we get: (u t + iv x ( (v ve v u 2 2 v (1.9 K(x, y(x(y(u(x u(y y + x Φ =. By aing an subtracting u, we iscover that (v ve v u 2 2 v = (u ue = u u + I. Inserting this expression in (1.9 gives (1.6. v u (v u (v ue v u 2 2 v

4 4 KAPE, MELLET, AND TIVISA Organization of the paper: The rest of this paper is organize as follows: In Section 2, we recall some existence results for the kinetic flocking moel (1.1 (these results were prove in [19] an for the Euler-flocking moel (1.5-(1.6 (a proof of this result is provie in Appenix A. In Section 3 we present our main result, which establishes the convergence of weak solutions of the kinetic equation (1.4 to the strong solution of the Euler-flocking system (1.5-(1.6. The proof of the main theorem is then evelope in Section Existence theory The purpose of this section is to state some existence results upon which our result relies. More precisely, the proof of our main result (convergence of (1.4 to (1.5-(1.6 makes use of relative entropy arguments which require the existence of weak solutions of (1.4 satisfying an appropriate entropy inequality, an the existence of strong solutions to the Euler-Flocking system (1.5-(1.6 satisfying an entropy equality. Note that the latter result will be obtaine only for short time. Since entropies play a crucial role throughout the paper, we first nee to present the entropy equalities an inequalities satisfie by smooth solutions of (1.4 an (1.5-( Entropy inequalities. Solutions of (1.1 satisfy an important entropy equality, which was erive in [19]: We efine the entropy F(f = 2 f log f + f v fφ v x (2.1 an the issipations 1 D 1 (f = f vf f(u v 2 v x, 2 D 2 (f = 1 (2.2 K(x, yf(x, vf(y, w v w 2 w y v x. 2 The latter is the issipation associate with the CS operator L[ ]. There hols: Proposition 2.1. Assume that L is the alignment operator given by (1.2 with K symmetric an boune. If f is a smooth solution of (1.4, then f satisfies t F(f + 1 ɛ D 1(f + D 2 (f = K(x, yf(x, vf(y, w w y v x, (2.3 with F(, D 1 (, D 2 ( given by (2.1 an (2.2. Furthermore, if the confinement potential Φ is non-negative an satisfies e Φ(x x < +, (2.4 then there exists C, epening only on K, Φ an f (x, v x v, such that t F(f + 1 2ɛ D 1(f + 1 K(x, y(x(y u(x u(y 2 y x CɛF(f(t. 2 (2.5 The first inequality (2.3 shows that the nonlocal alignment term is responsible for some creation of entropy. The secon inequality (2.5 shows that this term can be controlle by D 1 (f an the entropy itself. This last inequality will play a key role in this paper.

5 HYDODYNAMIC LIMIT OF THE KINETIC CUCKE-SMALE FLOCKING MODEL 5 The Euler system of equations (1.5-(1.6 also satisfies a classical entropy equality. More precisely, if we efine E (, u = u2 2 x + log + Φx, then any smooth solution of (1.5-(1.6 satisfies t E (, u + 1 K(x, y(x(y u(x u(y 2 y x =. 2 Note that the entropy E an F are relate to each other by the relation ( v u F 2 e 2 = E (, u. (2π /2 Furthermore, we have the following classical minimization principle (consequence of Jensen inequality: E (, u F(f, if = f v, u = vf v. (2.6 This relation will be important in the upcoming analysis when consiering the relative entropy of solutions to the kinetic equation (1.4 an solutions to (1.5 - ( Global weak solutions of the kinetic equation. The existence of a weak solution for (1.1 is far from trivial because of the singularity in the efinition of u. We will say that a function f satisfying f C(, T ; L 1 ( 2 L ((, T 2, ( v 2 + Φ(xf L (, ; L 1 ( 2, is a weak solution of (1.1 if the following hols: for any ψ C c fψ t vf x ψ + f x Φ x ψ fl[f] v ψ vxt σ v f v ψ βf(u v v ψ vxt 2+1 = f ψ(, vx, 2 ([, T 2, where u is such that j = u. (2.7 emark 2.2. Note that the efinition of u is ambiguous if vanishes (vacuum. We resolve this by efining u pointwise as follows j(x, t if (x, t u(x, t = (x, t (2.8 if (x, t = This gives a consistent efinition of u as can be seen from the boun ( 1/2 j v 2 f(x, v, t v 1/2, yieling j = whenever = an so (2.8 implies j = u. Note also that u oes not belong to any L p space. However, we have uf 2 x v f L ( 2 v 2 f(x, v, t v x, 2 2 so the term uf in the weak formulation (2.7 makes sense as a function in L 2. The existence result we shall utilize in this paper was obtaine as the main result in [19] an is recalle in the following theorem:

6 6 KAPE, MELLET, AND TIVISA Theorem 2.3. Assume that L is the alignment operator (CS given by (1.2 with K symmetric an boune. Assume furthermore that f satisfies f L ( 2 L 1 ( 2, an ( v 2 + Φ(xf L 1 ( 2. Then, for all ɛ >, there exist a weak solution f ɛ of (1.4 satisfying F(f ɛ (t + t 1 ɛ D 1(f ɛ + D 2 (f ɛ s F(f + Ct, (2.9 where the constant C epens only on K, Φ an f (x, v x v. Furthermore, if Φ satisfies (2.4, then f ɛ also satisfies F(f ɛ (t + 1 t D 1 (f ɛ s + 1 t K(x, y ɛ (x ɛ (y u ɛ (x u ɛ (y 2 yxs 2ɛ 2 2 for all t >. F(f + Cɛ t F(f ɛ (s s ( Existence of solutions to the Euler-Flocking system. As usual with relative entropy methos, our main result will state that the solutions of (1.4 converge to a strong solution of the asymptotic system (1.5 - (1.6, provie such a solution exists. It is thus important to prove that (1.5 - (1.6 has a strong solution, at least for short time. This is the object of the next theorem (which we state in the case = 3: Theorem 2.4. Let (ρ, u H s ( 3 with s > 5/2 an ρ (x > in 3 an assume that x Φ H s ( 3. Then, there exist T > an functions (, u C([, T ]; H s ( 3 C 1 ((, T ; H s 1 ( 3, ρ(x, t >, such that (, u is the unique strong solution of (1.5 - (1.6 for t (, T. Moreover, (, u satisfies the equality t E (, u + 1 K(x, y(x(y u(x u(y 2 yx =. ( Since the proof of this theorem is rather long an inepenent of the rest of the paper, we postpone it to the appenix. Note in particular that the conition s > 5/2 implies that the solution satisfies u L ([, T ]; W 1, ( 3. (2.12 Furthermore, iviing the momentum equation by ρ, we also get: x log ρ L ([, T ] 3. (2.13 These two estimates is all the regularity we will nee in our main theorem below. 3. Main result With the existence results of the previous section, we are reay to state our main result concerning the convergence of weak solutions of (1.4 to the strong solution (, u of the Euler-flocking system (1.6-(1.5 as ɛ. Theorem 3.1. Assume that: (1 f is of the form with f = u (x (x v 2 (2π /2 e 2, f L ( 2 L 1 ( 2, an ( v 2 + Φ(xf L 1 ( 2.

7 HYDODYNAMIC LIMIT OF THE KINETIC CUCKE-SMALE FLOCKING MODEL 7 (2 f ɛ is a weak solution of (1.4 satisfying the entropy inequality (2.1 an with initial conition f ɛ (, = f (. (3 T > is the maximal time for which there exists a strong solution (, u to the Euler system of equations (1.5 - (1.6, with = f v an u = f v v an satisfying (2.12 an (2.13 (Theorem 2.4 gives in particular T > if ρ an u are regular enough. There exists a constant C > epening on F(f, K L, Φ, T, u L (,T ;W 1, (, an log ρ L ((,T, such that T ɛ T C ɛ, 2 uɛ u 2 + ɛ ɛ z z z xt (3.1 K(x, y ɛ (x ɛ (y [(u ɛ (x u(x (u ɛ (y u(y] 2 where ɛ = f ɛ v, ɛ u ɛ = vf ɛ v. Moreover, any sequence of functions satisfying (3.1 satisfies: f ɛ ɛ e v u 2 2 a.e an L 1 loc(, T ; L 1 (, ɛ ɛ a.e an L 1 loc(, T ; L 1 (, ɛ u ɛ ɛ u a.e an L 1 loc(, T ; L 1 (, ɛ u ɛ 2 ɛ u 2 a.e an L 1 loc(, T ; L 1 (. xyt This theorem will be a irect consequence of Proposition 4.1 below, the proof of which is the object of Section Proof of Theorem 3.1 To reuce the amount of notations neee in the proof of Theorem 3.1 it will be preferable to write the Euler-Flocking system in terms of the conservative quantities. In our case, the conservative quantities are the ensity an the momentum P = u. If we enote ( U =, P we can rewrite the system (1.5-(1.6 as The flux an source term are then given by ( ( P A(U =, F (U = P P U t + iv x A(U = F (U. (4.1 P P x Φ where we have introuce the notation g = K(x, yg(y y. corresponing to (4.1 reas an the relative entropy is the quantity E(U = P 2 + log + Φ, 2 E(V U = E(V E(U E(U(V U,, The entropy E

8 8 KAPE, MELLET, AND TIVISA where stans for the erivation with respect to the variables (, P. For the system (1.5-(1.6, a simple computation yiels ( ( P 2 E(U(V U = 2 + log Φ q 2 P Q P = q u 2 2 u ( q(log Φ + u 2 quv. Conseqently, the relative entropy can alternatively be written E(V U = E(V E(U E(U(V U = q v 2 2 u 2 + q log q log + Φ(q 2 + q u 2 u 2 + ( q(log Φ + u 2 quv 2 2 v u 2 = q + p(q, 2 where we have introuce the relative pressure p(q = q log q log + ( q(log + 1 = q q z z z. (4.2 Note that the relative pressure controls the L 2 norm of the ifference p(q 1 { } 1 2 min q(x, 1 (q(x (x 2. (4.3 (x With the newly introuce notation, Theorem 3.1 can be recast as a irect consequence of the following proposition: Proposition 4.1. Uner the assumptions of Theorem 3.1, let ( U = u enote the strong solution to the Euler system of equations (1.5-(1.6 an let ( U ɛ ɛ = ɛ u ɛ, ɛ = f ɛ v, ɛ u ɛ = f ɛ v v, be the macroscopic quantities corresponing to the weak solution of the kinetic equation (1.4. The following inequality hols: E(U ɛ U(t x + 1 t K(x, y ɛ (x ɛ (y [(u ɛ (x u(x (u ɛ (y u(y] 2 x y s 2 t C E(U ɛ U x s + C ɛ. (4.4 The proof of this proposition relies on several auxiliary results which will be state an prove throughout this section. At the en of the section, in Section 4.6 we close the arguments an conclue the proof. However, before we continue, let us first convince the reaer that Proposition 4.1 actually yiels Theorem 3.1.

9 HYDODYNAMIC LIMIT OF THE KINETIC CUCKE-SMALE FLOCKING MODEL 9 Proof of Theorem 3.1. Let us for the moment take Proposition 4.1 for grante. Then, the main inequality (3.1 follows from (4.5 an Gronwall s lemma. We now nee to show that (3.1 implies the state convergence. First, we note that the entropy estimate (2.9 implies that f ɛ is boune in L log L an thus converges weakly to some f. Next, in view of (4.2 an (4.3, the main inequality (3.1 yiels T { 1 min ρ ɛ, 1 } ρ ɛ ρ 2 x t ρ an T ρ ɛ u ɛ u 2 x t. In particular, we get: { 1 ρ ɛ ρ x = min ρ ɛ, 1 } 1/2 max{ρ ɛ, ρ} 1/2 ρ ɛ ρ x ρ ( { 1 min ρ ɛ, 1 } 1/2 ( ρ ɛ ρ 2 x max{ρ ɛ, ρ}x ρ ( { 1 min ρ ɛ, 1 } 1/2 ρ ɛ ρ x 2 (2M 1/2. ρ Hence, we can conclue that ɛ ɛ a.e an L 1 loc(, T ; L 1 (, 1/2 Similary, we see that ɛ u ɛ u x ɛ (u ɛ u + ( ɛ u x ( M 1 2 ɛ u ɛ u 2 x ( { 1 + min ρ ɛ, 1 } 1/2 ( 1/2 ρ ɛ ρ 2 x max{ρ ɛ, ρ}u 2 x. ρ Consequently, also Moreover, by writing we easily euce ɛ u ɛ ɛ u a.e an L 1 loc(, T ; L 1 (. ρ ɛ u ɛ2 ρu 2 = ρ ɛ (u ɛ u 2 + 2u(ρ ɛ u ɛ ρu + u 2 (ρ ρ ɛ ɛ u ɛ 2 ɛ u 2 a.e an L 1 loc(, T ; L 1 (. At this stage, it only remains to prove that f has the state maxwellian form. For this purpose, we first sen ɛ in the entropy inequality (2.9 an use the convergence of ɛ u ɛ an ɛ u ɛ 2 to obtain lim f ɛ v2 ɛ 2 + f ɛ log f ɛ x E(, u, t K(x, y(x(y(u(y u(x 2 xyt (4.5

10 1 KAPE, MELLET, AND TIVISA where we have use that f = e (v u (2π /2 2 to conclue the last inequality. Next, we subtract the entropy equality (2.11 from (4.5 to iscover lim f ɛ v2 ɛ 2 + f ɛ log f ɛ xv u2 + log x, ( where the first inequality is (2.6. By convexity of the entropy, we conclue that f = (2π 2 which conclues the proof of Theorem e (u v2 2, 4.1. The relative entropy inequality. The funamental ingreient in the proof of Proposition 4.1 is a relative entropy inequality for the system (1.5 - (1.6 which we will erive in this subsection. However, before we embark on the erivation of this inequality, we will nee some aitional ientities an simplifications. First, we recall that E is an entropy ue to the existence an entropy flux function Q such that j Q i (U = k j A ki (U k E(U, i, j = 1,..., 2. (4.7 We then have E(U t + iv x Q(U = ( P + P P. (4.8 Since the confinement potential term Φ in the entropy is linear, it oes not play any role in the relative entropy. It is thus convenient to introuce the reuce entropy functional Ê(U = P 2 + log. 2 This allows us to treat the contribution of Φ as a forcing term (which is part of the F (U in the proof of Proposition 4.2. The reuce entropy flux Q(U is then efine by j Qi (U = k We note that Q satisfies j A ki (U k Ê(U, i, j = 1,..., 2. (4.9 iv x Q(u = Ê(U (iv x A(U, (4.1 while the total entropy flux, efine by Q(U = Q(U + P Φ, satisfies ( ( iv x Q(U = iv x A(U + E(U. x Φ We shall also nee the relative flux: A(V U = A(V A(U A(U(V U, where the last term is to be unerstoo as [A(U(V U] i = A i (U (V U, i = 1....,. The key relative entropy inequality is given by the following proposition: ( Proposition 4.2. Let U = be a strong solution of (4.1 satisfying (2.11 ( u q an let V = be an arbitrary smooth function. The following inequality Q = qv

11 HYDODYNAMIC LIMIT OF THE KINETIC CUCKE-SMALE FLOCKING MODEL 11 hols t E(V U x + 1 K(x, yq(xq(y [(v(x u(x (v(y u(y] 2 xy 2 [ t Ê(V + Q x Φ + 1 ] K(x, yq(xq(y[v(x v(y] 2 y x 2 x (Ê(U : A(V U x Ê(U [V t + iv A(V F (V ] x + K(x, yq(x((y q(y[u(y u(x][v(x u(x] xy emark 4.3. When K =, such an inequality was establishe by Dafermos [11] for general system of hyperbolic conservation laws. In orer to prove Proposition 4.2, we will nee the following lemma (see Dafermos [11]. Lemma 4.4. The following integration by parts formula hols 2 Ê(U (iv x A(U (V U x = (A(U(V U : ( x Ê(U where : is the scalar matrix prouct. x, (4.11 Proof of Lemma 4.4. The proof of this equality can be foun at several places in the literature. For the sake of completeness we recall its erivation here (we follow[1] [p. 1812]. Differentiating (4.9 with respect to U l, we obtain the ientity l k Ê(U j A ki (U = l j ˆQi (U k=1 Using this ientity, we calculate lij=1 k=1 l j A ki (U k Ê(U. k=1 2 Ê(U (iv x A(U (V U ( U j = l k Ê(U j A ki (V l U l x i = lij=1 l j ˆQi (U U j x i (V l U l lijk=1 l j A ki (U k Ê(U U j x i (V l U l = iv( l ˆQ(U(Vl U l iv x ( l A k (U(V l U l k Ê(U l kl + [ ] ( x V l x U l l A k (U k Ê(U l ˆQ(U. l k

12 12 KAPE, MELLET, AND TIVISA Now, we observe that (4.9 implies that the last term is zero. Thus, we can conclue 2 Ê(U (iv x A(U (V U = iv x ( ˆQ(U(V U iv x (A(U(V U Ê(U Integrating this ientity over yiels (4.11. Proof of Proposition 4.2. First of all, we recall that E(V U = E(V E(U E(U(V U = Ê(V Ê(U Ê(U(V U. We euce: E(V U x = t Ê(V t Ê(UU t 2 Ê(UU t (V U Ê(U(V t U t x = t Ê(V 2 Ê(UU t (V U Ê(UV t x := I 1 + I 2 + I 3. (4.12 Since 2 E(U = 2 Ê(U, formula (4.11 provies I 2 = 2 Ê(U [iv x A(U F (U] (V U x = x Ê(U : A(U(V U x 2 Ê(UF (U(V U x. By aing an subtracting, an integrating by parts, we fin I 3 = Ê(U [V t + iv A(V F (V ] x ( x Ê(U : A(V + Ê(UF (V x Consequently, I 2 + I 3 = x (Ê(U : A(V U + Ê(U [V t + iv A(V F (V ] x x (Ê(U : A(U x ( Ê(UF (U(V U + Ê(UF (V x := J 1 + J 2 + J 3. Now, using (4.1, we get: J 2 = x (Ê(U : A(U x = Ê(U iv x A(U x = iv x Q(U x =. (4.14 It only remains to compute the term J 3, which is the only non stanar term, since it inclues all the contributions of the forcing term F (U. We now insert our specific expression of Ê an U. A simple computation yiels: ( ( Ê(U = Ê(U ( P 2 = 2 + log + 1 P 2 P, 2 Ê(U = 2 P Ê(U P 1. 2

13 HYDODYNAMIC LIMIT OF THE KINETIC CUCKE-SMALE FLOCKING MODEL 13 Let us also introuce two functions (q, Q such that V = [q, Q] T an efine v = Q q. J 3 = 2 Ê(UF (U(V U + Ê(UF (V x = P ( 2 P P x Φ (q P ( q Q qq q x Φ x q ( = P ( Q P q P = q ( u u (v u + u ( P P x Φ (Q P + P ( q Q qq Q x Φ x ( q Q qq Q x Φ x which yiels J 3 = K(x, yq(x(y[u(y u(x][v(x u(x] + K(x, yq(yq(xu(x[v(y v(x] yx. = K(x, yq(xq(y[u(y u(x][v(x u(x] xy + K(x, yq(xq(y[v(y v(x][u(x v(x] xy + K(x, yq(xq(y[v(y v(x]v(x xy + K(x, yq(x((y q(y[u(y u(x][v(x u(x] xy qv x Φ x. (4.15 Using the symmetry of K, we see that the first two terms can be written K(x, yq(xq(y[u(y u(x][v(x u(x] xy + K(x, yq(xq(y[v(y v(x][u(x v(x] xy (4.16 = 1 K(x, yq(xq(y[u(y u(x][(v(x u(x (v(y u(y] xy K(x, yq(xq(y[v(y v(x][(u(x v(x (u(y v(y] xy 2 = 1 K(x, yq(xq(y [(v(x u(x (v(y u(y] 2 xy. 2 From the symmetry of K, we also easily euce K(x, yq(xq(y[v(y v(x]v(x xy = 1 K(x, yq(xq(y[v(x v(y] 2 xy. 2 (4.17

14 14 KAPE, MELLET, AND TIVISA Hence, by setting (4.17 an (4.16 in (4.15, we iscover J 3 = 1 K(x, yq(xq(y[v(x v(y] 2 xy + qv x Φ x 2 1 K(x, yq(xq(y [(v(x u(x (v(y u(y] 2 xy 2 + K(x, yq(x((y q(y[u(y u(x][v(x u(x] xy We conclue the proof by combining the previous ientities (4.14, (4.13 an (4.12. In our proof Proposition 4.1, we will use the following immeiate corollary of Proposition 4.2: Corollary 4.5. Let f ɛ be a weak solution of (1.4 satisfying (2.1 an let U ɛ = ( ɛ, ɛ u ɛ, with ɛ = f ɛ v, ɛ u ɛ = vf ɛ v. Let U = (, u be the strong solution of (4.1 satisfying (2.11. Then the following inequality hols: E(U ɛ U x t + 1 K(x, y ɛ (x ɛ (y [(u ɛ (x u(x (u ɛ (y u(y] 2 xy 2 [ t Ê(U ɛ + P ɛ x Φ + 1 ] K(x, y ɛ (x ɛ (y[u ɛ (x u ɛ (y] 2 y x 2 x (Ê(U : A(U ɛ U x Ê(U [U t ɛ + iv A(U ɛ F (U ɛ ] x ( K(x, y ɛ (x((y ɛ (y[u(y u(x][u ɛ (x u(x] xy. In orer to euce Proposition 4.1 from this corollary, it remains to show that (1 The first term in the right han sie in (4.18 is of orer ɛ when integrate with respect to t (Lemma 4.6. (2 The secon term is controlle by the relative entropy itself (in fact we will show that the relative flux is controlle by the relative entropy, see Lemma 4.7. (3 The thir term is of orer O(ɛ (Lemma 4.8. (4 The last term can be controlle by the relative entropy (Lemma 4.9. The rest of this paper is evote to the proof of these 4 points (1 The first term. Lemma 4.6. Let f ɛ be the weak solution of (1.4 given by Theorem 2.3 an let U ɛ = ( ɛ, ɛ u ɛ, with ɛ = f ɛ v, ɛ u ɛ = vf ɛ v. Then t [ t Ê(U ɛ + P ɛ x Φ + 1 ] K(x, y ɛ (x ɛ (y[u ɛ (x u ɛ (y] 2 y x s 2 Cɛ t F(f ɛ (s s

15 HYDODYNAMIC LIMIT OF THE KINETIC CUCKE-SMALE FLOCKING MODEL 15 for all t. Proof. First, we write t t Ê(U ɛ s = Ê(U ɛ (t Ê(U x [Ê(U = ɛ (t F(f ɛ (t ] + [ F(f ɛ (t F(f ] + [ F(f Ê(U ]. The well-prepareness of the initial ata gives F(f Ê(U x =, an (2.6 implies Ê(U ɛ (t F(f ɛ (t x. Finally, we euce (using (2.1 t t Ê(U ɛ s F(f ɛ (t F(f x F(f ɛ (t F(f x ρ ɛ (tφ ρ Φ x 1 t K(x, y ɛ (x ɛ (y u ɛ (y u ɛ (x 2 yxs 2 t t P ɛ Φ x s + Cɛ F(f ɛ (s s 4.3. (2 Control of the relative flux. We note that x Ê(U is boune in L by u L (,T ;W 1,, log ρ L, so the secon term in (4.18 will be controlle if we prove the following stanar lemma: Lemma 4.7. The following inequality hols for all U, V : A(V U x E(V U x Proof. A straightforwar computation gives A(U(V U ( Q P = (q P P P (Q P + 1, (Q P P + (q an using the fact that P = u an Q = qv, we get: 1 q Q Q 1 P P 1 P (Q P 1 (q (Q P P + 2 P P = qv v + u u u qv qv u + (q u u = q(v u (v u. Since all the other terms in A( are linear, we euce ( A(V U =. q(v u (v u

16 16 KAPE, MELLET, AND TIVISA This implies A(V U x = q v u 2 x E(V U x, which conclues our proof (3 Kinetic approximation. Lemma 4.8. Let U be a smooth function, let f ɛ be a weak solution of (1.4 satisfying (2.1 an efine U ɛ = ( ɛ, ɛ u ɛ, with ɛ = f ɛ v, ɛ u ɛ = vf ɛ v. There exists a constant C epening on T, u L (,T ;W 1, an log ρ L such that t ( t 1/2 E(U [Ut ɛ + iv A(U ɛ F (U ɛ ] x C D 1 (f ɛ s C ɛ. Proof. By setting φ := φ(t, x in (2.7, we see that ɛ t + iv( ɛ u ɛ = in D ([, T Ω. (4.19 Setting φ := vψ(t, x, where Ψ is a smooth vector fiel, we fin that ( ɛ u ɛ t + iv x ( ɛ u ɛ u ɛ + x ɛ K(x, y ɛ (t, x ɛ (t, y(u ɛ (x u ɛ (y y + ɛ x Φ = iv x (u ɛ u ɛ v v + I f ɛ v, (4.2 in the sense of istributions on [, T Ω. Hence, we have that t E(U [Ut ɛ + iv A(U ɛ F (U ɛ ] xt t x E(U (u ɛ u ɛ v v + I f ɛ v xt t C (u ɛ u ɛ v v + I f ɛ v xt, (4.21 where the constant C epens on u L (,T ;W 1, an log ρ L. To conclue, we have to prove that the righthan sie can be controlle by the issipation. As in [23], we calculate (u ɛ u ɛ v v + If ɛ v = (u ɛ (u ɛ v + (u ɛ v v + I f ɛ v = u ɛ ( f ɛ (u ɛ v f ɛ 2 v f ɛ + u ɛ v f ɛ ( + (u ɛ v f ɛ 2 v f ɛ v f ɛ + v f ɛ v + If ɛ v. Using integration by parts, we see that u ɛ v f ɛ v =, v f ɛ v v = fi v.

17 HYDODYNAMIC LIMIT OF THE KINETIC CUCKE-SMALE FLOCKING MODEL 17 By applying this an the Höler inequality to (4.4 we fin t (u ɛ u ɛ v v + If ɛ v x s (4.22 t ( 1 ( f ɛ v 2 + f ɛ u ɛ 2 2 t 1 vx D1 (f ɛ 1 2 s C D 1 (f ɛ 2, where the last inequality follows from the entropy boun (Proposition 2.1 an ( ɛ u ɛ 2 = f ɛ vu ɛ v ( = 1 ( 1 f ɛ v 2 2 v f ɛ u ɛ 2 2 v 1 f ɛ v 2 2 ( v ɛ u ɛ We conclue by combining (4.21 an ( (4 The last term. Finally, we have: Lemma 4.9. Assuming that U = (, u an V = (q, qv are such that u L (, q, L 1 (, there exists a constant C such that K(x, yq(x((y q(y[u(y u(x][v(x u(x] xy (4.23 C u L ( L 1 + q L 1 E(V U x Proof. We have K(x, yq(x((y q(y[u(y u(x][v(x u(x] xy { } 1/2 1 2 u L K(x, yq(x min q(y, 1 (y q(y (y max {q(y, (y} 1/2 v(x u(x xy ( { } 1 2 u L K(x, yq(x min q(y, 1 ((y q(y 2 xy (y ( 1/2 K(x, yq(x max {q(y, (y} v(x u(x 2 xy An so using (4.3 an the fact that K(x, y C, we euce: K(x, yq(x((y q(y[u(y u(x][v(x u(x] xy C u L q 1/2 L 1 ( q L 1 + L 1 1/2 ( p(q (y y ( q(x[v(x u(x] 2 x 1/2 C u L q 1/2 L 1 ( q L 1 + L 1 1/2 E(V U x. 1/2 1/2

18 18 KAPE, MELLET, AND TIVISA 4.6. Proof of Proposition 4.1. We recall that E(U ɛ ( U( =. For any t (, t, we integrate (4.18 over (, t an apply Lemmas 4.7, 4.8 an 4.9, an, to obtain the inequality E(U ɛ U(t x + 1 t K(x, y ɛ (x ɛ (y [(u ɛ (x u(x (u ɛ (y u(y] 2 x y s 2 t t Ê(U ɛ + P ɛ x Φ + 1 K(x, y ɛ (x ɛ (y[u ɛ (x u ɛ (y] 2 y x s 2 + C t ɛ + C E(U ɛ U x t. Lemma 4.6 now implies: E(U ɛ U(t x + 1 t K(x, y ɛ (x ɛ (y [(u ɛ (x u(x (u ɛ (y u(y] 2 xyt 2 t Cɛ F(f ɛ (s s + C t ɛ + C E(U ɛ U xt. (4.24 an (2.1 implies We euce t F(f ɛ (s s e Cɛt. E(U ɛ U(t x + 1 t K(x, y ɛ (x ɛ (y [(u ɛ (x u(x (u ɛ (y u(y] 2 xyt 2 C(T t ɛ + C E(U ɛ U xt. (4.25 which completes the proof of Proposition 4.1.

19 HYDODYNAMIC LIMIT OF THE KINETIC CUCKE-SMALE FLOCKING MODEL 19 Appenix A. Local well-poseness of the Euler-flocking system The purpose of this appenix is to prove the existence of a local-in-time unique smooth solution of the Euler-flocking equations. In particular, the objective is to prove Theorem 2.4 which we relie upon to prove our main result (Theorem 3.1. First, we observe that the system (1.5-(1.6 is a 4 4 system of conservation laws which can be written in the following equivalent form when the solution is smooth: { t + u x + iv x u = (A.1 ( t u + u x u + x = F (, u, x Φ. Here, F (, u, Φ(x = K(x, y(x(y[u(y u(x] y x Φ. (A.2 In what follows, we shall nee the Sobolev space H s ( given by the norm g 2 s = D α g 2 x. For g L ([, T ]; H s, efine g s,t = α s sup g(, t s. t T Now, consier the Cauchy problem of (A.1 with smooth initial ata: The values of the vector (, u t= = (, u (x. ( w = u (A.3 lie in the state space G, which is an open set in 4. The state space G is introuce because certain physical quantities such as ensity shoul be positive. Inee, by invoking the metho of characteristics for the continuity equation the following lemma hols: Lemma A.1. If (, u C 1 ( 3 [, T ] is a uniformly boune solution of (1.5 with (x, >, then (x, t > on 3 [, t]. System (A.1 can be written in the form ( ( ( ( ivx u t + u u x + u x = 1 F (, u, xφ(x or equivalently ( ( ( P t + u x P P = 1 + log I 2 3 F (, u, xφ(x (A.4 with P = u an F given in (A.2. It turns out that (A.4 has the following structure of symmetric hyperbolic systems: For all w G, there is a positive efinite matrix A (w that is smooth in w an satisfies: c 1 I 4 A (w c I 4, with a constant c uniform for w G 1 Ḡ1 G such that A i (w = A (w f i (w is symmetric. Here, f i (w, i = 1, 2, 3 are the 4 4 Jacobian matrices an I 4 is the 4 4 ientity matrix.

20 2 KAPE, MELLET, AND TIVISA The matrix ( 1 A (w = I 3 is calle the symmetrizing matrix of system (A.4. Multiplying (A.4 by A we obtain A (w t w + A(w x w = G(w, x Φ (A.5 with smooth initial ata w = w(x,. (A.6 Here, A(w = (A 1 (w,..., A 3 (w enotes a matrix whose columns A i (w = A (w x f i (w are symmetric, an ( P f(w = P P + log I 2 3 with P = u an x f i (w, are the 4 4 Jacobian matrices an I 3 enotes the 3 3 ientity matrix. The 4 1 vector G in (A.5 is given ( ( G(w, x Φ = A (w 1 F (, u, = (A.7 xφ F (, u, x Φ which implies that ( G(w, x Φ =. K(x, y(x(y[u(y u(x] y x Φ We are now reay to prove the local existence of smooth solutions. Theorem A.2. Assume w = (, u H s L ( 3 with s > 5/2 an (x > an that x Φ H s. Then there is a finite time T (,, epening on the H s an L norms of the initial ata, such that the Cauchy problem (1.5-(1.6 an (A.3 has a unique boune smooth solution w = (, u C 1 ( 3 [, T ], with > for all (x, t 3 [, T ], an (, u C([, t]; H s C 1 ([, T ]; H s 1. Theorem A.2 is a consequence of the following theorem on the local existence of smooth solutions, with the specific state space G = {(, u : > } 4 for the inhomogeneous system (A.5. Theorem A.3. Assume that w : G is in H s L with s > Then, for the Cauchy problem (A.5-(A.6, there exists a finite time T = T ( w s, w L (, such that there is a classical solution w C 1 ( 3 [, T ] with w(x, t G for (x, t [, T ] an w C([, T ]; H s C 1 ([, T ]; H s 1. Proof. The proof of this theorem procees via a classical iteration scheme metho. An outline of the proof of Theorem A.3 (an therefore Theorem A.2 is given below. Consier the stanar mollifier η(x C ( 3, supp η(x {x; x 1}, η(x, η(xx = 1, 3 an set η ɛ = ɛ η(x/ɛ. Define the initial ata w k C ( 3 by w(x k = η ɛk w (x = η ɛk (x yw (yy, 3 where ɛ k = 2 k ɛ with ɛ > constant. We construct the solution of (A.5-(A.6 using the following iteration scheme. Set w (x, t = w(x an efine w k+1 (x, t, for k =, 2,..., inuctively as the solutions of linear equations: { A (w k t w k+1 + A(w k x w k+1 = G(w k, x Φ, w k+1 t= = w k+1 (x. (A.8

21 HYDODYNAMIC LIMIT OF THE KINETIC CUCKE-SMALE FLOCKING MODEL 21 By well-known properties of mollifiers it is clear that: w k w s, as k, an w k w C ɛ k w 1, for some constant C. Moreover, w k+1 C ( [, T k ] is well-efine on the time interval [, T k ], where T k > enotes the largest time for which the estimate w k w s,tk C 1 hols. We can then assert the existence of T > such that T k T (T = for k =, 1, 2,..., from the following estimates: w k+1 w s,t C 1, w k+1 t s 1,T C 2, (A.9 for all k =, 1, 2,..., for some constant C 2 >. From (A.8, we get A (w k t (w k+1 w k + A(w k (w k+1 w k = E k + G k, (A.1 where { E k = (A (w k A (w k 1 t w k (A(w k A(w k 1 w k, G k = G(w k, x Φ. From the stanar energy estimate for the linearize problem (A.1 we get (A.11 w k+1 w k,t Ce CT ( w k+1 w k + T E k,t + T G k,t. Taking into consieration the property of mollification, relation (A.9, we have that Note that, u k+1 u k C2 k, E k,t C w k w k 1,T. G k,t = sup G k (, t s t T = sup K(x, y k (x k (y[u k (y u k (x] y k x Φ(x t T M u k 2 u k (y 2 y + C = M u k y + C. (y Here we use the well known result that states: For any v H 1 ( 3, 3 v(x x 2 x M v 2 x. 3 For small T such that C 2 T exp CT < 1 one obtains w k+1 w k,t <, k=1 which implies that there exists w C([, T ]; L 2 ( 3 such that lim k wk w,t =. s (A.12 From (A.9, we have w k s,t + w k t s 1,T C, an w k (x, t belongs to a boune set of G for (x, t [, T ]. Then by interpolation we have that for any r with r < s, w k w l r,t C s w k w l 1 r/s,t w k w l r/s s,t wk w l 1 r/s,t. (A.13 From (A.12 an (A.13, lim k wk w r.,t = for any < r < s. Therefore, choosing r > , Sobolev s lemma implies w k w in C([, t]; C 1 ( 3. (A.14 From (A.1 an (A.14 one can conclue that w k w in C([, T ]; C( 3, w C 1 ( 3 [, T ], an w(x, t is a smooth solution of (A.5-(A.6. To prove u

22 22 KAPE, MELLET, AND TIVISA C([, T ]; H s C 1 ([, T ]; H s 1, it is sufficient to prove u C([, T ]; H s since it follows from the equations in (A.5 that u C 1 ([, T ]; H s 1. emark A.4. The proof of Theorem A.2 follows the line of argument presente by Maja [22], which relies solely on the elementary linear existence theory for symmetric hyperbolic systems with smooth coefficients (see also Courant-Hilbert [8]. Lemma A.5. Let (, u be a sufficiently smooth solutions to (1.5-(1.6. The energy E(u satisfy the following entropy equality t E(u x + 1 K(x, y(x(y u(x u(y 2 yx =. (A.15 2 Proof. The result is obtaine by the following stanar process: first we multiply the continuity equation (1.7 by ( log an the momentum equation (1.8 by the velocity fiel u, an we a the resulting relations. Next we integrate over the omain taking into account that (, u is sufficiently smooth an using that t Φx = t Φx = iv(uφ x = u x Φx. an u(x K(x, y(x(y[u(y u(x]yx = 1 K(x, y(x(y u(x u(y 2 y x 2 we arrive at (A.15. The result for a solution (, u with the regularity establishe in Theorem A.2 is establishe using a ensity argument. A.1. Proof of Theorem 2.4. The proof of Theorem A.2 can be further reuce to verifying that u(x, t is strongly right continuous at t =, since the same argument works for the strong right-continuity at any other t (, T ] an the strong rightcontinuity on [, T implies the strong left-continuity on (, T ] because the equations in A.5 are reversible in time. Taking this into consieration estimate (A.15 implies the result. eferences [1] F. Bertheln an A. Vasseur, From Kinetic Equations to Multiimensional Isentropic Gas Dynamics Before Shocks, SIAM J. Math. Anal. 36, , [2] F. Bolley, J.A. Cañizo, J.A. Carrillo, Stochastic Mean-Fiel Limit: Non-Lipschitz Forces & Swarming, Math. Mo. Meth. Appl. Sci. 21: , 211. [3] J.A. Carrillo an J. osao. Uniqueness of boune solutions to aggregation equations by optimal transport methos. European Congress of Mathematics, no. 3-16, Eur. Math. Soc., Zurich, 21. [4] J.A. Carrillo, M. Fornasier, J. osao an G. Toscani. Asymptotic flocking ynamics for the kinetic Cucker-Smale moel. SIAM J. Math. Anal. 42 no. 1, (21. [5] J.A. Carrillo, M. Fornasier, G. Toscani, an V. Francesco. Particle, kinetic, an hyroynamic moels of swarming. Mathematical moeling of collective behavior in socio-economic an life sciences , Moel. Simul. Sci. Eng. Technol., Birkhuser Boston, Inc., Boston, MA, 21. [6] J.A. Cañizo, J.A. Carrillo an J. osao. A well-poseness theory in measures for some kinetic moels of collective motion. Mathematical Moels an Methos in Applie Sciences Vol. 21 No. 3: , 211. [7]. Courant an K. O. Frierichs, Supersonic Flow an Shock Waves, Springer-Verlag: New York, [8]. Courant an D. Hilbert, Methos of Mathematical Physics, Interscience Publishers, Inc.: New York, A.4 [9] F. Cucker an S. Smale. Emergent behavior in flocks. IEEE Transactions on automatic control, 52 no. 5: , 27. 1

23 HYDODYNAMIC LIMIT OF THE KINETIC CUCKE-SMALE FLOCKING MODEL 23 [1] F. Cucker an S. Smale. On the mathematics of emergence. Japanese Journal of Mathematics, 2 no. (1: , 27. [11] C.M. Dafermos. The secon law of thermoynamics an stability. Arch. ational Mech. Anal., 7: , , 4.3, 4.1 [12] onal DiPerna, Uniqueness of Solutions to Hyperbolic Conservation Laws, Iniana Univ. Math. J. 28 No. 1: ( [13]. Duan, M. Fornasier, an G. Toscani. A Kinetic Flocking Moel with Diffusion. Comm. Math. Phys , 21. [14] P. Degon. Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in 1 an 2 space imensions. Ann. Sci. École Norm. Sup. (4 19 no. 4: , [15] P. Degon. Existence globale e solutions e l équation e Vlasov-Fokker-Planck, en imension 1 et 2. (French [Global existence of solutions of the Vlasov-Fokker-Planck equation in imension 1 an 2.] C.. Aca. Sci. Paris Sér. I Math. 31 no. 373, [16] P. Goncalves, C. Lanim, an C. Toninelli. Hyroynamic limit for a particle system with egenerate rates. Ann. Inst. H. Poincar Probab. Statist. 45, 4, , [17] S.-Y. Ha, an J.-G. Liu. A simple proof of the Cucker-Smale flocking ynamics an mean-fiel limit. Commun. Math. Sci. Vol. 7, No. 2.: [18] S.-Y. Ha, an E. Tamor. From particle to kinetic an hyroynamic escriptions of flocking. Kinet. elat. Moels 1 no. 3: , 28. [19] T. Karper, A. Mellet, an K. Trivisa. Existence of weak solutions to kinetic flocking moels. Preprint 212. (ocument, 1.1, 2.1, 2.2 [2] T. Karper, A. Mellet, an K. Trivisa. On strong local alignment in the kinetic Cucker-Smale equation, Preprint 212. (ocument, 1 [21] C. Lanim. Hyroynamic limit of interacting particle systems. School an Conference on Probability Theory, 571 (electronic, ICTP Lect. Notes, XVII, Abus Salam Int. Cent. Theoret. Phys., Trieste, [22] A. Maja, Compressible Flui Flow an Systems of Conservation Laws in Several Space Variables, Applie Mathematical Sciences 53, Springer-Verlag: New York, A.4 [23] A. Mellet an A. Vasseur. Asymptotic Analysis for a Vlasov-Fokker-Planck Compressible Navier-Stokes Systems of Equations. Commun. Math. Phys. 281, , [24] M. Mourragui. Comportement hyroynamique et entropie relative es processus e sauts, e naissances et e morts. Ann. Inst. H. Poincaré Probab. Statist , [25] S. Motsch an E. Tamor. A new moel for self-organize ynamics an its flocking behavior. Journal of Statistical Physics, Springer, 141 (5: , 211. (ocument, 1 [26] B. Perthame an P.E. Souganiis. A limiting case for velocity averaging. Ann. Sci. École Norm. Sup. (4 31 no. 4: , [27] H.T. Yau, elative entropy an hyroynamics of Ginzburg-Lanau moels, Lett. Math. Phys. 22 (1 ( (Karper Center for Scientific Computation an Mathematical Moeling, University of Marylan, College Park, MD aress: karper@gmail.com UL: folk.uio.no/~trygvekk (Mellet Department of Mathematics, University of Marylan, College Park, MD aress: mellet@math.um.eu UL: math.um.eu/~mellet (Trivisa Department of Mathematics, University of Marylan, College Park, MD aress: trivisa@math.um.eu UL: math.um.eu/~trivisa