SUPPRESSING CHEMOTACTIC BLOW-UP THROUGH A FAST SPLITTING SCENARIO ON THE PLANE SIMING HE AND EITAN TADMOR

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1 SUPPRESSING CHEMOTACTIC BLOW-UP THROUGH A FAST SPLITTING SCENARIO ON THE PLANE SIMING HE AND EITAN TADMOR Abstract. We revisit the question of global regularity for the Patlak-Keller-Segel (PKS) chemotaxis moel. The classical D parabolic-elliptic moel blows up for initial mass M > 8π. We consier more realistic scenario which takes into account the flow of the ambient environment inuce by harmonic potentials, an thus retain the ientical elliptic structure as in the original PKS. Surprisingly, we fin that alreay the simplest case of linear stationary vector fiel, Ax, with large enough amplitue A, prevents chemotactic blow-up. Specifically, the presence of such an ambient flui transport creates what we call a fast splitting scenario, which competes with the focusing effect of aggregation so that enough mass is pushe away from concentration along the x -axis, thus avoiing a finite time blow-up, at least for M < 6π. Thus, the enhance ambient flow oubles the amount of allowable mass which evolve to global smooth solutions. Contents. Introuction.. A fast splitting scenario. Local existence 5.. Weak formulation 5.. Regularize equation an local existence theorems 5 3. Proof of the main results The three-step battle-plan Step control of the cell ensity istribution Step proof of the main theorem with moerate mass constraint 3.4. Step 3 proof of the main theorem 5 Appenix A. 9 A.. Proof of proposition. 9 A.. Proof of proposition. 6 References 6. Introuction The Patlak-Keller-Segel (PKS) moel escribes the time evolution of colony of bacteria with ensity n(x, t) subject to two competing mechanisms aggregation triggere by the concentration of chemo-attractant riven by velocity fiel u := ( ) n(, t), an iffusion ue to run-antumble effects, n t (nu) = n, u := ( ) n. We focus on the two-imensional case where ( ) n takes the general form of a funamental solution together with an arbitrary harmonic function ( ) n(x, t) = (K n)(x, t) H(x, t), K(x) := log x, H(, t). π Date: October, 7.

2 SIMING HE AND EITAN TADMOR The resulting PKS equation then reas n t (n c) b n = n, x = (x, x ) R, (.) subject to prescribe initial conitions n(x, ) = n (x). Here the ivergence free vector fiel b( ) represents the environment of an backgroun flui transporte with velocity b(x, t) := H(x, t). When b, the system is the classical parabolic-elliptic PKS equation moeling chemotaxis in a static environment [Pat953, KS97]. We recall the large literature on the static case b =, referring the intereste reaer to the review [Hor3] an the follow-up representative works [Bil995, BDP6, BCM8, BCC8, CC6, CC8, CR, JL99, BM4, BK]. It is well-known that the large-time behavior of the static case (.) b epens on whether the initial total mass M := n crosses the critical threshol of 8π: the equation amits global smooth solution in the sub-critical case M < 8π an it experiences a finite time blow-up if M > 8π [BDP6] (when M = 8π, aggregation an iffusion exactly balance each other an solutions with finite secon moments form Dirac mass as time approach infinity [BCM8]). In this paper we stuy a more realistic scenario of the PKS moel (.) where we take into account an ambient environment ue to the flui transport by vector fiel b(, t). Surprisingly, we fin that alreay the simplest case of linear stationary vector fiel, b = A( x, x ), corresponing to H(x) = A(x x ), prevents chemotactic blow-up for M < 6π. As we shall see, the presence of such an ambient flui transport creates what we call a fast splitting scenario which competes with the focusing effect of aggregation so that enough mass is able to escape a finite time blow-up, at least for M < 6π. This scenario is likely to be enhance even further when larger amount of mass can be transporte by a more pronounce ambient fiel b(x, t) = H(, t) x q at x. We mention two other scenarios of non-static PKS with strong enough transport preventing blow-up for M > 8π. In [KX5] the author exploit relaxation enhancing of a vector fiel b with a large enough amplitue in orer to enforces global smooth solution. Here regularity follows ue to a mixing property over T an T 3. In [BH6] it was shown how to exploit the enhance issipation effect of non-egenerate shear flow with large enough amplitue, [BCZ5], in orer to suppress the blow up in (.) on T, T 3, T R, T R... A fast splitting scenario. Here, we exploit yet another mechanism that suppress the possible chemotactic blow up of the equation (.), where the unerlying flui flow splits the population of bacteria with ensity n exponentially fast, resulting in several isolate subgroups with mass smaller than the critical 8π. In this manner, an initial total mass greater than 8π is able to escape the finite-time concentration of Dirac mass. This provies a first no blow-up scenario over R, at least for M up to 6π. We now fix the vector fiel riving a hyperbolic flow as the strain flow in [KJCY7]): b(x) := A( x, x ). (.) Our aim is to show that a large enough amplitue, A, guarantees the global existence of solution of PKS (.) subject to initial mass M < 6π. Observe that an initial center of mass at the origin is an invariant of the flow. Intuitively, the large enough amplitue A is require so that the ambient fiel A( x, x ) pushes away highly concentrate mass near the x axis, namely, n (x)x. With this we can state the main theorem of the paper. x ɛ Theorem.. Consier the PKS equation (.),(.) subject to initial ata, n H s (s ) with total mass, M := n < 6π, such that ( x )n L (R ) an n log n L (R ). Assume n We let x enote the l -size of vector x an let f p enote the L p -norm of a vector function f( ).

3 SUPPRESSING CHEMOTACTIC BLOW-UP THROUGH A FAST SPLITTING SCENARIO 3 is symmetric about the x -axis, an that the y-component of its center of mass in the upper half plane y (t) := n(x, t)x x, M := n(x, t)x M M x x, is not too close to the x -axis in the sense that y () > M V (), V (t) := x n(x, t) x y x. (.3) Then there exists a large enough amplitue, A = A(M, y (), V ()), such that the weak solution of (.),(.) exists for all time an the free energy (n E[n](t) := log n cn H(x)n(x, t)) x, H(x) = A (x x ), (.4) satisfies the issipation relation E[n](t) t We conclue the introuction with three remarks. R n log n c b xs E[n ]. (.5) Remark. (Why large enough stationary fiel prevents blow-up). Our main theorem extens the amount of critical mass, so that global regularity of (.),(.3) prevails for M < 6π, provie A is large enough. To realize the how large the amplitue A shoul be an thus clarifying the reason behin this oubling the initial mass threshol for global regularity, we express (.3) as R := M y () V () >. Then we can choose A = M δ with small enough δ so that δ (R ) V () M. To gain further insight, we recall that the blow up phenomena in the static case, b is euce from the time evolution of the secon moment, V (t) := n(x, t) x x. Inee, a straightforwar computation ( R yiels, V (t) = 4M M ) <, which implies that the positive V (t) ecreases to zero in a finite 8π time an hence rules out existence of global classical solutions for M > 8π. In contrast, the secon moment of our non-static PKS equation (.),(.), oes not ecrease to zero if A is chosen large enough. Lemma.. Let n(x, t) be the solution of (.) with vector b(x) = A( x, x ), subject to initial ata n such that W := n (x)(x x )x is strictly positive. Then, if A is chosen large enough, R the secon moment of the (classical) solution V (t) = n(x, t) x x increases in time. R Proof. First, the time evolution of V (t) can be calculate as follows: ( t V = 4M M ) x b n(x, t)x 8π ( = 4M M ) AW, W (t) = ( x x 8π )n(x, t)x. R (.6)

4 4 SIMING HE AND EITAN TADMOR Next, we compute the time evolution of W (t) t W = ( x x ) ( n cn bn)x = x y n(x, t)( x, x ) n(y, t)xy AV π x y = (x y ) (x y ) π (x y ) n(x, t)n(y, t)xy AV, (x y ) where the last step follows by symmetrization. Since the first term on the right is boune from below by π M, we have t W π M AV. (.7) Finally, notice that since W (an hence V ) are assume strictly positive, we can choose A large such that AV ( 4π M, AW M M ). (.8) 8π Combining (.8), (.6) an (.7) yiels that W (t) >, V (t) >. Remark. (On the free energy). We note that when b =, E[n] becomes the classical issipative free energy F = n log nx ncx. (.9) R R Due to the importance of the property (.5), a weak solution of (.) satisfying (.5) will be calle a free energy solution. One of the important properties of the PKS equation (.) with backgroun flow velocity (.) is the issipation of its free energy E[n]. The formal computation, inicating the energy issipation in non-static smooth solutions, is the content of our last lemma in this section. Lemma.. Consier the PKS equation (.) with backgroun flui velocity (.). If the solution is smooth enough, the free energy E[n](t) is ecreasing. Proof. The time evolution of the free energy (.4) can be compute in terms of the potential H = A(x x ), t E[n](t) = n t (log n c H)x = n( log n c b) ( log n c H)x = n log n c b x. This completes the proof of the lemma. Remark.3. Arguing along the lines [EM6], one shoul be able to prove that the free energy solution is smooth for all positive time, n Cc ((, T ]; Cx ) for all T < an thus our global weak solution is in fact a global strong solution. Our paper is organize as follows. In section, we introuce the regularize problem to (.) which leas to the local existence results. In section 3, we prove the main theorem, an in the appenix, we give etaile proofs of the results state in section.

5 SUPPRESSING CHEMOTACTIC BLOW-UP THROUGH A FAST SPLITTING SCENARIO 5. Local existence.. Weak formulation. It is stanar to unerstan the Keller-Segel equation (.) with backgroun flui velocity (.) in the following weak formulation. Definition. (weak formulation). n is sai to be the weak solution to (.) if for ϕ Cc (), the following equation hol: ϕnx = ϕnx ( ϕ(x) ϕ(y)) (x y) t R R 4π R R x y n(x, t)n(y, t)xy (.) ϕ bnx. R Taking avantage of the assume symmetry across the x -axis, one can further simplify the notation of a weak formulation aapte to the upper half plane, = {(x, x ) x }. Theorem.. If n is a weak solution to the equation (.), then for ϕ Cc (), the following hols: ϕn x = ϕn x ( ϕ(x) ϕ(y)) (x y) t R 4π R x y n (x, t)n (y, t)xy R (.) c ϕn x ϕ bn x. { n := n Here x n := n x }, an c (x) := y R x y π x y n (y)y,. Proof. Rewrite (.) as follows: ϕn x = ϕn x ( ϕ(x) ϕ(y)) (x y) t R 4π R x y n (x, t)n (y, t)xy R ϕ(x) (x y) π R x y n (x, t)n (y, t)xy ϕ bn x. R The thir term can be rewritten as c ϕ n (x)x, an we get (.)... Regularize equation an local existence theorems. In this section we introuce the local existence theorem an the blow up criterion for the Keller-Segel equation with avection. The theorems are stanar, so the proofs are postpone to the appenix. The intereste reaer are referre to the papers [BCM8],[BDP6] for further etails. In orer to prove the local existence theorem an the blow up criterion for the Keller-Segel system with avection (.), we regularize the system as follows: n ɛ t (nɛ c ɛ ) b n ɛ = n ɛ, c ɛ := K ɛ n, x R, t >, (.3) with the regularize kernel, K ɛ, given by K ɛ (z) := K ( z ɛ ) π log ɛ, K (z) := { π log z, if z 4,, if z. Noting that K ɛ (z) C ɛ for all z R, it follows that the solutions to the equation (.3) exist for all time. The proof is similar to the corresponing proof in the classical case. We refer the intereste reaer to the paper [BDP6] for more etails. (.4)

6 6 SIMING HE AND EITAN TADMOR Before stating the local existence theorems, we introuce the entropy of the solution S[n] := n log nx. (.5) R Now the local existence theorems are expresse as follows: Proposition.. (Local Existence Criterion). Assume that b (x) C x, x R. Suppose {n ɛ } ɛ are the solutions of the regularize equation (.3) on [, T ). If {S[n ɛ ](t)} ɛ is boune from above uniformly in ɛ an in t [, T ), then the cluster points of {n ɛ } ɛ, in a suitable topology, are non-negative weak solutions of the PKS system with avection (.) on [, T ) an satisfies the relation (.5). Proposition.. (Maximal Free-energy Solutions). Assume the bouneness of the vector fiel b (x) x an the integrability of initial ata ( x )n L (R ), n log n L (R ). Then there exists a maximal existence time T > of a free energy solution to the PKS system with avection (.),(.5). Moreover, if T < then lim t T R n log nx =. We conclue that if the entropy S[n](t) = n log n is boune, then the free energy solution of (.) exists locally. Moreover, if S[n](t) < for all t <, the solution exists for all time. 3. Proof of the main results 3.. The three-step battle-plan. We procee in three steps. The first step carrie in section 3. below, is to control cell ensity istribution. From the last section, we see that an entropy boun is essential for erivation of local existence theorems for the PKS equation (.),(.). To this en, information about the istribution of cell ensity is crucial. The following lemma is the key to the proof of the main results. It shows that mass cannot concentrate along the the x -axis, since we can fin a thin enough strip along the x -axis with controlle amount of mass. Lemma 3.. Suppose a sufficiently smooth n is symmetric about the x axis an assume that R y := M () >. (3.) V () Fix a small enough < η. Then there exists δ = δ(y (), V (), M, η) such that if we choose A > M, the smooth solutions to the regularize (.3) δ ɛ satisfy, uniformly for small enough ɛ, n ɛ ( η) (x, t)x R M. (3.) x y f >r x δ Conition (3.) implies, at least for M < 6π, that the mass insie that δ-strip is less than 8π. On the other han, it inicates the reason for the limitation R > : for if R <, then the boun (3.) woul allow a concentration of mass M 8π insie the strip x R δ, which in turn leas to a finite-time blow-up. The proof of lemma 3. is base on the following simple observation. Given f with -center of mass at (, y f ) an variation V f = x y f f(x)x, we fin that its total mass outsie the strip S[y f, r] := {(x, x ) x y f r} with raius r = R V f /M f, oes not excee f(x)x = f(x) x y f x y f x M f R f(x) x y f x = M f V f R x y f >r If we can fin the δ such that our target strip S δ := { x δ} is lying below an outsie the strip S[y f, r], then the total mass in the strip S δ woul be smaller than M R f. When n ɛ (x, t) takes the role of f(x) with (y f, V f ) (y (t), V (t)), the aim is to boun the strip S[y (t), r(t)] with raius

7 SUPPRESSING CHEMOTACTIC BLOW-UP THROUGH A FAST SPLITTING SCENARIO 7 r(t) = R V (t)/m away from a fixe strip S δ. To this en we collect the necessary estimates on y (t), V (t) an complete the proof of the lemma in section 3.. The secon step, carrie in section 3.3, is to prove the main theorem with moerate mass constraint. Equippe with lemma 3. we can control the entropy an prove a weaker form of our main theorem for moerate size mass M (which is still larger than the 8π barrier): Theorem 3.. Consier the PKS equation (.) with backgroun flui velocity (.) subject to H s (s ) initial ata with mass M = n, such that ( x )n L (R ), n log n L (R ). Furthermore, assume n is symmetric about the x -axis, that (.3) hols. If the total mass oes not excee M < ( η) 6π, (3.3) /R then there exists an A = A(M, y (), V (), η) large such that the free energy solution to PKS (.),(.) exists for all time. Thus, as the ratio increases over the range < R <, (3.) yiels global existence with an increasing amount of mass 8π < M < 6π. Although theorem 3. is not as sharp as the main theorem, its proof is more illuminating an can be extene easily to prove the main theorem for the limiting case of any M < 6π. We therefore inclue its proof in section 3.3. Finally, the thir step carrie in section 3.4 presents the proof of the main theorem.. We turn to a etaile iscussion of the three steps. 3.. Step control of the cell ensity istribution. As pointe out before, the proof involves the calculation of y (t) an V (t), summarize in the following two lemmas. Here an below, we let A B enote the relation A CB with a constant C which is inepenent of δ. Lemma 3.. Consier the regularize PKS equation (.3) with backgroun flui velocity (.). Assume that the initial center of mass y () is separate from the x -axes in the sense that (3.) hols. Then, there exists a constant such that the time evolution of y (t) remains boune from below y (t) [y () Cδ] e At,. (3.4) Lemma 3.3. Consier the regularize PKS equation (.3) with backgroun flui velocity (.). Assume that the initial variation aroun the center of mass V () is not too large in the sense that (3.) hols. Then, there exists a constant C = C(V ()) such that the variation V (t) remains boune from above, V (t) [CM δ V ()] e At,. (3.5) We note that all the calculations mae below shoul be carrie out at the level of the regularize equation (.3), but for the sake of simplicity, we procee at the formal level using the weak formulation (.). We explicitly point when there is a technical subtlety in the erivation ue to ifference between the regularize an weak formations. We begin with the proof of Lemma 3.. First of all, as x / Cc (), we introuce an approximate test function ϕ to x : x x (δ, ), ϕ := x (, δ), smooth x (δ, δ). Note that there exists a constant C ϕ, inepenent of δ, such that ϕ δ, x δ an ϕ δ ϕ C ϕ. Here an below, we use C ϕ to enote ϕ-epenent constants.

8 8 SIMING HE AND EITAN TADMOR By replacing x nx with ϕnx, we lose information on the stripe {(x, x ) x δ}. However, the contribution of this part is small in the sense that: ϕn x x n x 4M δ. (3.6) x δ x δ Next, one can use ϕ an the weak formulation (.) to extract information about y : ϕn x = ϕn x ( ϕ(x) ϕ(y)) (x y) t R 4π R x y n (x, t)n (y, t)xy R n c ϕx ϕ bn x =I II III IV. (3.7) Now we estimate the right han sie of (3.7) term by term. The first an secon terms are relatively easy to control from above as follows: I = ϕn x C ϕm, (3.8) δ II 4π ϕ n (x)n (y)xy C ϕ R 4π δ M. (3.9) R In orer to estimate the thir term in (3.7), we nee a pointwise estimate on the x c : x c (x) = (x y) π R x y n (y)y x π x x y n (y)y M π x = M π x. (3.) Now we can use the above estimate to control the thir term in (3.7): III = x c n x ϕx C ϕ M n x C ϕ M π δ π δ. (3.) Here we have use the fact that supp(ϕ) is δ away from the x axis. As a result, x δ. Remark 3.. The only ifference in estimating the regularize solutions (.3) vs. the formal calculation we have one above is in terms II an III. In the calculation for the (.3), we will nee the estimate K ɛ (z) π z, z R. Here we show how to get a similar estimate for term II in (3.7) for the regularize equation (.3): II = ϕ(x) x [K ɛ ( x y )]n ɛ (y)n ɛ (x)xy R R = ( ϕ(x) ϕ(y)) (x y) K ɛ ( x y )n ɛ (x)n ɛ (y)xy R x y R ϕ(x) ϕ(y) n ɛ (x)n ɛ (y)xy 4π x y 4π ϕ M C ϕ 4π δ M. R

9 SUPPRESSING CHEMOTACTIC BLOW-UP THROUGH A FAST SPLITTING SCENARIO 9 The treatment of term III is similar to the one we gave above. Finally, we nee to aress aitional transport term IV in (3.7) to compete with the focusing effect. Recall that b = A( x, x ) with A = M. First we write IV own explicitly, δ IV = b n ϕx = M δ x x ϕn x. Next we replace the right han sie by ϕn x. Due to the fact that x x ϕ = x = ϕ for x > δ, the error introuce in this process originates from the thin δ-strip: (x x ϕ ϕ)n x x x ϕ ϕ n x C ϕ δm, δ<x <δ an we conclue that IV M δ ( δ<x <δ ϕn x C ϕ δm ) M δ ϕn x C ϕm. (3.) δ Combining the equation (3.7) an estimates (3.8), (3.9), (3.) an (3.) yiels the following M ϕn x C ϕ A ϕn x, t δ which implies that ϕn x ( ϕn x C ϕ M δ ) e At. (3.3) Finally, we calculate the center of mass of the upper half plane using the lower boun (3.3) an the error control (3.6): ( y (t) = ) x n x ϕn x ϕn x M x δ x δ M x n x ϕn x ϕn x x δ x δ M ( ) 4δ ϕn x C ϕ M δ e At M (y () C ϕ δ) e At. This completes the proof of lemma 3.. Next we aress the proof of Lemma 3.3. The main goal is to calculate time evolution of the variation V (t) := x y (t) n(x, t)x. We again use C to enote constants which may change from line to line but are inepenent of δ. The first obstacle is that we cannot choose x y as a test function ue to the fact that y (t) epens on the solution. However, by the efinition of y we can expan the V -integran, ening up with the usual V (t) = x n (x, t)x M y(t). (3.4) Since we alreay know y, it is enough to calculate the R x n(x, t)x. For simplicity, we plug x insie the weak formulation (.) an (3.4) to get the time evolution of V. Of course, what

10 SIMING HE AND EITAN TADMOR one really oes is to use a test function to approximate the x. Furthermore, when we use the weak formulation, we formally integrate by part twice, but since the value an the first erivative of the function x are zero on the bounary, we will not create extra angerous bounary term. First combining (.) an (3.4) yiels t V = x n (x, t)x M t t y (t) = ϕn x 4π R ( ϕ(x) ϕ(y)) (x y) x y n (x, t)n (y, t)xy c n ϕx ϕ bn x ( M (y ) ) R t =I II III IV M t y (t) (3.5) Next we estimate every term on the right han sie of (3.5). The first two terms are estimate as follows: I = (x )n x = M, (3.6) an II = 4π 4π R R ( (x ) (y )) (x y) x y n (x, t)n (y, t)xy n (x)n (y)xy π M. (3.7) Now for the thir term III in (3.5), applying (3.) yiels III = x c n x x x M π x n x π M. (3.8) Note that for the term II an III, we only estimate them formally above, one can prove the estimates explicitly using the same techniques as the one in Remark 3.. For the IV term in (3.5), we use the (3.4) again to obtain IV = x bn x = A x n x = A(V M y) (3.9) Collecting equation (3.5) an all the estimates (3.6), (3.7), (3.8) an (3.9) above, we have the following ifferential inequality, ( ) ( ) V (t) y t M (t) C( M ) A V (t) y M (t), which yiels ( V (t) y M (t) C ) δ e At V (t)e At y M M ()e At. (3.) Combining (3.4) with (3.) yiels V (t) [ (y () Cδ) e At] V (t) y M M (t) ( C ) δ e At V ()e At y M M ()e At.

11 SUPPRESSING CHEMOTACTIC BLOW-UP THROUGH A FAST SPLITTING SCENARIO By collecting similar terms, we finally have [ Cδy () C M V (t) which completes the proof of Lemma 3.3. ( ) δ ] V () e At. (3.) M M Equippe with the estimate on V (t), we can now conclue the proof of Lemma 3.. Proof. (Lemma 3.) Once < η was fixe, we can clearly choose a small enough δ such that by (3.5) δ, there hols V (t) ( η)v ()e At. (3.) Now recalling that R = y () M V () >, then we can use (3.4), (3.) an further choose δ small enough to get: y (t) [ R V (t) y () Cδ η M [( = δe At δ. R η η )y () Cδ ] V () Thus, the thin δ-strip along the x -axis, S δ := {(x, x ) x δ}, lies outsie the strip centere aroun y (t), uniformly in time, S δ {(x, x ) x y (t) > R η (t)}, R η (t) := R V (t). η M It follows that thanks to our choice of δ, the mass insie the δ-strip S δ oes not excee n (x, t)x n (x, t)x n (x, t) x y S δ { x y >R η} { x y >R η} x y x ( η) R n (x, t) x y ( η) x = V /M R V V /M ( η) R M. By symmetry, the mass insie the symmetric δ-strip, {(x, x ) x δ} is smaller than ( η) M ] e At e At ( η) R M = R M, uniformly in time, which completes the proof of Lemma 3.. Remark 3.. One can o a similar computation to get the evolution for the higher moment estimates n(x, t) x k x, an erive similar results to Lemma Step proof of the main theorem with moerate mass constraint. With the Lemma 3. at our isposal, we can now turn to the proof of theorem 3. along the lines of [BCM8]. Note that the actual calculation are to be carrie out with the regularize solutions n ɛ of (.3), though for the sake of simplicity, we only o the formal calculation on n(x) = n(, t). The key is to use the logarithmic Hary-Littlewoo-Sobolev inequality to get a boun on the entropy S[n].

12 SIMING HE AND EITAN TADMOR Theorem 3. (Logarithmic Hary-Littlewoo-Sobolev Inequality). [CL9] Let f be a nonnegative function in L (R ) such that f log f an f log( x ) belong to L (R ). If R fx = M, then f log fx f(x)f(y) log x y xy C(M) (3.3) R M R R with C(M) := M( log π log M). Remark 3.3. It is pointe out in [BCM8] that by multiplying f by inicator functions, one can prove that the inequality (3.3) remains true with R replace by any boune omains D R. The iea of the proof goes as follows. By observing that the mass in the upper half plane an lower half plane are subcritical ( n ± < 8π), we plan to use the logarithmic Hary-Littlewoo- Sobolev inequality on these sub-omains to get uniform boun on the entropy. However, without extra information concerning the cell ensity istribution, naive application of logarithmic Hary- Littlewoo-Sobolev inequality fails. For this approach to work, the ensity istribution constraint require is that the cells in the upper an lower half plane are well-separate by a cell clear strip in which the total number of cells is sufficiently small. The strip is constructe through applying Lemma 3.. Combining the logarithmic Hary-Littlewoo-Sobolev inequality an the cell seperation constraint, we can use a total entropy reconstruction trick introuce in [BCM8] to obtain entropy boun. Now let s start the whole proof. Proof. First we contruct the cell clear strip. Define the following three regions: Γ = {x x > δ}, Γ = {x x < δ}, Γ 3 = R \(Γ Γ ). (3.4) Here region Γ contains points in the upper half plane which are δ away from the x axis, whereas region Γ contains points in the lower half plane with the same property. Region Γ 3 is a close neighborhoo of the x axis. The δ neighborhoo of the Γ, Γ region is enote as follows: We further ecompose Γ 3 into subomains: Γ (δ) = {x x > δ}, Γ (δ) = {x x < δ}. (3.5) S = {x δ < x δ}, S = {x δ > x δ}, S 3 = {x x δ}. (3.6) Applying Lemma 3. yiels that the total mass insie Γ 3 is small, i.e. ( η) nx = nx Γ 3 {x x δ} R M. (3.7) Therefore, the Γ 3 strip is the cell clear strip. Next, we estimate the entropy. First recall that the free energy E[n](t) (.4) is ecreasing, i.e, ( E[n ] E[n] = K ) n log nx ( K n log nx 8π 8π ) n(x)n(y) log x y xy Gnx R R ( =: K ) S[n] T T. (3.8) 8π To obtain the entropy boun, we nee to estimate T from below for some K < 8π an estimate T from above. We start by estimating T. Similar to [BCM8], we apply the Logarithmic

13 SUPPRESSING CHEMOTACTIC BLOW-UP THROUGH A FAST SPLITTING SCENARIO 3 Figure. Regions Γ, Γ, Γ 3 Hary-Littlewoo-Sobolev inequality in the three regions Γ (δ), Γ(δ), Γ 3 an obtain: Γ (δ) Γ (δ) n(x)x n(x)x Γ (δ) Γ (δ) n log nx n log nx Γ (δ) Γ(δ) Γ (δ) Γ(δ) n(x)n(y) log x y xy C, n(x)n(y) log x y xy C, n(x)x Γ 3 n log nx Γ 3 n(x)n(y) log x y xy C. Γ 3 Γ 3 Combining the above inequalities yiels C K n log nx R n(x)n(y) log x y xy R R 4 n(x)n(y) log x y xy Here the number K is efine as { K := max ((Γ (δ) )c Γ ) (Γ (S S 3 )) (S S ) (S S ) n(x)n(y) log x y xy =:I I I 3 I 4. (3.9) Γ (δ) Combining the efinition (3.3) an (3.7) yiels K } nx nx, nx nx. (3.3) Γ 3 Γ (δ) Γ 3 ( ) ( η) R M.

14 4 SIMING HE AND EITAN TADMOR By the moerate mass constraint (3.3), we have K < 8π. Next applying the fact that x y δ for all (x, y) in the integral omain of I 3, we estimate the I 3 an I 4 terms in (3.9) as follows I 3 4M log δ, I 4 C (S S ) (S S ) (S S ) (S S ) n(x)n(y) log x y xy n(x)n(y)( x y )xy C(M M x n(x)x). (3.3) Combining (3.3) with (3.9) yiels K n log nx n(x)n(y) log x y xy R R R 4M log δ C( M M Recall the well-known upper boun on the negative part of the entropy: n x x). (3.3) Lemma 3.4. ([BCM8]) For f positive function, the following estimate hols f log fx x fx log(π) fx C( M x nx). (3.33) R R R e Proof. For the proof of this lemma, we refer the intereste reaers to the papers [BDP6],[BCM8]. Combining (3.33) an (3.3) yiels the lower boun of T in (3.8) ( ) T C(M ) M n x x 4M log δ (3.34) for < K < 8π. For the T term in (3.8), it is boune above by Hnx A x nx. Therefore it is enough to show that the secon moment is boune for any finite time. The time evolution of the secon moment can be estimate as follows: n x x 4AM 4A Hn(x)x 4AM A x nx. t Gronwall inequality yiels that the secon moment is boune for all finite time: n x x C(A, T ) <, T <. (3.35) Therefore T C(A, T ). Combining this with (3.8) an (3.34), an recalling that K < 8π yiel ( S[n](T ) ( K 8π ) E[n ] C(M, A, T ) ) π M log δ, T <. (3.36) As a result, we see from (3.36) that the entropy S[n ɛ ] is uniformly boune inepenent of ɛ for any finite time interval [, T ], T <. Now by the Proposition.,., we have that the free energy solution exists on any time interval [, T ], T <.

15 SUPPRESSING CHEMOTACTIC BLOW-UP THROUGH A FAST SPLITTING SCENARIO Step 3 proof of the main theorem. In the proof of Theorem 3., we see that the cell population is separate by a cell clear zone near the x axis. Since total mass in the cell clear zone is small, we can heuristically treat the total cell population as a union of two subgroups with subcritical mass (< 8π). However, since we lack sufficiently goo control over the total number of cells near the x axis, we cannot use this iea to prove the optimal result as state in Theorem.. The iea of proving Theorem. is that instea of consiering the total cell population as the union of two subgroups separate by one fixe cell clear zone, we treat it as the union of three subgroups with subcritical mass, namely, the cells in the upper half plane, the lower half plane an the neighborhoo of the x axis, respectively. These three subgroups of cells are separate by two cell clear zones varying in time. The main ifficulty in the proof is setting up the three new regions such that:. mass insie each region is smaller than 8π;. the total mass of cells near their bounaries is well-controlle. Once the construction is complete, the remaining steps will be similar to step. Proof of Theorem.. We start by constructing the three regions. First we note that the Lemma 3. implies that there exists δ > such that the following estimate is satisfie for a fixe R > an η chosen small enough: x δ nx ( η) R nx M, t >. (3.37) R Now the region L = {(x, x ) x δ} have total mass less than M = M < 8π for all time. Seconly, we subivie the region L into J pieces: L = J L i, L i :={(x, x ) δ J (i) > x δ (i )}. J Here J = J(M), to be etermine later, epens on M. By the pigeon hole principle, there is at least three strips L i such that L i n(x)x J M. Suppose there are only two strips with mass smaller than J M, then total mass in L will be bigger than (J ) J M > M, a contraiction. Now we pick from these three strips the one which is neither L nor L J. As a result, this strip L i oes not touch the x axis nor the bounary of L. We enote this i by i. The L i is the cell clear zone. Notice that here i = i (n, t) epens on time. Finally, we use this i to efine the regions. First we efine the three regions, each of which has total mass smaller than 8π: Next we set Γ = { x δ } { J i, Γ = x δ } { J i, Γ 3 = x δ } J (i ). (3.38) ρ = δ 3J, (3.39) an efine the ρ neighborhoo of the above three regions: = {x > δ J (i { 3 )} Γ(ρ) = x < δ J (i } 3 ), 3 = { x < δ J (i )}. (3.4) 3

16 6 SIMING HE AND EITAN TADMOR Figure. Regions Γ, Γ, Γ 3 in the proof of the main theorem Now we efine the complement Γ 4 of the above three regions i, i =,, 3: { ( δ Γ 4 = i ) x δ ( i ) } = Γ 4 Γ 4, (3.4) J 3 J 3 Γ 4± =Γ 4 R ±. (3.4) Now we efine the complement 4 of 3 i= Γ i an ecompose it into subomains: 4 = ( 3 ) c ( (ρ)) ( (ρ)) ( (ρ)) (ρ) i= Γ i = Γ 4 Γ 4, Γ 4± = Γ 4 R ±, (3.43) 4 =Γ 4 S S S 3, (3.44) S = \Γ, S = \Γ, S 3 = 3 \Γ 3. (3.45) Remark 3.4. It is important to notice that the regions we are constructing is changing with respect to the given time t. Therefore, by oing the argument below, we can only show that the entropy is boune at time t, but since t is an arbitrary finite time, we have the boun on entropy for t [, T ], T <. We start estimating the entropy. By the free energy issipation, we obtain ( E[n ] K ) n log nx ( K n log nx 8π 8π ) n(x)n(y) log x y xy Hnx R R ( = K ) S[n(T )] T T. (3.46) 8π

17 SUPPRESSING CHEMOTACTIC BLOW-UP THROUGH A FAST SPLITTING SCENARIO 7 To erive entropy boun, we nee the estimate T form below for K < 8π an estimate T from above. We start by estimating T. Combining the efinition of L i an (3.37) yiels nx M = M < 8π, (3.47) nx M = M < 8π, (3.48) 3 4 nx M = M < 8π, (3.49) nx nx L i J (M ). (3.5) Now by the log-hary-littlewoo-sobolev inequality (3.3), we have that n(x)x n(x) log n(x)x n(x)n(y) log x y xy C, i =,, 3, i i 4± i i (3.5) n(x)x n(x) log n(x)x n(x)n(y) log x y xy C. 4± 4± Γ(ρ) 4± (3.5) Same as in subsection 3.3, we use these estimates to reconstruction the entropy an the potential on the whole R as follows: C K n(x) log n(x)x n(x)n(y) log x y xy R R R n(x)n(y) log x y xy R n(x)n(y) log x y xy (S S ) (S S ) (S 3 S 3 ) (S 3 S 3 ) =:I I I 3 I 4. (3.53) The region R an the integral omain of I 4 is inicate in Figure 3. The K in (3.53) can be estimate using (3.5) as follows ( K := M nx ) M. (3.54) J 4 By the assumption M < 8π, we can make J big such that K < 8π. This is where we choose the J = J(M). Applying the fact that x y δ 3J, (x, y) R, the I 3 an I 4 terms in (3.53) can be Region R is the union of the following nine regions: )Γ ( )c, )S (Γ ( 4 ) ) c, 3)Γ 4 (( 4 ) ) c, 4)S 3 ( 3 ( 4 ) ) c, 5)Γ 3 ( 3 )c, 6)S 3 ( 3 ( 4 ) ) c, 7)Γ 4 (( 4 ) ) c, 8)S ( ( 4 ) ) c, 9)Γ ( )c.

18 8 SIMING HE AND EITAN TADMOR Figure 3. Region R in R R estimate as follows: I 3 CM log δ 3J, I 4 n(x)n(y) log x y xy (S S ) (S S ) (S 3 S 3 ) (S 3 S 3 ) C n(x)n(y)( x y )xy (S S ) (S S ) (S 3 S 3 ) (S 3 S 3 ) C(M M x nx). (3.55) Combining (3.53), (3.54) an (3.55) yiels K n(x) log n(x)x n(x)n(y) log x y xy R R CM log δ 3J C( M M x nx). Moreover, applying (3.33) yiels T C(M )( M x nx) CM log δ 3J. (3.56) Combining (3.56), (3.35), (3.46) yiels ( S[n](T ) ( K 8π ) E[n ] C(M, A, T ) CM log δ ) <, T <. 3J Once the entropy is boune for any finite time, the existence is guarantee by Proposition. an Proposition..

19 SUPPRESSING CHEMOTACTIC BLOW-UP THROUGH A FAST SPLITTING SCENARIO 9 Appenix A. In the appenix, we prove the two local existence theorems state in section.. The proof follows the same line as the analysis in [BCM8]. A.. Proof of proposition.. Proof. For any fixe positive ɛ, following the argument as in section.5 of [BDP6], we obtain the global solution in L ([, T ], H ) C([, T ], L ) for the regularize Keller-Segel system with avection (.3). The goal is to use the Aubin-Lions Lemma to show that the solutions {n ɛ } ɛ is precompact in certain topology. Same as in the paper [BCM8], we ivie the the proof in steps. Step. A priori estimates on n ɛ an c ɛ. In this step, we erive several estimates which we will nee later. First we estimate the secon moment V := x n ɛ x. R The time evolution of V can be estimate using the regularise equation (.3) an the fact that the graient of the regularize kernel K ɛ is boune K ɛ (z) π z : t V =4M n ɛ (x, t)n ɛ (y, t)(x y) K ɛ (x y)xy x bn ɛ (x)x R R 4M C x n ɛ x. By Gronwall, we have that V (t) 4M C ect V C V (T ) <, t T, (A.) from which we obtain that ( x )n ɛ L ([, T ], L ) uniformly in ɛ. Next, we estimate n ɛ log n ɛ L t ([,T ];L ) an n ɛ c ɛ x L t. Combining the assumption of the proposition. an (3.33) yiels n ɛ log n ɛ x n ɛ (log n ɛ x )x log(π)m e. (A.) Therefore, we prove that n ɛ log n ɛ L ([, T ], L ) uniformly in ɛ. Recalling the bouneness of the secon moment (A.) an the representation of c ɛ as c ɛ = K ɛ n ɛ, we euce the following estimate using the Young s inequality, c ɛ (x) = K ɛ (x y)n ɛ (y)y C(M, V ) C(M) log( x ) (A.3) uniformly in ɛ. Combining this with the mass conservation of n ɛ an the secon moment control (A.), we euce that n ɛ c ɛ x C, (A.4) L ([,T ]) where C is inepenent of ɛ.

20 SIMING HE AND EITAN TADMOR Next we erive the main a priori estimate, namely, the L ([, T ] R ) estimate of n ɛ c ɛ. First we calculate the time evolution of n ɛ c ɛ x: n ɛ c ɛ x = n ɛ t tc ɛ x = ( n ɛ ( c ɛ n ɛ ) b n ɛ )c ɛ x = n ɛ c ɛ x n ɛ c ɛ x bn ɛ c ɛ x. Integrating this in time yiels: T n ɛ c ɛ xt = n ɛ c ɛ x(t ) T n ɛ c ɛ x() T n ɛ c ɛ xt n ɛ b c ɛ xt (A.5) Now we estimate the right han sie of (A.5). We see from (A.4) that the first two terms on the right han sie is boune. Next we estimate the thir term on the right han sie of (A.5), which requires information erive from the entropy boun. By the property that b =, we formally calculate the time evolution of S[n](t) as follows, n K S[n](t) = 4 t n x n(t) x. (A.6) The intereste reaer is referre to [BDP6] for more etails. We nee to estimate the secon term in (A.6). Before oing this, note that for K >, nx n log nx C =: η(k) log(k) log(k) (A.7) can be mae arbitrarily small. Now we can use the Gagliaro-Nirenberg-Sobolev inequality together with (A.7) to estimate the secon term in (A.6) as follows: ( ) / n x MK n x MK nx n 3/ 3 n K MK η(k) / CM / n. Combining this with (A.6) an (A.7), we have t S[n](t) = (4 η(k)/ CM / ) n K n x MK. (A.8) The factor (4 η(k) / CM / ) can be mae non-positive for K large enough an therefore we have that T n S[n]() S[n](T ) MKT xt (4 η(k) / M /. C) It follows that n is boune in L ([, T ] R ). The erivation for n ɛ L ([,T ];L ) C is similar but more technical, an the intereste reaers are referre to [BDP6] for more etails. As a consequence of the L ([, T ] R ) estimate on n ɛ an of the computation we have the estimate t S[nɛ ](t) = 4 T n ɛ x n ɛ ( c ɛ )xt C. n ɛ ( c ɛ )x, (A.9)

21 SUPPRESSING CHEMOTACTIC BLOW-UP THROUGH A FAST SPLITTING SCENARIO This completes the treatment of the thir term on the right han sie of (A.5). Next we estimate the last term T bnɛ c ɛ xt in (A.5). First we calculate the time evolution of Gn ɛ x: Gn ɛ x = G ( n ɛ c ɛ n ɛ bn ɛ )x t = Gn ɛ x G c ɛ n ɛ x b n ɛ x = b c ɛ n ɛ x b n ɛ x. Now integrating in time, we obtain T b c ɛ n ɛ x Gn ɛ x() Gn ɛ T x(t ) b n ɛ xt. (A.) From the assumption b x, we have that the right han sie of (A.) can be boune in terms of the secon moment V : Gn ɛ x b n ɛ x C x n ɛ x. (A.) Since the secon moment V is boune (A.), we have that T b c ɛ n ɛ x C. (A.) Applying estimates (A.4), (A.9) an (A.) to (A.5), we obtain T n ɛ c ɛ xt C <. (A.3) This conclues our first step. Step - Passing to the limit. As in [BCM8], the following Aubin-Lions compactness lemma is applie: Lemma A. (Aubin-Lions lemma). [BCM8] Take T > an < p <. Assume that (f n ) n N is a boune sequence of functions in L p ([, T ]; H) where H is a Banach space. If (f n ) n is also boune in L p ([, T ]; V ) where V is compactly imbee in H an f n / t L p ([, T ]; W ) uniformly with respect to n N where H is imbee in W, then (f n ) n N is relatively compact in L p ([, T ]; H). Our goal now is to fin the appropriate spaces V, H, W for n ɛ. We subivie the proof into steps, each step etermines one space in the lemma. Space H: Boun on n ɛ L ([,T ],L ): We can estimate the n ɛ by applying the following ecomposition trick: n ɛ = (n ɛ K) min{n ɛ, K}. The secon part in (A.4) is boune in L p, p. The first part is boune in L by (n ɛ K) nɛ log n ɛ log K (A.4) = η(k), (A.5)

22 SIMING HE AND EITAN TADMOR which can be mae arbitrary small. Now we estimate the time evolution of (n ɛ K) p x, < p < as follows: (n ɛ K) p p t x = (n ɛ K) p ( nɛ ( c ɛ n ɛ ) b n ɛ )x 4(p ) = p (n ɛ K) p/ x (n ɛ K) p ( cɛ n ɛ )x (n ɛ K) p b (nɛ K) x 4(p ) p (n ɛ K) p/ x p (n ɛ K) p p cɛ x K (n ɛ K) p cɛ x 4(p ) =: p (n ɛ K) p/ x T T. (A.6) Note that since b, the term involving b vanishes. For the T term in (A.6), using the facts that c ɛ = K ɛ n ɛ an K ɛ is uniformly boune, we can estimate it as follows: T (n ɛ K) p Kɛ (n ɛ K) x K (n ɛ K) p x Kɛ (n ɛ K) p p Kɛ (n ɛ K) p CK (n ɛ K) p x C (n ɛ K) p p CK (nɛ K) p p. (A.7) Similarly, we can estimate the T term in (A.6) as follows: T CK (n ɛ K) p p CK (n ɛ K) p p. Combining (A.6), (A.7) an (A.8), we obtain (n ɛ K) p ) x 4(p p t p (n ɛ K) p/ x C (n ɛ K) p p CK (nɛ K) p p CK (n ɛ K) p p. (A.8) (A.9) For the highest orer term C (n ɛ K) p p in (A.9), we use the following Gagliaro-Nirenberg- Sobolev inequality: f p x C (f p/ ) x fx, f (A.) R R R together with (A.5) to estimate it as follows (n ɛ K) p p C ((nɛ K) p/ ) (n ɛ K) Cη(K) ((n ɛ K) p/ ). (A.) We can take K big such that it is absorbe by the negative issipation term in (A.9). Now applying Höler s inequality, Young s inequality an Gronwall inequality to (A.9), we have that (n ɛ K) L p(t) C(T ) <, t [, T ], p (, ). Applying stanar argument, see e.g., [BDP6] proof of Proposition 3.3, we obtain the estimate In particular, we set p = an obtain that We conclue this step by setting H := L (R ). n ɛ L ([,T ];L p ) C(T ), p (, ). (A.) n ɛ L ([,T ];L ) C(T ). (A.3)

23 SUPPRESSING CHEMOTACTIC BLOW-UP THROUGH A FAST SPLITTING SCENARIO 3 Space V : Boun on n ɛ L ([,T ] R ): First we calculate the time evolution of the quantity n ɛ : t n ɛ x = = n ɛ x n ɛ x n ɛ c ɛ n ɛ x n ɛ c ɛ n ɛ x. (b(n ɛ ) )x Now integrating in time, we obtain the estimate n ɛ x(t ) n ɛ x() T = T T n ɛ xt ( T n ɛ xt n ɛ ( n ɛ c ɛ )xt ) / ( T n ɛ c ɛ xt n ɛ xt) / (A.4) The terms on the left han sie of (A.4) are boune ue to (A.). For the last term on the right han sie, we can estimate it as follows. The Hary-Littlewoo-Sobolev inequality yiels which implies c ɛ 4 C n ɛ 4/3, n ɛ c ɛ n ɛ 4 c ɛ 4 C n ɛ 4 n ɛ 4/3. (A.5) Combining this an the L p boun (A.) yiels the bouneness of n ɛ c ɛ in L ([, T ], L ). Applying this fact an (A.) in (A.4) an set X = ( T n ɛ xt) /, we obtain X n ɛ c ɛ L ((,T ) R )X n ɛ x(t ) n ɛ x() C. As a result, n ɛ L ([,T ] R ) C. (A.6) Same as in [BCM8], we set V := H (R ) {n x n L }, which is shown to be compactly imbee in H there. Thanks to the boun (A.3), (A.6) an the (A.), we have that n ɛ L ([, T ]; V ). Space W : Boun for the t n ɛ : In orer to estimate the L ([, T ]; H ) norm of the function t n ɛ, we first nee to get a boun on the fourth moment of the solution V 4 := n(x 4 x4 )x. The time evolution of V 4 can be formally estimate as follows: t V 4 = ( n ( cn) b n)(x 4 x 4 )x = n x x 4(x 3 y 3)(x y ) 4(x 3 y3 )(x y ) 4π x y n(x)n(y)xy bn (4x 3, 4x 3 )x C(M) n x x C n(x 4 x 4 )x. (A.7) Combining the secon moment estimate (A.) an the Gronwall inequality, we have that n(x, t) x 4 x C, t [, T ]. (A.8) One can aapt this calculation to the regularize solution n ɛ without much ifficulty. We leave the etails to the intereste reaer.

24 4 SIMING HE AND EITAN TADMOR Now we can estimate the L ([, T ]; H ) norm of the t n ɛ. Combining the L p boun on n ɛ (A.), the boun on c (A.5) an the fourth moment control (A.8) an testing the equation (.3) with f L ([, T ], H (R )), we have t n ɛ, f L ([,T ] R ) n ɛ L ([,T ];L ) f L ([,T ];H ) c ɛ n ɛ L ([,T ];L ) f L ([,T ];H ) bn ɛ L ([,T ];L ) f L ([,T ];H ) n ɛ L ([,T ];L ) f L ([,T ];H ) T / c ɛ L ([,T ];L 4 ) n ɛ L ([,T ];L 4 ) f L ([,T ];H ) CT / sup V /4 4 n ɛ 3/4 L t ([,T ];L 3 ) f L ([,T ];H ) C f L ([,T ];H ). As a result, we have that t n ɛ is uniformly boune in L ([, T ]; H ). Combining the results from all the steps above an the Aubin-Lions lemma, we have that (n ɛ ) ɛ is precompact in L ([, T ]; L ). We enote n as the limit of one converging subsequence (n ɛ k) ɛk. Moreover, combining (A.5) an the L 4/3 ([, T ] R ) boun on n ɛ which can be erive from (A.), we have that n ɛ k c ɛ k converge to n c in istribution sense. Step 3- Free energy estimates. By the convexity of the functional n R n x, the fact that n ɛ k c ɛ k converge to n c in istribution sense an weak semi-continuity, we have n xt lim inf n ɛ k xt, [,T ] R k [,T ] R n c xt lim inf n ɛ k c ɛ k xt. [,T ] R k [,T ] R Moreover, it can be checke that S[n ɛ ](t) S[n](t) for almost every t, whose proof is similar to the one use in [BDP6] Lemma 4.6. Next we show the free energy estimate (.5) using the strong convergence of {n ɛ k} in L ([, T ] R ). The key is to show the following entropy issipation term is lower semi-continuous for the sequence (n ɛ k): T n ɛ k log n ɛ k c ɛ k b xt =4 n ɛ k xt n ɛ k c ɛ k xt n ɛ k b xt [,T ] R [,T ] R [,T ] R (n ɛ k ) xt n ɛ k b c ɛ k xt [,T ] R [,T ] R =:T T T 3 T 4 T 5. (A.9) For the sake of simplicity, later we use n ɛ to enote n ɛ k. First, we estimate the T 3 term in (A.9). By the Fatou Lemma, we have the following inequality: This finises the treatment of T 3. n b xt lim inf [,T ] R ɛ k [,T ] R n ɛ k b xt. (A.3)

25 SUPPRESSING CHEMOTACTIC BLOW-UP THROUGH A FAST SPLITTING SCENARIO 5 Next, we show that the term T 5 in (A.9) actually converges as ɛ k. We ecompose the ifference between T 5 an its formal limit into two parts T b cn b c ɛ n ɛ xt R T c bn c ɛ bn nb c ɛ b c ɛ n ɛ xt R T T c c ɛ bn xt bn bn ɛ c ɛ xt R R =:T 5 T 5. (A.3) For the first term T 5 in (A.3), applying Höler an Hary-Littlewoo-Sobolev inequality, we can estimate it as follows ( T ) /4 ( T T 5 (c c ɛ ) 4 4t b ) / ( T ) n /4 4 xt n xt R R T T T C( n n ɛ 4 4/3 t)/4 ( x nxt) / ( n xt) /4 R R ( T ) /4 ( T ) / ( T ) /4 C n n ɛ n n ɛ t x nxt n xt R R ( T /4 ( T ) / ( T ) /4 C n n ɛ t) sup n n ɛ / x nxt n xt. t R R From the L ([, T ] R ) strong convergence of (n ɛ ) ɛ, we have that the first factor goes to zero. Other factors are boune ue to (A.), Fatou s Lemma, L boun on n ɛ, n an n L ([, T ] R ). As a result, T 5 converges to zero. Next we estimate the T 5 term in (A.3). Applying the fact that c a c a, the Höler an Hary-Littlewoo-Sobolev inequality, we have T T 5 = b( n n ɛ ) c ɛ xt R T b n( n T n ɛ ) c ɛ xt b n ɛ ( n n ɛ ) c ɛ xt R R c ɛ L 4 ([,T ] R ) b ( T ) /4 n L ([,T ] R ) n n ɛ 4 xt R c ɛ L 4 ([,T ] R ) b T n ɛ L ([,T ] R )( n n ɛ 4 xt) /4 R ( T ) / C sup n ɛ L (R ) / n ɛ / (n ɛ n) x xt n n ɛ / L t ([,T ] R ) L ([,T ] R ) By the same reasoning as in T 5, we have that this term goes to zero. This finishes the treatment for the term. Now for the T, T, T 4 terms in (A.9), we can hanle them in the same way as in [BDP6]. Combining all the estimates above we have that T T n log n c b xt lim inf n ɛ k log n ɛ k c ɛ k b xt (A.3) ɛ k The remaining part of the proof is the same as the one in [BDP6].

26 6 SIMING HE AND EITAN TADMOR A.. Proof of proposition.. The proof of the proposition follows along the lines of [BCM8]. Acknowlegment. Research was supporte by NSF grants DMS6-39, RNMS-7444 (KI- Net), ONR grant N4-594 an the Sloan research fellowship FG SH thanks the Ann G. Wylie Dissertation Fellowship an Jacob Berossian an Scott Smith for many fruitful iscussions. We thank ETH Institute for Theoretical Stuies (ETH-ITS) for the support an hospitality. References [BCZ5] J. Berossian an M. Coti Zelati, Enhance issipation, hypoellipticity, an anomalous small noise invisci limits in shear flows, ArXiv:5.898, 5. [BH6] J. Berossian, S. He, Suppression of blow-up in Patlak-Keller-Segel via shear flows, arxiv: [BK], J. Berossian an I. Kim, Global existence an finite time blow-up for critical Patlak-Keller-Segel moels with inhomogeneous iffusion, SIMA 45(3), , 3. [BM4] J. Berossian an N. Masmoui, Existence, Uniqueness an Lipschitz Depenence for Patlak-Keller-Segel an Navier-Stokes in R with Measure-value initial ata, Arch. Rat. Mech. Anal, 4(3), 77 8, 4. [Bil995] P. Biler, The Cauchy problem an self-similar solutions for a nonlinear parabolic equation, Stuia Math., 4():89, 995. [BCC8] A. Blanchet, V. Calvez an J. A. Carrillo, Convergence of the mass-transport steepest escent scheme for the subcritical Patlak-Keller-Segel moel, SIAM J. Numer. Anal., 46 (8), [BDP6] A. Blanchet an J. Dolbeault an B. Perthame, Two-Dimensional Keller-Segel Moel: Optimal Critical Mass an Qualitative Properties of the Solutions, EJDE 44, 3, 6. [BCM8] A. Blanchet, J. A. Carrillo an N. Masmoui, Infinite Time Aggregation for the Critical Parlak-Keller- Segel moel in R, Comm. Pure Appl. Math., , 8. [CL9] E. Carlen an M. Loss, Competing symmetries, the logarithmic HLS inequality an Onofri s inequality on S n, Geom. Func. Anal.(99) : [CC6] V. Calvez an J.A. Carrillo, Volume effects in the Keller-Segel moel: energy estimates preventing blow-up, JMPA, (6), volume 86, pages [CC8] V. Calvez an L. Corrias, The parabolic-parabolic Keller-Segel moel in R, Commun. Math. Sci., 6() , 8. [CCLW] J. A. Carrillo, L. Chen, J.-G. Liu an J. Wang, A Note on the Subcritical Two Dimensional Keller-Segel System, Acta Appl Math, V. 9, Issue, (), [CR] J.A. Carrillo an J. Rosao, Uniqueness of boune solutions to Aggregation equations by optimal transport methos, Proc. 5th Euro. Congress of Math. Amsteram, (8), European Mathematical Society. [Del99] J.M. Delort, Existence e nappes e tourbillon en imension eux. J. Am. Math. Soc. 4, (99) [DS9] J. Dolbeault an C. Schmeiser, The two-imensional Keller-Segel moel after blow-up, Discrete an Continuous Dynamical Systems, 5 (9), pp. 9-. [DLM] R.-J. Duan, A. Lorz an P. Markowich, Global Solutions to the Couple ChemotaxisFlui Equations, Comm. Partial Differential Equations, Vol. 35, (), page [EM6] G. Egaa an S. Mischler, Uniqueness an long time asymptotic for the Keller-Segel equation: the parabolicelliptic case, Arch. Ration. Mech. Anal. (6), no. 3, [FLM] M. Di Francesco, A. Lorz an P. Markowich, Chemotaxisflui couple moel for swimming bacteria with nonlinear iffusion: global existence an asymptotic behavior, Discrete Contin. Dyn. Syst. Ser. A, Vol. 8, (), page [Hor3] D. Hortsmann, From 97 until present: the Keller-Segel moel in chemotaxis an its consequences, I, Jahresber. Deutsch. Math.-Verein, 5(3), 3 65, 3. [JL99] W. Jäger an S. Luckhaus, On Explosions of Solutions to a System of Partial Differential Equations Moelling Chemotaxis, Trans. Am. Math. Soc., (99), volume 39, pages [KS97] E. F. Keller an L. A. Segel, Initiation of slie mol aggregation viewe as an instability, J. Theor. Biol., 6 (97). [KJCY7] S. Khan, J. Johnson, E. Cartee, an Y. Yao, Global Regularity of Chemotaxis Equations with Avection, to appear in Involve. [KX5] A. Kiselev an X. Xu, Suppression of chemotactic explosion by mixing, Archive for Rational Mechanics an Analysis, -36, 5. [LL] J.-G. Liu an A. Lorz, A couple chemotaxis-flui moel: global existence, Annales e linstitut Henri Poincare. Analyse Non Lineaire, Vol. 8, (), page [L] A. Lorz, Couple chemotaxis flui moel, Math. Moels Methos Appl. Sci., Vol., (), page

27 SUPPRESSING CHEMOTACTIC BLOW-UP THROUGH A FAST SPLITTING SCENARIO 7 [L] A. Lorz, A couple Keller-Segel-Stokes moel: global existence for small initial ata an blow-up elay, Communications in Mathematical Sciences, Vol., (), page [Pat953] C. S. Patlak, Ranom walk with persistence an external bias, Bull. Math. Biophys., 5 (953), pp [Po] F. Poupau, Diagonal efect measures, ahesion ynamics an Euler equations, Meth. Appl. Anal. 9 (), pp [TS] Takasi Senbaa, Takashi Suzuki, Weak Solutions to a Parabolic-Elliptic System of Chemotaxis J. Functional. Analysis 47 (), 7-5. Department of Mathematics an Center for Scientific Computation an Mathematical Moeling (CSCAMM), University of Marylan, College Park aress: simhe@cscamm.um.eu Department of Mathematics, Center for Scientific Computation an Mathematical Moeling (CSCAMM), an Institute for Physical Sciences & Technology (IPST), University of Marylan, College Park aress: tamor@cscamm.um.eu