Coupled Keller Segel Stokes model: global existence for small initial data and blow-up delay

Transcription

1 Couple Keller Segel Stokes moel: global existence for small initial ata an blow-up elay Alexaner Lorz Department of Applie Mathematics an Theoretical Physics University of Cambrige Wilberforce Roa, Cambrige CB3 WA, UK Abstract We stuy a system consisting of the elliptic-parabolic Keller Segel equations couple to Stokes equations by transport an gravitational forcing. We show globalin-time existence of solutions for small initial mass in 2D. In 3D we establish global existence assuming that the initial L 3/2 -norm is small. Moreover, we give numerical evience that for this extension of the Keller Segel system in 2D, solutions exist with mass above 8π, which is the critical mass for the system without flui. 1 Motivation The Keller Segel system moelling chemotaxis is a very well stuie moel in mathematical biology. In the following, we investigate a system consisting of the elliptic-parabolic Keller Segel equations couple to Stokes equations u c = c + n a 1 c, n t + u n = n (χn c), (1.1) a 2 u t + P η u + n φ =, u =. Here c enotes the concentration of a chemical, n acellensityanu afluivelocity fiel escribe by Stokes equations. The flui couples to n an c through transport an gravitational forcing moelle by φ. ThepressureP can be seen as the Lagrange multiplier enforcing the incompressibility constraint. The chemical c iffuses, it is prouce by the cells an it egraes. The cell ensity iffuses an it moves in the irection of the chemical graient. The constant a 1 measuresself-egraationofthechemical an the constants a 2, η>eterminetheevolutionunergonebyu. The motivation for this moel comes from experiments escribe in [21, 29, 32] for the case of bacteria consuming the chemical. There the authors observe large-scale convection patterns in a water rop sitting on a glass surface containingoxygen-sensitive bacteria, oxygen iffusing into the rop through the flui-air interfaceantheypropose this moel: c t + u c = c nf(c), n t + u n = n (nχ(c) c), (1.2) u t + u u + P η u + n φ =, u =. 1

2 The iea behin this paper is to take these equations an change consumption of the chemical to prouction i.e. make it the mathematical more interesting case of Keller Segel chemotaxis. An example in biology coul be E. coli which canswimanexcrete aspartate [1]. The ultimate aim is to stuy if an how flui coupling is changing the blow-up behaviour. Assuming a vanishing flui velocity fiel u, werecovertheelliptic-parabolic Keller Segel equations, see [3]. The problem of global existence vs. blow-up is very well unerstoo for the elliptic-parabolic Keller Segel moel { c = n, (1.3) t n + (χn c n) =, in.[5]summarizestheresults,i.e. there is a critical mass M crit,startingwithinitial mass below M crit,thereisglobal-in-timeexistenceanstartingwithinitial mass above M crit,thesolutionblowsupinfinitetime. OnabouneomainΩ the analysis is more involve because bounary effects play an important role. With zero Dirichlet bounary conitions for c an corresponing no-flux conitions on n, theresultsare the same as in full space, see [3]. For homogeneous Neumann bounary conitions, the Keller-Segel system has to be change to { c = n n, (1.4) t n + (χn c n) =, because we nee the right-han sie of the c-equation to have zero average. In this case, there are several threshol values: if Ω is regular then solutions are global if χm < 4π an may blow up above this threshol, either on the bounary or insie the omain. For more information about this type of bounary conitions, see [22]. For the parabolic-parabolic Keller Segel moel recent progress has been achieve in [1]. For more references on the general Keller Segel system, the intereste reaer can refer to recent work [4, 5, 1]. The Keller Segel system in higher space imensions has been investigate in [13, 14] an for the system with nonlinear iffusion to prevent blow-up, results can be foun in [9, 24]. Other ways to moel prevention of overcrowing can be foun in [6,7,2]. Kinetic moels for chemotaxis can be foun in [12]. Numerics for the Keller Segel moel have been performe in [17,28]. For the system (1.2) an relate systems there is a local existence result [27]. Moreover, in [16] the authors prove global existence for a simplifie version of (1.2) with weak potential or small initial c. In[15],globalexistencetothesystem(1.2)withnon- linear iffusion is shown. To our knowlege, these are the only resultson(1.2). However, attention has recently been focuse on couple kinetic flui systemsfirstlyintrouce in [8] which have a similar mathematical flavor; also refer to [11, 19] about the stuies of the Vlasov-Fokker-Planck equation couple with the compressible or incompressible Navier-Stokes or Stokes equations, where the main tool use to prove the global existence of weak solutions or hyroynamic limits is an existing entropy inequality. For the Navier-Stokes an Stokes equations see [25, 26] an references therein for etaile mathematical theory. The paper is structure as follows. In section 2 we state the problem in etail an give our results on global existence. In section 3 we prove the global existence of solutions to (1.1) in 2D for small mass in the following steps: obtaining anentropy, usingthe regularizing effect an passing to the limit. In section 4, we give the main technical ifferences to aress the existence issues in 3D for (1.1). Finally, in section 5 we give 2

3 numerical evience that solutions to (1.1) exists for initial mass larger than 8π, but for even larger mass blow-up seems to occur. So the blow-up is elaye. 2 Preliminaries We consier the system (1.1) in full space R, =2, 3. The system has to be supplie with initial ata n(t =,x)=n (x), for a 2 > alsou(t =,x)=u (x). φ is assume to be in L (R ). Following [5] we call a triple (c, n, u) aweaksolutionto(1.1)ifthefollowingtwo conitions hol: (i) for every T>, t T, n(t, x), c(t, x), x R, c L 2 (,T; H 2 (R )); c(,t) H 1 (R )fora.e.t, (2.1) n(1 + x + ln n ) L (,T; L 1 (R )), n L 2 (,T; L 2 (R )), (2.2) u L 2 (,T; H 2 (R )),a 2 u t L 2 (,T; L 2 (R )); (2.3) (ii) it is further require that for ψ 1,ψ 2,ψ 3 C (R, R) an ψ C (R, R )with ψ =, ψ 1 uc x = ( ψ 1 c + ψ 1 n a 1 ψ 1 c) x, (2.4) R R ψ 2 nx= ( ψ 2 un ψ 2 n χ ψ 2 n c) x, (2.5) t R R ( a 2 ψu x + η t ψ u + ψ ) n φ x =, (2.6) R R ψ 3 ux=. R (2.7) The main iea for showing existence is to first establish a boun on use the regularizing effect to achieve L p -bouns. n ln(n) anthen R Let us efine E(t) := n(t) ln(n(t)) x + u(t) 2 2, D := λ n η u 2 2. (2.8) an L q σ(r )for1 q as the closure of { } v C (R ) v = (2.9) in L q (R ). Let us recall (see [18]) that each vector f L q (R )isuniquelyecompose as f = f + Q 3

4 with some f L q σ(r ), Q L q loc (R ), Q L q (R )an Q q C f q an Q L q (B ) C Q q where C is inepenent of f an B is an open ball in R.Themappingf f efines acontinuousprojectionp q from L q (R ) onto L q σ(r ). Now we can efine the Stokes operator A q = P q analsothespace D α,s q := { ( v L q (Ω); v q + t 1 α A q e taq v s q ) t 1/s < }. t More information on the Stokes operator an this efinition can be foun in [18]. The following result hols for all a 1,a 2 >. For simplicity, let us fix a 2 =1. Theorem 2.1 (2D) Assume n, χ, η >, u D 2/3,3 3/2 an There exists a M exist > such that if n ln(n )+n x + n x <. (2.1) n x < M exist, (2.11) then there is a global in time weak solution for (1.1) an we have an entropy inequality E(t)+ t where E an D are given in (2.8) an t. D t C + Ct + C t E t, (2.12) Remark 2.2 (Justification of the assumption a 1,a 2 > ) Assuming a 1 > gives nx= a 1 cx an this together with a graient boun achieve below enables us to control a range of L q -norms of c which we coul not obtain in full space just from the L 2 -boun on the graient. The reason for a 2 =1is similar: for the stationary Stokes system in we only obtain a L 2 -graient boun, which oes not allow us to control any L q -norm of u itself. Remark 2.3 (Reusing techniques from classical Keller Segel) We recall that the entropy for the classical Keller Segel reas as E KS := n ln(n) χ nc x. 2 Alreay when calculating the time evolution of the secon term, we see χ nc x = χ c t n + n t cx t 2 = χ c n + χ 2 n c n + n tcx. But there is no way to hanle the time erivative on c. 4

5 At least for a large viscosity η an sufficiently large mass, one woul expect blow-up because in this case u is small an the system therefore close to the Keller-Segel system. But regaring the aitional terms in the computation of the secon moment of n: x 2 nx=4 nx+2 x un x +2 nx cx t =4M +2 x un x +2 (u c c + a 1 c)x cx, alreay for the term x un x, theregularitythatwecanobtainforu by using n in L 1 is not enough, to estimate this term. 3 Existence in Here we prove theorem 2.1: We will establish a-priori estimates. We first establish postivity of n an c. In the expression for n ln(n) theterm nu c occurs. In t orer to boun it, we use an estimate of c 2.Weconcluethissectionbyemploying aregularizingeffecttopasstothelimit. 3.1 Positivity By some stanar argument e.g. maximum principle we can show that for n, we have n(x, t) foralmostallt, cf.[2]. Theequation(1.1) 2 leas to nx=. t So the L 1 -norm of n also calle mass is conserve. Moreover, also by maximum principle we obtain c(x, t) for almost all t. Therefore, the L 1 -norm of c is also conserve 3.2 c 2 boun a 1 c 1 = n 1. Multiplying the equation (1.1) 1 by c an using the Gagliaro-Nirenberg inequality c q C c 1 1/q 2 c 1/q 1, we obtain (cu c) x + c 2 2 = nc x a 1 c 2 2, c 2 2 n q c q C n q c 1 1/q 2 c 1/q 1 C n q c 1/q 2 c 1 1/q 1, (3.1) with 1/q +1/q =1 an 1 < q,q <. Nirenberg inequality for 1 q 2 n q = n 2 2q C Therefore it follows using the Gagliaro- ( n 1 1/q 2 ) n 1/q 2 2 2/q 2 = C n 2 n 1/q 1, (3.2) 5

6 that c 2 C n q/(2q 1) q c (q 1)/(2q 1) 1 C a (q 1)/(2q 1) 1 C a (q 1)/(2q 1) 1 ( n 2 2/q 2 n 1/q 1 = C a (q 1)/(2q 1) estimate of the L 1 -norm of nu c n q/(2q 1) q n (q 1)/(2q 1) 1 ) q/(2q 1) n (q 1)/(2q 1) 1 n (2q 2)/(2q 1) 2 n q/(2q 1) 1. (3.3) Let us work on the term (u c)n: Using(3.3)withq =9/8 antheinequality we arrive at u C u 2/5 2 D 2 u 3/5 3/2, u c nx u c 2 n 2 C u 2/5 2 D 2 u 3/5 3/2 n 6/5 2 n 7/5 1. Integrating over time an applying Höler inequality, we obtain t u c nxt ( t ) 1/5 ( t ) 1/5 ( t C u 2 2 t D 2 u 3 3/2 t Now we recall a regularity result for the Stokes equations: n 2 2 t ) 3/5 n 7/5 1. Theorem 3.1 (from [18]) Assume Ω=R or Ω R asmoothbouneomain. Let 1 <s,q<. Then for every f L s ((,T); L q (Ω)) an u Dq 1 1/s,s there exists a unique solution of the Stokes system { u t + P η u = f, satisfying t D 2 u s q t + t u =, ( t t u s q t C f s q t + u s D 1 1/s,s q Using this, we estimate the secon term above (by means of (3.2)) as follows t ( t ) t D 2 u 3 3/2 t C n 3 3/2 t + u 3 C n 2 D 2 n 2 2/3,3 1 t + C. (3.4) 3/2 Therefore with we have t u c nxt C(δ) ab C(δ)a 5 + δb 5/4, t u 2 2 t + δ 6 t ). n 2 2 t + C(n,u ). (3.5)

7 3.4 n ln(n) estimate Now we have all necessary tools to establish the boun on n ln(n). Multiplying the equation (1.1) 2 by ln(n), integrating over,integrationbypartsanusingequation (1.1) 1,gives R2 n ln(n) x = n t ln(n) x = n 2 + χ n cx t n R2 = n 2 R2 χn cx= n 2 χn(u c n + a 1 c) x. (3.6) n n Using the Gagliaro-Nirenberg inequality n 2 2 K n 1 n 2 2, an integrating over (,t), gives t n(t)ln(n(t)) x + (4 χk n 1 ) n 2 2 t n ln(n ) x χ t (u c)nxt. (3.7) Multiplying the equation (1.1) 3 with u an integrating over,gives 1 2 t u η u 2 2 = n φ ux φ n 2 u 2 χδ 2 n C u 2 2. Integrating also this inequality over (,t), aing it to (3.7) an inserting (3.5), we obtain t n(t)ln(n(t)) x + u(t) (4 χk n 1 χδ) n 2 2 t 3.5 Moment control + η t u 2 2 t C t u(t) 2 2 t + C(n,u ). (3.8) Since we work in full space Ω =,wehavetoboun R n(t)ln(n(t)) x from below. In 2 orer to o that, we have to control the behavior of n as x + similarly to [16]. To perform this task, we multiply (1.1) 2 by the smooth function ξ = 1+ x 2,integrate an use (3.3) with q =3/2: t ξnx = nu ξx+ n ξx+ χ n c ξx n 1 u ξ + n 1 ξ + n 2 c 2 ξ C u 2/5 2 D 2 u 3/5 3/2 + C n 2 n 1/2 1 n 1/2 2 n 3/4 1 + C δ u 2 D 2 u 3/2 3/2 + δ n C(δ ). (3.9) We use that ξ an ξ are boune. Integrating the inequality over (,t), gives ( t ) 1/2 ( t ) 1/2 t ξn(t) x δ u 2 2 t D 2 u 3 3/2 t + δ n 2 2 t + Ct. (3.1) 7

8 Moreover we have ( ) 1 n ln 1 n 1 x n ( ) 1 n Combining (3.11), (3.1) an (3.4), it follows that n(t)ln(n(t))1 n 1 x χδ 2 1 e ξ n x + n ln R 2 n ln 1 n 1 n e ξ x ξnx + C n 1/2 1 n e ξ x C + ξnx. (3.11) t n 2 2 t + C t u 2 2 t + Ct + C. (3.12) Since n ln(n) x = n ln(n) x 2 n ln(n)1 n 1 x, we obtain from (3.8) an (3.12), that n(t) ln(n(t)) x + u(t) Now let us efine t (4 χk n 1 2χδ) n 2 2 t + η M exist := 4 χk. C t t u 2 2 t u 2 2t + C + Ct. (3.13) Therefore if n 1 <M exist,wecanchooseδ small enough, such that λ := 4 χk n 1 2χδ > anweobtaintheentropyinequality(2.12) E(t)+ t D t C(n,u )+Ct + C t E t. Remark 3.2 This is the same mass threshol that can be obtaine with the methos from [23]. With the entropy inequality at han, we have 1. n ln(n) L ((,T),L 1 ( )), 2. n L 2 ((,T) )), 3. n x L ((,T),L 1 ( )), 4. u L ((,T),L 2 ( )) L 2 ((,T),H 1 ( )). Moreover, we can prove the following lemma: Lemma 3.1 c L 2 ((,T) ) 8

9 Proof T From the equation (1.1) 1,wearriveatanestimatefor c: T c(t) 2 2 t ( u c 2 + n 2 + a 1 c 2 ) 2 t T Working on the first term, we have T C u 4/5 2 D 2 u 6/5 3/2 c C n 2 2 n 1 + C c 1 c 2 t. (3.14) u 4/5 2 D 2 u 6/5 3/2 c 2 2 t u 4/5 L t L2 x ( T ) 2/5 ( T D 2 u 3 3/2 t As in (3.8) an using (3.3) with q =7/4, it follows that T u 4/5 2 D 2 u 6/5 3/2 c 2 2 t C u 4/5 L t L2 x 3.6 Regularizing effect ( T c 1/3 2 t) 3/5. ) 2/5 ( T n 2 2 t + C(u ) 3/5 n 2 2 t). Theorem 3.3 Let t> an 1 <p<. With the hypothesis of Theorem 2.1, there exists a constant C(t) not epening on n p such that n p x C(t)(1 + (t ) 1 p ), <t t (3.15) i.e. the cell ensity n(,t ) belongs to L p for any positive time t. The proof works in the same way as in [1] an it uses the boun establishe in Lemma 3.1. For completeness, we give a sketch of the proof in the appenix. 3.7 Passing to the limit We approximate the system by u ɛ c ɛ = c ɛ + n ɛ ρ ɛ c ɛ, n ɛ t + uɛ n ɛ = n ɛ (χn ɛ c ɛ ), u ɛ t + P ɛ η u ɛ +(n ɛ φ) ρ ɛ =, u ɛ =, (3.16) with a stanar mollifier ρ ɛ an mollifie versions of n, u as initial ata n ɛ, uɛ. All a-priori estimates still hol e.g. equation (3.6) becomes R2 n ɛ ln(n ɛ ) x = nɛ 2 t n ɛ χn(u ɛ c ɛ n ɛ ρ ɛ + c ɛ ) x Here we can estimate n ɛ (n ɛ ρ ɛ ) x n ɛ 2 n ɛ 2 ρ ɛ 1 = n ɛ 2 2. In (3.1), we can estimate instea c ɛ 2 2 n ɛ ρ ɛ q c ɛ q n ɛ q ρ ɛ 1 c ɛ q. Similarly, the estimate for regularity of u still hols. To be able to pass to the limit, we show sufficient compactness. Similar to [4]: 9

10 Boun on n ɛ 2 from Theorem 3.3. Boun on n ɛ 2 For every p<, wehaven ɛ L ((δ, T ),L p ( )) for any δ (,T) We have n ɛ c ɛ 2 n ɛ 3 c ɛ 6 C n ɛ 3 c ɛ 3/2, (3.17) an working on the secon term we obtain: c ɛ 3/2 u ɛ 6 c ɛ 2 + n ɛ 3/2 + c ɛ 3/2. Therefore n ɛ c ɛ is boune in L ((δ, T ),L 2 ( )). Now n ɛ 2 x = 2 n ɛ 2 x +2χ n ɛ n ɛ c ɛ x, t shows that β := n ɛ L 2 ((δ,t ) ) satisfies the estimate 2β 2 2χ n ɛ c ɛ L ((δ,t ),L 2 ( ))β 2 n ɛ L ((δ,t ),L 2 ( )). This implies that n is boune in L 2 ((δ, T ) ). We efine V := { v H 1 ( ): x 1/4 v L 2 ( ) }.Since v 2 R R 1/2 x v 2, 2 x >R VembescompactlyinL 2 ( ). Moreover, the estimate ( ) 1/2 ( x (n ɛ ) 2 x n ɛ (n ɛ ) 3 shows that n ɛ is boune in L 2 ((δ, T ),V). Therefore, Aubin-Lions lemma gives a strongly convergent subsequence n ɛ n in L 2 loc ((δ, T ) R2 ). Also from Theorem 3.3, we obtain enough regularity to make all terms in the efinition of weak solutions well-efine. 4 Existence in R 3 ) 1/2 Ẽ(t) := n 3/2 x, D(t) :=λ n 3/4 2 x. (4.1) R 3 R 3 Theorem 4.1 (3D) Assume a 2 =, Ω=R 3, n, χ, η >, an R 3 n 3/2 + n x + n x <. (4.2) There exists a M exist ( n 1 ) > such that if n 3/2 <M exist, (4.3) then there is a global in time weak solution for (1.1) an we have an entropy inequality where Ẽ an D are given in (4.1). tẽ(t)+ D(t), (4.4) 1

11 In 3D, the critical norm for the Keller-Segel equations is L 3/2 see [14]. Let us apply the metho from before an unerstan why it oes not work in 3D: If wejustconsierthe case without the flui we have t R 3 n 3/2 x [ 1 2 K 4χ n 3/2 4 ] n 3/4 2 x. 3 R 3 Details can be foun in the calculations below. So for n 3/2 small enough initially, we have n(t) 3/2 n 3/2. With the non-stationary Stokes system, there is a aitive constant on the right-han sie that basically comes from the initial ata of the flui (see the calculations in the 2D-case). So there can be linear growth of n 3/2,whichwill eventually estroy the smallness require. Therefore we restrict ourselves to system (1.1) with a 2 =. Proof of theorem 4.1 We first establish an entropy. Multiplying the equation (1.1) 2 by n 1/2,integratingover R 3 an using the equation (1.1) 1,gives n 3/2 x = 4 t R 3 R 3 3 n3/ χ n3/2 cx = 4 R 3 3 n3/ χn3/2 cx = 4 R 3 3 n3/4 2 x 1 2 χn3/2 (u c n + a 1 c) x. (4.5) Now working on the equation for c to obtain an estimate for c 2 : (cu c) x + c 2 2 = nc x a 1 c 2 2, R 3 R 3 c 2 2 n 6/5 c 6. Moreover, it follows with v 6 K 1 v 2 that c 2 K 1 n 6/5. (4.6) The term n 3/2 (u c)canbeestimateusing v 6 K 1 v 2 an u 2 K 2 D 2 u 6/5 n 3/2 (u c) 1 n 3/2 3 u 6 c 2 K 2 1 n3/4 2 2 K 1K 2 D 2 u 6/5 K 1 n 6/5. Now applying the regularity estimates for the stationary Stokes equation D 2 u 6/5 K 3 n 6/5 (see [25, 26]) an the Höler inequality n 2 6/5 n 1 n 3/2,itfollows Using n 3/2 (u c) 1 K 4 1K 2 K 3 n 3/4 2 2 n 2 6/5 K4 1 K 2 K 3 n 3/4 2 2 n 1 n 3/2. (4.7) R 3 n 5/2 x K 4 R 3 n 3/4 2 x n 3/2, (4.8) we obtain n 3/2 x = 4 t R 3 R 3 3 n3/ χn3/2 (u c n + a 1 c) x [ 1 2 (K 4 + K1 4 K 2K 3 n 1 )χ n 3/2 4 ] n 3/4 2 x. (4.9) 3 R 3 11

12 Now let us efine M exist( n 1 ):= 8 3χ(K 4 + K 4 1 K 2K 3 n 1 ). Therefore if n 3/2 <M exist,wesetλ := (K 4 + K 4 1 K 2K 3 n 1 )χ n 3/2 > an we obtain the entropy inequality (4.4) tẽ(t)+ D(t). With the entropy inequality at han, we obtain the following: n is boune in L 5/2 ((,T) R 3 ) ue to (4.8). Therefore, u is boune in L 5/2 ((,T); W 2,5/2 (R 3 )) an c is boune in L 5/2 ((,T); W 2,5/2 (R 3 )). As in the 2D case, we can use a regularising effect (for etails see [14]) an pass to the limit. 5 Numerics Finally we woul like to illustrate the behavior of the Keller Segel flui system with ifferent numerical examples. In particular, we give evience that above the critical mass of 8π solutions still exist. All computations have been implemente using the software package FreeFem++ [33]. We woul like to solve the following system u c = c + n, n t + u n = n (χn c), (5.1) P η u + n φ =, u =, with zero Dirichlet bounary conitions for c an u an corresponing mass-preserving Neumann conitions for n. This system has the avantage that without the flui there is a mass threshol to separate global existence an finite-time blow-up. This is quite ifferent from (1.4), where e.g. all constant states are steay i.e. for constant initial n there is always global existence inepenent of its mass. Moreover, for (5.1) the existence result for small mass hols as for (1.1). We solve system (5.1) in an iterative manner: 1. Solve the Stokes equations (5.1) 3 an (5.1) 4 with a penalty metho cf. [31] an the Solver Crout. We use a classical Taylor-Hoo element technic i.e. the velocity u is approximate by P 2 finite elements, an the pressure P is approximate by P 1 finite elements. 2. Approximate the chemoattractant c by P 2 finite elements an solve equation (5.1) 1 with UMFPACK. 3. Perform an implicit Euler finite ifference approximation in time for equation (5.1) 2,approximatethecellensityn by P 2 finite elements an solve equation (5.1) 1 with UMFPACK. We start with a Gaussian as initial istribution n (x, y) =C mass exp[ 5(x x ) 2 5(y y ) 2 ]. 12

13 (a) t =.1 (b) t =.1 (c) t =.2 () t =.5 (e) t =.75 (f) t =1 Figure 1: Evolution of the ensity n with symmetric initial ata an mass M = 27. Although the mass is larger than 8π, sowithoutthefluiweexpectblow-up,withthe flui we see convergence to a steay state. The constant C mass is chosen initially such that a esire mass in the computational omain is obtaine an we set χ =1,η =1, φ =(;1). Thetestgeometryisasquare [, 1] [, 1] with a mesh consisting of 25 squares. In the first two examples, we use the mass M = 27 > 8.5π > 8π in the computational omain i.e. above the critical mass. 13

14 (a) t =.1 (b) t =1 Figure 2: Evolution of the concentration c corresponing to Figure 1. Notice, the orer of magnitue of c stays the same. In the first example, we set t =.1 an choose a Gaussian with x = y =.5 as initial atum. We observe that the solution oes not blow up, although the mass is above the critical value of 8π cf. Figure 1. Moreover, we see that the solution converges to a steay state. The ensity maximum moves ownwars because of gravity but stops since c vanishes at the bounary. The istribution of the chemical c is shown in Figure 2. Now we investigate various quantities for t = 1,becausethisseemstobefairlyclose to the steay state: Figure 3a shows the velocity of the flui. So the flui is constantly transporting n an c. ThisisillustrateinFigures3b-3.Figure3brepresentsthe flux resulting from the chemotaxis / iffusion terms ( n χn c) whereasfigure3cshows the flux coming from the flui contribution un. Thetotalflux ( n χn c un) is given in Figure 3. It shoul be notice that the flui counteracts the chemotactic flux especially in the high concentration region. To highlight the effect of the flui, the -level set of the scalar prouct of chemotactic flux an the total flux is plotte i.e. insie this region the flui changes the chemotactic flux vector by more than ±9. Moreover, ifferent initial configurations with the same massseemtoconvergetothe same steay state, cf. Figure 4. This is illustrate by the secon example with t =.1 an x =.8; y =.7. These two results above are stable uner mesh an time step refinement. Moreover, if we efine M := M n(t), Ω an take its maximum over the first 1 time step, it is less than 1 9. In a final step, for even larger mass M =4,weobservetheevelopmentofvery high concentration, which might inicate blow-up, see Figure 5. Here we set t =.5 an x = y =.5. These first numerical results illustrate the interesting behavior of the Keller-Segel- Flui system an can be regare as a starting point for further research. In particular we woul like to use numerical schemes, which are able to couple the Stokes equations with the chemotactial system an capture blow-up e.g. [28] or [17](forschemesesigne 14

15 (a) flui velocity (b) chemotactic flux (c) flui flux () total flux Figure 3: Flui velocity fiel, chemotactic flux ( n χn c), flui flux un an total flux ( n χn c un) atthesteaystate. Tohighlighttheeffectoftheflui,the -level set of the scalar prouct of chemotactic flux an the total flux is plotte in Figure 3 i.e. insie this region the flui changes the chemotactic flux vector by more than ±9.Soitcanbeseenthatthefluihasthestrongesteffectinthehigh concentration region of n. to resolve the blow-up for the Keller-Segel system). 6 Appenix The proof for the regularising effect is taken from [1]: Proof of Theorem 3.3: Since R n ln(n)(t) x C(1 + t) wehave 2 (n(x, t) k) + ln(n(x, t)) x 1 ln(k) (n(x, t) k) + 1 ln(k) n(x, t)(ln(n(x, t))) + x. (6.1) This means, that there exists a moulus of equi-integrability ω(t,k), T>ank> such that (n(x, t) k) + ω(t ; k) an lim ω(t ; k) =. k 15

16 (a) t =.1 (b) t =.1 (c) t =.2 () t =.5 Figure 4: Evolution of the ensity n with non-symmetric initial ata an mass M =27. For the same mass but ifferent initial ata we observe again blow-up prevention an convergence to the same steay state. In this section we follow the by now classical iea to obtain L p bouns for n by using the equi-integrability property (6.1). First step gives Multiplying the equation (1.1) 2 by (n k) p 1 + an integrating over R2, (n k) p 1 + x = 4p (n k) p/2 + t p 2 x (p 1) (n k) p + cx pk (n k) p 1 + cx. (6.2) Using the Galiaro-Nirenberg-Sobolev inequality for the secon term v 4 x C v 2 x v 2 x (6.3) 16

17 (a) t =.5 (b) t =.75 (c) t =.11 () t =.145 Figure 5: Evolution of the ensity n with symmetric initial ata an mass M =4. So for higher mass, we observe the formation of very high concentration that might inicate blow-up. Notice that the pictures are taken at much smaller times compare the plots shown before. leas to ( (n k) p + cx ( ) 1/2 ( (n k) p + x δc(p) c 2 2 (n k) 2p + x ) 1/2 c 2 1/2 (n k) p/2 + x) 2 c 2 (n k) p + x + 2 δp 17 (n k) p/2 + 2 x. (6.4)

18 Moreover, by interpolation an the same Galiaro-Nirenberg-Sobolev inequality as above, we have for p 3/2 that (n k) p 1 + cx ( 1/2 (n k) 2(p 1) + x) c 2 ) 1/2 (n k) 2p + x c 2 ( C(M,p)+ C(M,p) c 2 + δ c 2 2 (n k) p + x + p 1 δp 2 (n k) p/2 + k 2 x. (6.5) Using the estimate ( ) (n k) p+1 + x = (n k) (p+1)/2 2 ( ) 2 + x C (n k) (p+1)/2 + x R ( 2 ) 2 C(p) (n k) 1/2 + (n k) p/2 + x R 2 C(p) (n k) + x (n k) p/2 + 2 x, (6.6) an the fact that R (n k) 2 + x can be mae small when choosing k large, we obtain for p 2 ( ) (n k) p 1 + x =(p 1) 1 (n k) p+1 + t pc(p)ω(t,k) x + C(1 + c 2 2 ) (n k) p + x + C c pk2 M + C. (6.7) For fixe p, wechoosek = k(p, T )sufficientlylargesuchthat Using 1 1 <. (6.8) pc(p)ω(t,k) ( ) 1/p ( ) 1 1/p (n k) p + x (n k) + x (n k) p+1 + x R ( 2 ) 1/p ( ) 1 1/p (n k) + x (n k) p+1 + x, (6.9) we achieve the following ifferential inequality for Y p (t), p 2an<t T t Y p(t) (p 1)M 1/(p 1) δyp β (t) +C ( 1+ c(t) 2 2) Yp (t) +C ( 1+ c(t) 2 ) 2 with β = p p 1. Secon step see [1] for etails. Using ODE theory, we obtain that Y p (t) C(T ) 1, (6.1) tp 1 18

19 Thir step Using x p 2 p (x k) p for x 2k, weobservethat n p x = n p x + n p x {n 2k} {n>2k} (2k) p 1 M +2 p (n k) p x (2k) p 1 M +2 R p (n k) p + x. 2 {n>2k} Therefore together with (6.1), we obtain (3.15) n p x C(t)(1 + t 1 p ), <t t (6.11) for p 2. For 1 <p<2, the theorem hols by interpolation. Acknowlegement This publication is base on work supporte by Awar No. KUK-I1-7-43, mae by King Abullah University of Science an Technology (KAUST). A.Lorzwoulliketo thank K. Fellner, P. Markowich, T. Peley, B. Perthame an M.-T. Wolfram for useful iscussions. Moreover, A. Lorz acknowleges support from the ANR Samovar (France). References [1] H. C. Berg, E. coli in Motion, Springer-Verlag,23. [2] P. Biler, Existence an asymptotics of solutions for a parabolic-elliptic system with nonlinear no-flux bounary conitions, NonlinearAnalysisT.M.A.,19(1992), pp [3] P. Biler an T. Nazieja, Existence an nonexistence of solutions for a moel of gravitational interaction of particles, Colloq.Math.,66(1994),p [4] A. Blanchet, J. A. Carrillo, an N. Masmoui, Infinite time aggregation for the critical Patlak Keller Segel moel in,comm.pureappl.math.,61(28), pp [5] A. Blanchet, J. Dolbeault, an B. Perthame, Two-imensional Keller-Segel moel: optimal critical mass an qualitative properties of the solutions, Electron.J. Differential Equations, (26), pp. No. 44, 32 pp. (electronic). [6] M. Burger, M. Di Francesco, an Y. Dolak-Struß, The Keller Segel moel for chemotaxis with prevention of overcrowing: linear vs. nonlinear iffusion, SIAM J. Math. Anal., 38 (26), pp [7] H. M. Byrne an M. R. Owen, AnewinterpretationoftheKeller-Segelmoel base on multiphase moelling, J.Math.Biol.,49(24),pp [8] R. Caflisch an G. Papanicolaou, Dynamic theory of suspensions with brownian effects, SIAMJ.Appl.Math.,43(1983),pp [9] V. Calvez an J. A. Carrillo, Volume effects in the Keller Segel moel : energy estimates preventing blow-up, J.Math.PuresAppl.,86(26),pp

20 [1] V. Calvez an L. Corrias, The parabolic-parabolic Keller Segel Moel in, Comm. Math. Sci., 6 (28), pp [11] J. Carrillo an T. Gouon, Stability an asymptotic analysis of a flui-particle interaction moel, Comm.PartialDifferentialEquations,31(26),pp [12] F. Chalub, Y. Dolak-Struss, P. Markowich, D. Oelz, C. Schmeiser, an A. Soreff, Moel hierarchies for cell aggregation by chemotaxis, Math. Moels Methos Appl. Sci., 16 (26), pp [13] L. Corrias an B. Perthame, Critical space for the parabolic-parabolic Keller Segel moel in R n,c.r.aca.sci.paris,ser.i,342(26),pp [14] L. Corrias, B. Perthame, an H. Zaag, Global solutions of some chemotaxis an angiogenesis systems in high space imensions, MilanJ.Math.,72(24),pp [15] M. Di Francesco, A. Lorz, an P. Markowich, Chemotaxis flui couple moel for swimming bacteria with nonlinear iffusion: global existence an asymptotic behavior, DiscreteContin.Dyn.Syst.Ser.A,28(21),pp [16] R.-J. Duan, A. Lorz, an P. Markowich, Global solutions to the couple chemotaxis flui equations, Comm.PartialDifferentialEquations,35(21), pp [17] F. Filbet, AfinitevolumeschemeforthePatlak Keller Segelchemotaxis moel, Numer. Math., 14 (26), pp [18] Y. Giga an H. Sohr, Abstract L p estimates for the Cauchy problem with applications to the Navier Stokes equations in exterior omains, J.Funct.Anal.,12 (1991), pp [19] K. Hamache, Global existence an large time behaviour of solutions for the Vlasov Stokes equations, JapanJ.Inust.Appl.Math.,15(1998),pp [2] T. Hillen an K. Painter, Global existence for a parabolic chemotaxis moel with prevention of overcrowing, Av.inAppl.Math.,26(21),pp [21] A. J. Hilleson, T. J. Peley, an J. O. Kessler, The evelopment of concentration graients in a suspension of chemotactic bacteria, Bull.Math.Bio.,57 (1995), pp [22] D. Horstmann, From 197 until present: the Keller Segel moel in chemotaxis an its consequences. ii., Jahresber.Deutsch.Math.-Verein.,16(24),pp [23] W. Jäger an S. Luckhaus, On explosions of solutions to a system of partial ifferential equations moelling chemotaxis, Trans.Amer.Math.Soc.,329(1992), pp [24] R. Kowalczyk, Preventing blow-up in a chemotaxis moel, J.Math.Anal.Appl., 35 (25), pp [25] O. A. Layzhenskaya, Mathematical theory of viscous incompressible flow, Goron an Breach Science Publishers Inc.,

21 [26] P.-L. Lions, Mathematical topics in flui mechanics. Vol. 1, vol. 3ofOxforLecture Series in Mathematics an its Applications, The Clarenon Press Oxfor University Press, New York, Incompressible moels, Oxfor Science Publications. [27] A. Lorz, Couple chemotaxis flui moel, Math. MoelsMethosAppl. Sci., 2 (21), pp [28] A. Marrocco, Numerical simulation of chemotactic bacteria aggregation via mixe finite elements, M2ANMath.Moel.Numer.Anal.,37(23),pp [29] T. J. Peley an J. O. Kessler, Hyroynamic phenomena in suspensions of swimming microorganisms, Annu.Rev.FluiMech.,24(1992),pp [3] B. Perthame, Transport Equation in Biology, Birkhaeuser, Basel, 27. [31] J. E. Roberts an J.-M. Thomas, Mixe an Hybri Methos, Hanbook of Numerical Analysis, North-Hollan,1993. [32] I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler, an R. E. Golstein, Bacterial swimming an oxygen transport near contact lines, Proc.Natl.Aca.Sci.USA,12(25),pp [33] 21