HO¨LDER CONTINUITY OF KELLER-SEGEL EQUATIONS OF POROUS MEDIUM TYPE COUPLED TO FLUID EQUATIONS 7 1 YUN-SUNGCHUNG,SUKJUNGHWANG,KYUNGKEUNKANG,ANDJAEWOOKIM 0 2 Abstract. We consider a coupled system consisting of a degenerate porous n medium type of Keller-Segel system and Stokes system modeling the motion a ofswimmingbacterialivinginfluidandconsumingoxygen. Weestablishthe J globalexistenceofweaksolutionsandH¨oldercontinuoussolutionsindimension 1 three, under the assumption that the power of degeneracy is above a certain 1 number depending on given parameter values. To show H¨older continuity of weak solutions, we consider a single degenerate porous medium equation ] P with lower order terms, and via a unified method of proof, we obtain H¨older regularity,whichisofindependent interest. A . h t a m 1. Introduction [ WestudyaKeller-Segelmodelcoupledtothefluidequations,wheretheequation of biological cells is of porous medium type. To be more precise, we consider 1 v ∂ n ∆n1+α+u n= (χ(c)nq c), t 3 − ·∇ −∇· ∇ ∂ c ∆c+u c= κ(c)n, 2 (1.1) (KS-PME)  t − ·∇ − ∂ u ∆u+ p= n φ, 28  t u−=0, ∇ − ∇ ∇· .0 where α > 0 and q ≥1 are given constants. Here, the unknowns n, c, u and p 1 denote the density of bacteria,the oxygenconcentration,the velocity vector of the 0 fluid and the associated pressure, respectively. In addition, the locally bounded 7 functions χ : R R and κ : R R represent the chemotactic sensitivity and 1 → → consumption rate of oxygen. Moreover, φ = φ(x) is a given potential function. It : v is knownthatthe abovesystemmodels the motionofswimming bacteria,so called i X Bacillus subtilis, which live in fluid and consume oxygen. This system has been proposedbyTuvaletal. in[55]forthecaseα=0andq =1,whichcanbeextended r a tothe caseα>0whenthe diffusionofbacteriais viewedlikemovementinaporus medium. Inthismanuscript,wecalltheabovesystemaKeller-Segelporousmedium equation(KS-PME),since fluid equationsarerestrictedto the Stokes systemunder our considerations. The main purpose of this paper is to establishthe existence of weak and Ho¨lder continuous solutions globally in time for the Cauchy problem of (KS-PME) under general conditions of χ and κ and more extended range of α and q ever known. We first introduce local Ho¨lder regularity results for a scalar equation under proper conditions on the lower order term, that contributes later obtaining Ho¨lder continuity of a weak solution of system (KS-PME). In the domainΩ Rd [0,T]for d 2,we considerparabolic porousmedium T ⊂ × ≥ type equations in the form of (1.2) n =∆n1+α+ (B(x,t)n) t ∇· 1 2 YUN-SUNGCHUNG,SUKJUNGHWANG,KYUNGKEUNKANG,ANDJAEWOOKIM for α 0 under proper conditions on B where B : Rd Rd is a vector field. ≥ → Roughly speaking, if we are able to obtain regularity results of (1.2) under the condition on B that is expected from u,χ(c), and c of (KS-PME), then Ho¨lder ∇ continuity of a weak solution n of (1.2) yields the same regularity for n, a weak solution of (KS-PME). In fact, our method of showing Ho¨lder continuity of (1.2) works under the con- ditions on B and B such that ∇ (1.3) B L2qˆ1,2qˆ2(Ω ) and B Lqˆ1,qˆ2(Ω ) ∈ loc T ∇ ∈ loc T where positive constants qˆ,qˆ >1 satisfy 1 2 2 d (1.4) + =2 dκ qˆ qˆ − 2 1 for some κ (0,2/d). By letting ∈ 2qˆ (1+κ) 2qˆ (1+κ) 1 2 q = , q = , 1 2 qˆ 1 qˆ 1 1 2 − − the admissible range of constants are obtained from Proposition 2.4 when p=2. There are many papers working on the continuity of weak solutions to porous medium type equations (refer [15], [3], [5], et al. for a general porous medium equation and special classes of equations). Focusing the main term of (1.2), we share some common mathematical approaches. For the system (KS), we refer recent paper [38] carrying Ho¨lder regularity and uniqueness results (when α > 0) relying on technical proofs originated from [9] and [18]. Compare to similar Ho¨lder regularity results on [38], we play with a scalar equation (1.2) to obtain the same results under the weaker assumptions on B and B that belongs to scaling invariant class. By following natural behaviour ∇ of a solution using a more geometrical approach(refer [32] and [33]), as a separate interestofitsown,weprovideaunifiedmethodofproofinthesensethatthemethod has no limitation including α=0 (usually it is important to have α>0 in [38],[9], and[18]andshowingstabilitywhenα 0isregardedasananothercomputational → issue). Besides simplicity of computations in this manuscript, our method of proof carries potentials to provide significant common elements to the similar proofs for singular type of equations(when 1 < α 0) and even for generalized structured − ≤ equations (refer Remark 3.4 for details). Here we provide the definition of a weak solution of (1.2). Definition 1.1. Let Ω be an open set in Rd, B L2((0,T) Ω), and T >0. ∈ × n∈Cloc(0,T;L2loc(Ω)), nα+22 ∈L2loc(0,T;Wl1o,c2(Ω)) is a local weak solution to (1.2) with α 0 if for every compact set K Ω and ≥ ⊂ every subinterval [t ,t ] (0,T] 1 2 ⊂ t1 t1 (1.5) nϕdx + nϕ + n1+α ϕ+Bn ϕ dxdt=0 t − ∇ ∇ ∇ ZK (cid:12)t0 Zt0 ZK (cid:12) (cid:8) (cid:9) for all nonnegative(cid:12)testing functions (cid:12) ϕ W1,2(0,T;L2(K)) L2 (0,T;W1,2(K)). ∈ loc ∩ loc o KELLER-SEGEL-FLUID MODEL 3 Fromthedefinitionofweaksolutions,wecomputetwotypesofenergyestimates, provided in Propositions 4.1 and 4.2 which is called local and logarithmic energy estimates, respectively. Due to the difference of the nature of porous medium and p Laplacian equations, we modify the method of proof in [32], for example, − considering the a weak solution directly rather sub or super solutions, also cutting off a weak solution when u may stay near zero for DeGiorgi iteration. Moreover, anothertechnicalissuefollowsbecausethelowerordertermin(1.2)doesnotfollow the structure of main term (not given in the form of n1+α but n). By imposing conditions of B and B in scaling invariant class, we can provide simpler proof ∇ comparetocomputationin[38]. Alsoconditionson B doesfollowglobalestimates ∇ from (KS-PME). Before we deliver the local Ho¨lder continuity results, we make comments on in- trinsic scaling due to the nonhomogeneity of the equation (1.2). More precisely, the local energy estimate derived from (1.2) appears in Proposition 4.1 is non- homegenous unless α=0. Roughly speaking, in an intrinsically (rescaled with the behaviour of a solution) scaled cylinder, a weak solution behaves like a solution to the heat equation. That is, more specifically, rescaling the time length (1.6) T =θω−αρ2 ω,ρ for some constant θ and ρ and (1.7) ω :=ess oscn=µ µ :=ess supn ess infn. + − ΩT − ΩT − ΩT SinceΩT isopen,therearepositiveconstantsrandssuchthatKrx0×(t0−s,t0)⊂ Ω . If we set T 1 ωα/2s1/2 R= min r, , 4 θ1/2 (cid:26) (cid:27) then we conclude that Qx0,t0(θ)=Kx0 (t θω−αR2,t ) Ω . ω,4R 4R× 0− 0 ⊂ T Thenfor anypositive constantsθ andω, wecanfit the cylinderQx0,t0(θ) inΩ by ω,4R T selecting R properly. Basically,we aregoing to workwith the cylinder Qx0,t0(θ) to ω,4R finda propersubcylinder where a solutionhas less oscillationeventuallyleading to Ho¨lder continuity. Due to the intrinsic scaling (1.6), we define a time scale in terms of the function n and the set Ω on which n is defined. For any real number τ, we define (1.8) τ =ωα/2 τ 1/2. I | | | | With this time scale, we define the parabolic distance between two sets such 1 K and by 2 K dist( ; )= inf max x y , t s 1 2 ∞ I p K K (x,t)∈K1 {| − | | − | } (y,s)∈K2 s≤t with (which is defined by x y =max xi yi ). ∞ ∞ 1≤i≤d |·| | − | | − | Now we state the Ho¨lder continuity of a bounded weak solution of (1.2). Theorem 1.2. (Ho¨lder continuity of n) Let n be a nonnegative bounded weak solution of (1.2) under (1.3) with α 0 in Ω . Then n is locally continuous. T ≥ Moreover, there exist positive constant β (0,1) and γ depending on data(that ∈ is, d,Ω ,Ω′ ,α, B , B for some qˆ ,qˆ > 1 satisfying (1.4)) such T T k k2qˆ1,2qˆ2 k∇ kqˆ1,qˆ2 1 2 4 YUN-SUNGCHUNG,SUKJUNGHWANG,KYUNGKEUNKANG,ANDJAEWOOKIM that, for any two distinct points (x ,t ) and (x ,t ) in any subset Ω′ of Ω with 1 1 2 2 T T dist(Ω′ ;∂ Ω ) positive, we have T p T x x +ωα/2 t t 1/2 β 1 2 1 2 (1.9) n(x ,t ) n(x ,t ) γω | − | | − | . | 1 1 − 2 2 |≤ dist (Ω′ ;∂ Ω ) (cid:18) p T p T (cid:19) TheproofofthistheoremisgiveninSection3consideringtwoalternatives. Then the proofs of two alternatives are shown in Section 4 as combinations of DeGorgi iterations and the expansion of positivity along the time axis and the spatial axis. Now we state results on the existence of global-intime weak solution of (KS- PME) and global Ho¨lder continuity of the Cauchy problems of (KS-PME) as well. For the notational convenience, we denote 9q 8 A:= (q,α) α>2q 2,q 1 , B := (q,α) α> − ,q 1 , { | − ≥ } | 6 ≥ (cid:26) (cid:27) 10q 9 C := (q,α) α> − ,q 1 . | 8 ≥ (cid:26) (cid:27) We introduce the notions of weak solutions and Ho¨lder continuous solutions. We start with the definition of weak solutions. Definition 1.3. (Weak solutions) Let q 1and 0 < T < . A triple (n,c,u) is ≥ ∞ said to be a weak solution of the system (1.1) if the followings are satisfied: (i) n and c are non-negative functions and u is a vector function defined in R3 (0,T) such that × n(1+ x + logn) L∞(0,T;L1(R3)), n1+2α L2(0,T;L2(R3)), | | | | ∈ ∇ ∈ c L∞(0,T;H1(R3)) L2(0,T;H2(R3)), c L∞(R3 [0,T)), ∈ ∩ ∈ × u L∞(0,T;L2(R3)), u L2(0,T;L2(R3)), ∈ ∇ ∈ (ii) (n,c,u) satisfies the system (1.1) in the sense of distributions, namely, T nϕ n1+α ϕ+nu ϕ+nqχ(c) c ϕ dxdt= n ϕ(,0) dx, t 0 Z0 ZRd −∇ ·∇ ·∇ ∇ ·∇ −ZRd · (cid:0) T (cid:1) (cϕ c ϕ+cu ϕ+nκ(c)ϕ)dxdt= c ϕ(,0) dx, t 0 Z0 ZRd −∇ ·∇ ·∇ −ZRd · T (u ψ u ψ+(τ(u ψ))u n φ ψ)dxdt= u ψ(,0) dx t 0 Z0 ZRd · −∇ ·∇ ·∇ − ∇ · −ZRd · · for any ϕ C∞ R3 [0,T) and ψ C∞ R3 [0,T),R3 with ψ =0. ∈ 0 × ∈ 0 × ∇· Next we define Ho¨lder(cid:0)continuous(cid:1)solutions. For(cid:0)convenience, w(cid:1)e denote Q := T (0,T) R3. × Definition 1.4. (Ho¨lder continuous solutions) Let q 1, (q,α) (A B) and ≥ ∈ ∩ 0 < T < . A triple (n,c,u) is said to be a Ho¨lder continuous solutions of the ∞ system(1.1)if(n,c,u)isaweaksolutioninDefinition1.3andfurthermoresatisfies the following: there exists β >0 such that (1.10) n, ∂ c, ∂ u, 2c, 2u β(Q ). t t T ∇ ∇ ∈C KELLER-SEGEL-FLUID MODEL 5 Before stating our result precisely, we first recall some essential conditions for χ and κ. To preserve the non-negativity of the density of bacteria n(x,t) and the oxygenc(x,t)for0<t<T,itisnecessarytoassumethatκ(0)=0. The condition κ() 0isalsoessentialsincethebacteriaconsumetheoxygen. Thus,thefollowing · ≥ hypotheses are compulsory: κ() 0 and κ(0)=0. Furthermore, we suppose that · ≥ χ′ L∞. Summing up, throughout this thesis, we assume that ∈ loc (P ) χ′ L∞, κ L∞, κ() 0 and κ(0)=0. 1 ∈ loc ∈ loc · ≥ To obtain more extended range of α, we sometimes make further assumptions on κ, which are given by (P ) κ′ L∞ with κ′() κ for some constant κ >0. 2 ∈ loc · ≥ 0 0 We nowpresenttwodifferenttypes ofassumptionsonχ, κtogetherwith the range of α and q. The first one is reserved for weak solutions. Assumption 1.5. χ,κ and α satisfy (P ) and one of the following holds: 1 (i) (q,α) B. ∈ (ii) (q,α) A B and κ satisfies (P ). 2 ∈ ∪ Next assumption is prepared for Ho¨lder continuous solutions. Assumption 1.6. χ,κ and α satisfy (P ) and one of the following holds: 1 (i) (q,α) A B. ∈ ∩ (ii) (q,α) (A B) C and κ satisfies (P ). 2 ∈ ∪ ∩ We recall some known results related to our concerns. Firstly, we compare the system(KS-PME)totheclassicalKeller-Segelmodel(KS)ofporousmediumtype, which is given as ∂ n=∆n1+α (χn c), t (1.11) (KS) −∇· ∇ ( τ∂tc=∆c c+n, − where χ is a positive constant, q = 1 and τ = 0 or 1(for example, [36,37]). We remarkthattheequationofcin(1.11)ismodeledbythechemicalsubstance,which isproducedbybiologicalorganism,butinourcasetheequation(1.1) indicatesthe 2 dynamics of oxygen, which is consumed by a certain type of bacteria. That’s the reason opposite sign of the right side of each equation appears, which causes main difference regarding global existence or blow-up for the value on α. In case that (1.11), the equation of c, is of elliptic type, i.e. τ = 0, existence of bounded weak solutions was shown in [52] globally in time, provided that q 1 and α>q 2. If ≥ − d 0<α q 2, blow-up may occur in a finite time. Later, in [34], the result of [52] ≤ − d was extended to the case that the equation of c is of parabolic type, i.e. τ >0. Forthechemotaxisfluidsystem(1.1)withq =1intwodimensions,itwasknown that bounded weak solutions exist globally in time under some assumptions on κ and χ for sufficiently regular data. We remark that results in dimension two are evenvalidinreplacementwiththeNavier-Stokesequationsforfluidequations(refer to [12] and [53]) In three dimensions, it was shown in [43] that if α = 1, then the chemotaxis- 3 Stokes system (1.1) with q = 1 has global-in-time bounded weak solutions. For the special case that χ = 1 and κ(x) = x, existence of bounded weak solutions was proved in [54] for (KS-PME) with q = 1, provided that α > 1/7. In [12], for (1.1) with q =1, it was proved that global-in-time existence of weak solutions and 6 YUN-SUNGCHUNG,SUKJUNGHWANG,KYUNGKEUNKANG,ANDJAEWOOKIM Table 1. Relations between parameters and conditions weaksolutions H¨oldercontinuoussolutions (P1) α> 9q6−8 α>maxn2q−2, 9q6−8o (Theorem1.7) (Theorem1.9) (P1)and(P2) α>minn2q−2,9q6−8o α>maxnminn2q−2,9q6−8o, 10q8−9o (Theorem1.8) (Theorem1.10) bounded weak solutions under the same conditions as (P1) and (P2), if α > 1/6 and α>1/4, respectively. The range of α was improved in [11]. More precisely, if α> 1, bounded weak solutions exist under only the condition (P1). Furthermore, 6 it was also proved that if χ or κ satisfy (P1) and (P2), and if α > 1, then there 8 exists bounded weak solutions for the system (1.1) with q =1. As mentioned earlier, our main goal is to study more general Keller-segel-fluid system(1.1)withq 1andobtainglobalexistence ofweakandHo¨ldercontinuous ≥ solutions for extended range of α and q. Our results are summarized in the Table 1. We remark that in case that q =1, our results recover those of [11]. Nowwearereadytostateourmainresultsofthe system(1.1),andthe firstone is about existence of weak solutions, which reads as follows: Theorem 1.7. (Weak solutions) Let α belong to B i.e., α> 9q−8 and initial data 6 (n ,c ,u ) satisfy 0 0 0 (1.12) n (1+ x + logn ) L1(R3), c L∞(R3) H1(R3) and u L2(R3). 0 0 0 0 | | | | ∈ ∈ ∩ ∈ Suppose that χ, κ satisfy the hypothesis (P ). Then, there exists a weak solution 1 (n,c,u) for the system (1.1). Furthermore, for any p with 1 p α q+2 ≤ ≤ − n L∞(0,T;Lp(R3)), np+2α L2(0,T;L2(R3)), ∈ ∇ ∈ and the following inequality is satisfied: sup n(logn + x )+ nα−q+2+ c 2 + u2 0≤t≤T(cid:18)ZR3 | | h i ZR3 ZR3 |∇ | ZR3| | (cid:19) +C T n1+2α 2+ n2α−2q+2 2+ ∆c 2+ u 2 Z0 (cid:18)(cid:13)∇ (cid:13)2 (cid:13)∇ (cid:13)2 k k2 k∇ k2(cid:19) (cid:13) (cid:13) (cid:13) (cid:13) ≤C T,kc0kL∞(cid:13)∩H1,kn0(cid:13)(1+(cid:13)|x|+|logn(cid:13)0|)k1,kn0kα−q+2,ku0k2 , (cid:16) (cid:17) where x =(1+ x2)12. h i | | If the condition (P2) is additionally assumed, the range of α is a bit expanded, compared to that of Theorem 1.7. More precisely, we have the following: Theorem1.8. (Weak solutions)LetαbelongtoA B i.e.,α>min 2q 2,9q−8 . ∪ − 6 Suppose that χ, κ satisfy the hypothesis (P ), (P ) and initial data (n ,c ,u ) sat- 1 2 0 0 0 (cid:8) (cid:9) isfies (1.13) n (1+ x + logn ) L1(R3), c L∞(R3) H1(R3) and u H1(R3). 0 0 0 0 | | | | ∈ ∈ ∩ ∈ Then, there exists a weak solution (n,c,u)for the system (KS-PME). Furthermore, for any p with 1 p α 2q+3 ≤ ≤ − n L∞(0,T;Lp(R3)), n12 c L2(0,T;L2(R3)), np+2α L2(0,T;L2(R3)), ∈ ∇ ∈ ∇ ∈ KELLER-SEGEL-FLUID MODEL 7 and the following inequality is satisfied: sup n(logn + x )+ nα−2q+3+ c2+ u2 0≤t≤T(cid:18)ZR3 | | h i ZR3 ZR3|∇ | ZR3| | (cid:19) +C T n1+2α 2+ n2α−22q+3 2+ ∆c 2+ u 2 Z0 (cid:18)(cid:13)∇ (cid:13)2 (cid:13)∇ (cid:13)2 k k2 k∇ k2(cid:19) (cid:13) (cid:13) (cid:13) (cid:13) ≤C T,k∇c(cid:13)0k2,kn0(cid:13)logn(cid:13)0k1,kn0kα−(cid:13)2q+3,ku0k2 , (cid:16) (cid:17) where x =(1+ x2)12. h i | | Next, if α is greater than a certain value depending on q, we prove existence of Ho¨lder continuous solutions for (KS-PME) under the condition (P ). To be more 1 precise, the result reads as follows: Theorem 1.9. (Ho¨lder continuous solutions) Let α belongs to A B i.e., ∩ α> max 2q 2, 9q−8 . Suppose that χ, κ satisfy the hypothesis (P ) and initial − 6 1 data (n ,c ,u ) satisfies (1.12) as well as 0 0 0 (cid:8) (cid:9) (1.14) n L∞(R3), c W1,m(R3), u W1,m(R3), for any m< . 0 0 0 ∈ ∈ ∈ ∞ Then, there exists a Ho¨lder continuous solution (n,c,u) for the system (KS-PME). Furthermore, we assume the condition (P ) and we then see that the restriction 2 of α is relaxed for the existence of Ho¨lder continuous solutions. Theorem 1.10. (Ho¨lder continuous solutions) Let α belongs to (A B) C i.e., ∪ ∩ α > max min 2q 2,9q−8 , 10q−9 . Suppose that χ, κ satisfy the hypothesis − 6 8 (P ), (P ) and initial data (n ,c ,u ) satisfies (1.13) as well as 1 2 0 0 0 (cid:8) (cid:8) (cid:9) (cid:9) (1.15) n L∞(R3), c W1,m(R3), u W1,m(R3), for any m< . 0 0 0 ∈ ∈ ∈ ∞ Then, there exists a Ho¨lder continuous solution (n,c,u) for the system (KS-PME). Remark 1.11. There are some known results regarding uniqueness of Ho¨lder con- tinuous solutions for Keller-Segel system of porous medium type (see e.g. [46] and [38]). As for us, uniqueness of solutions in Theorem 1.9 and Theorem 1.10 doesn’t seem to be obvious, in particular, due to presence of the fluid velocity field. There- fore, we leave it as an open question. This paper is organized as follows: In section 2, we introduce some notations and review known results. Section 3 is devoted for the proof of Theorem 1.2 with the crucial aid of two alternatives, whose are clarified in section 4. In section 5, we present the proofs of existence for weak solutions of (KS-PME) in dimension three. We also provide the proofs of Theorem 1.9 and Theorem 1.10 in section 6. In appendix, proofs of Propositions 4.1 and 4.2 are given. 2. Preliminaries 2.1. Notations and useful inequalities. In this subsection, We introduce the notationsthroughoutthispaperandrecallsomeusefulinequalitiesforourpurpose. Let Ω be an open domain in Rd, d 1 and I a finite interval. ≥ Lp(Ω)= f :Ω R f is Lebesgue measurable , f < , { → | k kLp(Ω) ∞} 8 YUN-SUNGCHUNG,SUKJUNGHWANG,KYUNGKEUNKANG,ANDJAEWOOKIM where 1 p f = f pdx , (1 p< ). k kLp(Ω) | | ≤ ∞ (cid:18) ZΩ (cid:19) We will write f := f , unless there is any confusion to be expected. For k kLp(Ω) k kp 1 p , Wk,p(Ω) denotes the usual Sobolev space, i.e., ≤ ≤∞ Wk,p(Ω)= u Lp(Ω):Dαu Lp(Ω),0 α k . { ∈ ∈ ≤| |≤ } We also write the mixed norm of f in spatial and temporal variables as f = f = f . k kLpx,,qt(Ω×I) k kLqt(I;Lpx(Ω)) k kLpx(Ω) Lq(I) (cid:13) (cid:13) t Let m and p be positive constants greater than 1(cid:13)and consi(cid:13)der the Banach spaces (cid:13) (cid:13) Vm,p(Ω ):=L∞(0,T;Lm(Ω)) Lp(0,T;W1,p(Ω)) T ∩ and Vm,p(Ω ):=L∞(0,T;Lm(Ω)) Lp(0,T;W1,p(Ω)), 0 T ∩ 0 both equipped with the norm v Vm,p(Ω ), T ∈ v :=ess sup v(,t) + v . k kVm,p(ΩT) k · km,Ω k∇ kp,ΩT 0<t<T When m=p, we set Vp,p(Ω )=Vp(Ω ). Note that both spaces are embedded in T T Lq(Ω )forsomeq >p. We denotebyC =C(α,β,...) aconstantdepending onthe T prescribed quantities α,β,..., which may change from line to line. Now we introduce basic embedding inequalities and auxiliary lemmas for fast geometric convergence. (Refer Chapter I in [18]) Theorem 2.1. (Gagliardo-Nirenberg multiplicative embedding inequality) Let v ∈ W1,p(Ω), p 1. For every fixed number s 1 there exists a constant C depending 0 ≥ ≥ only upon d, p and s such that v C v α v 1−α, k kq,Ω ≤ k∇ kp,Ωk ks,Ω where α [0,1], p,q 1, are linked by ∈ ≥ −1 1 1 1 1 1 α= + , s − q d − p s (cid:18) (cid:19)(cid:18) (cid:19) and their admissible range is q [s, ], α [0, p ], if d=1, ∈ ∞ ∈ p+s(p−1) q [s, dp ], α [0,1], if 1 p<d, s dp , ∈ d−p ∈ ≤ ≤ d−p q ∈[dd−pp,s], α∈[0,1], if 1≤p<d, s≥ dd−pp, q [s, ), α [0, dp ), if 1<d p. ∈ ∞ ∈ dp+s(p−d) ≤ Theorem 2.2. (Sobolev embedding theorem) There exists a constant C depending only upon d,p,m such that for every v L∞(0,T;Lm(Ω)) Lp(0,T;W1,p(Ω)), ∈ ∩ 0 p/d v(x,t)q dxdt Cq ess sup v(x,t)m dx v(x,t)p dxdt | | ≤ | | |∇ | ZZΩT (cid:18)0<t<T ZΩ (cid:19) (cid:18)ZZΩT (cid:19) where q = p(d+m). Moreover, d v C v =C ess sup v(,t) + v . k kq,ΩT ≤ k kVm,p(ΩT) k · km,Ω k∇ kp,ΩT (cid:18)0<t<T (cid:19) KELLER-SEGEL-FLUID MODEL 9 When p=m, we apply Ho¨lder’s inequality to obtain the following corollary. Corollary 2.3. Let p > 1. There exists a constant C depending only upon d and p, such that for every v Vp(Ω ), ∈ 0 T v p C v >0 d+pp v p . k kp,ΩT ≤ |{| | }| k kVp(ΩT) Proposition 2.4. There exists a constant C depending only upon d and p such that for every v Vp(Ω ), ∈ 0 T v C v k kq1,q2;ΩT ≤ k kVp(ΩT) where the numbers q ,q 1 are linked by 1 2 ≥ 1 d d + = , q pq p2 2 1 and their admissible range is q (p, ), q (p2, ); for d=1, 1 2 ∈ ∞ ∈ ∞ q (p, dp ), q (p, ); for 1<p<d,  1 ∈ d−p 2 ∈ ∞ q (p, ), q (p2, ); for 1<d p. 1 ∈ ∞ 2 ∈ d ∞ ≤ Proof. Let v ∈V0p(ΩT) and let r ≥ 1 to be chosen. From Theorem 2.1 with s = p follows that 1/r T v(,τ) q2 dτ Z0 k · kq1,Ω ! 1/r T C v(,τ) αq2dτ ess sup v(,τ) 1−α. ≤ Z0 k∇ · kp ! 0≤r≤T k · kp,ΩT Choose α such that αq =p. (cid:3) 2 Westatealemmaconcerningthegeometricconvergenceofsequencesofnumbers. Lemma 2.5. Let Y and Z , n=0,1,2,..., be sequences of positive numbers, n n { } { } satisfying the recursive inequalities Y Cbn Y1+α+Z1+κYα n+1 ≤ n n n (Zn+1 ≤Cbn(cid:0)Yn+Zn1+κ (cid:1) where C,b>1 and κ,α>0 are given(cid:0)numbers. If(cid:1) Y0+Z01+κ ≤(2C)−1+σκb−1σ+2κ, where σ =min{κ,α}, then Y and Z tend to zero as n . n n { } { } →∞ The following lemma is introduced in [19]; it states that if the set where v is bounded away from zero occupies a sizable portion of K , then the set where v is ρ positivecluster aboutatleastone pointofK . Here wename the inequality asthe ρ isoperimetric inequality. Lemma 2.6. (Isoperimetric inequality) Let v W1,1(Kx0) C(Kx0) for some ∈ ρ ∩ ρ ρ > 0 and some x Rd and let k and l be any pair of real numbers wuch that 0 ∈ 10 YUN-SUNGCHUNG,SUKJUNGHWANG,KYUNGKEUNKANG,ANDJAEWOOKIM k < l. Then there exists a constant γ depending upon N,p and independent of k,l,v,x ,ρ, such that 0 ρd+1 (l k)Kx0 v >l γ Dv dx. − | ρ ∩{ }|≤ |Kρx0 ∩{v ≤k}|ZKρx0∩{k<v<l}| | We consider the following heat equation: (2.1) v ∆v =f, in Q :=R3 (0,T), t T − × with initial data v(x,0) = v (x). Next, we recall maximal estimates of the heat 0 equation in terms of mixed norms. Lemma 2.7. Let 1<l,m< . Suppose that f Ll,m(Q ) and v W2,l(R3). If ∞ ∈ x,t T 0 ∈ v is the solution of the heat equation (2.1), then the following estimate is satisfied: (2.2) v + 2v C f + v . k tkLlx,,mt(QT) ∇ Llx,,mt(QT) ≤ k kLlx,,mt(QT) k 0kW2,l(R3) (cid:13) (cid:13) (cid:16) (cid:17) We also recall the follow(cid:13)ing S(cid:13)tokes system, which is the linearized Stokes equa- tions: (2.3) v ∆v+ p=f, div v =0 in Q :=R3 (0,T), t T − ∇ × with initial data v(x,0) = v (x). The maximal estimates of the Stokes system is 0 given as follows. Lemma 2.8. Let 1<l,m< . Suppose that f Ll,m(Q ) and v W2,l(R3). If ∞ ∈ x,t T 0 ∈ v is the solution of the Stokes system (2.3), then the following estimate is satisfied: (2.4) v + 2v + p C f + v . k tkLlx,,mt(QT) ∇ Llx,,mt(QT) k∇ kLlx,,mt(QT) ≤ k kLlx,,mt(QT) k 0kW2,l(R3) (cid:13) (cid:13) (cid:16) (cid:17) (cid:13) (cid:13) 3. Proof of Theorem 1.2 For notational convention, we take ν to be the constant from Proposition 4.3 0 (DeGiorgi type of iteration) corresponding to θ = 1 and, with ω and R given positive constants, we set ω −α (3.1) ∆= (2R)2. 2 (cid:16) (cid:17) Ourfirstalternativeisthat, ifaboundedweaksolutionn staysclosetoits max- imum on most of one suitable small subcylinder, then n is awayfrom its minimum on a suitable subcylinder centered at (0,0), the target point. Moreover,for given constants k (in analysis it denotes the level of solution) and ρ (usually it means the spacial radius), we assume that (3.2) k−α−2α(q12+κ)ρdκ <1. If (3.2) fails, we have k ρǫ for some ǫ > 0 which directly implies the Ho¨lder ≤ continuity of a solution. Lemma3.1. (Thefirstalternative)Letθ >1beagivenconstant,ν beaconstant 0 0 in Proposition 4.3 (when θ = 1) and ∆ be in (3.1). Suppose n is a nonnegative bounded weak solution of (1.2) in (3.3) Q=K ( θ ∆,0) 2R 0 × −