Global Solutions to the Coupled Chemotaxis-Fluid Equations

Transcription

1 Global Solutions to the Couple Chemotaxis-Flui Equations Renjun Duan Johann Raon Institute for Computational an Applie Mathematics Austrian Acaemy of Sciences Altenbergerstrasse 69, A-44 Linz, Austria Alexaner Lorz, Peter Markowich Department of Applie Mathematics an Theoretical Physics University of Cambrige, Wilberforce Roa Cambrige CB3 WA, UK May 5, 1 Abstract In this paper, we are concerne with a moel arising from biology, which is a couple system of the chemotaxis equations an the viscous incompressible flui equations through transport an external forcing. The global existence of solutions to the Cauchy problem is investigate uner certain conitions. Precisely, for the Chemotaxis-Navier-Stokes system over three space imensions, we obtain global existence an rates of convergence on classical solutions near constant states. When the flui motion is escribe by the simpler Stokes equations, we prove global existence of weak solutions in two space imensions for cell ensity with finite mass, first-orer spatial moment an entropy provie that the external forcing is weak or the substrate concentration is small. Keywors: chemotaxis-flui interaction; chemotaxis; Stokes equations; global solution; energy metho; a priori estimates. Contents 1 Introuction The couple Chemotaxis-Navier-Stokes equations 5.1 Global classical solutions near constant states Uniform a priori estimates Proof of local an global existence Proof of rates of convergence The couple Chemotaxis-Stokes equations Global weak solutions with large ata Uniform a priori estimates Proof of global existence

2 R.-J. Duan, A. Lorz an P. Markowich References 3 1 Introuction Chemotaxis is a biological process, in which cells (e.g. bacteria) move towars a chemically more favorable environment. For example, bacteria often swim towars higher concentration of oxygen to survive. The chemical substrate can be prouce or consume by the cells, where we are only intereste in the latter in this paper which correspons to the repulsive case for the substrate. A typical moel escribing chemotaxis are the Keller-Segel equations erive by Keller an Segel [1] which have become one of the best-stuie moels in mathematical biology. On the other han, in nature cells often live in a viscous flui so that cells an chemical substrates are also transporte with flui, an meanwhile the motion of the flui is uner the influence of gravitational forcing generate by aggregation of cells. Generally, the motion of the flui is etermine by the well-known incompressible Navier-Stokes equations or Stokes equations. Thus, this kin of cell-flui interaction becomes more complicate since it not only consists of chemotaxis an iffusion, but also inclues transport an viscous flui ynamics. In particular, it is interesting an important in biology to stuy some phenomenon of seimentation on the basis of the couple cell-flui moel. Recently, to escribe the couple biological phenomena mentione above, the authors in [3] propose the following moel: t n + u n = δ n (χ(c)n c), t c + u c = µ c k(c)n, (1.1) t u + u u + P = ν u n φ, u =, t >, x Ω. Here, the unknowns are n = n(t, x) : R + Ω R +, c = c(t, x) : R + Ω R +, u(t, x) : R + Ω R 3 or R an P = P (t, x) : R + Ω R, enoting the cell ensity, substrate concentration, velocity an pressure of the flui, respectively. Ω R 3 or R is a spatial omain where the cells an the flui move an interact. Constants δ, µ an ν are the corresponing iffusion coefficients for the cells, substrate an flui. χ(c) is the chemotactic sensitivity an k(c) is the consumption rate of the substrate by the cells. φ = φ(t, x) is a given potential function. We remark that the moel (1.1) has been compare numerically with the experiment in [3], an it was also use in [7]. Similar moels were investigate in [8] an [17]. As usual, in orer for the system (1.1) to be well-pose, it shoul be supplemente with some initial conitions (n, c, u) t= = (n (x), c (x), u (x)), x Ω, (1.) an some proper bounary conitions. See [3] an references therein for the iscussions on the bounary conitions if Ω is a boune omain. In the sequel we shall take Ω = R or R 3 in orer to consier the well-poseness for the Cauchy problem (1.1)-(1.) over the whole space. Moreover, we shall also consier the simplifie Chemotaxis-Stokes system taking the following form t n + u n = δ n (χ(c)n c), t c + u c = µ c k(c)n, (1.3) t u + P = ν u n φ, u =, t >, x Ω, where compare with (1.1), the nonlinear convective term u u is ignore in the flui equation of (1.3).

3 Chemotaxis-Flui Equations 3 It can be seen from (1.1) or (1.3) that the coupling of chemotaxis an flui is realize through both the transport of cells an chemical substrates u n, u c an the external force n φ exerte on the flui by cells. Here, for the external force, it can be prouce by ifferent physical mechanism such as gravity, centrifugal, electric or magnetic forces. One example in the case of gravity is φ = ax 1, for a constant a R epening on the ratio of the flui mass ensity to the cell ensity an the gravity acceleration. Another example is the centrifugal force φ satisfying φ(x) = φ( x ) as x. In this paper, we are only intereste in the case of the centrifugal force with φ ecaying in x with some rates at infinity. Throughout this paper, we also assume the following conitions: (i) δ >, µ >, ν > ; (A) (ii) n (x), c (x), u (x) = for all x Ω; (iii) χ( ), k( ) an φ(, ) are smooth with k() = an k (c) for all c R. Notice that the above assumption implies that c preserves the nonnegativity of the initial ata by the maximum principle, an moreover, k(c) hols for c, that is the case of consumption of chemical substrates. Thus, the interaction force cause by the oxygen is repulsive but not attractive as in the classical Keller-Segel system. Later, we shall point out some special properties that the Keller-Segel system enjoys when k(c) takes the negative sign. We further remark that if the flui coupling is neglecte, i.e. u =, then the system obtaine is similar to a rift-iffusion moel in the context of semiconuctor, see [4]. The aim of this paper is to obtain some global existence results for the Cauchy problem on the above systems (1.1) an (1.3) uner the assumption (A) an some aitional conitions on the initial ata an on the functions χ(c), k(c), φ. It shoul be pointe out that these results coul also hol over the smooth boune omain Ω with homogeneous bounary conitions. Precisely, the first result of this paper is Theorem.1 which states the global existence of classical solutions to the Cauchy problem on the system (1.1) an (1.) over Ω = R 3 provie that the initial atum (n, c, u ) is a small smooth perturbation of the constant state (n,, ) with n, an the potential function φ epening on t, x ecays in x at infinity uniformly in time. The proof is base on some uniform a priori estimates combine with the local existence as well as the stanar continuity argument. Moreover, the timeecay rates in L p -norms for perturbations are obtaine in Theorem., where L p energy methos an spectral analysis are use in the proof. Here, the metho of proving the optimal time-ecay rate by combining the energy estimates an spectral analysis is actually the one recently evelope by [11] in the stuy of the nonlinear Boltzmann equation. The secon result is Theorem 3.1 about the global existence of weak solutions to the Cauchy problem on the system (1.3) with large initial ata (1.) in some functional spaces over Ω = R. In fact, the existence of global weak solution is obtaine when initial ensity n has finite mass, firstorer spatial moment an entropy uner the aitional conition that the strength of the nonnegative time-inepenent potential function φ is weak or c is small in some sense. The main part in the proof is to erive some uniform a priori estimates by constructing some proper free energy functionals. To explain this more, let us here skip the coupling of flui, that is to consier the repulsive chemotaxis system in the case of the consumption of chemical substrate: { t n = δ n (χ(c)n c), t c = µ c k(c)n, t >, x Ω. (1.4)

4 4 R.-J. Duan, A. Lorz an P. Markowich Define the instant free energy functional E(n(t), c(t)) by E(n(t), c(t)) = n ln nx + 1 Ω where Ψ(c) is given by Ψ(c) = c Ω ( ) 1/ χ(s) s. k(s) Then, for any smooth solution (n, c) to the system (1.4), Ψ(c) x, E(n(t), c(t)) + D(n(t), c(t)) = (1.5) t hols for any t, where the issipation rate D(n(t), c(t)) is given by n D(n(t), c(t)) = δ Ω n x + χ (c)k(c) + χ(c)k (c) n Ψ x Ω χ(c) +µ Ω Ψ k(c) Ψ Ψ x c χ(c) µ ( ) k(c) c Ψ 4 x. χ(c) Thus, uner the conition that χ(c) >, Ω c (χ(c)k(c)) >, c ( ) k(c) <, χ(c) D(n(t), c(t)) becomes a nonnegative functional an hence the free energy E(n(t), c(t)) is non-increasing in time t. When the equations are couple with the flui, the aitional terms from the transport u n, u c an the force n φ turn out to be controlle by the total issipation rate D total (n(t), c(t), u(t)) if the potential function φ is small in an appropriate sense. Furthermore, the bouneness of the entropy n ln n L 1 in finite time is prove together with the bouneness of firstorer spatial moment 1 + x n L 1. Instea of smallness of external forcing φ, if it is assume that k () > an c L is small enough, then by taking proper linear combination, one can also fin a non-increasing free energy functional an thus uniform a priori estimates also follow. Now we mention some work relate to issues iscusse in this paper. Firstly, since the moel system (1.1) is propose by [3] for the stuy of the biology of chemotaxis, let us recall some important mathematical characters on the classical chemotaxis system, the Keller-Segel moel, for which one of the simple versions takes the form { t n = n χ (n c), (1.6) c = n, t >, x Ω = R, where χ has been suppose to be a positive constant, k(c) 1 takes the minus sign, an other constants are normalize to have unit values. Reaers can refer to the surveys [18, 19] for a summary results on the above system an even more general Keller-Segel system. Here, it shoul be emphasize that compare with the system (1.4) uner the assumption (A) which particularly implies k(c), the classical Keller-Segel system (1.6) exposes some ifferent phenomena especially such as critical mass an blow-up. Precisely, the elementary computations for (1.6) show t R x n(t, x)x = 4 n L 1 ( 1 χ n L 1 8π ).

5 Chemotaxis-Flui Equations 5 Thus, the situations leaing to the ifferent properties of solutions are basically ivie into three cases: n L 1 < 8π χ (subcritical), n L 1 = 8π χ (critical), n L 1 > 8π χ (supercritical), where uner some conitions, solutions to (1.6) blow up in finite time for the supercritical case while solutions exist globally in time for the subcritical case. The further literature on the Keller-Segel moel is inee huge an thus out of the scope of this paper; see recent work [1,, 5] an references therein. Next, let us mention some work [31, 9, 1, 9, 1, 15], which are more connecte to the results obtaine in this paper. The ientity (1.5), which plays a key role in the erivation of the uniform a priori estimates, is inspire by the proof of [31] in the case of boune omain over R. Notice that from the late proof, the erivation of the ientity (1.5) oes not epen on the spatial imension. For a chemotaxis moel motivate by angiogenesis, [9] an [1] prove global existence of weak solutions in high space imensions for the case when k(c) = c m with m 1 by controlling the L p -norm of ensity n(t, x). For the same moel in one-imensional space, the global existence of classical solutions was stuie in [9] an [1] over the boune interval of R when χ(c) = c α an k(c) = c m with α an m taking ifferent values, an was later generalize in [15] to the full line R with χ(c) = 1 1+c an k(c) = c. On the other han, more attention has recently been focuse on some couple kinetic-flui system firstly introuce in [4] which escribe the interaction of two phases: the isperse phase of particles an ense phase of flui. For that, refer to [16, 6, 6, 5, 13, 3, 14] for the stuy 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 hyroynamical limit is the existing entropy inequality. It shoul also be pointe out that the moels for chemotaxis can be justifie as an asymptotic limit of some kinetic moels, cf. [8, 7]. Thus, it is interesting to establish the asymptotic valiity of the couple chemotaxis-flui system from the kinetic level [13]. Finally, the flui ynamic equations such as the Navier-Stokes or Stokes equations have a much longer history, see [, 3] an references therein for the etaile mathematical theory. The rest of this paper is organize as follows. In Section, we consier the chemotaxis system couple with the Navier-Stokes equations, an we prove global existence an convergence rates of classical solutions near constant states. In Section 3, we prove global existence of weak solutions to the couple chemotaxis-stokes equations uner two ifferent assumptions for weak forces or small initial concentration of c. The couple Chemotaxis-Navier-Stokes equations.1 Global classical solutions near constant states As mentione before, we consier in this section the Cauchy problem on the couple Chemotaxis- Navier-Stokes equations in R 3 : t n + u n = δ n (χ(c)n c), t c + u c = µ c k(c)n, (.1) t u + u u + P = ν u n φ, u =, t >, x R 3, with initial ata where it is suppose to hol that (n, c, u) t= = (n (x), c (x), u (x)), x R 3, (.) (n (x), c (x), u (x)) (n,, ) as x,

6 6 R.-J. Duan, A. Lorz an P. Markowich for some constant n. The goal of this section is to prove the global existence of classical solutions to the above Cauchy problem when initial ata is a small smooth perturbation near the constant steay state (n,, ), an moreover, obtain the rates of convergence of solutions towars this steay state in some functional spaces. For simplicity of presentation later, let us take change of variables n = σ + n an P = P + n φ so that the Cauchy problem (.1) an (.) is reformulate as t σ + u σ δ σ = (χ(c)σ c) n (χ(c) c), t c + u c µ c + k ()(σ + n )c = (k(c) k ()c)(σ + n ), t u + u u + P (.3) ν u = σ φ, u =, t >, x R 3, with (σ, c, u) t= = (σ (x), c (x), u (x)) (,, ) as x, (.4) where σ = n n. The first result of this section about the global existence of classical solutions is state as follows. Here an in the sequel, for simplicity, α means the spatial erivatives x α with multi-inex α, j means all the spatial erivatives of j-orer for integer j, an always enotes the L -norm without confusion. Theorem.1. Let n be a constant, an the assumption (A) hol with n (x) σ (x) + n for x R 3, an sup(1 + x ) φ(t, x) + t,x 1 α 3 sup α φ(t, x) <. (.5) t,x Furthermore, suppose that (σ, c, u ) H 3 is sufficiently small. Then, the Cauchy problem (.3) an (.4) amits a unique classical solution (σ, c, u) satisfying that n(t, x) σ(t, x) + n, c(t, x), t, x R 3, an there are constants λ >, C such that t (σ, c, u)(t) H + λ (σ + n 3 ) k(c)c + k () R 3 hols for any t. +λ t 1 α 3 α c(s) xs (σ, c, u)(s) H 3s C (n, c, u ) H 3 (.6) Remark.1. The assumption (.5) in Theorem.1 shows that the potential function φ(t, x) of the external force fiel epening on both t an x nee not be small an also nee not ecay in time, which is essentially cause by the fact that φ(t, x) only appears in the linear coupling term n φ in the flui equation. From the later proof, the spatial ecay of φ with rate (1 + x ) 1 can be replace the integrability conition φ L 3 (R 3 ). Finally, we shoul emphasize again that the conition in (A) that k(c) is non-ecreasing in c plays a key role in obtaining the Lyapunov-type inequality (.6) for the stability of solution, an as shown in the next section later, it also plays an important role in consiering the global existence of weak solutions (cf. Theorem 3.1) or global classical solutions with large smooth ata (cf. Proposition 3.1). The secon result of this section is concerne with the time-ecay rates of the obtaine classical solutions near constant steay states.

7 Chemotaxis-Flui Equations 7 Theorem.. Let n =, an let all conitions in Theorem.1 hol. The solution (σ, c, u) to the Cauchy problem (.3)-(.4) obtaine in Theorem.1 enjoys the following time-ecay estimates. Assume that σ, c L 1 (R 3 ). Then, for any 1 p <, it hols that for any t. If it is further assume that u L q (R 3 ) an with 1 < q < 6/5, then it hols that for any t, where K is efine by σ(t) L p C σ L 1 L p(1 + t) 3 (1 1 p ), (.7) c(t) L p C c L 1 L p(1 + t) 3 (1 1 p ), (.8) φ L (R + ; L q/( q) (R 3 )) u(t) C( u Lq H 3 + K )(1 + t) 3 ( 1 q 1 ), (.9) K = (σ, c ) L1 H 3 + σ L1 L c L1 L. Remark.. Due to time-ecay estimates on the high-orer erivatives of (n, c, u) (cf. (.41) an (.47)) in the proof of Theorem., the time-ecay rates of L -norms for (n, c, u) can also follow from Sobolev inequalities. When n >, the proof of Theorem. can be moifie to obtain similar time-ecay estimates, which we shall not pursuit in this paper for brevity. For that, a further remark will be given at the en of this section in orer to explain it only at the level of linearization.. Uniform a priori estimates Theorem.1 will be prove by combining the local existence an some uniform a priori estimates as well as the continuation argument. In this subsection, we evote ourselves to the proof of uniform a priori estimates. For this purpose, for any < T, let us enote the space X(, T ) = {(σ, c, u) : σ, c, u C ([, T ); H 3 (R 3 )) C 1 ([, T ); H 1 (R 3 )), σ, c, u L ([, T ]; H 3 (R 3 ))}. Lemma.1 (a priori estimates). Let all conitions in Theorem.1 hol. Suppose that the Cauchy problem (.3) an (.4) has a solution (σ, c, u) in X(, T ) with sup (σ, c, u)(t) H 3 ɛ (.1) t T for < ɛ 1. Then n(t, x) σ(t, x) + n, c(t, x) hol for any t T, x R 3. Furthermore, there are ɛ >, C > an λ > such that for any ɛ ɛ, t (σ, c, u)(t) H + λ 3 (σ + n ) k(c)c + k () α c(s) xs R 3 hols for any t T. 1 α 3 t +λ (σ, c, u)(s) H 3s C (n, c, u ) H (.11) 3 Proof. By the Sobolev imbeing theorem, from (.1), one has sup (σ, c, u) W 1, Cɛ. (.1) t T

8 8 R.-J. Duan, A. Lorz an P. Markowich The original equations (.1) 1 -(.1) can be rewritten as t n + (δ n + n(u + χ(c) c)) =, t c + u c = µ c k (ξ)nc, where ξ = ξ(t, x) is between an c(t, x). Thus, by the assumption A1, the maximum principle implies n(t, x), c(t, x) c L Cɛ C, for any t T, x R 3. To prove (.11), we ivie it into two steps. In the first step, we eal with the zero-orer estimates. For (.3) on c, it hols that 1 c x + µ c x + k(c)c(σ + n )x =. (.13) t R 3 R 3 R 3 For (.3) 1 on n, it follows from integration by parts an the Cauchy-Schwarz inequality that 1 σ x + δ σ x 1 t R 3 R δ ( n L + n 3 x ) χ(c) L c x x R 3 C δ (1 + n ) max c C χ(c) c x. (.14) R 3 For (.3) 3 on u, one can use the Hary s inequality [] to get R3 n x C n x 1 u x + ν u x C ( ) sup x φ(t, x) n x. (.15) t R 3 R ν 3 t,x R 3 By choosing the constants 1 >, > large enough such that 1 µ C δ (1 + n ) max c C χ(c), δ 4 C ν ( sup x φ(t, x) ), t,x the linear combination of (.13), (.14) an (.15) yiels 1 ( u + σ + 1 c ) x + ν u x + δ σ x t R 3 R 4 3 R µ c x + 1 k(c)c(σ + n )x. (.16) R 3 R 3 For the secon step, we make estimates on the high-orer erivatives of (σ, c, u). Take α with 1 α 3. Applying α to (.3), multiplying by α c an then integrating, one has 1 α c x + µ α c x + k () (σ + n ) α c x t R 3 R 3 R 3 = Cβ α α β u α c β cx k () Cβ α α β σ β c α cx β<α R 3 β<α R 3 α [(k(c) k ()c)σ] α cx n α (k(c) k ()c) α cx, R 3 R 3

9 Chemotaxis-Flui Equations 9 where from the Cauchy-Schwarz inequality, (.1) an (.1), the right han sie is boune by Cɛ α c x. R 3 1 β 4 Then, since ɛ > can be small enough, after summation over 1 α 3, one has 1 α c x + µ α c x t 1 α 3 R 3 1 α 3 R 3 + k () (σ + n ) α c x Cɛ c. (.17) R 3 1 α 3 Similarly, it follows from (.3) 1 that 1 α σ x + δ α σ x t R 3 R 3 = Cβ α α β u α σ β σx + α (χ(c)σ c) α σx β<α R 3 R 3 which is boune by δ α σ x + Cɛ 4 R 3 Then one has +n R 3 α (χ(c) c) α σx, 1 β 4 β σ x + C(n + 1) R 3 β 3 1 α σ x + δ α σ x t 1 α 3 R 3 1 α 3 R 3 Cɛ σ x + C(n + 1) R 3 β 3 R 3 β c x. R 3 β c x. (.18) For α u, it follows from (.3) 3 that 1 α u x + ν α u x t R 3 R 3 = Cβ α α β u i α i u j β u j x + α ( σφ) α ux, β<α ij R 3 R 3 which is boune by ν ( α u + α u )x + Cɛ 4 R 3 Thus one has 1 β 3 + C sup φ(t) Wx 3, t R 3 β u x β 3 R 3 β σ x. 1 α u x + ν α u x t 1 α 3 R 3 1 α 3 R 3 ( ν 4 + Cɛ ) u x + C sup φ(t) β σ x. (.19) W 3, R 3 x t R 3 β 3

10 1 R.-J. Duan, A. Lorz an P. Markowich Now, by choosing 3, 4 large enough such that 3 4 C sup t φ(t), Wx 3, µ 4 4 C 3(n + 1), then the linear combination (.17) 4 + [(.18) 3 + (.19)] leas to 1 t C α α (σ, c, u) x + λ α (σ, c, u) x R 3 α 4 R 3 +λk () (σ + n ) α c x C (σ, c, u) x, (.) R 3 R 3 1 α 3 1 α 3 for some positive constants C α an λ. Therefore (.11) follows from the further linear combination of (.16) an (.) an the time integration over [, T ]. This completes the proof of Proposition.1..3 Proof of local an global existence This subsection is evote to the proof of Theorem.1. We construct the solution sequence (n j, c j, u j ) j by solving iteratively the Cauchy problems on the following linear equations t n j+1 + u j n j+1 = δ n j+1 (χ(c j )n j c j+1 ), t c j+1 + u j c j+1 = µ c j+1 k ()n j c j+1 [k(c j ) k ()c j ]n j+1, t u j+1 + u j u j+1 + P j+1 = ν u j+1 n j φ, u j+1 =, t >, x R 3, with (n j+1, c j+1, u j+1 ) t= = (n, c, u ), x R 3, for j, where (n, c, u ) (n,, ) is set at initial step. In terms of the perturbation σ j = n j n, the above Cauchy problems are reformulate as t σ j+1 δ σ j+1 + n χ() c j+1 = u j σ j+1 (χ(c j )σ j c j+1 ) n ([χ(c j ) χ()] c j+1 ), t c j+1 µ c j+1 + n k ()c j+1 = u j c j+1 k ()σ j c j+1 [k(c j ) k ()c j ](n + σ j+1 ), t u j+1 + P j+1 ν u j+1 = u j u j+1 σ j φ, u j+1 =, t >, x R 3, (.1) with (σ j+1, c j+1, u j+1 ) t= = (σ, c, u ), x R 3, for j,, where (σ, c, u ) (,, ) hols. In what follows, let us write W j = (n j, c j, u j ) an W = (σ, c, u ) for simplicity. Then, one has Lemma.. Let all conitions in Theorem.1 hol. There are constants ɛ 1 >, T 1 >, M 1 > such that if W H 3 ɛ 1, then for each j, W j C([, T 1 ]; H 3 ) is well-efine an sup W j (t) H 3 M 1, j. (.) t T 1

11 Chemotaxis-Flui Equations 11 Furthermore, (W j ) j is a Cauchy sequence in the Banach space C([, T 1 ]; H 3 ), the corresponing limit function enote by W belongs to C([, T 1 ]; H 3 ) with sup W (t) H 3 M 1, (.3) t T 1 an W = (σ, c, u) is a solution over [, T 1 ] to the Cauchy problem (.3)-(.4). Finally, the Cauchy problem (.3)-(.4) amits at most one solution in C([, T 1 ]; H 3 ) satisfying (.3). Proof. First of all, we prove (.) by inuction. The trivial case is j = since W = by the assumption at initial step. Suppose that it is true for j with M 1 > small enough to be etermine later. To prove it for j + 1, we nee some energy estimates on W j+1. From (.1) 1, it hols for any α 3 that 1 t α σ j+1 + δ α σ j+1 = α (u j σ j+1 ) α σ j+1 x + n χ() α c j+1 α σ j+1 x R 3 R 3 + α [χ(c j )σ j c j+1 ] α σ j+1 x + n α [(χ(c j ) χ()) c j+1 ] α σ j+1 x, R 3 R 3 where the r.h.s. is further boune by C u j H 3 σ j+1 H 3 + C α c j+1 α σ j+1 +C σ j H 3 c j+1 H 3 σ j+1 H 3 + C c j H 3 c j+1 H 3 σ j+1 H 3. Then, after taking summation over α 3 an using the Cauchy inequality, one has Similarly from (.1) an (.1) 3, one has an 1 t σj+1 H + 3 λ σj+1 H 3 C c j+1 H + C (σ j, c j, u j ) 3 H (σ j+1, c j+1 ) 3 H. (.4) 3 1 t cj+1 H + 3 λ cj+1 H 3 C c j H + 3 C (σj, c j, u j ) H 3 (σj+1, c j+1 ) H3, (.5) 1 t uj+1 H + 3 λ uj+1 H 3 C σj H + 3 C uj H 3 uj+1 H3. (.6) Taking the linear combination of inequalities (.4), (.5) an (.6), one has 1 t ( σj+1 H + 3 cj+1 H + 3 uj+1 H 3) + λ (σj+1, c j+1, u j+1 ) H 3 C (σ j, c j ) H + 3 C (σj, c j, u j ) H 3 (σj+1, c j+1, u j+1 ) H3, (.7) for some constant > large enough. Further, after taking time integration, it hols that t t W j+1 (t) H + λ W j+1 (s) 3 H 3s C W H + C W j (s) 3 H 3s +C t W j (s) H 3 W j+1 (s) H 3s,

12 1 R.-J. Duan, A. Lorz an P. Markowich which from the inuctive assumption implies t W j+1 (t) H + λ W j+1 (s) 3 H 3s Cɛ 1 + CM1 T 1 + CM1 t W j+1 (s) H 3s, for any t T 1. Now, one can take properly small constants ɛ 1 >, T 1 > an M 1 > such that t W j+1 (t) H + λ W j+1 (s) 3 H 3s M 1, (.8) for any t T 1. This implies that (.) hols true for j + 1 if so for j. Hence (.) is prove for all j. Next, efine the equivalent energy functional E(W j+1 (t)) by E(W j+1 (t)) = σ j+1 H 3 + cj+1 H 3 + uj+1 H 3, where the constant > is given in (.7). Similar to prove (.7), one has E(W j+1 (t)) E(W j+1 (s)) = C t s t s θ E(W j+1 (θ))θ t W j (θ) H 3θ + C (1 + W j (θ) H 3) W j (θ) H 3θ CM 1 (t s) + C(1 + M 1 ) s t s W j (θ) H 3θ, for any s t T 1. Here, the time integral in the secon term on the r.h.s. is finite ue to (.8), an hence E(W j+1 (t)) is continuous in t for each j. On the other han, from (.5) an (.6), the same argument can be applie to c j+1 an u j+1 so that both c j+1 (t) an u j+1 (t) are continuous in t, an so is σ j+1 (t). Therefore, W j (t) H is continuous in time for each j 1. 3 In orer to consier the convergence of the sequence (W j ) j, let us take the ifference of (.1) for j an j 1 so that it gives t (σ j+1 σ j ) δ (σ j+1 σ j ) + n χ() (c j+1 c j ) = u j (σ j+1 σ j ) (u j u j 1 ) σ j [χ(c j )σ j (c j+1 c j )] [(χ(c j )σ j χ(c j 1 )σ j 1 ) c j ] n ([χ(c j ) χ()] (c j+1 c j )) n ([χ(c j ) χ(c j 1 )] c j ), t (c j+1 c j ) µ (c j+1 c j ) + n k ()(c j+1 c j ) = u j (c j+1 c j ) (u j u j 1 ) c j k ()σ j (c j+1 c j ) k ()(σ j σ j 1 )c j c j η k (θ)θη(σ j+1 σ j ) c j η c j 1 k (θ)θησ j, t (u j+1 u j ) + P j+1,j ν (u j+1 u j ) = u j (u j+1 u j ) (u j u j 1 ) u j (σ j σ j 1 ) φ, (u j+1 u j ) =, t >, x R 3. By using the same energy estimates as before, one has 1 t E(W j+1 W j ) + λ (W j+1 W j ) H C W j W j 1 3 H 3 +C W j W j 1 H 3 (W j, W j 1 ) H + C (W j, W j 1 ) 3 H 3 (W j+1 W j ) H 3.

13 Chemotaxis-Flui Equations 13 The further time integration gives t W j+1 (t) W j (t) H + λ (W j+1 (s) W j (s)) 3 H 3s C(1 + M1 )T 1 sup W j (s) W j 1 (s) H + CM 3 1 s T 1 t (W j+1 (s) W j (s)) H 3s, for any t T 1, which by smallness of T 1 an M 1 implies that there is a constant λ < 1 such that sup W j+1 (t) W j (t) H 3 λ sup W j (t) W j 1 (t) H 3, (.9) t T 1 t T 1 for any j 1. It can be seen from the above inequality that (W j ) j is a Cauchy sequence in the Banach space C([, T 1 ]; H 3 ). Thus the limit function W = W + lim inee exists in C([, T 1 ]; H 3 ), an satisfies m j= m (W j+1 W j ) sup W (t) H 3 sup lim inf W j (t) H 3 M 1. t T 1 t T 1 j Finally, suppose that W an W are two solutions in C([, T 1 ]; H 3 ) satisfying (.3). By applying the same process as in (.9) to prove the convergence of (W j ) j, one has sup W (t) W (t) H 3 λ sup W (t) W (t) H 3 t T 1 t T 1 for the constant λ < 1. Hence, W = W hols. This proves the uniqueness an thus completes the proof of Lemma.. Proof of Theorem.1: Choose a positive constant M = min{ɛ, ɛ 1 }, where ɛ > an ɛ 1 > are given in Lemma.1 an Lemma., respectively. Further choose initial ata W = (σ, c, u ) such that M W H 3, 1 + C where C > is given in Lemma.1. Define the lifespan of solutions to the Cauchy problem (.3)-(.4) by T = sup{t; sup W (s) H 3 M}. s t Since M W H 3 M 1 + C < M ɛ 1, then T > hols true from the local existence result Lemma. an the continuation argument. If T is finite, it follows from the efinition of T that sup W (s) H 3 = M, s T which is a contraiction to the fact from uniform a priori estimates that sup W (s) H 3 C W M C s T M 1 + C.

14 14 R.-J. Duan, A. Lorz an P. Markowich Therefore, T =. This implies that the local solution W (t) obtaine in Lemma. can be extent to infinite time. Thus, the Cauchy problem (.3)-(.4) amits a solution W = (σ, c, u) in C([, ); H 3 ) with σ + n an c, an uniqueness of solutions also follows from Lemma.. Finally, (.6) follows from (.11). This completes the proof of Theorem.1..4 Proof of rates of convergence Throughout this subsection, we suppose that n =, all conitions in Theorem.1 hol an also t (σ, c, u)(t) H + λ k(c)c + k () α c(s) xs 3 +λ σ R 3 t 1 α 3 (σ, c, u)(s) H3s Cɛ for any t, where ɛ > is a small constant. For later use, let us efine the rate inex γ p,q by γ p,q = 3 ( 1 q 1 ), p for any 1 q p. Proof of Theorem.: It is ivie into three steps corresponing to the estimates of the timeecay rates for c, σ an u, respectively. Time-ecay of c. When n =, (.3) also reas t c + u c µ c + k(c)σ =. (.3) Take p <. Multiplying the above equation by pc p 1 an then integrating, one has c p 4µ(p 1) x + c p/ x. (.31) t R p 3 R 3 Notice that since k(c) an σ, (.3) also gives c(t) L 1 c L 1 (.3) for any t. Thus (.8) can be prove by using the metho in []. Here, for completeness, we give the proof of (.8). Let α >. Multiplying (.31) by (1 + t) α an taking integration over [, t], one has Recalling the interpolation inequality t (1 + t) α c(t) p 4µ(p 1) Lp + (1 + s) α c p/ (s) s p t c p L + α (1 + s) α 1 c(s) p p Lps. (.33) f L p for any p < an 1 q p, one has α t C η η (1 + s) α 1 c(s) p L ps t t t C f p/ γp,q 1+pγp,q f 1 1+pγp,q L q (1 + t) α 1 c p/ pγp,1/(1+pγp,q) c(s) p/(1+pγp,1) L 1 (1 + s) α c p/ (s) s + C η t s (1 + s) α pγp,1 1 c(s) p L 1 s (1 + s) α c p/ (s) s + C η c p L 1 (1 + t) α pγp,1, (.34)

15 Chemotaxis-Flui Equations 15 for any η >, where the Young inequality pγ p,1 /(1 + pγ p,1 ) + 1/(1 + pγ p,1 ) = 1 an (.3) were use an α > pγ p,1 was suppose. Therefore, (.8) for p < follows from (.33) an (.34) by taking η > small enough. (.8) for 1 p < follows from the interpolation inequality c L p c /p 1 L 1 c (p 1)/p. Time-ecay of σ. Recall the equation of σ when n = : Similar as in (.31), for p <, one has t σ(t) p L t σ + u σ δ σ = (χ(c)σ c). 4δ(p 1) p + p which implies by the smallness of ɛ that σ p/ (t) p(p 1) χ(c) c σ p 1 σx R 3 C c L 3 σ p/ L 6 σ p/ 1 σ C c H 1 σ p/ Cɛ σ p/, 4δ(p 1) t σ(t) p Lp + σ p/ (t). p Since σ(t) L 1 σ L 1 for any t, (.7) for any 1 p < similarly follows. Time-ecay of u. In orer to obtain the time-ecay rate of u, we first consier the time-ecay of high-orer erivatives of (σ, c). The high-orer energy estimates (.17) an (.18) give t (σ, c) H + λ (σ, c) H 3 C (σ, c). (.35) One can use the spectral analysis of the classical heat operator to get the time-ecay of (σ, c). For this purpose, one can write σ, c as σ(t) = e δ t σ + c(t) = e µ t c + t t e δ (t s) ( u σ (χ(c)σ c))s, (.36) e µ (t s) ( u c k(c)σ)s. (.37) Recall m e λ t f C f Lp H m(1 + t) γ,p m/, (.38) for 1 p an integer m. Then, for 1 p, applying (.38) to (.36) an (.37), one has σ(t) C σ L p H1(1 + t) γ,p 1/ +C t (1 + t s) 5/4 ( u σ L 1 H 1 + χ(c)σ c L 1 H )s C σ L p H1(1 + t) γ,p 1/ +Cɛ t (1 + t s) 5/4 (σ, c)(s) H 1s, (.39)

16 16 R.-J. Duan, A. Lorz an P. Markowich an c(t) C c Lp H1(1 + t) γ,p 1/ +C t (1 + t s) 5/4 ( u c L 1 H 1 + k(c)σ L 1 H 1)s C c L 1 H 1(1 + t) γ,p 1/ + Cɛ t (1 + t s) 5/4 c(s) H 1s +C(1 + t) 5/4 c L 1 L σ L 1 L. (.4) Define the high-orer energy E(t) an its weighte norm E (t) by Then, (.39) an (.4) implies E(t) = (σ, c)(t) H, E (t) = sup (1 + s) γ,p+1 E(s). s t (σ, c)(t) C(1 + t) γ,p 1/ (K p + ɛ E (t)), where K p is enote by K p = (σ, c ) Lp H 1 + σ L1 L c L1 L. Therefore, by the Gronwall inequality, it follows from (.35) that which implies E(t) E()e λt + C t e λ(t s) (σ, c)(s) s C(E() + K p + ɛe (t))(1 + t) γ,p 1, E (t) C(E() + K p + ɛe (t)), for any t. Since ɛ > is small enough, E (t) is uniformly boune in time an hence, from the efinition of E (t), it hols that (σ, c)(t) H C( (σ, c ) H + K p )(1 + t) γ,p 1/, (.41) for any 1 p an any t. Next, we use the same metho to eal with the time-ecay of high-orer erivatives of u. Define the projection operator P by P = I 1, where I is the ientity. Applying P to (.3) 3 yiels Notice that P (σφ) =, an hence (.4) can be written as u(t) = e ν t u + t u ν u = P(u u) P(σ φ). (.4) P(σ φ) = P(φ σ). t e ν (t s) ( P(u u) + P(φ σ))s. Let 1 < p an 1 < q p with q close to 1. Then, it hols that +C t u(t) C u L p H1(1 + t) γ,p 1/ (1 + t s) γ,q 1/ ( P(u u) Lq H 1 + P(φ σ) L q H1)s (.43)

17 Chemotaxis-Flui Equations 17 From the Riesz inequality, Pf L r C f L r hols for any 1 < r <, an thus one has P(u u) Lq H 1 + P( σφ) L q H 1 C u u L q H 1 + C φ σ L q H 1. Here, for H 1 -norm, it hols that u u H 1 Cɛ u H 1, φ σ H 1 C σ H 1 sup φ(t) W 1,, t an for L q -norm, it follows from the Höler inequality that which imply that u u L q u u L q/( q) C u u H Cɛ u, φ σ L q σ sup φ(t) L q/( q), t P(u u) Lq H 1 + P(φ σ) L q H 1 Cɛ u H 1 + C φ σ H 1, (.44) where C φ only epens on the L (R + ; L q/( q) (R 3 ) W 1, (R 3 ))-norm of φ. Thus, (.43) together with (.41) an (.44) give u(t) C( u L p H 1 + K )(1 + t) γ,p 1/ +Cɛ t From the high-orer energy estimates, it hols that (1 + t s) γ,q 1/ u(s) H 1s. (.45) t (σ, c, u)(t) H + λ (σ, c, u)(t) H 3 C (σ, c, u). (.46) Therefore, similarly as before, by using (.41), it follows from (.45) an (.46) that Then, similarly to (.45), one has (σ, c, u)(t) H C( u L p H 3 + K )(1 + t) γ,p 1/. (.47) u(t) C u Lp L(1 + t) γ,p +C t (1 + t s) γ,q (ɛ u(s) + C φ σ(s) )s. (.48) Since 1 < p < 6/5 implies γ,p + 1/ > 1, then (.9) follows from (.47) an (.48). This completes the proof of Theorem.. We conclue this section with a remark on the time-ecay rates of solutions when n > as mentione in Remark.. In fact, the linearize part of the system (.3) reas t σ δ σ + n c =, t c µ c + n k ()c =, t u + P ν u = σ φ, u =, for t an x R 3. Thus, if k () > hols, then c ecays exponentially in time, an otherwise it ecays like a heat kernel. Moreover, the time-ecay of c in turn leas us to obtain the time-ecay of σ an then u for which the ecay rate in time is algebraic.

18 18 R.-J. Duan, A. Lorz an P. Markowich 3 The couple Chemotaxis-Stokes equations 3.1 Global weak solutions with large ata In this section, we are concerne with the Cauchy problem on the couple Chemotaxis-Stokes equations in Ω = R : t n + u n = δ n (χ(c)n c), t c + u c = µ c k(c)n, (3.1) t u + P = ν u n φ, u =, t >, x R, with initial ata (n, c, u) t= = (n (x), c (x), u (x)), x R. (3.) Notice that throughout this section, we shall suppose that the spatial omain is two imensional an the stuy in the case of three space imensions are left for future. Instea of the perturbation theory in the presence of convective terms evelope in Section, in what follows we shall consier the nonperturbation existence theory for the above Cauchy problem when the initial cell ensity n (x) has finite mass. Actually, uner some conitions, the global existence of weak solutions can be establishe provie that the strength of the potential function φ is weak or the initial substrate concentration c (x) is small in some sense. First of all, let us give the efinition of weak solutions. Definition 3.1. (n, c, u) is calle a weak solution to the Cauchy problem (3.1)-(3.) if the following two conitions hol: (i) n(t, x), c(t, x), t, x R, an for any T >, n(1 + x + ln n ) L (, T ; L 1 (R )), n L (, T ; L (R )), c L (, T ; L 1 (R ) L (R ) H 1 (R )), c L (, T ; L (R )), n c L (, T ; L (R )), u L (, T ; L (R )), u L (, T ; L (R )); (3.3) (ii) ϕ C ([, ) R ), n( t ϕ + δ ϕ)xt + nu ϕxt R R + χ(c)n c ϕxt + n (x)ϕ(, x)x =, (3.4) R R c( t ϕ + µ ϕ)xt + cu ϕxt R R k(c)nϕxt + c (x)ϕ(, x)x =, (3.5) R R an ϕ C ([, ) R ) with ϕ =, R u ( t ϕ + ν ϕ)xt n φ ϕxt + R u (x) ϕ(, x)x =. R (3.6) Next, to state the global existence results, the following assumptions are neee: ( ) (B1) χ(c) >, χ (c), k (c) >, k(c) c χ(c) < ;

19 Chemotaxis-Flui Equations 19 (B) φ = φ(x), x R, φ L (R ), an is small, where w(x) = (1 + x )(1 + ln(1 + x )); (B3) (n, c, u ) satisfies sup w(x) φ(x) + sup w (x) φ(x) (3.7) x R x R n (1 + x + φ(x) + ln n ) L 1 (R ), c L 1 (R ) L (R ), Ψ(c ) L (R ), u L (R ), an c L 4 is small, where Ψ(c) = c χ(s) s. (3.8) k(s) If the size of φ is not small as escribe in the assumption (B), we postulate another class of assumptions in which smallness of c in L is essential: (C1) k (c) > ; (C) φ = φ(x), x R, φ L (R ), an where w(x) is given in (3.7); (C3) (n, c, u ) satisfies sup w(x) φ(x) + sup w (x) φ(x) <, x R x R n (1 + x + φ(x) + ln n ) L 1 (R ), c L 1 (R ) L (R ) H 1 (R ), u L (R ), an c L is small. Now, the main result of this section is state in the following Theorem 3.1. Uner the assumptions (A)(B1)(B)(B3) or (A)(C1)(C)(C3), the Cauchy problem (3.1)-(3.) amits a global weak solution (σ, c, u). The proof of Theorem 3.1 will be carrie out as follows. In the next subsection, on the basis of two classes of ifferent assumptions given in Theorem 3.1, for any smooth solutions to the Cauchy problem (3.1)-(3.), we erive some Lyapunov inequalities in orer to obtain the natural a priori bouns uniformly in time escribe in (i) of Definition 3.1. An then, Theorem 3.1 is prove in Subsection 3.3 ue to the usual approximation an compactness argument. Precisely, to establish a complete proof, we firstly regularize initial ata (n ɛ, c ɛ, u ɛ ) with n ɛ L (R ), c ɛ H 1 (R ) L (R ) an u ɛ H (R ) for ɛ >. Then, we obtain the global existence of solutions to the corresponing Cauchy problem with the prescribe regularize initial ata by esigning a mapping over a proper functional space an further applying the Schauer fixe point theorem with the help of the Aubin- Lions compactness metho. Finally, base on the obtaine a priori estimates, we eal with the existence of weak solutions by passing to the limit ɛ so as to conclue the proof of Theorem 3.1. Here, we remark that Lemma 3.1 an Lemma 3. about the uniform a priori estimates also hol for higher space imensions, an two space imensions play a key role in the compactness argument when passing to the limit.

20 R.-J. Duan, A. Lorz an P. Markowich 3. Uniform a priori estimates The main goal of this subsection is to provie some uniform a priori estimates on any smooth solution to the Cauchy problem (3.1)-(3.) uner the assumptions (A) an (B1)(B)(B3) or (C1)(C)(C3). One can achieve this goal if the strength of the potential function φ is weak or L -norm of c is small. Lemma 3.1 (case of weak force). Uner assumptions (A)(B1)(B)(B3), any smooth solution (n, c, u) to the Cauchy problem (3.1)-(3.) satisfies that an n(t, x), c(t, x), t, x R ; (3.9) n(t) L 1 n L 1, t ; sup c(t) L p c L p, for any 1 p, (3.1) t E 1 (t) + t D 1 (s)s E 1 (), (3.11) for any t, where the temporal free energy functional E 1 (t) an its issipation rate D 1 (t) are given by ( E 1 (t) = n ln n + 1 R Ψ(c) + 1 λ 1 µν nφ + 1 ) λ 1 µν u x, D 1 (t) = δ R n n x + λ n Ψ x + λ 1µ Ψ 4 x + 1 u x R R λ 1 µ R +µ ij R i j Ψ k(c) c χ(c) iψ j Ψ x, with constants λ > an λ 1 > epening only c. Moreover, (n, c, u) satisfies (3.3) for any finite T >. Proof. Firstly, it is straightforwar to obtain (3.9) an (3.1) as before. inequality (3.11), we claim that the following two ientities hol: (n ln n + 1 ) R n t Ψ(c) x + δ R n x an + R χ (c)k(c) + χ(c)k (c) n Ψ x + µ χ(c) ij R µ R ( ) k(c) c Ψ 4 x = χ(c) ij i j Ψ c k(c) To prove the Lyapunov χ(c) iψ j Ψ x R i u j i Ψ j Ψx, (3.1) (nφ + 1 ) t u x + ν u x = δ n φx + k(c)χ(c)n Ψ φx, (3.13) R R R R for any t. In fact, the ientity (3.13) irectly follows from the equations (3.1) 1 an (3.1) 3 an integration by parts. Here, notice that φ is inepenent of time t, an the efinition (3.8) of Ψ was use. To prove (3.1), by the efinition (3.8) an the equation (3.1) of c, Ψ = Ψ(c) satisfies [ ] t Ψ + u Ψ = µ Ψ + k(c) c χ(c) Ψ n k(c)χ(c).

21 Chemotaxis-Flui Equations 1 Applying to the above equation, multiplying it by Ψ an then integrating, one has 1 Ψ x + µ Ψ k(c) x + µ t R R R c χ(c) Ψ Ψx = i u j i Ψ j Ψx + k(c)χ(c)n Ψx. (3.14) ij R R On the other han, as before, it follows from (3.1) 1 that R n n ln nx + δ t R n x = χ(c) c nx R = k(c)χ(c) Ψ nx, R which by the further integration by parts, gives Aing (3.15) to (3.14), one has R n n ln nx + δ t R n x = k(c)χ(c)n Ψx R (n ln n + 1 ) R n t Ψ x + δ R = ij +µ R R k(c) c χ(c) Ψ Ψx + k(c) k(c)χ(c) c χ(c) n Ψ x, (3.15) n x + µ Ψ x R R k(c) k(c)χ(c) c χ(c) n Ψ x R i u j i Ψ j Ψx, (3.16) where from the assumptions (A) an (B1), it hols that Noticing the ientity it follows from integration by parts that k(c) k(c)χ(c) c χ(c) = χ (c)k(c) + χ(c)k (c) >. χ(c) ( Ψ Ψ) = Ψ Ψ + ( Ψ ) Ψ, k(c) R µ R c χ(c) Ψ Ψx = µ c µ ij R k(c) χ(c) c k(c) χ(c) Ψ 4 x k(c) χ(c) iψ j Ψ i j Ψx.

22 R.-J. Duan, A. Lorz an P. Markowich Therefore, it hols that µ Ψ k(c) x + µ R R c χ(c) Ψ Ψx = µ ( i j Ψ k(c) ij R c χ(c) iψ j Ψ i j Ψ c = µ ij R i j Ψ k(c) c χ(c) iψ j Ψ x 1 µ where we use the ientity ( k(c) c χ(c) ) + c k(c) k(c) χ(c) χ(c) = 1 k(c) χ(c) ) k(c) χ(c) iψ j Ψ x R c k(c) χ(c) Ψ 4 x, (3.17) c ( ) k(c). χ(c) Thus, putting (3.17) into (3.16) yiels the esire ientity (3.1). Now, base on the ientities (3.1) an (3.13), one can prove the Lyapunov inequality (3.11). In fact, let us enote c M = c L an efine λ = min c c M λ 1 = min 1 c c M c χ (c)k(c) + χ(c)k (c), χ(c) ( ) k(c), χ(c) where by the assumptions (A) an (B1), λ > an λ 1 > hol true. Thus from the ientity (3.1) an the Cauchy-Schwarz inequality, one has (n ln n + 1 ) R n t Ψ(c) x + δ R n x + λ n Ψ x R +µ ij R i j Ψ k(c) c χ(c) iψ j Ψ x + λ 1µ Ψ 4 x R 1 u x. (3.18) λ 1 µ R Take ɛ φ > an suppose sup x w(x) φ(x) + sup w (x) φ(x) ɛ φ. x Then similarly, from (3.13), one has (nφ + 1 ) t u x + ν u x R R ( ) δɛ φ n 1/ w(x) x + ɛ φ sup k(c)χ(c) c c M ( R δɛ φ + sup c c M k(c)χ(c) ɛ φ λ λ 1 µν where one use the Hary inequality over R R n w(x) ) R n R n x + λ λ 1 µν n w(x) n Ψ x x C R n n x C R n x R n Ψ x, (3.19)

23 Chemotaxis-Flui Equations 3 for w(x) = (1 + x )(1 + ln(1 + x )). One can choose ɛ φ > small enough such that ( 1 δɛ φ + sup ) c c M k(c)χ(c) ɛ φ λ 1 µν λ λ 1 µν Therefore, (3.11) follows by multiplying (3.19) by 1/(λ 1 µν) an then aing it to (3.18). Thus (3.11) is prove. Finally, we prove that (n, c, u) satisfies (3.3) for any finite T >. The key point is to obtain the boun of the first-orer spatial moment of n(t, x). For this purpose, multiplying (3.1) 1 by the smooth function x = (1 + x ) 1/ an taking integration, one has x nx = nu x x + δ n x x + k(c)χ(c)n Ψ x x. t R R R R Next, we estimate each term on the r.h.s. of the above ientity. Notice that x L an x L are finite constants. For the first term, it follows from the Cauchy-Schwarz inequality, Sobolev inequality an the mass conservation for n that δ. R nu ϕx x L n L u L C n L 4 u L C n L n L u L ɛ n + C n L 1 u, ɛ where ɛ > is a small constant to be chosen later. For the secon term, it is straightforwar to use the mass conservation for n to obtain δ n x x δ ϕ L n L 1 Cδ n L 1. R For the thir term, it follows from the Cauchy-Schwarz inequality that k(c)χ(c)n Ψ x x ɛ n Ψ + C ( ) sup χ(c) k(c)n. R ɛ c c M Collecting the above estimates, it therefore hols that x nx Cδ n L 1 + C ( sup χ(c) t R ɛ c c M ) k(c)n L 1 + C n L 1 u L ɛ +ɛ ( n L + n Ψ ). (3.) On the other han, recall from the proof of (3.11) that for the free energy functional E 1 (t) an the corresponing non-negative issipation rate D 1 (t), one has t E 1(t) + D 1 (t), (3.1) for any t. Now, let us efine a moifie temporal functional E 1 + (t) from E 1(t) by E 1 + (t) = E 1(t) + Λ x nx, (3.) R where Λ > is a large constant to be chosen later. We claim that as long as Λ > is large enough, it hols that E + 1 (t) R [ n( ln n + x + φ(x)) + Ψ(c) + u ] x, (3.3)

24 4 R.-J. Duan, A. Lorz an P. Markowich where A B means that C 1 B A C B hols for two generic constants C 1 > an C >. In fact, this follows from the ientity n ln nx = n ln n x n ln 1 R R R n χ n 1x, an the estimate n ln 1 R n χ n 1x = n ln 1 R n χ e nx + n ln 1 x R n χ n e x x x nx + C n 1/ χ n e x x R R x nx + C R ( 1 + C ) x nx, n L 1 R where n L 1 > has been assume without loss of generality an the mass conservation law for n was also use. Then, one can fix the constant Λ > in (3.) large enough, epening only on the total mass n L 1 of cells, such that (3.3) hols. Recall the efinition of D 1 (t) in Lemma 3.1. Then, by letting ɛ > small enough, the linear combination of (3.) an (3.1) yiels CΛδ n L 1 + CΛ ɛ ( t E 1 + (t) + 1 D 1(t) ) sup χ(c) c c M k(c)n + CΛ n L 1 E 1 + (t), (3.4) ɛ for any t, where (3.3) was use. Recall that from the equation (3.1), it follows that c(t) L 1 + t k(c)n L 1s c L 1, for any t. Then, applying the Gronwall inequality to (3.4), one has E + 1 (t) E ()e CΛT ɛ n L 1 t [ CΛδ n L 1 + CΛ ɛ ( sup χ(c) c c M [ E + 1 () + CΛδ n L 1T + CΛ ɛ c L 1 ) k(c)n L 1 sup c c M χ(c) for any t T. Therefore, the time integration of (3.4) gives ] ] s e CΛT ɛ n L 1 e CΛT ɛ n L1, t E 1 + (t) + D 1 (s)s C(n, c, u, T ) (3.5) for any t T, where C(n, c, u, T ) is a finite constant epening only on T an bouns of (n, c, u ) appearing in the assumption (B3). Finally, similarly as before, from (3.1), one has for any t. Notice c(t) + µ t c(s) s c, (3.6) Ψ(c) = χ(c) k(c) c λ 3 c, (3.7)

25 Chemotaxis-Flui Equations 5 where from the assumptions (A) an (B1), λ 3 is a positive constant efine by λ 3 = min c c M χ(c) max c cm k(c) >. Hence, (3.3) follows from (3.5), (3.6) an (3.7). This completes the proof of Lemma 3.1. Lemma 3. (case of small c ). Uner assumptions (A)(C1)(C)(C3), any smooth solution (n, c, u) to the Cauchy problem (3.1)-(3.) satisfies that an n(t, x), c(t, x), t, x R ; n(t) L 1 n L 1, t ; sup c(t) L p c L p, for any 1 p, t E (t) + λ t D (s)s C( n ln n L 1 + c H + u 1 L), (3.8) for any t, where the free energy functional E (t) an its issipation rate D (t) are given by E (t) = n(ln n + λφ)x + λ( c H + 1 u ) R D (t) = n + c H + nc + n c + u, 1 with λ > a small constant. Moreover, (n, c, u) also satisfies (3.3) for any finite T >. Proof. It suffices to prove the Lyapunov inequality (3.8) since the rest estimates can be obtaine similarly as in the proof of Lemma 3.1. To the en, we still enote c M = c L. Firstly, similarly as before, from (3.1) 1, it hols that R n n ln nx + δ t R n x = χ(c) c nx R δ R n n x + 1 max χ(c) n c x, δ c c M R which implies that n ln nx + δ R n t R n x 1 max χ(c) n c x. (3.9) δ c c M R From (3.1), it follows that 1 c x + µ c x + k(c)cnx =, (3.3) t R R R an 1 c x + µ c x + min k (c) n c x t R R c c M R η n c x + 1 R k(c) n R 4η n x + µ c x + 1 u c x, R µ R for any η >, where it is notice that min k (c) > c c M

26 6 R.-J. Duan, A. Lorz an P. Markowich ue to the assumption (C1). By taking η = 1 min c c M k (c), one has c x + µ c x + min k (c) n c x t R R c c M R max c c M k(c) R n min c cm k (c) n x + c M u x. (3.31) µ R Let λ 4 > be such that λ 4 min k (c) = 1 max χ(c) + 1, c c M δ c c M an then let c M be small enough ue to the assumption (C3), such that λ 4 max c cm k(c) min c cm k (c) δ 4, where k() = from the assumption (A) was use. Thus, multiplying (3.31) by λ 4 an aing it to (3.9) gives (n ln n + λ 4 c )x + λ 4 µ c x + n c x t R R R + δ R n 4 n x λ 4c M u x, µ R which further combining with (3.3) yiels (n ln n + λ 4 c + c )x + µ min{λ 4, } ( c + c )x t R R + min{1, min k (c)} n( c + c )x + δ R n c c M R 4 n x λ 4c M u x. (3.3) µ R Similarly as before, from (3.1) 3, it hols that (nφ + 1 ) t u x + ν u x = δ n φx + χ(c)n c φx R R R R R δ sup w(x) n φ(x) x n x + 1 (R sup n ) w(x) φ(x) sup χ(c) x c c M n x + n c x. (3.33) R Therefore, multiplying (3.33) by a small constant λ 5 >, aing it to (3.3) an then taking c M small enough, one has [n ln n + λ 4 c + c + λ 5 (nφ + 1 t R u )]x +µ min{λ 4, } ( c + c )x + min{ 1 R, min k (c)} n( c + c )x c c M R + δ R n 8 n x + λ 5ν u x, R for any t. Hence, (3.8) follows by further taking the time integral of the above inequality. This completes the proof of Lemma 3..

27 Chemotaxis-Flui Equations Proof of global existence In this subsection, we evote ourselves to the proof of Theorem 3.1 by three steps. Step 1 Construction of the regularize solutions. Given initial ata (n, c, u ) satisfying the assumption of Theorem 3.1, let (n ɛ, c ɛ, u ɛ ) for ɛ > be a smooth approximation of (n, c, u ), where an n ɛ L (R ), c ɛ H 1 (R ) L (R ), u ɛ H (R ), (3.34) n ɛ (x), c ɛ (x), u ɛ (x) =, x R, (3.35) hol, an it is further suppose that one has the convergence n ɛ (1 + x + ln n ɛ ) L 1 (R ), (3.36) n ɛ n in L 1 (R ), c ɛ c in L p (R ) (1 p < ), c ɛ c in w*-l (R ), (3.37) c ɛ c in L (R ), u ɛ u in L (R ), (3.38) as ɛ tens to zero. Consier the Cauchy problem t n ɛ + u ɛ n ɛ = δ n ɛ (χ(c ɛ )n ɛ c ɛ ), t c ɛ + u ɛ c ɛ = µ c ɛ k(c ɛ )n ɛ, t u ɛ + P ɛ = ν u ɛ n ɛ φ, u ɛ =, t >, x R, (3.39) with the prescribe-above initial ata (n ɛ, c ɛ, u ɛ ) t= = (n ɛ (x), c ɛ (x), u ɛ (x)), x R. (3.4) Now, one can state the global existence result for the regularize solutions with large initial ata. Proposition 3.1. Suppose that either assumptions (A)(B1)(B)(B3) or (A)(C1)(C)(C3) hol. Fix ɛ >. Let (3.34), (3.35) an (3.36) hol. Then, for any finite T >, the Cauchy problem (3.39)-(3.4) amits one solution (n ɛ, c ɛ, u ɛ ) with n ɛ (t, x), c ɛ (t, x), t T, x R, an n ɛ C([, T ]; L (R )) L (, T ; H 1 (R )), c ɛ C([, T ]; H 1 (R )) L (, T ; H (R )) L ((, T ) R ), u ɛ C([, T ]; H (R )) L (, T ; H 3 (R )). Proof. Fix T >. For this time we shall skip the parameter ɛ > in (3.39)-(3.4) for simplicity. Denote H = L (R ) an V = {ρ H 1 (R ); x ρ L 1 < }, Y (, T ) = {ρ L (, T ; V ); t ρ L (, T ; V ), ρ(t, x), t T, x R }, where V is the ual of V. Define a mapping T : Y (, T ) L (, T ; H) L (, T ; H) as follows. Given ñ Y (, T ), n =: T [ñ] is obtaine by iteratively solving three Cauchy problems: (i) Solve u =: T 1 [ñ] to be the solution to the Cauchy problem t u + P = ν u ñ φ, u =, u t= = u.

28 8 R.-J. Duan, A. Lorz an P. Markowich Actually, u can be given by u(t) = G(t) u + t G(t s) P( ñ φ)s, where P is the ivergence-free projection operator in R, G(t, x) = (4πt) 1 e x 4t is the Green function associate to the heat equation in R an enotes the space convolution. By the stanar energy estimates on the Stokes system an the Hary inequality over R, it follows that t u(t) H + λ ( u H + tu )s u H + C(M φ) for any t T, where M φ enotes the boun of φ: M φ = sup x R w(x) φ(x) + sup x R w (x) φ(x), an C( ) is a nonecreasing continuous function with C() =. Therefore, one has t ñ H1s, (3.41) u ɛ C([, T ]; H (R )) L (, T ; H 3 (R )), (3.4) where the continuity follows from the similar proof as that of Lemma.. (ii) Solve c =: T [ñ] to be the solution to the Cauchy problem { t c + u c = µ c k(c)ñ, c t= = c, (3.43) where u = T 1 [ñ] is given in (i). For that, efine the iterative scheme by t c j+1 + u c j = µ c j+1 k(c j )ñ, for j, where c has been set. Notice that c j+1 can be written as c j+1 (t) = G(t) c + t G(t s) (u(s)c j (s))s t G(t s)(k(c j (s))ñ(s))s, for any t T. The convergence of the sequence (c j ) j is verifie as follows. Without loss of generality we may suppose c H (R ). Take T > small enough. Similar to the proof in Subsection.3, we use inuction to prove that there are constants λ >, M > such that c j (t) H + λ t c j H3s M (3.44) for any j an t T. In fact, suppose that (3.44) is true for some j. Firstly, the Sobolev inequality shows sup t T,x R c j (t, x) CM. Then, the zero-orer energy estimate on c j+1 gives 1 t cj+1 + µ c j+1 uc j c j+1 x + k(c j )ñc j+1 x R R µ cj+1 + C u L cj + c j+1 + C ñ,