Existence of Smooth Solutions to Coupled Chemotaxis-Fluid Equations
aa r X i v : . [ m a t h . A P ] F e b Existence of Smooth Solutions to Coupled Chemotaxis-FluidEquations
Myeongju Chae, Kyungkeun Kang and Jihoon Lee
Abstract
We consider a system coupling the parabolic-parabolic Keller-Segel equations to theincompressible Navier-Stokes equations in spatial dimensions two and three. We estab-lish the local existence of regular solutions and present some blow-up criterions. For twodimensional Navier-Stokes-Keller-Segel equations, regular solutions constructed locally intime are, in reality, extended globally under some assumptions pertinent to experimentalobservation in [20] on the consumption rate and chemotactic sensitivity. We also show theexistence of global weak solutions in spatially three dimensions with stronger restriction onthe consumption rate and chemotactic sensitivity. : 35Q30, 35Q35, 76Dxx, 76Bxx
Keywords : chemotaxis-fluid equations, Keller-Segel, Navier-Stokes system, global solu-tions, energy estimates
In this paper, we consider mathematical models describing the dynamics of oxygen diffusionand consumption, chemotaxis, and viscous incompressible fluids in R d , with d = 2 ,
3. Bacteriaor microorganisms often live in fluid, in which the biology of chemotaxis is intimately relatedto the surrounding physics. Such a model was proposed by Tuval et al.[20] to describe thedynamics of swimming bacteria,
Bacillus subtilis . We consider the following equations in [20]and set Q T = (0 , T ] × R d with d = 2 , ∂ t n + u · ∇ n − ∆ n = −∇ · ( χ ( c ) n ∇ c ) ,∂ t c + u · ∇ c − ∆ c = − k ( c ) n,∂ t u + u · ∇ u − ∆ u + ∇ p = − n ∇ φ, ∇ · u = 0 in ( x, t ) ∈ R d × (0 , T ] , (1.1)where c ( t, x ) : Q T → R + , n ( t, x ) : Q T → R + , u ( t, x ) : Q T → R d and p ( t, x ) : Q T → R denotethe oxygen concentration, cell concentration, fluid velocity, and scalar pressure, respectively.The nonnegative function k ( c ) denotes the oxygen consumption rate, and the nonnegativefunction χ ( c ) denotes chemotactic sensitivity. Initial data are given by ( n ( x ) , c ( x ) , u ( x )).To describe the fluid motions, we use Boussinesq approximation to denote the effect dueto heavy bacteria. The time-independent function φ = φ ( x ) denotes the potential functionproduced by different physical mechanisms, e.g., the gravitational force or centrifugal force.Thus, φ ( x ) = ax d is one example of gravity force, and φ ( x ) = φ ( | x | ) is an example of centrifugalforce. Experiments in [20] suggest that the functions k ( c ) and χ ( c ) are constants at large c andrapidly approach zero below some critical c ∗ . Hence, in [20], these functions are approximatedby step functions, e.g., k ( c ) = κ θ ( c − c ∗ ) and χ ( c ) = κ θ ( c − c ∗ ) for some positive constants1 and κ . Also in [2], numerical simulation of plumes was obtained for the same species ofbacteria in [20] in two dimensions. Furthermore, they assumed that the functions χ ( c ) and k ( c ) are constant multiples of each other, i.e., χ ( c ) = µk ( c ).The main goals of this paper are to show the local existence of smooth solutions in two andthree dimensions with the general condition on the oxygen consumption rate and chemotacticsensitivity, and to demonstrate global existence of smooth solutions in two dimensions andweak solutions in three dimensions with appropriate assumptions of χ ( c ), k ( c ), φ and initialdata. Here we mention the related works for the result in this paper. If we ignore the couplingof the fluids, we obtain the angiogenesis type system. The classical model to describe themotion of cells was suggested by Patlak[17] and Keller-Segel[11, 12]. It consists of a system ofthe dynamics of cell density n = n ( t, x ) and the concentration of chemical attractant substance c = c ( t, x ) and is given as n t = ∆ n − ∇ · ( nχ ∇ c ) ,αc t = ∆ c − τ c + n,n ( x,
0) = n ( x ) , c ( x,
0) = c ( x ) , (1.2)where χ is the sensitivity and τ − represents the activation length. The system in (1.2) hasbeen extensively studied by many authors(see [9, 10, 15, 16, 21] and references therein). Forthe chemical consumption model by the cell or bacteria, we refer to the following chemotaxismodel motivated by angiogenesis. n t = ∆ n − ∇ · ( nχ ( c ) ∇ c ) ,c t = − c m n,n ( x,
0) = n ( x ) , c ( x,
0) = c ( x ) . (1.3)The global existence of the weak solution to the system in (1.3) was obtained by Corrias,Perthame and Zaag[3, 4] with a small data assumption of k n k L d . The bacterial movementtoward the concentration gradient model in the absence of the fluid, i.e., u = 0, was recentlystudied. When u ≡ χ ( c ) ≡ χ and k ( c ) ≡ c in (1.1), it was shown in [19] that there exists auniquely global bounded solution if 0 < χ ≤ d + 1) k c k L ∞ .If the flow of the fluid is slow, then the Navier-Stokes equations can be simplified to the Stokesequations. For the case χ ( c ) ≡ χ , Lorz[14] showed the local existence of the solution for theKeller-Segel-Stokes system. In two dimensions, Duan, Lorz, and Markowich[6] showed theglobal existence of the weak solution to the Keller-Segel-Stokes equations with the small dataassumptions on c , φ and the assumptions on the functions such that χ ( c ) > , χ ′ ( c ) ≥ , k ′ ( c ) > , d dc (cid:18) k ( c ) χ ( c ) (cid:19) < . (1.4)In [13], Liu and Lorz showed the global existence of a weak solution to the two-dimensionalKeller-Segel-Navier-Stokes equations with similar assumptions on k and χ to those in (1.4).The equation of n in (1.1) could have been replaced by a porous medium equation, i.e., ∆ n isreplaced by ∆ n m and the following Keller-Segel-Stokes system has been considered in [7]. ∂ t n + u · ∇ n − ∆ n m = −∇ · ( χ ( c ) n ∇ c ) ,∂ t c + u · ∇ c − ∆ c = − k ( c ) n,∂ t u − ∆ u + ∇ p = − n ∇ φ, ∇ · u = 0 , in ( x, t ) ∈ R d × (0 , T ] . (1.5)2n [7], Francesco, Lorz and Markowich showed the global existence of the bounded solution to(1.5) when m ∈ ( , m > in three dimensions. For the Keller-Segel-Navier-Stokes system (1.1), Duan, Lorzand Markowich[6] showed the global-in-time existence of the H ( R d )-solution, near constantstates, to (1.1), i.e., if the initial data k ( n − n ∞ , c , u ) k H is sufficiently small, then thereexists a unique global solution.As mentioned earlier, the aim of this paper is to obtain the local-in-time existence of thesmooth solution in two and three dimensions and the global-in-time existence of the classicalsolution to (1.1) in two dimensions under the minimal assumptions on the consumption rateand chemotactic sensitivity. Now we are ready to state our main results. The first result inthis article is the local existence in time of the smooth solutions to (1.1). Comparing with theresult in [6], we show the local-in-time existence without smallness of the initial data. Theorem 1 (Local existence) Let m ≥ and d = 2 , . Assume that χ , k ∈ C m ( R + ) and k (0) = 0 , k∇ l φ k L ∞ < ∞ for ≤ | l | ≤ m . There exists T > , the maximal time of existence,such that, if the initial data ( n , c , u ) ∈ H m − ( R d ) × H m ( R d ) × H m ( R d ) , then there existsa unique classical solution ( n, c, u ) of (1.1) satisfying for any t < T ( n, c, u ) ∈ L ∞ (0 , t ; H m − ( R d ) × H m ( R d ) × H m ( R d )) , ( ∇ n, ∇ c, ∇ u ) ∈ L (0 , t ; H m − ( R d ) × H m ( R d ) × H m ( R d )) . Remark 1
For simplicity, we denote k ( n ( t ) , c ( t ) , u ( t )) k X m := k n ( t ) k H m − ( R d ) + k c ( t ) k H m ( R d ) + k u ( t ) k H m ( R d ) . We remark that if T is the maximal time of existence with T < ∞ in Theorem 1, then lim sup t ր T k ( n ( t ) , c ( t ) , u ( t )) k X m + Z T k ( n ( t ) , c ( t ) , u ( t )) k X m +1 = ∞ , m ≥ . Secondly, we obtain two blow-up criteria for the system (1.1) depending on dimensions.
Theorem 2
Suppose that χ , k , φ and the initial data ( n , c , u ) satisfy all the assumptionspresented in Theorem 1. If T ∗ , the maximal time existence in Theorem 1, is finite, then oneof the following is true in each case of two or three dimensions, respectively: (2 D ) Z T ∗ k∇ c k L ∞ ( R ) = ∞ . (1.6)(3 D ) Z T ∗ k u k qL p ( R ) + Z T ∗ k∇ c k L ∞ ( R ) = ∞ , p + 2 q = 1 , < p ≤ ∞ . (1.7) Remark 2
Theorem 2 can be interpreted as follows: If R T k∇ c k L ∞ < ∞ in two dimensions orif R T ( k u k qL p + k∇ c k L ∞ ) dt < ∞ in three dimensions, then the local solution can persist beyondtime T , i.e., ( n, c, u ) ∈ L ∞ (0 , T + δ ; X m ) ∩ L (0 , T + δ ; X m +1 ) for some δ > . R . Motivated by experiments in [20] and [2], we assume that the oxygen con-sumption rate k ( c ) and chemotactic sensitivity χ ( c ) satisfy the following conditions: (A) There exists a constant µ such that sup | χ ( c ) − µk ( c ) | < ǫ for a sufficiently small ǫ > (B) χ ( c ) , k ( c ) , χ ′ ( c ) , k ′ ( c ) are all non-negative, i.e., χ ( c ) , k ( c ) , χ ′ ( c ) , k ′ ( c ) ≥ (A) plays a crucial role in obtaining LlogL × H × L typeestimates. Theorem 3 (Global existence in two dimensions) Let d = 2 . Suppose that χ , k , φ and theinitial data ( n , c , u ) satisfy all the assumptions presented in Theorem 1. Assume further that χ and k satisfy the assumptions (A) and (B) and φ ≥ . Then the unique regular solution ( n, c, u ) exists globally in time and satisfies for any T < ∞ ( n, c, u ) ∈ L ∞ (0 , T ; H m − ( R ) × H m ( R ) × H m ( R )) , ( ∇ n, ∇ c, ∇ u ) ∈ L (0 , T ; H m − ( R ) × H m ( R ) × H m ( R )) . Remark 3
If we approximate Heaviside functions using the smooth functions, then the exper-iments in [20] satisfy the assumptions (A) and (B) . Furthermore, the assumptions on φ aresatisfied by gravitational and centrifugal forces. Also we note that 2D numerical studies wereperformed under the assumption χ ( c ) = µk ( c ) in [2]. Our final main theorem is on the global-in-time existence of weak solutions in three dimen-sions. The notion of a weak solution of (1.1) is detailed in section 4 (see Definition 5). Forexistence of global weak solution, we need similar restrictions on k ( c ) and χ ( c ) as in Theorem3. More precisely, compared to (A) , we impose a slightly stronger assumption, denoted by (AA) , which is given as follows: (AA) There exists a constant µ such that χ ( c ) − µk ( c ) = 0.We are ready to state our last main result. Theorem 4
Let d = 3 and ( n , c , u ) satisfy n ∈ L ( R ) , c ∈ H ( R ) ∩ L ∞ ( R ) , u ∈ H ( R ) ,n ≥ , c ≥ , Z R (1 + | ln n | + | x | ) n dx < ∞ . (1.8) Assume further that χ and k ∈ C ( R + ) satisfy the assumption (AA) , (B) and k (0) = 0 and φ satisfies φ ≥ and k∇ l φ k L ∞ < ∞ for ≤ | l | ≤ . Then a weak solution ( n, c, u ) exists globallyin time. The rest of this paper is organized as follows. In Section 2, we prove local-in-time existenceof the smooth solution for the two and three dimensional chemotaxis system with incompress-ible Navier-Stokes equations and obtain some blow-up criteria for the solution. In Section 3,we show the global in time existence of the regular solution in two dimensions. In Section 4,we establish the existence of a weak solution in three dimensions.4
Local existence and blow-up criterion
We first consider the chemotaxis system coupled with the Navier-Stokes equations in two andthree dimensions. We show the local existence of solutions ( n, c, u ) in H m − × H m × H m spacewith m ≥ Proof of Theorem 1.
We construct the solution sequence ( n j , c j , u j ) j ≥ by iterativelysolving the Cauchy problems on the following linear equations ∂ t n j +1 + u j · ∇ n j +1 = ∆ n j +1 − ∇ · ( χ ( c j ) n j +1 ∇ c j ) ,∂ t c j +1 + u j · ∇ c j +1 = ∆ c j +1 − k ( c j ) n j ,∂ t u j +1 + u j · ∇ u j +1 + ∇ p j +1 = ∆ u j +1 − n j ∇ φ, ∇ · u j +1 = 0 . (2.1)We first set ( n ( x, t ) , c ( x, t ) , u ( x, t )) = ( n ( x ) , c ( x ) , u ( x )). Then, using the same ini-tial data to solve the linear Stokes type equations and the linear parabolic equations, we obtain u ( x, t ), c ( x, t ) and n ( x, t ), respectively. Similarly, we define ( n j ( x, t ) , c j ( x, t ) , u j ( x, t )) iter-atively. For this, we presume that c j and n j are nonnegative and show the existence and theconvergence of solutions in the adequate function spaces. We show the nonnegativity of c j and n j at the end of the proof.To prove the conclusion, i.e., to obtain contraction in adequate function spaces, we show theuniform boundness of the sequence of functions under our construction via energy estimates. • (Uniform boundedness) We here show that the iterative sequences ( n j , c j , u j ) are in X m := H m − × H m × H m space for all j ≥
0. Observing that X | α |≤ m − k ∂ α ( u j n j +1 ) k L ≤ C ( k u j k L ∞ k n j +1 k H m − + k n j +1 k L ∞ k u j k H m − ) , (cid:13)(cid:13) χ ( c j ) n j +1 ∇ c j (cid:13)(cid:13) H m − ≤ C (cid:13)(cid:13) n j +1 (cid:13)(cid:13) H m − (cid:16) (cid:13)(cid:13) c j (cid:13)(cid:13) mH m (cid:17) , (cid:13)(cid:13) k ( c j ) n j (cid:13)(cid:13) H m − ≤ C (cid:13)(cid:13) n j (cid:13)(cid:13) H m − (cid:16) (cid:13)(cid:13) c j (cid:13)(cid:13) m − H m (cid:17) , we have the following energy estimates:( i ) The estimate of n j +1 ddt (cid:13)(cid:13) n j +1 (cid:13)(cid:13) H m − + (cid:13)(cid:13) ∇ n j +1 (cid:13)(cid:13) H m − ≤ C (cid:13)(cid:13) u j (cid:13)(cid:13) L ∞ (cid:13)(cid:13) n j +1 (cid:13)(cid:13) H m − (cid:13)(cid:13) ∇ n j +1 (cid:13)(cid:13) H m − + C (cid:13)(cid:13) u j (cid:13)(cid:13) H m − (cid:13)(cid:13) n j +1 (cid:13)(cid:13) H m − (cid:13)(cid:13) ∇ n j +1 (cid:13)(cid:13) H m − + C (cid:16) (cid:13)(cid:13) c j (cid:13)(cid:13) mH m (cid:17) (cid:13)(cid:13) n j +1 (cid:13)(cid:13) H m − (cid:13)(cid:13) ∇ n j +1 (cid:13)(cid:13) H m − ≤ C (cid:16) (cid:13)(cid:13) u j (cid:13)(cid:13) H m + (cid:13)(cid:13) c j (cid:13)(cid:13) mH m (cid:17) (cid:13)(cid:13) n j +1 (cid:13)(cid:13) H m − + 12 (cid:13)(cid:13) ∇ n j +1 (cid:13)(cid:13) H m − . (2.2)( ii ) The estimate of c j +1 ddt (cid:13)(cid:13) c j +1 (cid:13)(cid:13) H m + (cid:13)(cid:13) ∇ c j +1 (cid:13)(cid:13) H m ≤ C (cid:13)(cid:13) u j (cid:13)(cid:13) L ∞ (cid:13)(cid:13) c j +1 (cid:13)(cid:13) H m (cid:13)(cid:13) ∇ c j +1 (cid:13)(cid:13) H m + C (cid:13)(cid:13) u j (cid:13)(cid:13) H m (cid:13)(cid:13) c j +1 (cid:13)(cid:13) H m (cid:13)(cid:13) ∇ c j +1 (cid:13)(cid:13) H m + C (cid:16) (cid:13)(cid:13) c j (cid:13)(cid:13) m − H m (cid:17) (cid:13)(cid:13) n j (cid:13)(cid:13) H m − (cid:13)(cid:13) ∇ c j +1 (cid:13)(cid:13) H m C (cid:16) (cid:13)(cid:13) c j (cid:13)(cid:13) m − H m (cid:17) (cid:13)(cid:13) n j (cid:13)(cid:13) H m − + C (cid:13)(cid:13) u j (cid:13)(cid:13) H m (cid:13)(cid:13) c j +1 (cid:13)(cid:13) H m + 12 (cid:13)(cid:13) ∇ c j +1 (cid:13)(cid:13) H m . (2.3)( iii ) The estimate of u j +1 ddt (cid:13)(cid:13) u j +1 (cid:13)(cid:13) H m + (cid:13)(cid:13) ∇ u j +1 (cid:13)(cid:13) H m ≤ C (cid:13)(cid:13) ∇ u j (cid:13)(cid:13) L ∞ (cid:13)(cid:13) u j +1 (cid:13)(cid:13) H m + C (cid:13)(cid:13) u j (cid:13)(cid:13) H m (cid:13)(cid:13) ∇ u j +1 (cid:13)(cid:13) L ∞ (cid:13)(cid:13) u j +1 (cid:13)(cid:13) H m + C (cid:13)(cid:13) n j (cid:13)(cid:13) H m − (cid:13)(cid:13) ∇ u j +1 (cid:13)(cid:13) H m ≤ C (cid:13)(cid:13) u j (cid:13)(cid:13) H m (cid:13)(cid:13) u j +1 (cid:13)(cid:13) H m + C (cid:13)(cid:13) n j (cid:13)(cid:13) H m − + 12 (cid:13)(cid:13) ∇ u j +1 (cid:13)(cid:13) H m , (2.4)where standard commutator estimates are used. We show that there exists a constant M > j , the following inequality holds for a small time interval [0 , T ] ( T will bespecified later): sup ≤ t ≤ T (cid:16)(cid:13)(cid:13) n j (cid:13)(cid:13) H m − + (cid:13)(cid:13) c j (cid:13)(cid:13) H m + (cid:13)(cid:13) u j (cid:13)(cid:13) H m (cid:17) + Z T (cid:13)(cid:13) ∇ n j (cid:13)(cid:13) H m − + (cid:13)(cid:13) ∇ c j (cid:13)(cid:13) H m + (cid:13)(cid:13) ∇ u j (cid:13)(cid:13) H m dt ≤ M. (2.5)Here M is a number with M ≥ k n k H m − + k c k H m + k u k H m ).We prove (2.5) via an inductive argument. Suppose (2.5) hold for j ≤ i . If we add (2.2),(2.3), and (2.4) and use Young’s inequality, then we have ddt ( (cid:13)(cid:13) n i +1 (cid:13)(cid:13) H m − + (cid:13)(cid:13) c i +1 (cid:13)(cid:13) H m + (cid:13)(cid:13) u i +1 (cid:13)(cid:13) H m ) + (cid:13)(cid:13) ∇ n i +1 (cid:13)(cid:13) H m − + (cid:13)(cid:13) ∇ c i +1 (cid:13)(cid:13) H m + (cid:13)(cid:13) ∇ u i +1 (cid:13)(cid:13) H m ≤ C (cid:16) (cid:13)(cid:13) u i (cid:13)(cid:13) H m + (cid:13)(cid:13) c i (cid:13)(cid:13) mH m (cid:17) (cid:13)(cid:13) n i +1 (cid:13)(cid:13) H m − + C (cid:16) (cid:13)(cid:13) c i (cid:13)(cid:13) m − H m (cid:17) (cid:13)(cid:13) n i (cid:13)(cid:13) H m − + C (cid:13)(cid:13) u i (cid:13)(cid:13) H m (cid:13)(cid:13) c i +1 (cid:13)(cid:13) H m + C (cid:13)(cid:13) u i (cid:13)(cid:13) H m (cid:13)(cid:13) u i +1 (cid:13)(cid:13) H m + C (cid:13)(cid:13) n i (cid:13)(cid:13) H m − ≤ C (1 + M + M m ) (cid:13)(cid:13) n i +1 (cid:13)(cid:13) H m − + CM (cid:13)(cid:13) c i +1 (cid:13)(cid:13) H m + CM (cid:13)(cid:13) u i +1 (cid:13)(cid:13) H m + C (1 + M m − ) M + CM.
In the last inequality, we use the induction hypothesis. Hence, we get ddt ( (cid:13)(cid:13) n i +1 (cid:13)(cid:13) H m − + (cid:13)(cid:13) c i +1 (cid:13)(cid:13) H m + (cid:13)(cid:13) u i +1 (cid:13)(cid:13) H m ) + (cid:13)(cid:13) ∇ n i +1 (cid:13)(cid:13) H m − + (cid:13)(cid:13) ∇ c i +1 (cid:13)(cid:13) H m + (cid:13)(cid:13) ∇ u i +1 (cid:13)(cid:13) H m ≤ C (1 + M + M m )( (cid:13)(cid:13) n i +1 (cid:13)(cid:13) H m − + (cid:13)(cid:13) c i +1 (cid:13)(cid:13) H m + (cid:13)(cid:13) u i +1 (cid:13)(cid:13) H m ) + C (1 + M m − ) M. (2.6)We choose time T such that max { C (1 + M + M m ) T, C (1 + M m − ) T } ≤ . Then fromGronwall’s inequality, we have (2.5). In short, we have ( n j +1 , c j +1 , u j +1 ) ∈ L ∞ (0 , T ; X m ) and( ∇ n j +1 , ∇ c j +1 , ∇ u j +1 ) ∈ L (0 , T ; X m ) and the uniform bound (2.5) for small T > c j +1 ) q − on the both sides of the second equation of (2.1) and integrateover spatial varaibles, then we obtain1 q ddt k c j +1 k qL q + 4( q − q k∇ ( c j +1 ) q k L ≤ . Thus, the L ∞ norm of c j +1 is uniformly bounded, which implies that χ ( c j ) and k ( c j ) are uni-formly bounded for all j ≥
0. 6 (Contraction) The estimate of this part is similar to that of the previous one. For con-venience, we denote δf j +1 := f j +1 − f j . Subtracting the j -th equations from the ( j + 1)-thequations, we have the following equations for δn j +1 , δc j +1 and δu j +1 : ∂ t δn j +1 + u j · ∇ δn j +1 − ∆ δn j +1 = − δu j · ∇ n j − ∇ · ( χ ( c j ) δn j +1 ∇ c j ) −∇ · ( χ ( c j ) n j ∇ c j ) + ∇ · ( χ ( c j − ) n j ∇ c j − ) ,∂ t δc j +1 + u j · ∇ δc j +1 − ∆ δc j +1 = − δu j · ∇ c j − k ( c j ) δn j + ( k ( c j ) − k ( c j − )) n j − ,∂ t δu j +1 + u j · ∇ δu j +1 + ∇ δp j +1 − ∆ δu j +1 = − δu j · ∇ u j − δn j ∇ φ, ∇ · δu j +1 = 0 . (2.7)( i ) The estimate of δn j +1 .Using the following standard commutator estimates X | α |≤ m − Z [ ∂ α ( u j · ∇ δn j +1 ) ∂ α δn j +1 − ( u j · ∇ ∂ α δn j +1 ) ∂ α δn j +1 ] ≤ C ( k∇ u j k L ∞ k δn j +1 k H m − + k u j k H m − k∇ δn j +1 k L ∞ k δn j +1 k H m − ) , we have the following estimate:12 ddt k δn j +1 k H m − + k∇ δn j +1 k H m − ≤ C k∇ u j k L ∞ k δn j +1 k H m − + C k u j k H m − k δn j +1 k H m k δn j +1 k H m − + C k δu j k L ∞ k n j k H m − k∇ δn j +1 k H m − + C k δu j k H m − k n j k L ∞ k∇ δn j +1 k H m − + C ( k c j k H m + k c j k m − H m ) k δn j +1 k H m − k∇ δn j +1 k H m − + C k n j k H m − (1 + k c j k m − H m + k c j − k m − H m ) k δc j k H m k∇ δn j +1 k H m − . ≤ C ( k u j k H m + k u j k H m ) k δn j +1 k H m − + C k δu j k H m k n j k H m − + C ( k c j k H m + k c j k m − H m ) k δn j +1 k H m − + C k n j k H m − (1 + k c j k m − H m + k c j − k m − H m ) k δc j k H m + 12 k∇ δn j +1 k H m − . ( ii ) The estimate of δc j +1 .12 ddt k δc j +1 k H m + k∇ δc j +1 k H m ≤ C k u j k L ∞ k δc j +1 k H m k∇ δc j +1 k H m + C k u j k H m − k δc j +1 k H m k∇ δc j +1 k H m + C k δu j k L ∞ k c j k H m k∇ δc j +1 k H m + C k c j k H m k δu j k H m − k∇ δc j +1 k H m + C (1 + k c j k m − H m ) k δn j k H m − k∇ δc j +1 k H m + C k n j − k H m − (1 + k c j k m − H m + k c j − k m − H m ) k δc j k H m k∇ δc j +1 k H m ≤ C k u j k H m k δc j +1 k H m + C k c j k H m k δu j k H m + C (1 + k c j k m − H m ) k δn j k H m − + C k n j − k H m − (1 + k c j k m − H m + k c j − k m − H m ) k δc j k H m + 12 k∇ δc j +1 k H m , where we used the Mean Value Theorem for the last term.( iii ) The estimate of δu j +1 .12 ddt k δu j +1 k H m + k∇ δu j +1 k H m ≤ C k∇ u j k L ∞ k δu j +1 k H m C k u j k H m k∇ δu j +1 k L ∞ k δu j +1 k H m + C k δu j k L ∞ k∇ u j k H m − k δu j +1 k H m +1 + C k δu j k H m − k∇ u j k L ∞ k δu j +1 k H m +1 + C k δn j k H m − k δu j +1 k H m +1 ≤ C k u j k H m k δu j +1 k H m + C k u j k H m k δu j k H m + C k δn j k H m − + 12 k δu j +1 k H m +1 , where similar standard commutator estimates are used as in the case of δn j +1 . Using Young’sinequality, we have ddt ( k δn j +1 k H m − + k δc j +1 k H m + k δu j +1 k H m ) + k∇ δn j +1 k H m − + k∇ δc j +1 k H m + k∇ δu j +1 k H m ≤ C k δn j +1 k H m − + C k δc j +1 k H m + C k δu j +1 k H m + C k δn j k H m − + C k δc j k H m + C k δu j k H m , where C depend on the H m − × H m × H m norm of ( n j , c j , u j ) and ( n j − , c j − , u j − ) and themaximum values of χ ( i ) and k ( i ) . Gronwall’s inequality gives ussup ≤ t ≤ T ( k δn j +1 k H m − + k δc j +1 k H m + k δu j +1 k H m ) ≤ CT exp ( CT ) sup ≤ t ≤ T ( k δn j k H m − + k δc j k H m + k δu j k H m ) . From the above inequality, we find that ( n j , c j , u j ) is a Cauchy sequence in the Banach space C (0 , T ; X m ) for some small T >
0, and thus we have the limit in the same space. • (Uniqueness) To show the uniqueness of the above local-in-time solution, we assume thatthere exist two local-in-time solutions ( c ( x, t ) , n ( x, t ) , u ( x, t )) and ( c ( x, t ) , n ( x, t ) , u ( x, t ))of (1.1) with the same initial data over the time interval [0 , T ], where T is any time before themaximal time of existence. Let ˜ c ( x, t ) := c ( x, t ) − c ( x, t ), ˜ n ( x, t ) := n ( x, t ) − n ( x, t ), and˜ u ( x, t ) := u ( x, t ) − u ( x, t ). Then (˜ c, ˜ n, ˜ u ) solves ∂ t ˜ n + u · ∇ ˜ n − ∆˜ n = − ˜ u · ∇ n − ∇ · (( χ ( c ) − χ ( c )) n ∇ c ) −∇ · ( χ ( c )˜ n ∇ c ) − ∇ · ( χ ( c ) n ∇ ˜ c ) ,∂ t ˜ c + u · ∇ ˜ c − ∆˜ c = − ˜ u · ∇ c − ( k ( c ) − k ( c )) n − k ( c )˜ n,∂ t ˜ u + u · ∇ ˜ u − ∆˜ u + ∇ ˜ p = − ˜ u · ∇ u − ˜ n ∇ φ, ∇ · ˜ u = 0 , in ( x, t ) ∈ R d × (0 , T ] . (2.8)Multiplying ˜ n to both sides of the first equation of (2.8) and integrating over R d , we have12 ddt k ˜ n k L + k∇ ˜ n k L ≤ k ˜ un k L k∇ ˜ n k L + k ( χ ( c ) − χ ( c )) n ∇ c k L k∇ ˜ n k L + k χ ( c )˜ n ∇ c k L k∇ ˜ n k L + k χ ( c ) n ∇ ˜ c k L k∇ ˜ n k L ≤ C k ˜ u k L k n k L ∞ + C k ˜ c k L k n k L ∞ k∇ c k L ∞ + C k ˜ n k L k∇ c k L ∞ + C k∇ ˜ c k L k n k L ∞ + ǫ k∇ ˜ n k L . Multiplying ˜ c and − ∆˜ c to both sides of the second equation of (2.8), we have12 ddt k ˜ c k L + k∇ ˜ c k L ≤ k ˜ uc k L k∇ ˜ c k L + C k ˜ c k L k n k L ∞ + C k ˜ n k L k ˜ c k L C k ˜ u k L k c k L ∞ + C k ˜ c k L k n k L ∞ + C k ˜ n k L + C k ˜ c k L + ǫ k∇ ˜ c k L , and 12 ddt k∇ ˜ c k L + k ∆˜ c k L ≤ k u · ∇ ˜ c k L k ∆˜ c k L + k ˜ u · ∇ c k L k ∆˜ c k L + k k ( c ) − k ( c ) k L k n k L ∞ k ∆˜ c k L + C k ˜ n k L k ∆˜ c k L ≤ C k∇ ˜ c k L k u k L ∞ + C k ˜ u k L k∇ c k L ∞ + C k ˜ c k L k n k L ∞ + C k ˜ n k L + ǫ k ∆˜ c k L . Multiplying ˜ u to both sides of the third equation of (2.8) and integrating over R d , we have12 ddt k ˜ u k L + k∇ ˜ u k L ≤ k ˜ u k L k u k L ∞ k∇ ˜ u k L + C k ˜ n k L k ˜ u k L ≤ C k ˜ u k L k u k L ∞ + C k ˜ n k L + C k ˜ u k L + ǫ k∇ ˜ u k L . Summing the above estimates, we obtain12 ddt ( k ˜ n k L + k ˜ c k L + k∇ ˜ c k L + k ˜ u k L ) ≤ C ( k |∇ c | + |∇ c | k L ∞ + k n + n k L ∞ + k n k L ∞ k∇ c k L ∞ + k c k L ∞ + k | u | + | u | k L ∞ + 1) × ( k ˜ n k L + k ˜ c k L + k∇ ˜ c k L + k ˜ u k L ) . Since all L ∞ norms of ( n i , c i , u i ) are controlled by H m − × H m × H m norm of ( n i , c i , u i ) with m ≥
3, and the initial data of (˜ c, ˜ n, ˜ u ) are all zero, (˜ c, ˜ n, ˜ u ) are all zero for T >
0. That impliesthe uniqueness of the local classical solution. • (Nonnegativity) For completeness, we briefly show that n j and c j are nonnegative for all j . To use induction, we assume c j and n j are nonnegative. If we apply the maximum principleto the equation of c j +1 in (2.1), we find that c j +1 is nonnegative ( k ( c j ) n j is nonnegative). Letus decompose n j +1 = n j +1+ − n j +1 − , where n j +1+ = ( n j +1 n j +1 ≥ n j +1 < , n j +1 − = ( − n j +1 n j +1 ≤ n j +1 > . Recall that the weak derivative of n j +1 − is −∇ n j +1 if n j +1 − < , otherwise zero. It holds that Z t Z R ∂ t n j +1 ( n j +1 ) − dxds = 12 ( k ( n j +1 ) − (0) k L − k ( n j +1 ) − (0) k L )since n j +1 − , ∂ t n j +1 ∈ L (0 , T ; L ( R )) (see e.g. [18]). Now multiplying the negative part ( n j +1 ) − on both sides of the first equation of (2.1) and integrating over [0 , t ] × R , we have12 Z t k ( n j +1 ) − k L + k∇ ( n j +1 ) − k L ds ≤ C Z t k ( n j +1 ) − k L k∇ c j k L ∞ + 12 k∇ ( n j +1 ) − k L ds. Using Gronwall’s inequality, we have k ( n j +1 ) − ( t ) k L ≤ k ( n j +1 ) − (0) k L exp (cid:18) C Z t k∇ c j k L ∞ ds (cid:19) . Since the initial data n j +10 is nonnegative, we conclude that n j +1 is nonnegative. This completesthe proof. 9 .2 Blow-up criterion Next, we observe a blow-up criterion for the fluid chemotaxis equations.
Proposition 1 (A Blow-up criterion) Suppose that χ , k , φ and the initial data ( n , c , u ) satisfy all the assumptions presented in Theorem 1. If T < ∞ is the maximal time of existence,then Z T (cid:16) k∇ u ( t ) k L ∞ ( R d ) + k∇ c ( t ) k L ∞ ( R d ) (cid:17) dt = ∞ . Proof.
At first, we consider the L estimate of n . Multiplying n to both sides of the equationof n and integrating, we have12 ddt k n k L + k∇ n k L ≤ C k χ ( c ) n ∇ c k L k∇ n k L . Since χ is continuous and c is uniformly bounded until the maximal time of existence, we have C k χ ( c ) n ∇ c k L k∇ n k L ≤ k∇ n k L + C k∇ c k L ∞ k n k L . For the estimates of c , we use the calculus inequality k∇ ( u · ∇ c ) − ( u · ∇ ) ∇ c k L ≤ C k∇ u k L ∞ k∇ c k L . Multiplying − ∆ c to both sides of the equation of c and integrating, we obtain12 ddt k∇ c k L + k ∆ c k L ≤ C k∇ u k L ∞ k∇ c k L + C k ( k ( c ) n ) k L + 14 k ∆ c k L . For the equations of u , multiplying − ∆ u to both sides of the equations and integrating byparts, we have 12 ddt k∇ u k L + k ∆ u k L ≤ C k∇ u k L ∞ k∇ u k L + C k n k L k ∆ u k L . Collecting all the estimates, we obtain ddt (cid:0) k n k L + k∇ c k L + k∇ u k L (cid:1) + (cid:0) k∇ n k L + k ∆ c k L + k ∆ u k L (cid:1) ≤ C (cid:0) k∇ c k L ∞ + k∇ u k L ∞ (cid:1) (cid:0) k n k L + k∇ c k L + k∇ u k L (cid:1) . From Gronwall’s inequality, we havesup (cid:0) k n k L + k∇ c k L + k∇ u k L (cid:1) + Z T (cid:0) k∇ n k L + k ∆ c k L + k ∆ u k L (cid:1) dt ≤ C ( k n k L + k∇ c k L + k∇ u k L ) exp (cid:18)Z T k∇ u k L ∞ + k∇ c k L ∞ dt (cid:19) . Note that k n k L ∞ (0 ,T ; L ) and k∇ n k L (0 ,T ; L ) are uniformly bounded if R T k∇ u k L ∞ + k∇ c k L ∞ dt is bounded. Moreover, we see that n ∈ L qx L ∞ t and ∇ n q/ ∈ L x L t for all 2 < q < ∞ . Indeed, ddt k n k qL q + (cid:13)(cid:13)(cid:13) ∇ n q (cid:13)(cid:13)(cid:13) L ≤ C Z R (cid:12)(cid:12) n ∇ c ∇ n q − (cid:12)(cid:12) dx ≤ C k∇ c k L ∞ k n k qL q + 12 (cid:13)(cid:13)(cid:13) ∇ n q (cid:13)(cid:13)(cid:13) L . k n ( t ) k L q ≤ C , where C is independent of q . Letting q → ∞ , we have n ∈ L ∞ x L ∞ t .Next, we consider the estimate in the space ( n, c ) ∈ H × H . We have12 ddt k∇ n k L + k ∆ n k L ≤ C k∇ u k L ∞ k∇ n k L + C k∇ n k L k∇ c k L ∞ (cid:13)(cid:13) ∇ n (cid:13)(cid:13) L + C k n k L ∞ k ∆ c k L (cid:13)(cid:13) ∇ n (cid:13)(cid:13) L + C k n k L ∞ k∇ c k L ∞ k∇ c k L (cid:13)(cid:13) ∇ n (cid:13)(cid:13) L . From Young’s inequality and Gronwall’s inequality, we havesup k∇ n k L + Z T (cid:13)(cid:13) ∇ n (cid:13)(cid:13) L dt ≤ (cid:18) k∇ n k L + C k n k L ∞ (0 ,T ; L ∞ ) (cid:18)Z T k ∆ c k L dt + k∇ c k L ∞ (0 ,T ; L ) Z T k∇ c k L ∞ dt (cid:19)(cid:19) × exp (cid:18)Z T k∇ u k L ∞ + k∇ c k L ∞ dt (cid:19) . Hence, n ∈ H x L ∞ t ∩ H x L t . For the H estimate of c , we have12 ddt k ∆ c k L + k∇ ∆ c k L ≤ C k∇ u k L ∞ k ∆ c k L + C k ∆ u k L k c k L ∞ k∇ ∆ c k L + C k∇ c k L k n k L ∞ k∇ ∆ c k L + C k∇ n k L k∇ ∆ c k L . By Gronwall’s inequality, we have c ∈ H x L ∞ t ∩ H x L t . Similarly, u ∈ H x L ∞ t ∩ H x L t . Then,we consider the estimate in the space ( n, c, u ) ∈ H × H × H . Proceeding similarly to theabove, we obtain12 ddt k n k H + k∇ n k H ≤ C k u k L ∞ k n k H k∇ n k H + C k∇ u k L ∞ k n k H k∇ n k H + 14 k∇ n k H + C k χ ( c ) n ∇ c k H . In the above, the last term can be controlled by k χ ( c ) n ∇ c k H ≤ C k n k H k χ ( c ) ∇ c k H , and (cid:13)(cid:13) ∇ ( χ ( c ) ∇ c ) (cid:13)(cid:13) L ≤ C (cid:13)(cid:13) ∇ c (cid:13)(cid:13) L + C (cid:13)(cid:13) ∇ c (cid:13)(cid:13) L k∇ c k L ∞ + C k∇ c k L . We already obtained c ∈ H x L ∞ t ∩ H x L t . Hence, if we use Young’s inequality and Gronwall’sinequality, we havesup k n k H + Z T k∇ n k H dt ≤ k n k H exp (cid:18) C + Z T k∇ u k L ∞ + k∇ c k L ∞ dt (cid:19) . Similarly, we estimate c as12 ddt k c k H + k∇ c k H ≤ C k∇ u k L ∞ k c k H k∇ c k H + C k u k H k∇ c k L ∞ k∇ c k H C k ( k ( c ) n ) k H + 14 k∇ c k H . We can control the term k ( k ( c ) n ) k H by C ( k c k H k n k H + k c k H k∇ c k L ∞ k n k H ). For theestimate of u , we have12 ddt k u k H + k∇ u k H ≤ C k∇ u k L ∞ k u k H k∇ u k H + 14 k∇ u k H + C k n k H . Using Gronwall’s inequality, we have ( c, u ) ∈ ( H x L ∞ t ∩ H x L t ) × ( H x L ∞ t ∩ H x L t ). Let usconsider H m − × H m × H m estimates. The case m = 2 , m ≥ ∂ α ( | α | ≤ m −
1) and multiplying ∂ α n to both sides ofthe equation n and integrating and summing, we have12 ddt k n k H m − + k∇ n k H m − ≤ C k∇ u k L ∞ k n k H m − k∇ n k H m − + C k u k H m − k∇ n k L ∞ k∇ n k H m − + 14 k∇ n k H m − + C k χ ( c ) n ∇ c k H m − . We already obtained the estimate for the case m = 4, thus k∇ c k L ∞ (0 ,T ; L ∞ ) is bounded. Hence,we have k∇ χ ( c ) k H m − ≤ C (1 + k∇ c k L ∞ ) k∇ χ ′ ( c ) k H m − . Using the classical product lemma on each step of iteration, we can control k∇ χ ( c ) k H m − ≤ C (1 + k∇ c k L ∞ ) m − . Then we have k χ ( c ) n ∇ c k H m − ≤ C (1 + k c k H m + k∇ c k mL ∞ ) k n k H m − using the product lemma. For the H m estimate of c , we proceed similarly to have12 ddt k c k H m + k∇ c k H m ≤ C k∇ u k L ∞ k c k H m k∇ c k H m + C k u k H m k∇ c k L ∞ k∇ c k H m + C k ( k ( c ) n ) k H m − + 14 k∇ c k H m . As is shown for the term k χ ( c ) n ∇ c k H m − , we control the term k ( k ( c ) n ) k H m − by C (1 + k∇ c k m − L ∞ ) k n k H m − . For the estimate of u , we have12 ddt k u k H m + k∇ u k H m ≤ C k∇ u k L ∞ k u k H m k∇ u k H m + 14 k∇ u k H m + C k n k H m − . Thus, by collecting all the above estimates and using Gronwall’s inequality, we have ( n, c, u ) ∈ ( H m − x L ∞ t ∩ H mx L t ) × ( H mx L ∞ t ∩ H m +1 x L t ) × ( H mx L ∞ t ∩ H m +1 x L t ). This completes the proof.We are ready to present the proof of Theorem 2. Proof of Theorem 2.
In the proof of Proposition 1, we notice that k∇ c k L ∞ is solelyresponsible for n ∈ L x L ∞ t and ∇ n ∈ L x L t . Indeed, ddt k n k L + k∇ n k L ≤ C Z R d | n ∇ c ∇ n | dx ≤ C k∇ c k L ∞ k n k L + 12 k∇ n k L . (2.9)12his implies u ∈ L x L ∞ t and ∇ u ∈ L x L t by ddt k u k L + k∇ u k L ≤ C k n k L k u k L . (2.10)Moreover, we have n ∈ L qx L ∞ t and ∇ n q/ ∈ L x L t for all 2 < q < ∞ ; ddt k n k qL q + (cid:13)(cid:13)(cid:13) ∇ n q (cid:13)(cid:13)(cid:13) L ≤ C q Z R d (cid:12)(cid:12) n ∇ c ∇ n q − (cid:12)(cid:12) dx ≤ C q k∇ c k L ∞ k n k qL q + 12 (cid:13)(cid:13)(cid:13) ∇ n q (cid:13)(cid:13)(cid:13) L . Next, we see that ∇ c ∈ L x L ∞ t and ∇ c ∈ L x L t . Indeed, ddt k∇ c k L + (cid:13)(cid:13) ∇ c (cid:13)(cid:13) L ≤ C k∇ c k L ∞ k u k L (cid:13)(cid:13) ∇ c (cid:13)(cid:13) L + C k n k L (cid:13)(cid:13) ∇ c (cid:13)(cid:13) L . (2.11)We first consider the two-dimensional case. • (
2D case ) For convenience, we denote vorticity as ω := ∇ × u ; that is ω = ∂ u − ∂ u in two dimensions. Next, we consider the vorticity equation ω t − ∆ ω + u ∇ ω = −∇ ⊥ n ∇ φ, where ∇ ⊥ n = ( − ∂ n, ∂ n ). We note that ω ∈ L x L ∞ t and ∇ ω ∈ L x L t , since ddt k ω k L + k∇ ω k L ≤ C k∇ n k L k ω k L . (2.12)Furthermore, we observe that ∇ ω ∈ L x L ∞ t and ∇ ω ∈ L x L t . Indeed, testing − ∆ ω , we get ddt k∇ ω k L + (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L ≤ k u k L k∇ ω k L k ∆ ω k L + k∇ n k L k ∆ ω k L ≤ C k u k L k∇ u k L k∇ ω k L (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L + k∇ n k L k ∆ ω k L . Therefore, via embedding, we have Z T k∇ u k L ∞ dt ≤ Z T k∇ u k H dt ≤ C Z T k ω k H dt < ∞ . This completes the proof of the 2D case. • (
3D case ) We will show this case by contradictory arguments. We suppose that the con-dition (1.7) is not true. We first recall the vorticity equation ω t − ∆ ω + u ∇ ω = ω ∇ u − ∇ ⊥ n ∇ φ. Under the condition (1.7) we have ω ∈ L x L ∞ t and ∇ ω ∈ L x L t as follows. We denote Q ∗ = R × ( T ∗ − δ, t ) for T ∗ − δ < t < T ∗ . For any given p, q satisfying 3 /p + 2 /q = 1, 3 < p ≤ ∞ ,we choose l, m such that 1 /p + 1 /l = 1 / /q + 1 /m = 1 /
2. We then remind that, due tothe Gargliardo-Nirenberg’s inequality, k u k L l,mx,t ≤ C k u k θL , ∞ x,t k∇ u k − θL , x,t , ≤ l ≤ , l + 2 m = 32 , θ = (6 − l ) / l and 1 − θ = (3 l − / l . Then we have ddt k ω k L + k∇ ω k L ≤ k u k L p k ω k L l k∇ ω k L + C k n k L k∇ ω k L . Next, integrating in time over ( T ∗ − δ, t ), k ω ( t ) k L + Z tT ∗ − δ k∇ ω k L ≤ k ω ( T ∗ − δ ) k L + C k u k L p,qx,t k ω k L l,mx,t k∇ ω k L , x,t + C k n k L , x,t k∇ ω k L , x,t ≤ k ω ( T ∗ − δ ) k L + k u k L p,qx,t k ω k θL , ∞ x,t k∇ ω k − θL , x,t + C k n k L , x,t k∇ ω k L , x,t . Note that θ >
0. By Young’s inequality, we have k ω ( t ) k L + Z tT ∗ − δ k∇ ω k L ≤ k ω ( T ∗ − δ ) k L + C k u k θ L p,qx,t k ω k L , ∞ x,t + C k n k L , x,t + 12 k∇ ω k L , x,t . Since k u k L p,qx,t can be sufficiently small in ( T ∗ − δ, T ∗ ) × R by decreasing δ , we have k ω ( t ) k L + 12 Z tT ∗ − δ k∇ ω k L ≤ k ω ( T ∗ − δ ) k L + C k n k L , x,t , (2.13)which is bounded by (2.9). Since t is arbitrary for all t < T ∗ , this estimate is uniform.Next, we observe that ∇ c ∈ L x L ∞ t and ∇ c ∈ L x L t . Indeed, we estimate ddt (cid:13)(cid:13) ∇ c (cid:13)(cid:13) L + (cid:13)(cid:13) ∇ c (cid:13)(cid:13) L ≤ C ( k∇ n k L + k∇ c k L ∞ k∇ u k L ) (cid:13)(cid:13) ∇ c (cid:13)(cid:13) L + k u k L (cid:13)(cid:13) ∇ c (cid:13)(cid:13) L (cid:13)(cid:13) ∇ c (cid:13)(cid:13) L ≤ C ( k∇ n k L + k∇ c k L ∞ k∇ u k L ) (cid:13)(cid:13) ∇ c (cid:13)(cid:13) L + k ω k L (cid:13)(cid:13) ∇ c (cid:13)(cid:13) L (cid:13)(cid:13) ∇ c (cid:13)(cid:13) L , and use (2.9), (2.10). Similarly, we show that n ∈ L ∞ t H x ∩ L t H x by estimating ddt k∇ n k L + (cid:13)(cid:13) ∇ n (cid:13)(cid:13) L ≤ k u k L k∇ n k L k∇ n k L + k∇ c k L ∞ k∇ n k L k ∆ n k L + k u k L (cid:13)(cid:13) ∇ c (cid:13)(cid:13) L (cid:13)(cid:13) ∇ n (cid:13)(cid:13) L + k∇ c k L ∞ k n k L k∇ c k L k ∆ n k L . Finally, we show that ω ∈ H x L ∞ t ∩ H x L t . Testing − ∆ ω to the equations, we have ddt k∇ ω k L + (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L ≤ k u k L p k∇ ω k L l (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L + k∇ u k L k ω k L (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L + C k∇ n k L (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L , where 3 < p ≤ ∞ and 1 /p + 1 /l = 1 /
2. Note that, via the Gargliardo-Nirenberg’s inequality, k∇ u k L k ω k L (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L ≤ C k ω k L (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L ≤ C k ω k L (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L . We treat the term k u k L p k∇ ω k L l (cid:13)(cid:13) ∇ ω (cid:13)(cid:13) L similarly to k u k L p k ω k L l k∇ ω k L in the estimationof (2.13). Therefore, since ∇ ω ∈ L x L t , we have Z T k∇ u k L ∞ dt ≤ Z T k∇ u k H dt ≤ C Z T k ω k H dt < ∞ . This completes the proof. 14
Global solutions in two dimensions
In this section, we provide the proof of global existence of smooth solutions in time with largeinitial data in two dimensions. For the proof of Theorem 3, we show some a priori estimates,which are uniform until the maximal time of existence. Moreover, such estimates imply thatthe blow-up condition quantity in Theorem 2 is uniformly bounded up to the maximal timeof existence. Therefore, the maximal time cannot be finite. Now we present the proof ofTheorem 3.
Proof of Theorem 3.
We first present the following estimates for the solutions to thetwo-dimensional chemotaxis system coupled with the Navier-Stokes equations. n (1 + | x | + | ln n | ) ∈ L ∞ (0 , T ; L ( R )) , ∇√ n ∈ L (0 , T ; L ( R )) , (3.1) c ∈ L ∞ (0 , T ; L ( R ) ∩ L ∞ ( R ) ∩ H ( R )) , ∇ c ∈ L (0 , T ; L ( R )) , (3.2) u ∈ L ∞ (0 , T ; L ( R )) , ∇ u ∈ L (0 , T ; L ( R )) . (3.3)We have the mass conservation for n ( t, x ) as Z R n ( t, x ) dx = Z R n ( x ) dx. (3.4)Multiplying c q − ( t, x ) to both sides of the second equation of (1.1) and integrating over R ,we have 1 q ddt k c k qL q + 4( q − q k∇ c q k L + Z R k ( c ) nc q − dx = 0 . (3.5)Hence, we have c ∈ L ∞ (0 , T ; L q ) for any 1 < q ≤ ∞ and ∇ c q ∈ L (0 , T ; L ) for any 1 < q < ∞ .Multiplying ln n to both sides of the first equation of (1.1) and integrating over R , we have ddt Z R n ln ndx + 4 Z R |∇√ n | dx + Z R χ ′ ( c ) |∇ c | ndx = − Z R χ ( c )∆ cndx. (3.6)Multiplying − ∆ c to both sides of (1.1) and integrating over R , we obtain ddt k∇ c k L + k ∆ c k L = Z R k ( c )∆ c ndx + X j,k Z R c∂ k u i ∂ i ∂ k cdx ≤ Z R k ( c )∆ c ndx + C k∇ u k L k c k L ∞ k ∆ c k L . (3.7)Multiplying µ to both sides of (3.7) and then adding (3.6), we have ddt Z R n ln n + µ |∇ c | dx + Z R |∇√ n | + µ | ∆ c | dx ≤ ǫ k ∆ c k L k√ n k L + C k c k L ∞ k∇ u k L + 14 µ k ∆ c k L ≤ µ k ∆ c k L + ǫC k∇√ n k L + C k c k L ∞ k∇ u k L , (3.8)15here we used the condition (A). Here we choose ǫ to be so small that ǫC < λ := C k c k L ∞ . On the other hand, multiplying u to both sides of the thirdequations of (1.1) and integrating over R , we have12 ddt k u k L + k∇ u k L = − Z R n ∇ φudx. (3.9)Multiplying φ to both sides of the first equation of (1.1) and integrating over R , we have ddt Z R nφdx = − Z R u · ∇ nφdx − Z R ∇ n · ∇ φdx + Z R χ ( c ) n ∇ c · ∇ φdx ≤ − Z R u · ∇ nφdx + C k∇√ n k L k√ n k L + C k n k L k∇ c k L . (3.10)Summing (3.9) and (3.10), we have ddt (cid:18) k u k L + Z R nφdx (cid:19) + k∇ u k L ≤ C k√ n k L k∇√ n k L + C k√ n k L k∇√ n k L k∇ c k L . (3.11)Multiplying λ to both sides of (3.11) and adding (3.8), we obtain ddt Z R n ln n + µ |∇ c | + λ | u | + λ nφdx + Z R |∇√ n | + µ | ∆ c | + λ |∇ u | dx ≤ λ C k n k L k∇√ n k L + λ C k n k L k∇√ n k L k∇ c k L ≤ λ C k n k L + λ C k n k L k∇ c k L + k∇√ n k L . (3.12)Using Gronwall’s inequality, we havesup ≤ t ≤ T (cid:18)Z R n ln n + µ |∇ c | + λ | u | + λ nφdx (cid:19) + Z T Z R |∇√ n | + µ | ∆ c | + λ |∇ u | dxdt ≤ C ( T ) . Next, we show that n | ln n | ∈ L ∞ (0 , T ; L ( R )), following a typical argument for dealing withkinetic entropy (see e.g. [5]). We first note that Z R n (ln n ) − ≤ C + C Z R n h x i , (3.13)where (ln n ) − is a negative part of ln x and h x i = (1 + | x | ) . Indeed, setting D = { x : n ( x ) ≤ e −| x | } and D = { x : e −| x | < n ( x ) ≤ } , we have Z R n (ln n ) − = − Z D n ln n − Z D n ln n ≤ C Z D √ n + Z D n h x i ≤ C Z R e − | x | + Z R n h x i . (3.14)16his deduces the estimate (3.13). Next, integrating (3.12) in time t , we get Z R n ( · , t ) ln n ( · , t ) + µ k∇ c ( t ) k L + λ k u ( t ) k L + λ k n ( t ) φ k L + Z t Z R (cid:18) |∇√ n | + µ | ∆ c | + λ |∇ u | (cid:19) dxdτ ≤ C + C t + C Z t k∇ c k L dxdτ, (3.15)where C = R R n ln n + µ k∇ c k L + λ k u k L + λ k n φ k L . Remembering (3.13), we compute ddt Z R h x i ndx = Z R nu ∇h x i dx + Z R n ∆ h x i dx + Z R χ ( c ) n ∇ c ∇h x i dx. (3.16)The term R R nu ∇h x i dx is bounded as follows: (cid:12)(cid:12)(cid:12)(cid:12)Z R nu ∇h x i dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ k√ n k L k u k L ≤ k∇√ n k L + C k n k L k u k L . Noting that |∇h x i| + | ∆ h x i| ≤ C , we get (cid:12)(cid:12)(cid:12)(cid:12)Z R n ∆ h x i dx (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z R χ ( c ) n ∇ c ∇h x i dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C + C (cid:13)(cid:13) ∇√ n (cid:13)(cid:13) L k∇ c k L , where we used that k n k L ≤ C k n k L k∇√ n k L . In summary, we obtain ddt Z R h x i ndx ≤ δ k∇√ n k L + C k u k L + C δ k∇ c k L + C, (3.17)where δ is sufficiently small, which will be specified later. Therefore, integrating (3.17) in time, Z R h x i n ( · , t ) dx ≤ Z R h x i n dx + δ Z t k∇√ n k L + C Z t k u k L + C δ Z t k∇ c k L + Ct. (3.18)Now adding 2 R n (ln n ) − to both sides of (3.15), we obtain Z R n ( · , t ) | ln n ( · , t ) | + µ k∇ c ( t ) k L + λ k u ( t ) k L + λ k n ( t ) φ k L + Z t Z R (cid:0) |∇√ n | + µ | ∆ c | + λ |∇ u | (cid:1) dxdτ ≤ C + Ct + C Z t k∇ c k L dxdτ + C Z t k u k L dxdτ, (3.19)where δ in (3.17) is so small that term R t k∇√ n k L is absorbed to the left hand side of (3.15).Since (3.19) holds for all t until the maximal time of existence, due to Gronwall’s inequality,we obtain n | ln n | ∈ L ∞ (0 , T ; L ( R )). Moreover, again via the inequality (3.19), we deduce(3.1)-(3.3).We note that from the blow-up criterion in two dimensions in Theorem 2, it suffices to showthat ∇ c ∈ L (0 , T ; L ∞ ( R )) for global existence of smooth solutions in R . We first considerthe vorticity equation of velocity fields. Taking curl, we have ∂ t ω + ( u · ∇ ) ω − ∆ ω = −∇ ⊥ n · ∇ φ, ∇ ⊥ = ( − ∂ , ∂ ). If we multiply ω to both sides of the above equation and integrate over R , then we have12 ddt k ω k L + k∇ ω k L = Z R n ∇ φ ∇ ⊥ ωdx ≤ C k n k L k∇ ⊥ ω k L . Hence, we have k ω k L ∞ (0 ,T ; L ) + k∇ ω k L (0 ,T ; L ) ≤ C k n k L (0 ,T ; L ) . Since k n k L ≤ k√ n k L ≤ C k√ n k L k∇√ n k L , we have ω ∈ L ∞ (0 , T ; L ) ∩ L (0 , T ; H ). Nextwe consider the equation of n . Multiplying n and integrating over R , we have12 ddt k n k L + k∇ n k L = Z R χ ( c ) n ∇ c ∇ ndx = − Z R ∇ · ( χ ( c ) ∇ c ) n dx ≤ C Z R |∇ c | n dx + C Z R |∇ c | n dx ≤ C ( (cid:13)(cid:13) ∇ c (cid:13)(cid:13) L + k∇ c k L ) k n k L ≤ C (cid:13)(cid:13) ∇ c (cid:13)(cid:13) L k n k L k∇ n k L , where we used that χ is C and c ∈ L ∞ (0 , ∞ ; L ∞ ), i.e., χ ( c ) and χ ′ ( c ) are bounded. Due toYoung’s inequality, we have ddt k n k L + k∇ n k L ≤ C k n k L k∇ c k L . Therefore, via Gronwall’s inequality, we have n ∈ L ∞ (0 , T ; L ) ∩ L (0 , T ; H ). Multiplying∆ c to both sides of the equation of c and integrating over R , we have12 ddt k ∆ c k L + k∇ ∆ c k L ≤ k∇ u k L k∇ c k L k∇ ∆ c k L + k u k L ∞ k∇ c k L k∇ ∆ c k L − Z R k ( c ) n ∆ cdx. We note that the last term above is controlled as follows: (cid:12)(cid:12)(cid:12)(cid:12)Z R k ( c ) n ∆ cdx (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z R k ′ ( c ) ∇ c · ( ∇ ∆ c ) ndx (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z R k ( c ) ∇ n · ( ∇ ∆ c ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k∇ ∆ c k L k n k L k∇ c k L + C k∇ n k L k∇ ∆ c k L ≤ ǫ k∇ ∆ c k L + C k n k L k∇ c k L + C k∇ n k L . Hence, we have12 ddt k ∆ c k L + k∇ ∆ c k L ≤ C k∇ u k L k∇ c k L + C k u k L ∞ k ∆ c k L + C k n k L k∇ c k L + C k∇ n k L . ≤ C k ω k L k∇ ω k L k∇ c k L (cid:13)(cid:13) ∇ c (cid:13)(cid:13) L + C k∇ ω k L k ∆ c k L + C k n k L k∇ n k L k∇ c k L (cid:13)(cid:13) ∇ c (cid:13)(cid:13) L + C k∇ n k L . Gronwall’s inequality gives c ∈ L ∞ (0 , T ; H ) ∩ L (0 , T ; H ), which implies via embedding that ∇ c ∈ L (0 , T ; L ∞ ). This completes the proof.18 Global weak solution in three dimensions
In this section we will show the global existence of the weak solutions for (1.1) in three dimen-sions. We start with notations. H ( R ) is used to indicate the closure of compactly supportedsmooth functions in H ( R ) and H − ( R ) means the dual space of H ( R ). We also introducethe function spaces V ( R ) , V σ ( R ) , H ( R ) defined as follows: V ( R ) = { u = ( u , u , u ) | u i ∈ H ( R ) } , V σ ( R ) = { u ∈ V ( R ) | div u = 0 } , H ( R ) = the closure of V σ ( R ) in ( L ( R )) . The dual space of V ( R ) is denoted by V ′ ( R ) = { u = ( u , u , u ) | u i ∈ H − ( R ) } . Theduality h w, v i for w ∈ V ′ ( R ) , v ∈ V ( R ) is, as usual, given as h w, v i = P i =1 h w i , v i i H − × H and we denote V ◦ σ ( R ) = { w ∈ V ′ ( R ) | h w, v i = 0 for all v ∈ V σ ( R ) } .Next, we define the notion of a weak solution for the system (1.1). Definition 5
Let < T ≤ ∞ . A triple ( n, c, u ) is called a weak solution to the Cauchyproblem (1.1) in R × [0 , T ) if the following conditions are satisfied:(a) The functions n and c are non-negative and ( n, c, u ) satisfy n (1 + | x | + | ln n | ) ∈ L ∞ (0 , T ; L ( R )) , ∇√ n ∈ L (0 , T ; L ( R )) ,c ∈ L ∞ (0 , T ; L ( R ) ∩ L ∞ ( R ) ∩ H ( R )) , ∇ c ∈ L (0 , T ; L ( R )) ,u ∈ L ∞ (0 , T ; L ( R )) , ∇ u ∈ L (0 , T ; L ( R )) . (b) The functions n, c , and u solve the chemotaxis-fluid equations (1.1) in the sense of dis-tributions, namely for any Ψ ∈ C ([0 , T ]; ( C ∞ c ( R )) ) with ∇ · Ψ = 0 Z R ( u · Ψ)( · , T ) + Z T Z R u · ( ∂ t Ψ + ∆Ψ) + Z T Z R u ⊗ u : ∇ Ψ − Z ∞ Z R n ∇ φ · Ψ + Z R u · Ψ(0 , x ) = 0 , where u ⊗ u : ∇ Ψ = P j,k =1 u j u k ∂ j Ψ k and Z ∞ Z R n ( ∂ t ϕ + ∆ ϕ ) + Z ∞ Z R nu · ∇ ϕ + Z ∞ Z R χ ( c ) n ∇ c · ∇ ϕ + Z R n ( x ) ϕ (0 , x ) = 0 , Z ∞ Z R c ( ∂ t ϕ + ∆ ϕ ) + Z ∞ Z R cu · ∇ ϕ − Z ∞ Z R k ( c ) nϕ + Z R c ( x ) ϕ (0 , x ) = 0 for any ϕ ∈ C ([0 , T ]; ( C ∞ c ( R ))) with ϕ ( · , T ) = 0 .(c) The functions n , c and u satisfy the following energy inequality: Z R ( | u | nφ + n | ln n | + |∇ c | h x i n ) dx + Z T k∇ u k L + k∇√ n k L + k ∆ c k L dt ≤ C, with C = C ( T, k χ ( c ) k L ∞ , kh x i n k L , k∇ c k L , k n | ln n |k L , k ∆ φ k L ∞ , k∇ φ k L ∞ , k φ k L ∞ ) . (AA) and (B) . We note first, by maximum principle, that n ( t, x ) ≥ , c ( t, x ) ≥ , k c ( t ) k L p ≤ k c k L p for t ≥ , ≤ p ≤ ∞ . It is straightforward that k n ( t ) k L = k n k L for t ≥ ddt (cid:18)Z R | u | dx + Z R nφdx (cid:19) + Z R |∇ u | dx = Z R n ∆ φdx + Z R χ ( c ) n ∇ c ∇ φdx, (4.1) ddt Z R n ln ndx + Z R |∇ n | n dx + Z R χ ′ ( c ) |∇ c | ndx = − Z R χ ( c )∆ cndx, (4.2) ddt Z R |∇ c | dx + Z R | ∆ c | dx = Z R k ( c )∆ cndx + X i,j =1 Z R c∂ i ∂ j c∂ i u j dx. (4.3)Multiplying µ to the last equation (4.3) and adding it to the second equation (4.2), we have ddt (cid:18)Z R n ln ndx + µ |∇ c | dx (cid:19) + Z R |∇√ n | dx + µ Z R | ∆ c | dx + Z χ ′ ( c ) |∇ c | ndx ≤ − Z R ( χ ( c ) − µk ( c )) | {z } =0 ∆ cndx + µ k c k L ∞ k∇ u k L k ∆ c k L ≤ C k∇ u k L + µ k ∆ c k L (4.4)for some C , which can be taken bigger than 1, i.e. C >
1. Also it holds that ddt Z R h x i ndx = Z R nu ∇h x i dx + Z R n ∆ h x i dx + Z R χ ( c ) n ∇ c ∇h x i dx. (4.5)Since the term R R nu ∇h x i dx is bounded as follows: k n k L k u k L ≤ C k n k L k∇√ n k L k∇ u k L ≤ k∇√ n k L + 12 k∇ u k L + C ( k n k L ) , we can have ddt Z R h x i ndx ≤ k∇√ n k L + 12 k∇ u k L + Z R χ ( c ) n ∇ c ∇h x i dx + C. (4.6)We estimate the term R R χ ( c ) n ∇ c ∇h x i dx similarly as above. Z R χ ( c ) n ∇ c ∇h x i dx ≤ C k√ n k L k∇ c k L ≤ C k√ n k L k∇√ n k L k∇ c k L k ∆ c k L ≤ C k∇ c k L k ∆ c k L + 14 k∇√ n k L ≤ C k∇ c k L + 14 k ∆ c k L + 14 k∇√ n k L . (4.7)Multiplying C to (4.1) and adding it together with (4.4) and (4.6), we have ddt (cid:18)Z R C ( | u | nφ ) + n ln n + |∇ c | h x i ndx (cid:19) + C − k∇ u k L + 14 k∇√ n k L + 14 k ∆ c k L ≤ C ( k∇ c k L + k u k L ) + C. (4.8)20hen, by Gronwall’s inequality, we have Z R ( | u | nφ + n ln n + |∇ c | h x i n ) dx + Z T k∇ u k L + k∇√ n k L + k ∆ c k L dt ≤ C, (4.9)where C ( T, k χ ( c ) k L ∞ , k n k L , kh x i n k L , k ∆ φ k L ∞ , k∇ φ k L ∞ ). By same reasoning for treating n (ln n ) − term in (3.14), it follows that Z R ( | u | + nφ + n | ln n | + |∇ c | + h x i n ) dx + Z T k∇ u k L + k∇√ n k L + k ∆ c k L dt ≤ C. (4.10)Streamline of constructing global weak solutions, as in usual steps for the Navier-Stokes equa-tions, is the following: · regularizing the system for which we prove the existence of smooth solutions · finding uniform estimates for the solutions of the regularized system · passing to the limit on the regularized parameters. In this subsection, we intend to construct approximate solutions of the system. For theincompressible Navier-Stokes equations defined on a general bounded domain, the global weaksolutions are constructed by using the spectral projections ( P k ) k ∈ Z , associated to the inhomo-geneous Stokes operator ([1, Chapter 2]). A number of useful properties of the family ( P k ) k ∈ Z are listed as follows: For any u ∈ H (Ω) ,P k P k ′ u = P min ( k,k ′ ) u, lim k →∞ k P k u − u k H (Ω) = 0 , (4.11) k∇ P k u k L (Ω) ≤ √ k k u k L (Ω) , k ∆ P k u k L (Ω) ≤ k k u k L (Ω) , (4.12) k (1 − P k ) u k L ≤ √ k k u k V σ . (4.13)In particular, (4.12) implies P k u ∈ L ∞ (Ω) for u ∈ L (Ω) in three dimensions. Definition 6
The bilinear map Q is defined by Q : V × V → V ′ , ( u, v )
7→ − div ( u ⊗ v ) . From now on we denote by H k ( R ) the space P k H ( R ). We regularize (1.1) by a frequencycut-off operator P k and a mollifier σ ǫ : ∂ t n k,ǫ ( t ) = − u k,ǫ · ∇ n k,ǫ + ∆ n k,ǫ − ∇ · ( n k,ǫ [( χ ( c k,ǫ ) ∇ c k,ǫ ) ∗ σ ǫ ]) ,∂ t c k,ǫ ( t ) = − u k,ǫ · ∇ c k,ǫ + ∆ c k,ǫ − k ( c k,ǫ )( n k,ǫ ∗ σ ǫ ) ,∂ t u k,ǫ ( t ) = − P k Q ( u k,ǫ , u k,ǫ ) + P k ∆ u k,ǫ − P k ( n k,ǫ ∇ φ ) , (4.14)with initial data ( n k,ǫ , c k,ǫ , u k,ǫ ) = ( n ∗ σ ǫ , c ∗ σ ǫ , P k u ∗ σ ǫ ) , n , c , u is the initial data of (1.1) satisfying the condition (1.8) in Theorem 4. Themollifier is defined as usual such that σ ǫ ( x ) = ǫ − σ ( ǫ − x ) for σ ∈ C ∞ ( R ). Apart from thefrequency cut-off the regularization is same one for a chemotaxis-fluid model studied in [13].Repeating similar arguments in Theorem 1, we obtain the local solution of (1.1) in the class n k,ǫ ∈ L ∞ (0 , T ; H m − ( R )) ∩ L (0 , T ; H m ( R )) c k,ǫ ∈ L ∞ (0 , T ; H m − ( R )) ∩ L (0 , T ; H m ( R )) u k,ǫ ∈ L ∞ (0 , T ; H m − ( R ) ∩ H k ( R )) ∩ L (0 , T ; H m ( R )) (4.15)for some time T and for all m >
3. It turns out that due to the regularization of nonlinearterms and smoothing properties of P k (see (4.12)), the local solution of (1.1) can be extendedup to infinite time. Proposition 2
The regularized system (4.14) has the unique global solution ( n k,ǫ , c k,ǫ , u k,ǫ ) ina class (4.15) for any time T < ∞ . Before presenting the proof we observe that the approximating solution ( n k,ǫ , c k,ǫ , u k,ǫ ) of (4.14)satisfies an energy inequality. Proposition 3
The solution ( n k,ǫ , c k,ǫ , u k,ǫ ) of (4.14) satisfies the following inequality. Z R ( | u k,ǫ | n k,ǫ φ ) + n k,ǫ | ln n k,ǫ | + |∇ c k,ǫ | h x i n k,ǫ dx + Z T k∇ u k,ǫ k L + k∇√ n k,ǫ k L + k ∆ c k,ǫ k L dt ≤ C, (4.16) where C = C ( T, k χ ( c ) k L ∞ , kh x i n k L , k∇ c k L , k n | ln n |k L , k ∆ φ k L ∞ , k∇ φ k L ∞ , k φ k L ∞ ) . Proof.
We note that the same cancellation as in (4.4) holds for the regularized system (4.14),hence ( n k,ǫ , c k,ǫ , u k,ǫ ) satisfying (4.15) satisfy the energy inequalities (4.1)-(4.5). Moreover thefollowing moment bound holds by similar estimates as (4.6), (4.7), ddt Z R h x i n k,ǫ dx = Z R n k,ǫ u k,ǫ ∇h x i dx + Z R n k,ǫ ∆ h x i dx + Z R n k,ǫ [( χ ( c k,ǫ ) ∇ c k,ǫ ) ∗ σ ǫ ] ∇h x i dx ≤ C k u k,ǫ k L k∇ u k,ǫ k L + C k∇ c k,ǫ k L k ∆ c k,ǫ k L + 12 k∇√ n k,ǫ k L + k n k L . Then we have (4.16) with T depending on k∇ c k,ǫ k L , kh x i n k,ǫ k L , k n k,ǫ | ln n k,ǫ |k L . It isimmediate to have k∇ c k,ǫ k L + kh x i n k,ǫ k L ≤ k∇ c k L + kh x i n k L . Note that x ln x is convex and dµ = σ ǫ ( y ) dy provide a probability measure. Then by Jensen’sinequality, we have n k,ǫ (ln n k,ǫ ) + ≤ ( n (ln n ) + ) ∗ σ ǫ . Integrating the above in x and observing that lim ǫ → k ( n | ln n | ) ∗ σ ǫ k L = k n | ln n |k L , wehave k n k,ǫ (ln n k,ǫ ) + k L ≤ k n | ln n |k L . (4.17)22or the k n k,ǫ (ln n k,ǫ ) − k L , proceeding similarly as (3.14), we have k n k,ǫ (ln n k,ǫ ) − k L ≤ C + Z R n k,ǫ h x i dx ≤ C (cid:18) Z R n h x i dx (cid:19) , from which we deduce the proposition.Now we give the proof of Proposition 2. Proof of Proposition 2
We first observe that the regularity criterion in Theorem 2 holdtrue for the system (4.14). Since its verification is tedious repetition of that of Theorem 2,we omit its details. If we consider the second equation of (4.14), then we have the followingenergy estimates.12 ddt k c k,ǫ k H + k∇ c k,ǫ k H ≤ C k∇ u k,ǫ k L k c k,ǫ k L + C k∇ c k,ǫ k L k n k,ǫ k L + 12 k∇ c k,ǫ k H ≤ Ck k u k,ǫ k L k c k,ǫ k H + C k n k,ǫ k L k c k,ǫ k H + 12 k∇ c k,ǫ k H . By using Gronwall’s inequality, we have (cid:13)(cid:13) ∇ c k,ǫ (cid:13)(cid:13) L ∞ x L t < ∞ . Since k u k,ǫ ( t ) k L is bounded and k∇ u k,ǫ k L ≤ C √ k k u k,ǫ k L , we can also demonstrate that the Serrin condition in Theorem 2 issatisfied for u k,ǫ . This completes the proof. In this subsection, we give the proof of Theorem 4.
Proof of Theorem 4.
We consider an approximating sequence ( n l,ǫ , c l,ǫ , u l,ǫ ) to ( n , c , u ).Note that Z R | n l,ǫ − n | + |∇ c l,ǫ − ∇ c | dx + Z R | u l,ǫ − u | dx → ., and Z R h x i n l,ǫ dx + Z R n l,ǫ | ln n l,ǫ | ≤ C Z R h x i n dx + Z R n | ln n | dx + C. We denote by ( n l,ǫ , c l,ǫ , u l,ǫ ) the approximating solution constructed in the previous section forthe system (4.14) with initial data ( n l (0 , · ) , c l (0 , · )) = ( n l (0 , · ) , c l (0 , · )) and u l (0 , · ) = P l u ( · ).Several uniform estimates hold for the approximating solutions: k c l,ǫ k L ∞ (0 ,T ; L p ( R )) ≤ C for 1 ≤ p ≤ ∞ , (4.18) k c l,ǫ k L ∞ (0 ,T ; H ( R )) + k ∆ c l,ǫ k L (0 ,T ; L ( R )) ≤ C, (4.19) k√ n l,ǫ k L ∞ (0 ,T ; L ( R )) + k∇√ n l,ǫ k L (0 ,T ; L ( R )) ≤ C, (4.20) k u l,ǫ k L ∞ (0 ,T ; L ( R )) + k∇ u l,ǫ k L (0 ,T ; L ( R )) ≤ C. (4.21)Then there exists subsequences n l,ǫ , c l,ǫ , u l,ǫ and some functions n, c, u such that √ n l,ǫ ⇀ √ n L ∞ (0 , T ; L ( R )) − weak ∗ ,c l,ǫ ⇀ c L ∞ (0 , T ; L p ( R )) ∩ L ∞ (0 , T ; H ( R )) − weak ∗ ,u l,ǫ ⇀ u L ∞ (0 , T ; L ( R )) − weak ∗ ∩ L (0 , T ; V σ ( R )) − weak23or 1 ≤ p ≤ ∞ . Let us show that n, c, u is a weak solution in the sense of Definition 5. ByGagliardo-Nirenberg inequality and (4.20), we have Z R | n l,ǫ | p dx ≤ C k n k − p L ( R ) k∇√ n l,ǫ k p − L ( R ) , and therefore, k n l,ǫ k L q (0 ,T ; L p ( R )) < C ( T ) , ≤ q ≤ p p −
1) (4.22)for 1 ≤ p ≤
3. Some strong convergences are necessary. We note that (4.22) implies the sourceterm of the Navier Stokes equation n l,ǫ ∇ φ is in L ([0 , T ]; V σ ′ ( R )) uniformly with respect to l ;for any w ∈ L ([0 , T ]; V σ ( R )), it holds that Z T Z R P l ( n l,ǫ ∇ φ ) wdxdt ≤ k∇ φ k L ∞ ( R ) k n l,ǫ k L (0 ,T ; L ( R )) k w k L (0 ,T ; L ( R )) . It proves that ∂ t u l,ǫ is uniformly bounded in L (0 , T ; V σ ′ ( R )). Note that u k is uniformlybounded in L ∞ (0 , T ; H ( R ))) ∩ L (0 , T ; V σ ( R )) due to (4.16). Combining these facts and(4.12), (4.13) we have compactness result for ( u l,ǫ ) (see [1, Proposition 2.7] for detailed proof):there exists u in L (0 , T ; V σ ( R )) such that up to subsequencelim l →∞ ,ǫ → Z T Z K | u l,ǫ ( t, x ) − u ( t, x ) | dxdt = 0 , (4.23)for any T > K of R . In addition, for Ψ ∈ L ([0 , T ]; V ( R )) andΦ ∈ L ([0 , T ] × R )lim l →∞ ,ǫ → Z T Z R ∇ u l,ǫ ( t, x ) ∇ Ψ( t, x ) dxdt = Z T Z R ∇ u ( t, x ) ∇ Ψ( t, x ) dxdt, lim l →∞ ,ǫ → Z T Z R u l,ǫ ( t, x )Φ( t, x ) dxdt = Z T Z R u ( t, x )Φ( t, x ) dxdt. (4.24)Furthermore, For any ψ ∈ C ( R + ; V σ ( R ))lim l →∞ ,ǫ → sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12)Z R ( u l,ǫ ( t, x ) − u ( t, x )) ψ ( t, x ) dx (cid:12)(cid:12)(cid:12)(cid:12) = 0 . (4.25)Applying a test function Ψ in C ([0 , T ]; V σ ( R )), we obtain ddt h u l,ǫ ( t ) , Ψ( t ) i = h ∆ u l,ǫ ( t ) , P l Ψ( t ) i + h Q ( u l,ǫ ( t ) , u l,ǫ ( t )) , P l Ψ( t ) i + h ( n l,ǫ ∇ φ ) , P l Ψ( t ) i + h u l,ǫ ( t ) , ddt Ψ( t ) i . (4.26)Following the arguments in [1], that is, using (4.23)-(4.25) and the factlim l →∞ sup t ∈ [0 ,T ] k P l Ψ( t ) − Ψ( t ) k V ( R ) = 0 , (4.27)24e can pass to the limit with respect to l so that Z R u · Ψ( T, x ) dx + Z T Z R ( ∇ u : ∇ Ψ − u ⊗ u : ∇ Ψ − u · ∂ t Ψ)( s, x ) dxds = Z R u ( x )Ψ(0 , x ) dx + lim l →∞ ,ǫ → Z T h n l,ǫ ∇ φ, Ψ i dt. For the strong convergence of ( n l,ǫ ) we have √ n l,ǫ → √ n strongly in L loc ( R ) for a.e. t ∈ [0 , T ]by Sobolev embedding. Since k√ n l,ǫ ( t ) k L ( R ) is continuous in time, we redefine n ( t ) such that k√ n l,ǫ − √ n k L ( R ) → t ∈ [0 , T ]. Then by (4.22) and Lebesgue Dominated convergencetheorem, it follows that k n l,ǫ − n k L q (0 ,T ; L ploc ( R )) → , ≤ q ≤ p p −
1) (4.28)for 1 ≤ p ≤
2. For the convergence of ( c l,ǫ ) we have c l,ǫ ( t ) → c ( t ) strongly in L loc ( R ) for all t ∈ [0 , T ] and therefore, k c l,ǫ − c k L ploc ((0 ,T ) × R ) → , ≤ p < ∞ (4.29)by the uniform boundedness (4.18). Moreover we have k∇ c l,ǫ − ∇ c k L (0 ,T ; L ploc ( R )) → , ≤ p < . (4.30)By (4.19), k∇ c l,ǫ k L (0 ,T ; H ( R )) is uniformly bounded. For any ∇ g ∈ L (0 , T ; L ( R )), we have Z T Z R ∂ t ∇ c l,ǫ gdxdt ≤ Z T Z R u l,ǫ ∇ c l,ǫ ∇ g + ∆ c l,ǫ ∇ g + k ( c l,ǫ )( n l,ǫ ∗ σ ǫ ) ∇ gdxdt. We estimate Z T Z R u l,ǫ ∇ c l,ǫ ∇ gdxdt ≤ C Z T k u l,ǫ k L ( R ) k∇ c l,ǫ k L ( R ) k ∆ c l,ǫ k L ( R ) k∇ g k L ( R ) dt ≤ C Z T k∇ u l,ǫ k L ( R ) k ∆ c l,ǫ k L ( R ) k∇ g k L ( R ) dt ≤ C k∇ u l,ǫ k L (0 ,T ; L ( R )) k ∆ c l,ǫ k L (0 ,T ; L ( R )) k∇ g k L (0 ,T ; L ( R )) , Z T Z R k ( c l,ǫ )( n l,ǫ ∗ σ ǫ ) ∇ gdxdt ≤ C k n l,ǫ k L (0 ,T ; L ( R )) k∇ g k L (0 ,T ; L ( R )) . Thus we have ∂ t c l,ǫ ∈ L (0 , T ; H − ( R )). The strong convergences (4.28)-(4.30) are enough topass to the limit for nonlinear terms in the chemotaxis part. For instance, testing a Ψ ∈ C ∞ c ( R )to the worst nonlinear term ∇ · ( n l,ǫ ( χ ( c l,ǫ ) ∇ c l,ǫ ) ∗ σ ǫ ), we have Z T Z R ∇ · ( n l,ǫ [( χ ( c l,ǫ ) ∇ c l,ǫ ) ∗ σ ǫ ])Ψ − ∇ · ( nχ ( c ) ∇ c )Ψ dxdt = Z T Z R ( n l,ǫ − n )[( χ ( c l,ǫ ) ∇ c l,ǫ ) ∗ σ ǫ ] ∇ Ψ dxdt + Z T Z R n [( χ ( c l,ǫ ) ∇ c l,ǫ ) ∗ σ ǫ − χ ( c ) ∇ c ] ∇ Ψ dxdt. Z T Z R [( n ∇ Ψ) ∗ σ ǫ − n Ψ] χ ( c l,ǫ ) ∇ c l,ǫ + n ∇ Ψ( χ ( c l,ǫ ) ∇ c l,ǫ − χ ( c ) ∇ c ) dxdt = Z T Z R [( n ∇ Ψ) ∗ σ ǫ − n Ψ] χ ( c l,ǫ ) ∇ c l,ǫ + n ∇ Ψ([( χ ( c l,ǫ ) − χ ( c )] ∇ c l,ǫ + χ ( c )( ∇ c l,ǫ − ∇ c )) dxdt. The integrals go to zero by the uniform estimates (4.18)-(4.21) and (4.28)-(4.30) with the Lip-schitz continuous assumption on χ ( · ).Lastly, we consider the approximated energy inequality (4.16) replacing n l,ǫ | ln n l,ǫ | with n l,ǫ ln n l,ǫ .Taking the limit and uing the convexity of x ln x we deduce Z R ( | u | nφ + n ln n + |∇ c | h x i n ) dx + Z T k∇ u k L + k∇√ n k L + k ∆ c k L dt ≤ C, with C = C ( T, k χ ( c ) k L ∞ , kh x i n k L , k∇ c k L , k n | ln n |k L , k ∆ φ k L ∞ , k∇ φ k L ∞ , k φ k L ∞ ). Bythe same reasoning for treating n (ln n ) − term in (3.14) we show the weak solutions ( n, c, u )satisfy the energy inequality in Definition 5 (c). This completes the proof of Theorem 4. Acknowledgments
M. Chae’s work was supported by the National Research Foundation of Korea(NRF No. 2009-0069501). K. Kang’s work was partially supported by KRF-2008-331-C00024 and NRF-2009-0088692. J. Lee’s work was partially supported by NRF-2009-0072320. We appreciate professorDmitry Vorotnikov for valuable comments.
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