Global Well-posedness and Long Time Behaviors of Chemotaxis-Fluid System Modeling Coral Fertilization

We consider generalized models on coral broadcast spawning phenomena involving diffusion, advection, chemotaxis, and reactions when egg and sperm densities are different. We prove the global-in-time existence of the regular solutions of the models as well as their temporal decays in two and three dimensions. We also show that the total masses of egg and sperm density have positive lower bounds as time tends to infinity in three dimensions.


Introduction
In this paper, we study the interaction between reactions and chemotaxis in the mathematical model of the broadcast spawining phenomenon. Broadcast spawning is a fertilization strategy used by many sea animals, like sea urchins and corals(see [6,7,17]). In contrast with the numerical simulations based on the turbulent eddy diffusivity, the field measurements indicate that fertilization rates are often extremely as high as 90%(see [8,9] and references therein) and it seems plausible that the chemotaxis emitted by the egg gametes play an important role in these high fertilization rates.
The simplest and most classical models of chemotaxis equations describing the collective motion of cells or bacterias have been introduced by Patlak [18] and Keller-Segel [13,14]. The logistic source type of reaction term is also considered in many studies for the mathematical modeling of chemotaxis equations in a bounded domain with Neumann boundary conditions(see [19,20,21] and references therein).
In [15,16], Kiselev and Ryzhik initiated mathematical study on the phenomenon of broadcast spawning when males and females release sperm and egg gametes into the surrounding fluid. There is experimental evidence that eggs release a chemical that attracts sperm. The authors in [15] and [16] in particular have proposed the following chemotaxis model regarding the fertilization process (assuming that the densities of egg and sperm gametes are identical): ∂ t n + (u · ∇)n − ∆n = χ∇ · (n∇(∆) −1 n) − ǫn q , in (x, t) ∈ R d × (0, T ), (1.1) where n is the density of egg (sperm) gametes, u is the smooth divergence free sea fluid velocity, and χ denotes the positive chemotactic sensitivity constant. Also, −ǫn q denotes the reaction (fertilization) phenomenon. In [15], the global-in-time existence of the solution to (1.1) is presented under suitable conditions. Additionally, in R 2 , they showed that the total mass m 0 (t) = R 2 n(x, t)dx approaches a positive constant whose lower bound is C(χ, n 0 , u) as t → ∞ when q is an integer larger than 2. They also provided that C(χ, n 0 , u) → 0 as χ → ∞. This implies that if the chemotactic sensitivity increases, then more eggs can be fertilized. The critical case of d = q = 2 was studied in [16]; the total mass can go to zero with a reaction term only, but not faster than a logarithmic rate when the initial data is in the Schwartz class. If chemotaxis is present, the total mass is diminished in a power of 1/χ, which gives a faster decay rate than 1/ log t in a certain time scale. Recently, the existence and total mass behaviors have been studied in [1] when the chemical concentration is governed by the parabolic equation. Espejo and Suzuki [10] considered parabolic-parabolic Keller Segel equations with reaction term coupled with Stokes equations in R 2 . They obtained the globalin-time existence of solution. Kiselev and Ryzhik [15] also presented the following model of sperm and egg densities ∂ t s + (u · ∇)s = κ 1 ∆s − (se) q 2 , s(x, 0) = s 0 (x), ∂ t e + (u · ∇)e = κ 2 ∆e − (se) q 2 , e(x, 0) = e 0 (x). (1.2) Here, s and e denote the densities of sperm and egg gametes. From [15], it is obtained that if q > max{ d+2 d , 2}, then there exists an absolute positive constant µ 1 such that s(·, t) L 1 (R d ) + e(·, t) L 1 (R d ) ≥ µ 1 > 0 for all t. In this paper we consider more general mathematical models by allowing that egg density can differ from sperm density in R d (d = 2, 3) with q = 2 considering the chemotaxis effect in the s equation in (1.2). Our first model reads as follows : ∂ t e + (u · ∇)e − ∆e = −ǫ(se), where e ≥ 0 , s ≥ 0, and u denote the density of egg gametes, sperm gametes and the divergence free sea velocity of sea fluid, respectively. In the above, χ and ǫ are positive constants representing chemotactic sensitivity and fertilization rate, respectively. We also assume that u is in C ∞ (R d+1 ) and div u = 0. We will obtain the apriori estimates in section 2. Initial data are given by (e 0 (x), s 0 (x)) with e 0 (x), s 0 (x) ≥ 0.
From now on, we denote L q,p t,x = L q (0, T ; L p (R d )) and L p t,x = L p (0, T ; L p (R d )) with any given time T in the context. We mostly omit the spatial domain R d in L p (R d ) if there is no ambiguity. We also denote a norm and Banach space K m,n defined by the norm f Km,n = f Mn + f H m . We also denote a function space X T m,n ≡ C([0, T ]; K m,n ) and X ∞ m,n ≡ C([0, ∞); K m,n ) . Let m s (t) and m e (t) denote the total mass of sperm and egg gametes, respectively : m s (t) = R d s(x, t)dx and m e (t) = R d e(x, t)dx.
Our first main result is the global-in-time existence of smooth solutions to (1.3). We also obtain the positive lower bound of the total mass for 3-dimensional case and the decay estimates of e L p and s L p . Compared to the case of e, the temporal decay of s is a bit tricky, due to the presence of the chemotatic effect, i.e. χ∇ · (s∇∆ −1 e). It turns out that the reaction term −ǫ(se) in the egg equation, in particular in two dimenstions, plays a crucial role in controlling the chemotatic term. See the argument around (2.8).
(iii) When d = 2, 3, we have the following temporal decay estimates

4)
and Remark 1 In Theorem 1 (ii), the fact that lower bound C(χ, ǫ, s 0 , ∇e 0 ) → 0 as χ → ∞ implies that if the chemotactic sensitivity is dominant, then total mass of egg or sperm density may vanish, hence perfect fertilization may occur.
Next, we consider the following egg-sperm chemotaxis model coupled with the incompressible fluid equations(Navier-Stokes or Stokes equations): where e, s, c ≥ 0, and u denote the density of egg gametes, sperm gametes, chemicals and the divergence free sea velocity of sea fluid governed by the fluid equations, respectively. φ denotes potential function, which is given by gravitational force, centrifugal force, etc. We will set κ = 1(Navier-Stokes equations) when d = 2 and κ = 0 (Stokes system) when d = 3. Chemotaxis equation coupled with the fluid equations have been considered in many studies, especially for describing the dynamics of Bacillus Subtilis in the water droplet. For recent mathematical developments in the model, please refer to [2,4,5,10] and references therein.
For the system (1.6) our main aim is to establish global well-posedness of solutions. To be more precise, in two dimensions, we prove that unique regular solutions exist globally in time for large initial data, provided that the data are regular enough. On the other hand, for three dimensional case, global well-posedness can be obtained under smallness condition of L 1 -norm of intial data of s, i.e. s 0 L 1 (more specifically, it suffices to assume that χ 2 s 0 is small). It is worth mentioning that L 1 -norm of s 0 is a super-critical qunatity in 3D under the scaling invariance (3.1) (L 3/2 -norm of s 0 is indeed scaling invariant in 3D). In this sense, our result is beyond scaling invariance but we do not know if the smallness assumption can be removed or not. Now we are ready to state our second result, where temporal decays of solutions are also shown as well.
(iii) We have the following decay estimates Furthermore, when d = 2 and ω is the vorticity of u, if we assume that where 1 < p ≤ ∞ and 1 < γ < 2.
Remark 2 Formally integrating both sides of (1.3) (or (1.6)) over R d and subtracting the first equation from the second equation, we deduce that Hence the difference of the total mass of sperm and egg cells is conserved.
On the other hand, in the 2D case, Kiselev and Ryzhik [16,Theorem 1.1] showed that if ρ 0 ∈ S (Schwartz class) and ρ satisfy then, for any σ > 0 and t ≥ 1, there exists a constant C(σ, ρ 0 ) > 0 such that Note that (1.11) corresponds to (1.1) when the chemotaxis is absent and q = 2.
Taking into account (1.10), we infer that, in 2D, an egg cell can be perfectly fertilized if the initial sperm cell density is much larger than that of the egg cell.

Remark 3
After completing this work, we are informed that Espejo and Winkler [11] obtained classical solvability and stabilization in a chemotaxis-Navier-Stokes system modeling coral fertilization in a smooth bounded two-dimensional domain. Our result has an essential difference from their work in the asymptotic behaviour in the whole domain.
The rest of this paper is as follows : In Section 2, we provide the proofs for the global-in-time existence of the smooth solution to (1.3) and also provide the proofs of the positive lower bounds of the total mass and decay estimates. In Section 3, we consider the global wellposedness of the system (1.6) and provide the proof of Theorem 2 and especially consider the decay properties of the solutions to (1.6) with the small initial data.

Global Well-posedness and Asymptotic Behavior of Total Mass
In this section, we provide some apriori estimates of solutions to (1.3). Also we provide the proof of global well-posedness of (1.3) (Theorem 1 (i)) and lower bound of the total mass(Theorem 1 (ii)). Using the standard method(contraction mapping principle), the local-in-time existence of regular solution can be shown, which reads as follows: Proposition 1 Let d = 2, 3 and n be a positive integer and initial data (e 0 , s 0 ) as in Theorem is divergence free and any of its spatial derivatives is uniformly bounded for all (x, t) ∈ R d × (0, ∞). Then there exists a maximal time of existence T * , such that for t < T * , a pair of unique regular solution (e, s) of (1.3) exists and satisfies (e, s) ∈ X t m,n × X t m,n .
The proof of the proposition is quite standard, hence we omit it. It can be found in [15,Theorem 5.4].
In this section and throughout the paper we use the maximal L p − L q estimates or maximal regularity estimates for the heat equations: let 1 < p, q < ∞. If v is the solution of the heat equation there exist a constant C > 0 (see [12]) such that In what follows, we derive some a priori estimates of (e, s) to prove Theorem 1.
• (L 1 estimates) First, we have the following decreasing properties for the total mass Integrating with respect to time, we have • (L p -estimates) By multiplying e p−1 both sides of e equation, and integrating over R d , we obtain that For the sperm density, we have the following We note that if ǫ ≥ χ, then the righthand side can be absorbed to the left. Hence it is direct that s ∈ L ∞ (0, ∞; L p ) and ∇s p/2 ∈ L 2 (0, ∞; L 2 ) for p ∈ (1, ∞).
• (H 1 estimates) Next, we consider H 1 estimates of s : By use of the maximal regularity of heat equation, we easily deduce that Therefore, together with (L p -estimates) we obtain ∂ t e ∈ L 2 t,x , and e ∈ L 2 (0, T ; H 2 ).
Taking L 2 inner product of −∆s with s equation, we find that In the above, δ can be chosen as a sufficiently small positive constant which can be absorbed in the lefthand side. Using the Gronwall type inequality, we have for any T > 0.
• (H 2 estimates) For the higher norm estimates, we proceed as follows. We estimate similarly with the above For the estimates of solution s, we have Using Young's inequality, the righthand side high order term ∇∆s 2 L 2 can be absorbed in the lefthand side. By integrating with respect to time, we find • (H 3 estimate) Finally, we can obtain the following H 3 estimates for s. By the use of maximal regularity of the heat equation, we have Similarly to the previous H 2 estimates, we obtain We are ready to prove Theorem 1.
Proof of Theorem 1 (i) From the previous apriori estimates, only remaining estimates are about the estimates in M n . As in [15,Theorem 5.4.], the only nontrivial part is that the contraction constant depends on H m norm of (s 0 , e 0 ) and not on M n norm of (s 0 , e 0 ). In a different way, we provide the following direct estimates for any integer k ≥ 1 inductively : By using Young's inequality and Gronwall's inequality, we can have for any T > 0, Similarly, we can have |x| k ∇(s, e) L 2,∞ t,x (Q T ) < ∞. This together with the previous L 1 -estimates proves for any n > 0 and T > 0 (s, e) Km,n < ∞. This completes the proof of Theorem 1 (i).
Proof of Theorem 1 (ii) For this regular solution obtained in Theorem 1 (i), we can investigate the asymptotic behaviors of the total mass m s (t) and m e (t), especially in To be concrete, we will show that .
We have 1 p For convenience, we denote y p (t) := e(t) L p (R d ) . We show that for sufficiently large t > T and Indeed, for k = 1, we have d dt y 2 2 (t) + Cy Solving the above differential inequality, we have Suppose that the above is true up to k = m − 1 with m > 1. Then we obtain Solving the above inequality, we have (1.7). Then we have Letting p → ∞, we have .
• (Total mass behavior of m s (t)) It is ready to prove the lower bound of mass of the sperm cell density. Consider the case that d = 3. We have the differential inequality Then integrating with respect to time from t 0 until t and setting y = R 3 s(t)dx, we have dy y ≥ − Cdt t 3 2 , and thus, Since t ≥ t 0 , we have m s (t) ≥ m s (t 0 ).
• (L 2 decay estimate of s(t)) To prove the lower bound of the mass for the egg cell density, we should obtain L 2 decay estimates for the sperm cell density. Similarly, we obtain The right hand side of the above equality can be estimated by Hölder's and Sobolev's inequality as follows : .
Since e(t) , we choose t 0 so large that Cχ We infer that s(t) L 2 ≤ C t 3 4 .
• (Total mass behavior of m e (t)) Finally, we deduce that .
Similarly, we have m e (t) ≥ m e (t 0 ). In the above, C has the order 1 χ , it implies that lower bound approaches 0 as χ → ∞.
Proof of Theorem 1 (iii) We already obtained the temporal decay of e, that is, (1.4), hence we only consider the temporal decay of s.
• (2D case) We recall that the solution e to (1.3) 2 satisfies the equation Multiplying a large constant M on both sides of (2.5) (M will be specified later), we have Note first that the following interpolation inequality holds (see [15]) We compute The solution s(t) satisfies that Adding (2.6) and (2.7), we have This gives the decay estimate Due to (1.7), for any given p > 1 and sufficiently small δ > 0, there exists t 0 such that Hence we deduce that for t ≥ t 0 , Since δ is a sufficiently small positive constant, we immediately have This yields that This completes the proof of Theorem 1.

Remark 4
In two dimensions, we have e(t) L ∞ ≤ C t . Then via similar computations as above, we obtain Hence, in two dimensions, we can not obtain the positive lower bound of the total mass via same method in three dimensions and leave as an open problem.
3 Global well-posednes for the model (1.6) In this section, we prove the global well-posedness of solutions to the system (1.6).
Note that the solution (e, s, c, u, p) satisfies the scaling invariant property if φ has the following scaling property : φ(x, t) = φ λ (x, t) := φ(λx, λ 2 t). That is, (e λ (x, t), s λ (x, t), c λ (x, t), u λ (x, t), p λ (x, t)) = (λ 2 e(λx, λ 2 t), λ 2 s(λx, λ 2 t), c(λx, λ 2 t), λu(λx, λ 2 t), λ 2 p(λx, λ 2 t)) Hence it holds that To obtain other L p and higher norm estimates we first consider the estimates of u; Let us denote the Stokes operator by G t . Namely G t * u 0 is the solution of the free Stokes equations (f = 0) ∂ t u − ∆u + ∇p = f, ∇ · u = 0 with initial data u 0 . It is well known that G t satisfies that (see e.g. [12]) in two dimensions. For the inhomogeneous Stokes equations the following maximal regularity estimate is known [12]; Proof. We remind that total masses of s and e are preserved. Thus, s∇φ, e∇φ belong to We decompose the solution u to the equations (3.2) to v + w in Q, where v satisfies the Stokes system: and w satisfies a perturbed homogeneous Navier-Stokes equations with zero initial data: For convenience, we denote f := −(s + e)∇φ.
Therefore, using the convolution inequality, we have For I 3 , using w ∈ L 4 (Q T ), we observe that Using the convolution inequality again, we obtain Finally, we compute Similarly we get . Summing up estimates, we obtain that ∇w ∈ 1≤q<2 L 2 (0, T ; L q (R 2 )). This completes the proof.
We proceed other L p and higher order estimates to conclude the global well-posedness part of Theorem 2. We treat spatial two and three dimensional cases separately.
where the second inequality is due to L p esitmates of c, e and Lemma 3. Multiplying both sides of the equation of s by s p−1 and integrating over R 2 , we deduce that Therefore, s ∈ L ∞ (0, T ; L p ) and ∇s p 2 ∈ L 2 (0, T ; L 2 ) for all p ∈ [2, ∞). On the other hand, we have It gives us that u ∈ L ∞ (0, T ; L 2 ) and ∇u ∈ L 2 (0, T ; L 2 ).
Therefore, we also have ∇u ∈ L ∞ (0, T ; L 2 ) and ∆u ∈ L 2 (0, T ; L 2 ), that is In general the maximal regularity of the heat equation and the L p estimates of c, e yield that for all p ∈ [2, ∞) and q > p. We can replace c with e in the above. Applying the maximal regularity of the heat equation to s equation together with the previous estimates, we have (3.14) Then by the bootstraping argument, we complete the proof of the Case I. Indeed (3.12) and (3.14) yields L p estimate for ∇c, ∇e. Then L p estimate of ∇s follows from the boundedness of ∆c L p t,x in (3.13) as is obtained s L p in (3.11). Those L p estimates are used to yield ∇u ∈ L ∞ (0, T ; L 2 ), ∇ 2 u ∈ L ∞ (0, T ; L 2 ), ∇ 3 u ∈ L 2 (0, T ; L 2 ), which closes the H 1 estimate of e, c, s, u. Maximal regularity estimates for ∇c, ∇e, ∇s prove the boundedness of ∇c, ∇e, ∇s L p t W 2,p x for all p ∈ [2, ∞), which corresponds to one more derivative version of (3.13) and (3.14). The H 2 estimates can be similarly done.
In the three dimensional case the regularity of u obtained in Lemma 3 is not enough to prove (3.9) and (3.10) as is in two dimensions. We need to prove an entropy type inequality for s (3.19) for three dimensions. Taking log s as a test function for the equation (1.6) Also we note that − Hence we deduce that Integrating in time gives us that Considering the equation of c c t − ∆c = −∇ · (uc) + e, and by the fact that e ∈ L ∞ t,x , we have where the last inequality is from (3.4). Since we have s L 3 x , we deduce that Therefore, from the assumption that C * χ 2 ∇φ 2 Integration by parts gives us that x se.
, for sufficiently small δ > 0. Also from the equation ∂ t c − ∆c = −∇ · (uc) + e, we have Considering (3.16) and adding 2 R 3 s(log s) − on the both sides of (3.15), we obtain From the equation of u, we deduce that Multiplying 4C * * on the both sides of the above inequality and integrating with respect to time, we have If we add (3.17) and (3.18), then we have Hence we have ∇ √ s ∈ L 2 (0, t; L 2 (R 3 )) i.e., s ∈ L 1 (0, t; L 3 (R 3 )).
From the interpolation, it gives us that and hence u ∈ L 5 t,x by the Sobolev embedding. Also we have Hence we have ∇c ∈ L ∞,2 t,x and ∆c ∈ L 2 t,x .
Also from the equation ∂ t c − ∆c = −∇ · (uc) + e, we have Hence we have 1 2 By using Gronwall's inequality, we have s ∈ L ∞,2 t,x and ∇s ∈ L 2 t,x . The higher order estimates can be obtained in a similar fashion. A brief sketch of the proof is as follows : as in [3, Theorem 1], we can show that ∇u L 5 t,x ≤ C( s and x + 1). Also we can show that if T * is a finite maximal existence time, then But, by the previous estimates and the standard continuation argument, we can complete the proof of existence of solution in (ii). For the positive lower bound of the total mass, we can obtain the lower bounds in (ii). The proof of the last part in (ii) is parallel to the proof of Theorem 1 (ii) and we omit the details.

Decay estimates in Theorem 2
In this section we prove the part (iii) of Theorem 2. From the equation of e, we have Following the same proof for Theorem1 (ii) (see (2.4) below), we have (1.7) Next we will obtain the decay estimate of c L q when d = 3. Noting first that c(t) L 1 ≤ Ct for sufficiently large t, we have . Setting x(t) = c(t) L 3 , the above inequality can be rewritten as x ′ (t) ≤ t −1 (C 1 − C 2 x 2 (t)) for some constants C 1 and C 2 . Since it is a separable form of 1st order ordinary differential inequality, direct computations show that c(t) L 3 is uniformly bounded. Its verification is rather standard, and thus we skip its details.
Using the similar methods with above and the estimates in [2], we have c N∞ e 0 L 1 + ǫ 2 1 + e K∞ + ǫ 1 c N∞ .