Existence and uniqueness of mild solutions to the chemotaxis-fluid system modeling coral fertilization

In this paper, we consider the egg-sperm chemotaxis model of coral with the incompressible fluid equations in the whole space. The existence of global mild solutions in scaling invariant spaces is proved with sufficient small initial data. Here the main tool we use is the implicit function theorem. Furthermore, we obtain the asymptotic stability of solutions when the time goes to infinity. Since the initial data could be in the weak Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{p}$\end{document}-spaces, we finally get the existence of self-similar solutions when the initial data are small homogeneous functions.


Introduction and main results
In this paper, we consider the following chemotaxis-fluid system modeling coral fertilization in R N with N ≥ 3: ∂ t e + (u · ∇)ee = -(se), ∂ t s + (u · ∇)ss = -χ∇ · (s∇c) -(se), ∂ t c + (u · ∇)cc = e, ∂ t u + k(u · ∇)u -u + ∇p = -(s + e)∇φ, div u = 0, (1.1) where e is the density of egg gametes, s denotes the density of sperm gametes, c represents the chemicals, u is the divergence free sea velocity of sea fluid, φ denotes the potential function, which is given by gravitational force and centrifugal force, the constants and χ are positive, and k ∈ R. To complete system (1.1), the initial data are given by e| t=0 = e 0 (x), s| t=0 = s 0 (x), c| t=0 = c 0 (x), and u| t=0 = u 0 (x). (1.2) Here we see that system (1.1) is invariant by the transformation e(x, t) = l 2 e lx, l 2 t, ,s(x, t) = l 2 s lx, l 2 t, , c(x, t) = c lx, l 2 t ,ū(x, t) = lu lx, l 2 t , (1.3) up to a change of the pressure lawp = l 2 p for all l > 0. And a function space is called critical space if the norm is invariant under transformation (1.3).
In the following, we will introduce the related works of this model. The most classical chemotaxis model describing the collective motion of cells or bacteria was first proposed by Patlak [1] and Keller and Segel [2,3]. In their papers, the partial differential equation of the random walk problem with orientation persistence and external bias was derived. In [4,5], Kiselev and Ryzhik found that the fertilization rate could be close to completion as long as chemotaxis was strong enough. So they considered the case of the weakly coupled quadratic reaction term ∂ t n + (u · ∇)nn = χ∇ · n∇( ) -1 nεn q , in (x, t) ∈ R d × (0, T). (1.4) Here, n represents the density of egg (sperm) gametes, u is the specified spiral current rate, and χ > 0 denotes the chemotactic sensitivity constant, the term εn q (q ≥ 2) represents fertilization phenomenon. In [6,7], under Neumann boundary condition, the authors provided a simpler proof of the nontrivial bounded classical solution of the decay profile for the following system: ⎧ ⎨ ⎩ ∂ t n + (u · ∇)nn = -χ∇ · (n∇c)εn 2 , (1.5) In [8], Espejo and Suzuki utilized the incompressible Navier-Stokes equation to describe the chemical c and the velocity field u of fluid, which include the pressure p, and used n∇φ to simulate gravity. Those equations read as follows: Now, we consider a more general mathematical model (1.1) by making the egg density different from the sperm density in R N . For the chemotaxis-fluid system (1.1), many people have done a lot of research. For example, Chae, Kyungkeun, and Lee [9] studied the global well-posedness of coral fertilization models. Li, Pang, and Wang [10] explored the global boundedness and decay property. Zheng [11] showed the global weak solution of this system in a three-dimensional space. Htwe [12] proved the global classical-small data solutions. For more models based on the Navier-Stokes equations, one can refer to [13][14][15][16] etc.
The aim of this paper is to prove the global existence of mild solutions to the chemotaxisfluid system (1.1) in R N (N ≥ 3) when the initial data are small in the scaling invariant spaces. Furthermore, based on our results concerning the existence and uniqueness of mild solutions (see Theorem 1.1), the global stability of those solutions is obtained under small initial perturbation. The main way we use is the implicit function theorem, which is inspired by Kozono, Miura, Sugiyama [17]. Let us mention that Tan, Wu, and Zhou [18] and Zhang, Deng, and Bie [19] applied the implicit function theorem to prove the existence and uniqueness of mild solutions to the magneto-hydro-dynamic equations and the nematic fluid crystals, respectively.
Before giving our results, we first introduce some usually used symbols and definitions of mild solutions. We denote BC w ([0, ∞); X) as the set of bounded weakly-star continuous functions on [0, ∞) with values in the Banach space X and L p w (R N ) as the weak L p -space.
hold for t ∈ (0, ∞), where exp(t ) denotes the heat semi-group defined by exp(-|x| 2 4t ) and P = {P jk } j,k=1,...,N denotes the projection operator onto the solenoidal vector fields with the expression Now our main results are as follows.
Suppose that the indexes p, q, and r satisfy and there exists a constant δ = δ (N, p, q, r) such that the initial data {e 0 , s 0 , c 0 , u 0 } satisfy the following condition: Then there exists a mild solution {e, s, c, u} of (1.1) which satisfies , and let the exponents p, q, and r be the same as in Theorem 1.1, and δ = δ(N, p, q, r) is the same constant as in (1.10). Suppose that the two initial data {e 0 , s 0 , c 0 , u 0 } and {ẽ 0 ,s 0 ,c 0 ,ũ 0 } and the two external forces φ andφ satisfy  w , ∇c 0 ∈ L N w , and u 0 ∈ L N w . Assume that, for all x ∈ R N and all λ > 0, If the initial data {e 0 , s 0 , c 0 , u 0 } and ∇φ satisfy condition (1.10), then the mild solution {e, s, c, u} given by Theorem 1.1 is a forward self-similar one, i.e., it holds that for all x ∈ R N , t > 0, and all λ > 0.

Key proposition
Firstly, we introduce two function spaces X and Y defined as follows: respectively.
Here, X and Y are Banach spaces, and they have the norms · X and · Y , respectively.
For {e 0 , s 0 c 0 , u 0 , φ} ∈ X and {e, s, c, u} ∈ Y , we define the map Then we have the following key proposition. Proposition 2.1 For N ≥ 3, we assume that the exponents p, q, and r satisfy the condition then we deduce that: Owing to the L q -L N w estimate of the heat semi-group, it holds that with C = C (N, p, q), and here B(s, t) denotes the beta function defined by for positive constants s and t.
where C = C (N, p, q). Next, we are going to prove that t In fact, it holds that for all t > 0 with C = C(N, p, q).
In the following, we will demonstrate that t for all t > 0 with C = C (N, p, q, r). Hence, it follows from (2.2) 3 and (2.13)-(2.14) that for all t > 0, where C = C (N, p, q, r). Finally, we deal with U(t). Similar to (2.4), one has for all t > 0. By the fact that 1 2 -N 2p > 0, 1 -N 2 ( 1 N + 1 q -1 p ) > 0 and the boundedness of the projection operator P in L p ( for all t > 0 with C = C (N, p, q).
for all t > 0, where C = C(N, p, q). where C = C(N, p, q, r). This means that F is a continuous map from X × Y to Y .
(ii) We need to prove that F is C 1 . For each {e, s, c, u} ∈ Y , we define a linear map Then it holds that Hence, it follows from (2.5) and (2.6) that for all t > 0.
In the same manner, we infer that Putting (2.9), (2.10), and (2.11) together yields Similarly, for C(t), it follows that Then one gets from (2.14) that Similar to (2.21), it holds that From (2.17), we get

Proof of the main results
This section is devoted to proving Theorems 1.