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Existence and uniqueness of mild solutions to the chemotaxis-fluid system modeling coral fertilization
Boundary Value Problems volume 2020, Article number: 72 (2020)
Abstract
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 \(L^{p}\)-spaces, we finally get the existence of self-similar solutions when the initial data are small homogeneous functions.
1 Introduction and main results
In this paper, we consider the following chemotaxis-fluid system modeling coral fertilization in \(\mathbb{R}^{N}\) with \(N\geq 3\):
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 ϵ and χ are positive, and \(k\in \mathbb{R}\). To complete system (1.1), the initial data are given by
Here we see that system (1.1) is invariant by the transformation
up to a change of the pressure law \(\bar{p}=l^{2}p\) 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
Here, n represents the density of egg (sperm) gametes, u is the specified spiral current rate, and \(\chi >0\) denotes the chemotactic sensitivity constant, the term \(\varepsilon n^{q} (q\geq 2)\) 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:
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 \(n\nabla \phi \) 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 \(\mathbb{R}^{N}\). 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 [13–16] etc.
The aim of this paper is to prove the global existence of mild solutions to the chemotaxis-fluid system (1.1) in \(\mathbb{R}^{N}\ (N\geq 3)\) 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 \(BC_{w}([0,\infty );X)\) as the set of bounded weakly-star continuous functions on \([0,\infty )\) with values in the Banach space X and \(L_{w}^{p}(\mathbb{R}^{N})\) as the weak \(L^{p}\)-space.
Definition 1
Let \(N\geq 3\), and assume that the initial data \(\{e_{0}, s_{0},c_{0},{\mathbf{u}}_{0} \}\) satisfy \(e_{0}\in L_{w}^{\frac{N}{2}}(\mathbb{R}^{N})\), \(s_{0}\in L_{w}^{\frac{N}{2}}(\mathbb{R}^{N})\), \(\nabla c_{0}\in L_{w}^{N}(\mathbb{R}^{N})\), and \({\mathbf{u}}_{0}\in L_{w}^{N}(\mathbb{R}^{N})\). We call measurable function \(\{e,s,c,{\mathbf{u}}\}\) on \(\mathbb{R}^{N}\times (0,\infty )\) a mild solution of (1.1) on \((0,\infty )\) if the identities
hold for \(t\in (0,\infty )\), where \(\exp (t\Delta )\) denotes the heat semi-group defined by
with \(G(x, t)=\frac{1}{(4\pi t)^{\frac{N}{2}}}\exp (-\frac{|x|^{2}}{4t})\) and \(P=\{P_{jk}\}_{j,k=1,\ldots ,N}\) denotes the projection operator onto the solenoidal vector fields with the expression
Now our main results are as follows.
Theorem 1.1
(Existence)
Let\(N\geq 3\). Suppose that the indexes\(p, q\), andrsatisfy
and there exists a constant\(\delta =\delta\ (N, p, q,r)\)such that the initial data\(\{e_{0},s_{0},c_{0},{\mathbf{u}}_{0}\}\)satisfy the following condition:
Then there exists a mild solution\(\{e,s,c,{\mathbf{u}}\}\)of (1.1) which satisfies
If the norms corresponding to spaces (1.11)–(1.14) are small enough, then the mild solution\(\{e,s,c,{\mathbf{u}}\}\)to (1.1) is unique. Moreover, when\(t\rightarrow \infty \), the mild solution\(\{e,s,c,{\mathbf{u}}\}\)has the following asymptotic behavior:
Theorem 1.2
(Stability)
Let\(N\geq 3\), and let the exponents\(p, q\), andrbe the same as in Theorem 1.1, and\(\delta =\delta (N, p, q,r)\)is the same constant as in (1.10). Suppose that the two initial data\(\{e_{0},s_{0},c_{0},{\mathbf{u}}_{0}\}\)and\(\{\tilde{e}_{0},\tilde{s}_{0},\tilde{c}_{0},\tilde{\mathbf{u}}_{0}\}\)and the two external forcesϕandϕ̃satisfy
Suppose that\(\{e,s,c,{\mathbf{u}}\}\)and\(\{\tilde{e},\tilde{s},\tilde{c},\tilde{\mathbf{u}}\}\)are mild solutions of (1.1) on\([0,\infty )\)given by Theorem 1.1with the initial data\(\{e_{0},s_{0},c_{0},{\mathbf{u}}_{0}\}\)and\(\{\tilde{e}_{0},\tilde{s}_{0},\tilde{c}_{0},\tilde{\mathbf{u}}_{0}\}\)in the class (1.11)–(1.14), respectively. Then, for any\(\varepsilon >0\), there is a constant\(\eta =\eta (N, p, q, r, \varepsilon )>0\)such that if
then we have
Corollary 1.1
(Self-similarity)
Let\(N\geq 3\), and suppose that the initial data\(\{e_{0},s_{0},c_{0},{\mathbf{u}}_{0}\}\)satisfy\(e_{0}\in L_{w}^{\frac{N}{2}}\), \(s_{0}\in L_{w}^{\frac{N}{2}}\), \(\nabla c_{0}\in L_{w}^{N}\), and\({\mathbf{u}}_{0}\in L_{w}^{N}\). Assume that, for all\(x\in \mathbb{R}^{N}\)and all\(\lambda >0\), there are\(e_{0}(\lambda x)=\lambda ^{-2}e_{0}(x)\), \(s_{0}(\lambda x)=\lambda ^{-2}s_{0}(x)\), \(c_{0}(\lambda x)=c_{0}(x)\), \({\mathbf{u}}_{0}(\lambda x)=\lambda ^{-1}{\mathbf{u}}_{0}(x)\), and\(\phi (\lambda x)=\phi (x)\). If the initial data\(\{e_{0},s_{0},c_{0},{\mathbf{u}}_{0}\}\)and ∇ϕsatisfy condition (1.10), then the mild solution\(\{e,s,c,{\mathbf{u}}\}\)given by Theorem 1.1is a forward self-similar one, i.e., it holds that
for all\(x\in \mathbb{R}^{N}, t>0\), and all\(\lambda >0\).
2 Key proposition
Firstly, we introduce two function spaces X and Y defined as follows:
with the norm
and
with the norm
respectively.
Here, X and Y are Banach spaces, and they have the norms \(\|\cdot \|_{X}\) and \(\|\cdot \|_{Y}\), respectively. For \(\{e_{0},s_{0} c_{0},{\mathbf{u}}_{0},\phi \} \in X\) and \(\{e,s,c,{\mathbf{u}}\} \in Y\), we define the map
where
Then we have the following key proposition.
Proposition 2.1
For\(N\geq 3\), we assume that the exponents\(p,q\), andrsatisfy the condition
then we deduce that:
- (i)
The mapFdefined by (2.1) is a continuous map from\(X\times Y\)intoY.
- (ii)
For each data\(\{e_{0},s_{0},c_{0},{\mathbf{u}}_{0},\phi \} \in X\), the map\(F(e_{0},s_{0},c_{0},{\mathbf{u}}_{0},\phi ,\cdot ,\cdot ,\cdot ,\cdot )\)is of class\(C^{1}\)fromYinto itself.
Proof
(i) First, we will verify that \(t^{\frac{N}{2}(\frac{2}{N}-\frac{1}{q})}E(t)\in BC_{w}([0,\infty );L^{q})\). Owing to the \(L^{q}\)-\(L_{w}^{N}\) estimate of the heat semi-group, it holds that
where \(C=C(N, q)\), which means that \(t^{\frac{N}{2}(\frac{2}{N}-\frac{1}{q})}\exp (t\Delta )e_{0}\in BC_{w}([0, \infty );L^{q})\).
According to (2.3), one has that \(\frac{1}{2}-\frac{N}{2p}>0\), \(\frac{1}{N}<\frac{1}{p}+ \frac{1}{q}\leq 1\). Then, for all \(t>0\),
with \(C=C(N, p, q)\), and here \(B(s,t)\) denotes the beta function defined by
for positive constants s and t.
Similarly, in view of \({1-\frac{N}{2q}>0}\), \({-1+\frac{N}{q}>0}\), \(\frac{1}{q}+\frac{1}{q}\leq 1\), we have
for all \(t>0\) with \(C=C(N, p, q)\).
Combining (2.2)1 and (2.4)–(2.6), we get \(t^{\frac{N}{2}(\frac{2}{N}-\frac{1}{q})}E(t) \in BC_{w}([0,\infty );L^{q})\), and with the estimate
where \(C=C(N,p,q)\).
Next, we are going to prove that \(t^{\frac{N}{2}(\frac{2}{N}-\frac{1}{q})}S(t)\in BC_{w}([0,\infty );L^{q})\). In fact, it holds that
Since \(\frac{1}{2}-\frac{N}{2p}> 0\), \(\frac{N}{2}(\frac{1}{p}+\frac{1}{q})-\frac{1}{2}>0\), \(\frac{1}{p}+\frac{1}{q}\leq 1\) and \(1-\frac{N}{2q}> 0\), \(-1+\frac{N}{q}> 0\), \(\frac{1}{q}+\frac{1}{q}\leq 1\), we infer
and
for all \(t>0\) with \(C=C(N, p, q)\).
By a similar way, since \(\frac{1}{2}-\frac{N}{2r}> 0\), \(\frac{N}{2}(\frac{1}{r}+\frac{1}{q})-\frac{1}{2}>0\), \(\frac{1}{q}+\frac{1}{r}\leq 1\), one gets
for all \(t>0\) with \(C=C(N,q, r)\).
Putting (2.2)2 and (2.8)–(2.11) together, one has \(t^{\frac{N}{2}(\frac{2}{N}-\frac{1}{q})}S(t) \in BC_{w}([0,\infty );L^{q})\), and with the estimate
for all \(t>0\), where \(C=C(N,p,q,r)\).
In the following, we will demonstrate that \(t^{\frac{N}{2}(\frac{1}{N}-\frac{1}{r})}\nabla C(t)\in BC_{w}(0, \infty ; L^{r})\). Indeed, it holds that
for all \(t>0\). From (2.3), we have
for all \(t>0\) with \(C=C(N, p, q,r)\). Hence, it follows from (2.2)3 and (2.13)–(2.14) that
with the estimate
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 \(\frac{1}{2}-\frac{N}{2p}>0\), \(1-\frac{N}{2}(\frac{1}{N}+\frac{1}{q}-\frac{1}{p})>0\) and the boundedness of the projection operator P in \(L^{p} (1< p<\infty )\), we have
and
for all \(t>0\) with \(C=C(N, p, q)\).
Combining (2.2)4 with (2.16)–(2.18), we have
with the estimate
for all \(t>0\), where \(C=C(N,p,q)\).
Based on (2.7), (2.12), (2.15), and (2.19), we conclude that
with the estimate
where \(C=C(N,p,q, r)\). This means that F is a continuous map from \(X\times Y\) to Y.
(ii) We need to prove that F is \(C^{1}\). For each \(\{e,s,c,{\mathbf{u}}\}\in Y\), we define a linear map \(L_{\{e,s,c,{\mathbf{u}}\}}(\tilde{e},\tilde{s},\tilde{c},\tilde{\mathbf{u}})=\{ \tilde{E}, \tilde{S},\tilde{C},\tilde{U}\}\) on Y by
We shall show that, for each fixed \(\{e_{0},s_{0},c_{0},{\mathbf{u}}_{0},\phi \}\in X\), \(L_{\{e,s,c,{\mathbf{u}}\}}\) is the Fréchet derivative of \(F(e_{0}, s_{0}, c_{0},{\mathbf{u}}_{0},\phi ,e,s,c,{\mathbf{u}})\) at \(\{e,s,c,{\mathbf{u}}\}\in Y\). We define \(\{\mathfrak{E}, \mathfrak{S},\mathfrak{C}, \mathfrak{U}\}\) by
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 \(\mathfrak{C}(t)\), it follows that
Then one gets from (2.14) that
Similar to (2.21), it holds that
From (2.17), we get
for all \(t>0\). Thus, by (2.22)–(2.25), we have
for each \(\{e_{0},s_{0},c_{0},{\mathbf{u}}_{0},\phi \}\in X\) and each \(\{e,s,c,{\mathbf{u}}\}\in Y\). This implies that the Fréchet derivative of F at point \(\{e_{0},s_{0},c_{0},{\mathbf{u}}_{0},\phi ,e,s,c,{\mathbf{u}}\}\in X\times Y\) in the direction to \(\{e,s,c,{\mathbf{u}}\}\) is equal to \(L_{\{e,s,c,{\mathbf{u}}\}}({\tilde{e}},{\tilde{s}},{\tilde{c}},{ \tilde{\mathbf{u}}})\). □
3 Proof of the main results
This section is devoted to proving Theorems 1.1–1.2 and Corollary 1.1.
Proof of Theorem 1.1
Firstly, we shall show bijectivity of the Fréchet \(L_{\{e,s,c,{\mathbf{u}}\}}\) at \(\{e,s,c,{\mathbf{u}}\}=\{0,0,0,0\}\). From (2.20), we have that \(L_{\{0,0,0,0\}}\{{\tilde{e}}, {\tilde{s}},{\tilde{c}},{\tilde{\mathbf{u}}}\}= \{{\tilde{E}}_{0}, {\tilde{S}}_{0},{\tilde{C}}_{0}, {\tilde{\mathbf{U}}_{0}} \}\), where
for \(\{{\tilde{e}}, {\tilde{s}},{\tilde{c}},{\tilde{\mathbf{u}}}\}\in Y\). Therefore, \({\tilde{E}}_{0}(t)={\tilde{S}}_{0}(t)={\tilde{C}}_{0}(t)={ \tilde{\mathbf{U}}}_{0}(t)=0\) implies that \({\tilde{e}(t)}={\tilde{s}(t)}={\tilde{c}(t)}={\tilde{\mathbf{u}}(t)}=0\), which means that \(L_{\{0,0,0,0\}}\) is injective.
For each \(\{{\tilde{E}}_{0}, {\tilde{S}}_{0},{\tilde{C}}_{0}, {\tilde{\mathbf{U}}}_{0} \}\in Y\), we can take \(\{{\tilde{e}}, {\tilde{s}},{\tilde{c}}, {\tilde{\mathbf{u}}}\}\in Y\) as
then we obtain
This means that \(L_{\{0, 0,0,0\}}\) is surjective from Y to Y.
Now, using the Banach implicit function theorem, we could see that there exists a \(C^{1}\)-map \(h:X_{\delta }\rightarrow Y_{\delta }\),
for some \(\delta (N, p, q, r)>0\) such that
for all \(\{e_{0},s_{0},c_{0},{\mathbf{u}}_{0},\phi \}\in X_{\delta }\).
As a result, by condition (1.10), we may find that the function \(h(e_{0},s_{0},c_{0},{\mathbf{u}}_{0},\phi )\) gives the unique solution of (1.7) satisfying properties (1.11)–(1.14).
The uniqueness of solutions \(\{e,s,c,{\mathbf{u}}\}\) of (1.7) with the small norms corresponding to the class (1.11)–(1.14) is a consequence of the existence of the \(C^{1}\)-map h from \(X_{\delta }\) to \(Y_{\delta }\). This shows that the asymptotic behaviors (1.15)–(1.18) follow from estimates (2.5)–(2.6), (2.9)–(2.11), (2.14), and (2.17)–(2.18), respectively. □
Proof of Theorem 1.2
The estimate of stability (1.22) under condition (1.21) is a consequence of the continuity of the map \(h: X_{\delta }\rightarrow Y_{\delta }\). The proof of Theorem 1.2 is complete. □
Proof of Corollary 1.1
Assume that \(\{e,s,c,{\mathbf{u}}\}\) is the solution of (1.7) by Theorem 1.1. Then one has
where
and
for \(i=1, 2, \ldots , N\), with
By the assumption on homogeneity of \(e_{0}\), it holds that
where in the third equality, we have used the fact \(z\overset {\mathrm {def}}{=}\frac{y}{\lambda }\). Similarly, we have \(s_{1}(\lambda x, \lambda ^{2}t)=\lambda ^{-2}s_{1}(x,t),c_{1}( \lambda x, \lambda ^{2}t)=c_{1}(x,t),{\mathbf{u}}_{1}(\lambda x, \lambda ^{2}t)= \lambda ^{-1}{\mathbf{u}}_{1}(x,t)\) for all \(x\in \mathbb{R}^{N}, t>0\).
Since the solution \(\{e,s,c,{\mathbf{u}}\}\) of system (1.7) in Theorem 1.1 is given by the mapping \(h: X_{\delta }\rightarrow Y_{\delta }\), in order to demonstrate Corollary 1.1, we will use the following proposition, and as its proof is quite standard, here we omit it. □
Proposition 3.1
Let\(N\geq 3\)and\(\{e,s,c,{\mathbf{u}}\}\)satisfy (1.23). Then one has
for all\(x\in \mathbb{R}^{N}, t>0\)and all\(\lambda >0\).
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The authors would like to thank the anonymous referees for giving us helpful suggestions and comments which led to improvement of the presentation.
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Research supported by the NNSF of China (Nos. 11871305, 11701325).
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Hu, D., Chen, P. & Ma, D. Existence and uniqueness of mild solutions to the chemotaxis-fluid system modeling coral fertilization. Bound Value Probl 2020, 72 (2020). https://doi.org/10.1186/s13661-020-01368-7
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DOI: https://doi.org/10.1186/s13661-020-01368-7