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.
Highlights
Introduction and main resultsIn this paper, we consider the following chemotaxis-fluid system modeling coral fertilization in RN with N ≥ 3: ⎧ ⎪⎪⎪⎪⎪⎨∂∂tt e s + + (u (u · ·∇)e ∇)s e =, s = –χ∇ · (s∇c),⎪⎪⎪⎪⎪⎩∂∂tt c + (u · ∇)c – u + k(u · ∇)u c = e, u+ ∇
The aim of this paper is to prove the global existence of mild solutions to the chemotaxisfluid system (1.1) in RN (N ≥ 3) when the initial data are small in the scaling invariant spaces
Based on our results concerning the existence and uniqueness of mild solutions, the global stability of those solutions is obtained under small initial perturbation
Summary
The aim of this paper is to prove the global existence of mild solutions to the chemotaxisfluid system (1.1) in RN (N ≥ 3) when the initial data are small in the scaling invariant spaces. 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.
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