Existence And Uniqueness Of Mild Solutions To The Chemotaxis-fluid System Modeling Coral Fertilization

In this paper,we consider the egg-sperm chemotaxis system with the incompressible fluid equations in the global space,and the existence and uniqueness of the equation are proved by the implicit function theorem.The system is as follows:(?) The main contents include the following two aspects:the existence and uniqueness of the mild solution of the above system with s variables in the same Lq space;The existence and uniqueness of the mild solution of the above system in different Lq,Lm spaces for e,s variables.We consider the first part is that when the initial value condition is small enough,the existence of global mild solutions in scale invariant space is proved by implicit function theorem.Furthermore,the asymptotic stability of the solution can be obtained when the time variable tends to infinity.And because the initial value condition belongs to the weak space,the existence of the self-similar solution is obtained when the initial value is a small homogeneous function.We promote the conclusions of the first part in the second part.Considering that the e,s variables selected in the first part of the proof are in the same space,in order to further promote the theorem,the second part considers changing the selection of the e,s variable space index,and then proves the existence and uniqueness of the mild solution of the incompressible egg-sperm chemotaxis fluid system of coral fertilization under the invariant space.Then compare the theorem ranges in the two different spatial index situations,and discuss which index selection situation is more reasonable and has a wider application range.