Study On Periodic Solutions Of Two Kinds Of Partial Differential Equations

Partial differential equations refer to differential equations that contain the partial derivatives of multivariable functions.They are usually associated with natural sciences such as physics,biology,and chemistry,and have become an important branch of modern mathematical research.Many mathematicians use the theory and methods of partial differential equations to explain and illustrate complex natural phenomena,which has strong research value.This article mainly studies the properties of periodic solutions for two types of equation groups: the chemotaxis-shallow water system and the compressible magneto-micropolar fluids equation group.(1)The existence and uniqueness of periodic solutions within a bounded domain for the chemotaxis-shallow water system are studied.By introducing a compact continuous operator when the external force is small enough,periodic and symmetrical,the existence of periodic solutions within the periodic domain for the quasi-geostrophic shallow water wave equation is obtained using energy estimates and topology theory.Furthermore,using approximation methods,we obtained the uniqueness of periodic solutions within the periodic domain for this equation.(2)The existence of periodic solutions for compressible magneto-micropolar fluids system is studied.By introducing a compact continuous operator and using techniques such as partial integration and commonly used inequalities,some uniform estimates are obtained.By letting the bounded region sequence tend towards the original unbounded region using topology theory,the existence of strong small periodic solutions for the corresponding solution sequence within a bounded domain is obtained.As a result,the existence of periodic solutions for compressible magnetohydrodynamic equations in three-dimensional space-time is established.The results of this paper enriches the theory of chemotaxis-shallow water system and compressible magneto-micropolar fluids equation group.Meanwhile they are the foundation for the future study of periodic solutions of other equations.