Research On The Regularity Of Solution To The Generalized Supercritical Quasi-geostrophic Equations

In the field of fluid equations,there are many challenging problems both in physics and mathematics.In this paper,we study commutator estimates and regu-larity of solution to the generalized supercritical quasi-geostrophic equations.This paper is divided into two parts.Firstly,we consider the new type of commutator estimates which only involve separately solution partial derivative components.Secondly,by using the new commutator estimate,we study the regularity criteria of solutions to the generalized supercritical quasi-geostrophic equations in Hilbert space for solving partial derivative components.The generalized quasi-geostrophic equations is as follows:In the first chapter,we introduce the research background,significance of this problem and the current research situation.We also present some conclusions and necessary preliminary knowledge.In the second chapter,with the use of the Fourier transform and Bony decom-position,we will obtain the new type of commutator estimates which only involve separately solution partial derivative components in Hilber space and Besov space.In the third chapter,by applying the commutator estimates,we consider the regularity criteria of solution of the generalized supercritical quasi-geostrophic equations.Finally,we conclude the problems to be solved in the future.