Long Time Existence Of Solutions To Initial Value Problems Of Fractional Fluid Equations

Hydrodynamica describes the dynamic behavior of fluid from the microscopic point of view,among which the Navier-Stokes equation is one of the hot research topics in fluid mechanics.Using Littlewood-Paley decomposition,high and low frequency decomposition method,fourier analysis and energy estimation,strichartz estimation,basing on the estimation of Lp-Lq of fractional thermal semigroups in Besov space,We study the adaptability and blasting criterion of the initial value problems of fractional fluid equations.Firstly,we consider the initial value problem of fractional incompressible NavierStokes equations in homogeneous Sobolev-Gevrey Spaces Ha,σs(R3)is studied.It is proved that there is a unique solution u ∈ C([0,T];Ha,σs(R3))for the equation with the initial value Ha,σs(R3).Then,it is proved that the exponential blasting criterion for the solution T*<∞ at that time.Secondly,We prove the initial value problem of fractional incompressible Navier-Stokes-Coriolis equations in Sobolev space HS(s>max(3+4α/2,4),α>0).The local existence of the solution is proved by energy estimation.Then,the existence of long time solutions of fractional incompressible Navier-Stokes-Coriolis equations is proved by high and low frequency decomposition techniques and Littlewood-Paley decomposition.Finally,We study the initial value problems of fractional incompressible Navier-Stokes-Coriolis equations with external forces f.The linear and nonlinear of fractional Navier-Stokes-Coriolis equations are estimated by Lp-Lq estimation of fractious thermal semigroups.Using the iterative approximation method and induction theory,it is proved that the periodic mild solution is the only one,and the period of the external force f is the same.