Determining Modes For The Critical Modified SQG Equation

As we all know,determining modes can describe the asymptotic behavior of infinite dimensional dissipative dynamic system.Specifically,for the solution of dissipative partial differential equation,the behavior of the exact solution can be described by the behavior of the approximate solution constructed by determining modes,thus describing the longtime dynamic behavior of the solution.In this paper,we study the determining modes and the estimation of the uniform upper bound of the solution to the critical modified quasi-geostrophic equation(SQG eauation)Where T2 is a periodic domain in R2,??(1,2),?=(-?)1/2.In this paper,we study the H(?)lder continuity of the solution orbit of the initial value in H1(T2)space,so as to effectively deal with the non-linear turbulence terms.We proved that for the critical modified SQG equation,the two solution orbits ?1(x,t)and ?2(x,t)are not less than a certain constant Q,they can be gradually forced in a certain Besov space(as t??),then the whole solution orbit ?1(t)is also asymptotically approaching ?2(t)in the corresponding Besov space(depending on the correction range).That is,the critical modified SQG equation has determining modes and uniform the upper bound estimate is obtained.