Global Regularity Of Solutions Of Tropical Climate Model

In this paper7 we dedicate to studying the global regularity of solutions of d-D tropical climate model with different dissipation and damping.We first introduce d-D tropical climate model with fractional dissipation:where u and v denote the barotropic mode and the first baroclinic mode of velocity flied,respectively.θ,p stand for temperature and pressure.In addition,μ,ν,η,α,β,γ≥0.The fractional Laplacian operator Λ=(-Δ)1/2 is defined via the Fourier transform as followsΛu=|ξ|u.Tropical climate model studies the interaction between water vapor and dynamics in the tropics.This model retains both barotropic and baroclinic modes of the velocity,which play an important role for the study of the tropical-extratropical interactions.In chapter three,we consider system(0.0.1)with α=γ=0,β=1.By utilizing some functions whose Fourier transform lives in a compact set away from the origin,we construct a class of special large initial data of equations in the 2-D and 3-D cases,respectively,and establish a unique global solution for them.In chapter four,we focus on the following 3-D tropical climate model with damping term:where damping coefficient σ>0 and real parameter α’≥1.We examine the effect of damping term on the global regularity of solution.When α’≥4,we use damping term to close the L2m(m>3/2)norm estimates of v,θ,and establish the global regular solution of system(0.0.4)in Sobolev space H1(R3)by these estimates.In chapter five,we study the global well-posedness issue on d-D tropical climate model without thermal diffusion(η=0).Due to the lack of the thermal diffusion,we have to construct a suitable Lyapunov functional to overcome the difficulty.When 0≤α<1,1/2≤β≤1 and α+β≥1,we finally prove the global existence and uniqueness of the solutions under small initial data.In chapter six,we establish the global regularity of two-dimensional tropical climate model in Sobolev space H1(R2).The model considered here has fractional dissipationΛ2αu and damping term(e|v|2-1)v or |v|r-1v on the equations of the barotropic mode and the first baroclinic mode of velocity,respectively.Where the basic requirement of fractional dissipation is α>1,and the range of α in more detail depends on the selection of damping term.Meanwhile,as a by-product of the main results,we also obtain the unique global solution of two-dimensional MHD equations with similar dissipation and damping term on the equations of velocity field u and magnetic field b,respectively.