非均质三维Navier-Stokes方程模型的整体正则性

Global Regularity for a Model of Inhomogeneous Three-dimensional Navier-Stokes Equations

考虑一个非均质三维Navier-Stokes方程模型,借助能量方法、Littlewood-Paley仿积分解技巧和Sobolev嵌入定理研究解的整体正则性.用-D2u近似替代经典非均质Navier-Stokes方程中的耗散项Δu,得到一个新的NavierStokes方程模型,其中D是一个傅里叶乘子,其特征是m(ξ)=|ξ|5/4,对于任意小的正常数ε和δ,当初值(ρ0,u0)∈H3/2+ε×Hδ时,证明了该模型解的爆破准则和整体正则性.

A model of inhomogeneous three-dimensional Navier-Stokes equations was studied in this paper. By using the energy method,Littlewood-Paley paraproduct decomposition techniques and Sobolev embedding theorem study of the global regularity of solutions were adopted. The dissipative term Δu in the classical inhomogeneous Navier-Stokes equations is replaced by- D^2 u and a new Navier-Stokes equations model was obtained,where D was a Fourier multiplier whose symbol is m（ ξ） = ｜ ξ ｜^5 /4. Blowup criterion and global regularity of this model were proved for the initial data（ ρ0,u0） ∈H^3 /2 ＋ ε× H^δ,where ε and δ are arbitrary small positive constants.