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

摘要: 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- D ² u近似替代经典非均质Navier-Stokes方程中的耗散项Δ u,得到一个新的Navier-Stokes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( ρ ₀, u ₀)∈H ³ ^/ ² ^+ε ×H ^δ 时,证明了该模型解的爆破准则和整体正则性.

Abstract: 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 Δ u 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. Blow-up criterion and global regularity of this model were proved for the initial data ( ρ ₀, u ₀)∈ H ³ ^/ ² ^+ε×H ^δ , where ε and δ are arbitrary small positive constants.