一个超临界高耗散三维navier-stokes方程的整体解

Global solution for super critical hyper - dissipative Three-dimensional navier-stokesequation

摘要: 研究了全空间r3中一个超临界高耗散N a v i e r - s t o k e s方程模型的整体正则性,此模型修正了经典N a v i e r - s t o k e s方程中的耗散项,方程中耗散项 δu被- d 2 u替代,这里d是一个傅里叶乘子,当d的特征是M (Ξ) =|ξ|Α,Α ≥5 / 4时, n a v i e r - s t o k e s方程临界和超临界高耗散情形的整体正则性已经得到了证明.考虑当D的特征m (Ξ) =|ξ|Α/G ( |ξ| ) ,α ≥5 / 4时的整体正则性,其中 ξ 任意充分大, g : R+→r +为非减函数,满足∫∞1 dS / (S g(S )4 ) = +∞.通过经典的能量方法,在更弱的条件下证明了该模型的整体正则性.

Abstract: The global regularity of the supercritical hyper-dissipative three-dimensional Navier-Stokes equation in the whole space was studied. This model modified the dissipation term in the classical Navier-Stokes equation, and the dissipative term Δu here is replaced by －D^2u, where D is a Fourier multiplier. The global regularity of the critical and supercritical high dissipative case of Navier-Stokes equation was proved when the symbol is m(ξ)=|ξ|α,α≥5/4. This slightly by establishing global regularity under the condition that m(ξ)=|ξ|α/g(|ξ|) for all sufficiently large ξ and some nondecreasing function g:R+→R+such that∫??1ds/(sg(s)4)=+?? were improved. By the classical energy method, the global regularity of the model was proved in a slightly weaker condition.