Global regularity for a model of Navier-Stokes equations with logarithmic sub-dissipation
Shuguang Shao Shu Wang Wen-Qing Xu
Kinetic & Related Models 2018, 11(1): 179-190 doi: 10.3934/krm.2018009

In this paper, we study the global regularity to a three-dimensional logarithmic sub-dissipative Navier-Stokes model. This system takes the form of ${\partial _t}u +(\mathcal {D}^{-1/2}u)·\nabla u + \nabla p =-\mathcal {A}^2u$, where $\mathcal {D}$ and $\mathcal {A}$ are Fourier multipliers defined by $\mathcal {D}=|\nabla|$ and $\mathcal {A}= |\nabla|\ln^{-1/4}(e + λ \ln (e + |\nabla|)) $ with $λ≥q0$. The symbols of the $\mathcal {D}$ and $\mathcal {A}$ are $m(ξ) =\left| ξ \right|$ and $h(ξ) = \left| ξ \right| / g(ξ)$ respectively, where $g(ξ) = {\ln ^{{1 / 4}}}(e + λ \ln (e + |ξ|))$, $λ≥0$. It is clear that for the Navier-Stokes equations, global regularity is true under the assumption that $h(ξ) =|ξ|^α$ for $α≥q 5/4$. Here by changing the advection term we greatly weaken the dissipation to $ h(ξ)={{\left| ξ \right|} / g(ξ)}$. We prove the global well-posedness for any smooth initial data in $H^s(\mathbb{R}^3)$, $ s≥q3 $ by using the energy method.

keywords: Navier-Stokes equations global regularity sub-dissipation energy estimates

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