论文标题
Navier-Stokes和液晶不平等的部分规律性，没有最大原则
Partial regularity for Navier-Stokes and liquid crystals inequalities without maximum principle
论文作者
论文摘要
1985年，V。Scheffer讨论了他所谓的“ Navier-Stokes不平等”解决方案的部分规律性结果。这些地图基本上满足了不可压缩的条件以及局部和全球能量不平等以及可以正式从Navier-Stokes方程系统中得出的压力方程，但是并不需要满足Navier-Stokes System本身。 我们将此概念扩展到了Fang-Hua Lin和Chun Liu在1990年代中期考虑的系统，该系统与列液晶流量的模型有关，当“导演场” $ d $被视为零时，其中包括Navier-Stokes系统。除了扩展的Navier-Stokes系统外，LIN-LIU模型还包括一个进一步的抛物线系统，该系统暗示了$ d $的先验最大原理，它们用于建立部分规律性（具体来说，$ \ Mathcal {p}^{1}} {1}（\ Mathcal {s}）= 0 $）。 对于类似的“不平等”，一个人都失去了最大原则，但是在这里我们仍然建立了某些部分规律性结果（即$ \ Mathcal {p}^{\ frac 92 +δ}（\ nathcal {s}）= 0 $，因此特别是假定的Singular Singular Set $ \ Mathcal量子{S} $ zere {sere）在$ d $的附加假设下，对于某个参数的任何固定值$σ\ in（5,6）$（$σ= 6 $ in Lin and Liu使用的$ d $的界限），我们获得了相同的部分规律性（$ \ \ \ \ \ \ \ \ \ \ {p}^{1}^{1}}（\ nlincal and dou and lincal and lincal and lincal and lin和lin）特别是，我们恢复了caffarelli-kohn-nirenberg（1982）的部分规则性结果（$ \ Mathcal {p}^{1}（\ Mathcal {s}）= 0 $）（1982），以实现Navier-Stokes系统的“合适的弱解决方案”，以及我们对Scheffer的Servention savers and and verne and verne and verne and nefutions and neffience nefuntion nefuntion nefuntion。
In 1985, V. Scheffer discussed partial regularity results for what he called solutions to the "Navier-Stokes inequality". These maps essentially satisfy the incompressibility condition as well as the local and global energy inequalities and the pressure equation which may be derived formally from the Navier-Stokes system of equations, but they are not required to satisfy the Navier-Stokes system itself. We extend this notion to a system considered by Fang-Hua Lin and Chun Liu in the mid 1990s related to models of the flow of nematic liquid crystals, which include the Navier-Stokes system when the "director field" $d$ is taken to be zero. In addition to an extended Navier-Stokes system, the Lin-Liu model includes a further parabolic system which implies an a priori maximum principle for $d$ which they use to establish partial regularity (specifically, $\mathcal{P}^{1}(\mathcal{S})=0$) of solutions. For the analogous "inequality" one loses this maximum principle, but here we nonetheless establish certain partial regularity results (namely $\mathcal{P}^{\frac 92 + δ}(\mathcal{S})=0$, so that in particular the putative singular set $\mathcal{S}$ has space-time Lebesgue measure zero). Under an additional assumption on $d$ for any fixed value of a certain parameter $σ\in (5,6)$ (which for $σ=6$ reduces precisely to the boundedness of $d$ used by Lin and Liu), we obtain the same partial regularity ($\mathcal{P}^{1}(\mathcal{S})=0$) as do Lin and Liu. In particular, we recover the partial regularity result ($\mathcal{P}^{1}(\mathcal{S})=0$) of Caffarelli-Kohn-Nirenberg (1982) for "suitable weak solutions" of the Navier-Stokes system, and we verify Scheffer's assertion that the same hold for solutions of the weaker "inequality" as well.
