论文标题
Navier-Stokes和液晶不平等的抛物线分形维度
Parabolic fractal dimension of forward-singularities for Navier-Stokes and liquid crystals inequalities
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。一个人可以将此概念扩展到F.-H的系统。 1990年代中期，Lin和C. Liu与列液晶流量的模型有关，其中包括Navier-Stokes系统，当“导演” $ d $被认为为零时。除了扩展的Navier-Stokes系统外，LIN-LIU模型还包括一个进一步的抛物线系统，该系统暗示了$ d $的先验最大原理，当人们认为类似的“不平等”时就会丢失。 在2018年，刘Q.证明了LIN-LIU模型的某些解决方案的部分规律性结果，从抛物线分形尺寸$ \ textrm {dim} _ {\ textrm {pf}} $方面，依赖于$ d $的边界来自最大原则。具体而言，Q. liu证明了$ \ textrm {dim} _ {\ textrm {pf}}}}（σ_{ - { - } \ cap \ mathcal {k}）\ leq \ leq \ leq \ tfrac {95} {63} {63} {63} {63} $ for任何compact $ \ nation $ \ nery ins went “前线”时空点附近，解决方案会及时向前吹。对于相应的“不平等”的解决方案，我们在这里证明，对于缺乏最大原则的任何补偿，一个人的$ \ textrm {dim} _ {\ textrm {\ textrm {pf}}（σ__{ - { - } \ cap \ cap \ cap \ mathcal {k}）我们还提供了一系列标准，包括一个示例$ d $的界限，其中任何一个都将暗示$ \ textrm {dim} _ {\ textrm {\ textrm {pf}}}（σ_{ - { - } \ cap \ cap \ cap {k}）\ leq \ leq \ tfrac} $不平等，正如刘Q.证明了用于LIN-LIU系统的解决方案一样。
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. One may extend this notion to a system considered by F.-H. Lin and C. 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 is lost when one considers the analogous 'inequality'. In 2018, Q. Liu proved a partial regularity result for certain solutions to the Lin-Liu model in terms of the parabolic fractal dimension $\textrm{dim}_{\textrm{pf}}$, relying on the boundedness of $d$ coming from the maximum principle. Specifically, Q. Liu proves $\textrm{dim}_{\textrm{pf}}(Σ_{-} \cap \mathcal{K}) \leq \tfrac{95}{63}$ for any compact $\mathcal{K}$, where $Σ_{-}$ is the set of 'forward-singular' space-time points, near which the solution blows up forwards in time. For solutions to the corresponding 'inequality', we prove here that, without any compensation for the lack of maximum principle, one has $\textrm{dim}_{\textrm{pf}}(Σ_{-} \cap \mathcal{K}) \leq \tfrac {55}{13}$. We also provide a range of criteria, including as just one example the boundedness of $d$, any one of which would furthermore imply that $\textrm{dim}_{\textrm{pf}}(Σ_{-} \cap \mathcal{K}) \leq \tfrac{95}{63}$ for solutions to the inequality, just as Q. Liu proved for solutions to the Lin-Liu system.
