论文标题
有限速度轴向对称的Navier-Stokes流通过锥体
Finite speed axially symmetric Navier-Stokes flows passing a cone
论文作者
论文摘要
令$ d $为球内部的圆锥体的外部，其高度角度最多为$π/6 $ in $ \ mathbb {r}^3 $，它触及了$ x_3 $ axis。对于任何初始值$ v_0 = v_0 = v_ {0，r} e_ {r} + v_ {0，θ}e_θ + v_ {0,3} e__ {3} $ in a $ c^2（\ overline {d} $ class）in $ c^2（\ overline {d}）$ class，它具有通常在$ x的$ x_3 $中的$ x $ symenty $ x_3 $ | r v_ {0，θ} | \ leq \ frac {1} {100} $，带有Navier-Hodge-Lions滑动边界条件的轴向对称的Navier-Stokes方程（ASNS）具有有限的能源解决方案，该解决方案始终保持限制。特别是，没有发生有限的流体速度爆炸。与初始速度上的标准小假设相比，在组件$ v_ {0，r} $和$ v_ {0,3} $上没有大小限制。从广义上讲，此结果似乎可以解决与上述对称性的解决方案类别中ASN的规律性问题的$ 2/3 $。等效地，该结果连接到一个普遍的开放问题，该问题要求初始速度的一个组成部分的绝对小度意味着全球平滑度，例如，请参见例如第873页in \ cite {czz17}。我们的结果似乎在特殊环境中给出了积极的答案。 作为副产品，我们还在具有有限能量的特殊尖式域中构建了强制Navier Stokes方程的无界解。缩放系数为$ -1 $的强迫术语在标准规律性类中。该结果证实了这样的直觉，如果流体的通道非常薄，可以在经典意义上任意高速，可以在轻度奇异的力下获得，这在物理上是合理的，鉴于牛顿重力和库仑力具有缩放因子$ -2 $。
Let $D$ be the exterior of a cone inside a ball, with its altitude angle at most $π/6$ in $\mathbb{R}^3$, which touches the $x_3$ axis at the origin. For any initial value $v_0 = v_{0,r}e_{r} + v_{0,θ} e_θ + v_{0,3} e_{3}$ in a $C^2(\overline{D})$ class, which has the usual even-odd-odd symmetry in the $x_3$ variable and has the partial smallness only in the swirl direction: $ | r v_{0, θ} | \leq \frac{1}{100}$, the axially symmetric Navier-Stokes equations (ASNS) with Navier-Hodge-Lions slip boundary condition has a finite-energy solution that stays bounded for all time. In particular, no finite-time blowup of the fluid velocity occurs. Compared with standard smallness assumptions on the initial velocity, no size restriction is made on the components $v_{0,r}$ and $v_{0,3}$. In a broad sense, this result appears to solve $2/3$ of the regularity problem of ASNS in such domains in the class of solutions with the above symmetry. Equivalently, this result is connected to the general open question which asks that if an absolute smallness of one component of the initial velocity implies the global smoothness, see e.g. page 873 in \cite{CZZ17}. Our result seems to give a positive answer in a special setting. As a byproduct, we also construct an unbounded solution of the forced Navier Stokes equation in a special cusp domain that has finite energy. The forcing term, with the scaling factor of $-1$, is in the standard regularity class. This result confirms the intuition that if the channel of a fluid is very thin, arbitrarily high speed in the classical sense can be attained under a mildly singular force which is physically reasonable in view that Newtonian gravity and Coulomb force have scaling factor $-2$.
