不可压缩纳维-斯托克斯方程：在 $R^3$ 与 $t=0$ 处无解的例子
流体动力学
2016-08-18 v1
摘要
我们给出一个光滑、无散度 ∇ ⋅ u ⃗ ( x ⃗ ) = 0 \nabla \cdot \vec{u}(\vec{x})=0 ∇ ⋅ u ( x ) = 0 u ⃗ ( x ⃗ ) \vec{u}(\vec{x}) u ( x ) R 3 R^{3} R 3 f ⃗ ( x ⃗ , t ) \vec{f}(\vec{x}, t) f ( x , t ) x ⃗ ∈ R 3 \vec{x}\in R^{3} x ∈ R 3 t = 0 t=0 t = 0 u i ( x ⃗ ) = 2 x h ( i − 1 ) − x h ( i + 1 ) ( 1 + ∑ j = 1 3 x j 2 ) 2 u_{i} (\vec{x})=2\frac{x_{h(i-1)} -x_{h(i+1)} }{\left(1+\sum _{j=1}^{3}x_{j}^{2} \right)^{2} } u i ( x ) = 2 ( 1 + ∑ j = 1 3 x j 2 ) 2 x h ( i − 1 ) − x h ( i + 1 ) i = { 1 , 2 , 3 } i=\{ 1,2,3\} i = { 1 , 2 , 3 } h ( l ) = { l ; 1 ≤ l ≤ 3 1 ; l = 4 3 ; l = 0 h(l)=\left\{\begin{array}{ccc} {l} & {;1\le l\le 3} & {} \\ {1} & {;l=4} & {} \\ {3} & {;l=0} & {} \end{array}\right. h ( l ) = ⎩ ⎨ ⎧ l 1 3 ; 1 ≤ l ≤ 3 ; l = 4 ; l = 0 u ( x ⃗ ) ∈ C ∞ u(\vec{x})\in C^{\infty } u ( x ) ∈ C ∞ ∫ R 3 ∣ u ⃗ ∣ 2 d x = π 2 \int _{R^{3} }\left|\vec{u}\right|^{2} dx=\pi ^{2} ∫ R 3 ∣ u ∣ 2 d x = π 2 ∣ x ⃗ ∣ → ∞ \left|\vec{x}\right|\to \infty ∣ x ∣ → ∞ f ⃗ ( x ⃗ , t ) = ( 0 , 0 , 1 1 + t 2 ( ∑ j = 1 3 x j ) 2 ) \vec{f}(\vec{x},t)=(0,0,\frac{1}{1+t^{2} (\sum _{j=1}^{3}x_{j} )^{2} )} f ( x , t ) = ( 0 , 0 , 1 + t 2 ( ∑ j = 1 3 x j ) 2 ) 1 f ( x ⃗ , t ) ∈ C ∞ f(\vec{x},t)\in C^{\infty } f ( x , t ) ∈ C ∞ ∣ x ⃗ ∣ → ∞ \left|\vec{x}\right|\to \infty ∣ x ∣ → ∞ u ⃗ ( x ⃗ ) \vec{u}(\vec{x}) u ( x ) f ⃗ ( x ⃗ , t ) \vec{f}(\vec{x}, t) f ( x , t ) p ( x ⃗ , t ) p(\vec{x}, t) p ( x , t ) t = 0 t=0 t = 0 0 0 \frac{0}{0} 0 0 x ⃗ ∈ R 3 \vec{x} \in R^{3} x ∈ R 3
引用
@article{arxiv.1608.04975,
title = {Incompressible Navier-Stokes Equations: Example of no solution at $R^3$ and $t=0$},
author = {Dejan Kovacevic},
journal= {arXiv preprint arXiv:1608.04975},
year = {2016}
}