L^1_tL^2_x强迫下纯旋度解的有界性丧失：精确混合范数范围
Pure-swirl loss of boundedness under $L^1_tL^2_x$ forcing: exact mixed-norm ranges
AI总结：
该研究构造了圆柱域受迫三维Navier--Stokes方程的显式纯旋度解，确定了力与速度的精确混合范数范围，其结果也适用于受迫Stokes系统。
AI中文摘要：
我们构造了圆柱域中受迫三维Navier--Stokes方程的显式纯旋度解。对任意T>0及满足1/p+1/q>1的1≤p,q<∞，该力属于L^q(0,T;L^p(D))且在t<T时光滑。此外，该构造总能给出附加的能量类性质f∈L^1(0,T;L^2(D))。该解在[0,T)上是经典解，在L^2(D)中强延拓为时刻T的唯一Leray--Hopf解，满足能量等式，而当t↑T时，||v(t)||_{L^∞(D)}→∞。我们确定了力和速度的精确混合范数范围，并得到力、速度上确界范数及涡量的双侧速率。该构造通过环形消去法改进了Zhang的剖面，保留了对称轴处的光滑性，是对我们先前加权构造k=1部分的直接、自包含的精细化，取代了其端点讨论。由于对流项被压力吸收，该例子也适用于受迫Stokes系统。
英文摘要：
We construct an explicit pure-swirl solution of the forced three-dimensional Navier--Stokes equations in a circular cylinder. For every $T>0$ and $1\le p,q<\infty$ with $1/p+1/q>1$, the force belongs to $L^q(0,T;L^p(D))$ and is smooth for $t<T$. Moreover, the same construction always yields the additional energy-class property $f\in L^1(0,T;L^2(D))$. The solution is classical on $[0,T)$, extends strongly in $L^2(D)$ to the unique Leray--Hopf solution at time $T$, and satisfies the energy equality, while $\|v(t)\|_{L^\infty(D)}\to\infty$ as $t\uparrow T$. We determine the exact mixed-norm ranges of both the force and the velocity and obtain two-sided rates for the force, the velocity supremum norm and the enstrophy. The construction refines Zhang's profile by an annular cancellation that preserves smoothness at the symmetry axis. It is a direct, self-contained sharpening of the $k=1$ part of our previous weighted construction and supersedes its endpoint discussion. Since the convection term is absorbed by the pressure, the same example applies to the forced Stokes system.
