AI 中文总结
研究具有特定耗散项的三维次耗散纳维 - 斯托克斯方程，通过将各向异性涡环混合机制用于分数耗散，证明存在初始数据在特定空间中任意小，但对应局部光滑解在短时间内于同一空间中任意大的情况。
AI 中文摘要
我们证明了具有耗散项\((-\Delta)^\alpha\)（\(0<\alpha<1\)）的三维次耗散纳维 - 斯托克斯方程的同空间范数膨胀。设\(2\leq p<\infty\)且\(0<s<1 - 2\alpha+\frac{3}{p}\)，在贝索夫情形下，还设\(1\leq q\leq\infty\)。存在无散度的\(C_c^\infty(\mathbb{R}^3)\)初始数据，在\(W^{s,p}\)或\(B^s_{p,q}\)中任意小，但相应的唯一局部光滑解在任意短时间内会在同一空间中变得任意大。证明将各向异性涡环混合机制应用于分数耗散；严格的缩放超临界间隙使曲率误差和非局部耗散误差都具有微扰性。
英文摘要
We prove same-space norm inflation for the three-dimensional hypodissipative Navier--Stokes equations with dissipation $(-Δ)^α$, $0<α<1$. Let $2\le p<\infty$ and \[ 0<s<1-2α+\frac3p. \] In the Besov case, let also $1\le q\le\infty$. There exist divergence-free $C_c^\infty(\mathbb R^3)$ initial data that are arbitrarily small in $W^{s,p}$, respectively in $B^s_{p,q}$, while the corresponding unique local smooth solution becomes arbitrarily large in the same space in arbitrarily short time. The proof adapts the anisotropic vortex-ring mixing mechanism to fractional dissipation; the strict scaling-supercritical gap makes both the curvature error and the nonlocal dissipative error perturbative.
