论文标题
在$ s^1 $上的略带干扰的de Gregorio模型上
On the Slightly Perturbed De Gregorio Model on $S^1$
论文作者
论文摘要
猜想是，由于Okamoto，Sakajo和Wunsch，Constantin-Lax-Majda型号（GCLM）$ω_t + auΩ_x= u_x = u_xω$ $可以从$ a <1 $的$ a <1 $的平滑初始数据中开发出有限的时间奇异性。对于$ a $接近且不到$ 1 $的端点情况，我们从一类平滑的初始数据中证明了在圆圈上对GCLM的有限时间自相似的爆炸。对于具有相同初始数据的圆圈的GCLM，如果thoot $ a $的强度略大于$ 1 $，我们证明该解决方案以$ ||的形式在全球范围内存在。 ω（t）|| _ {h^1} $在很大的时间内以$ o（t^{ - 1}）$的速率衰减。两种不同行为之间的过渡阈值是$ a = 1 $，与de gregorio模型相对应。
It is conjectured that the generalization of the Constantin-Lax-Majda model (gCLM) $ω_t + a uω_x = u_x ω$ due to Okamoto, Sakajo and Wunsch can develop a finite time singularity from smooth initial data for $a < 1$. For the endpoint case where $a$ is close to and less than $1$, we prove finite time asymptotically self-similar blowup of gCLM on a circle from a class of smooth initial data. For the gCLM on a circle with the same initial data, if the strength of advection $a$ is slightly larger than $1$, we prove that the solution exists globally with $|| ω(t)||_{H^1}$ decaying in a rate of $O(t^{-1})$ for large time. The transition threshold between two different behaviors is $a=1$, which corresponds to the De Gregorio model.
