二维Qausi-Geostrophic方程的几何约束与非爆炸性

研究二维Qausi-Geostrophic(QG)方程水平集的动态演化过程.在"涡线"上"涡度"的最大值与全局"涡度"最大值可比的假设下,得到若"涡线"的长度L(t)被O(ln ln ||▽~⊥θ(·,t)||_∞)控制(被O(ln ln ||▽~⊥θ(·,t)||_∞)控制),曲率最大值K(t)与L(t)的乘积被O(ln ln ln ||▽~⊥θ(·,t)||_∞)控制(被O(ln ln ||▽~⊥θ(·,t)||_∞)控制),则||▽~⊥θ(·,t)||_∞)的增长阶数最多可达四重指数阶,这样二维QG方程在有限时间内无爆炸发生.

Geometric Constraints and Non-blowup of 2D Quasi-Geostrophic Equations

We study the level set 9 dynamics of the 2D Qausi-Geostrophic(QG) equationsin this article.If the maximum "vorticity" along this closed "vortex line" is comparablewith the global maximum "vorticity",we find that,when the length of "vortex line"L(t)is bounded by O(1/(ln ln ||▽~⊥θ(·,t)||_∞))(O(1/(ln ln ||▽~⊥θ(·,t)||_∞))) and product of K(t)and L(t) isbounded byO(ln ln ||▽~⊥θ(·,t)||_∞)(O(ln ln ||▽~⊥θ(·,t)||_∞)),where K(t) is the maximum ofcurvature,and the growth rate of ||▽~⊥θ(·,t)||_∞is bounded by fourfold exponential,then nofinite time blowup can occur.