Geometric Constrains And Non-blowup Of 2D Quasi-Geostrophic Equation

There are two parts mainly discussed in this paper. Firstly, Under the assumption that there is a closed level set of the active scalarθin the 2D Quasi-Geostrophic(QG) equation, we prove the non-blowup of the 2D QG equation in a fixed time interval. If the arc/curvature of the boundary of the region is bounded all the time, and the maximumvorticity along this closed vortex line is comparable with the global maximum vorticity. We find that, when m(t) is bound by O((ln‖ω(·,t)‖_∞)~α) and l(t) is bound by O((ln‖ω(·,t)‖_∞)~β), where l(t) is the perimeter of the region and m(t) is the maximum of |▽·▽⊥θ/|▽⊥θ|| along the boundary of this region, then‖ω(t)‖_∞is bounded up to time T, thus no finite time blowup can occur up to time T.Secondly, we mainly discuss the 2D Quasi-Geostrophic(QG) equation and the 3D incompressible Euler equation in fluid mechanics. We give a LAX pair representation of the 2D QG equation through the similarity between the 2D QG equation and 2D Euler equation, then we give a Darboux Transformations corresponding to the LAX pair. Similarly, we get a Darboux Transformations for the 3D Euler equation. From what have been discussed above, we can further get the similar structure of the two equations.