Some Conditions of Non-Blow-Up of Generalized Inviscid Surface Quasigeostrophic Equation

Abstract

In this paper, we survey some non-blow-up results for the following generalized modified inviscid surface quasigeostrophic equation (GSQG) $\left\{\begin{array}{c}{\theta }_t+u·\nabla \theta =0,\hfill \\ u={\nabla }^{\perp }\psi ,\hfill \\ -{\Lambda }^{\beta }\psi =\theta ,\hfill \\ \theta \left(x,0\right)={\theta }_0\left(x\right).\hfill \end{array}\right.$ . This is a generalized surface quasigeostrophic equation (GSQG) with the velocity field $u$ related to the scalar $\theta$ by $u=-{\nabla }^{\perp }{\Lambda }^{-\beta }\theta$ , where $1\le \beta \le 2$ . We prove that there is no finite-time singularity if the level set of generalized quasigeostrophic equation does not have a hyperbolic saddle, and the angle of opening of the saddle can go to zero at most as an exponential decay. Moreover, we give some conditions that rule out the formation of sharp fronts for generalized inviscid surface quasigeostrophic equation, and we obtain some estimates on the formation of semiuniform fronts. These results greatly weaken the geometrical assumptions which rule out the collapse of a simple hyperbolic saddle in finite time.