In this paper we consider a family of active scalars with a velocity field given by $u={\mathrm{\Lambda }}^{-1+\alpha }{\mathrm{\nabla }}^{\perp }\theta$, for $\alpha \in (0,1)$. This family of equations is a more singular version of the two-dimensional Surface Quasi-Geostrophic (SQG) equation, which would correspond to $\alpha =0$.

We consider the evolution of sharp fronts by studying families of almost-sharp fronts. These are smooth solutions with simple geometry in which a sharp transition in the solution occurs in a tubular neighbourhood (of size δ). We study their evolution and that of compatible curves, and introduce the notion of a spine for which we obtain improved evolution results, gaining a full power (of δ) compared to other compatible curves.

在本文中，我们考虑一类活动标量，其速度场为 $ü={\mathrm{\Lambda }}^{-\mathrm{1个}+\alpha }{\mathrm{\nabla }}^{\perp }\theta$，对于 $\alpha \in （0，\mathrm{1个}）$。该方程组是二维表面准地转（SQG）方程的更奇异版本，它对应于$\alpha =0$。

我们通过研究几乎锋利的锋线的家庭来考虑锋利锋线的演变。这些是具有简单几何形状的平滑解决方案，其中解决方案在管状邻域（大小为δ）中发生急剧过渡。我们研究了它们的演化和兼容曲线的演化，并引入了脊柱的概念，对于该脊柱，我们获得了改进的演化结果，与其他兼容曲线相比，获得了全功率（δ）。