The generalized surface quasi-geostrophic (GSQG) equations are transport equations for an active scalar that depend on a parameter $0<\alpha \le 2$. Special cases are the two-dimensional incompressible Euler equations ($\alpha = 2$) and the surface quasi-geostrophic (SQG) equations ($\alpha = 1$). We derive contour-dynamics equations for a class of two-front solutions of the GSQG equations when the fronts are a graph. Scalar reductions of these equations include ones that describe a single front in the presence of a rigid, flat boundary. We use the contour dynamics equations to determine the linearized stability of the GSQG shear flows that correspond to two flat fronts. We also prove local-in-time existence and uniqueness for large, smooth solutions of the two-front equations in the parameter regime $1<\alpha\le 2$, and small, smooth solutions in the parameter regime $0<\alpha\le 1$.

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SQG方程的两前解及其推广

广义表面准地转 (GSQG) 方程是活动标量的输运方程，它取决于参数 $0<\alpha\le 2$。特殊情况是二维不可压缩欧拉方程 ($\alpha = 2$) 和表面准地转 (SQG) 方程 ($\alpha = 1$)。当前沿是图时，我们推导了一类 GSQG 方程的双前沿解的轮廓动力学方程。这些方程的标量约简包括在刚性、平坦边界存在的情况下描述单个前沿的方程。我们使用等高线动力学方程来确定对应于两个平坦前沿的 GSQG 剪切流的线性稳定性。我们还证明了参数范围 $1<\alpha\le 2$ 和小，