We consider the capillary–gravity water wave equation in two dimensions. We assume that the fluid is inviscid, incompressible, irrotational and the air density is zero. We construct an energy functional and prove a local wellposedness result without assuming the Taylor sign condition. When the surface tension \(\sigma \) is zero, the energy reduces to a lower order version of the energy obtained by Kinsey and Wu (Camb J Math 6(2):93–181, 2018) and allows angled crest interfaces. For positive surface tension, the energy does not allow angled crest interfaces but admits initial data with large curvature of the order of \(\sigma ^{-\frac{1}{3}+ \epsilon } \) for any \(\epsilon >0\).

我们从二维角度考虑毛细管重力水波方程。我们假设流体是无粘性的，不可压缩的，无旋转的，并且空气密度为零。我们构造了一个能量函数，并在不假设泰勒符号条件的情况下证明了局部适定性结果。当表面张力\（\ sigma \）为零时，能量减少到Kinsey和Wu获得的能量的低阶版本（Camb J Math 6（2）：93-181，2018），并允许成角度的波峰界面。对于正表面张力，能量不允许倾斜波峰接口，但与的顺序的大曲率承认初始数据\（\西格玛^ { - \压裂{1} {3} + \小量} \）对于任何\（\ epsilon> 0 \）。