The Green's function approach of Giles and Pierce is used to build the lift and drag based analytic adjoint solutions for the two-dimensional incompressible Euler equations around irrotational base flows. The drag-based adjoint solution turns out to have a very simple closed form in terms of the flow variables and is smooth throughout the flow domain, while the lift-based solution is singular at rear stagnation points and sharp trailing edges owing to the Kutta condition. This singularity is propagated to the whole dividing streamline (which includes the incoming stagnation streamline and the wall) upstream of the rear singularity (trailing edge or rear stagnation point) by the sensitivity of the Kutta condition to changes in the stagnation pressure.

翻译：采用Giles和Pierce的格林函数法，构建了基于升力和阻力的二维不可压缩欧拉方程绕无旋基流的解析伴随解。基于阻力的伴随解在流变量表达式中具有非常简单的闭合形式，且在整个流场区域内光滑分布；而基于升力的伴随解由于库塔条件，在后驻点和尖锐后缘处存在奇异性。该奇异性通过库塔条件对滞止压力变化的敏感性，向后奇点（后缘或后驻点）上游的整个分流流线（包括来流滞止流线和壁面）传播。