The entropy weak solution to a generalized Fornberg–Whitham equation

We investigate a nonlinear generalized Fornberg–Whitham equation. The key element is that we derive an $L^{2}(\mathbb{R})$ conservation law for solutions of the equation. We establish several estimates by utilizing the $L^{2}(\mathbb{R})$ conservation law. These estimates lead to the proof of the existence and uniqueness of entropy weak solution of the equation in the space $L^{1}(\mathbb{R})\cap L^{\infty}(\mathbb{R})$.