Well-posedness of the Fornberg–Whitham equation on the circle

In this paper, we show that the Fornberg–Whitham equation is Well-posed in Sobolev spaces HsHs, for s>3/2s>3/2, and in the periodic case. We then show that the Well-posedness is sharp in the sense that the continuity of the data-to-solution map is not better than continuous by using the method of approximate solutions. However, we also show that the solution map is Hölder continuous in a weaker topology. These results are based on the Well-posedness result, as well as the solution size and lifespan estimates.