On the fifth order KdV equation: local well-posedness and lack of uniform continuity of the solution map

In this paper we prove that the fifth order equation arising from the KdV hierarchy $ \partial_tu + \partial_x^5u + c_1\partial_x u\partial_x^2u + c_2u\partial_x^3u = 0 $ is locally well-posed in $ H^s(\mathbb{R}) $ for $ s> 5/2. Also, we prove the solution map of the equation is not uniformly continuous for $s>0$.