Global well-posedness for a fifth-order shallow water equation on the circle

The periodic initial value problem of a fifth-order shallow water equation ∂tu − ∂x2∂tu + ∂x3u − ∂x5u + 3u∂xu − 2∂xu∂x2u − u∂x2u = 0is shown to be globally well-posed in Sobolev spaces H˙s(T) for s > 2/3 by I-method. For this equation lacks scaling invariance, we first reconsider the local result and pay special attention to the relationship between the lifespan of the local solution and the initial data, and then prove the almost conservation law, and finally obtain the global well-posedness by an iteration process.