The global attractor for a weakly damped forced shallow water equation in Sobolev spaces of low regularity

We consider the global attractor for the weakly damped forced shallow water equation ∂tu+∂x3u+12∂xu2+(1−∂x2)−1∂x[u2+12(∂xu)2]+λu=f in Sobolev spaces Ḣs(T) for s<1s<1. Under the assumption that f∈Ḣ1(T), we prove the existence of the global attractor for the equation in Ḣs(T) for 2/3