Global well-posedness for a fifth-order shallow water equation in Sobolev spaces

AbstractThe Cauchy problem of a fifth-order shallow water equation∂tu−∂x2∂tu+∂x3u+3u∂xu−2∂xu∂x2u−u∂x3u−∂x5u=0 is shown to be globally well-posed in Sobolev spaces Hs(R) for s>(610−17)/4. The proof relies on the I-method developed by Colliander, Keel, Staffilani, Takaoka and Tao. For this equation lacks scaling invariance, we reconsider the local result and pay special attention to the relationship between the lifespan of the local solution and the initial data. We prove the almost conservation law, and combine it with the local result to obtain the global well-posedness