Rough solutions of the fifth-order KdV equations

We consider the Cauchy problem of the fifth-order equation arising from the Korteweg–de Vries (KdV) hierarchy{∂tu+∂x5u+c1∂xu∂x2u+c2u∂x3u=0,x,t∈R,u(0,x)=u0(x),u0∈Hs(R). We prove a priori bound of solutions for Hs(R)Hs(R) with s⩾54 and the local well-posedness for s⩾2s⩾2. The method is a short time Xs,bXs,b space, which was first developed by Ionescu, Kenig and Tataru [13] in the context of the KP-I equation. In addition, we use a weight on localized Xs,bXs,b structures to reduce the contribution of high–low frequency interaction where the low frequency has large modulation. As an immediate result from a conservation law, we obtain that the fifth-order equation in the KdV hierarchy,∂tu−∂x5u−30u2∂xu+20∂xu∂x2u+10u∂x3u=0 is globally well-posed in the energy space H2H2.