ENERGY SOLUTIONS FOR THE FIFTH-ORDER MODIFIED KORTEWEG DE-VRIES EQUATIONS

We consider the Cauchy problem for the fifth-order modified Korteweg-de Vries equation (mKdV) on T. The fifth-order mKdV is an asymptotic model for shallow surface waves, as is the second equation in the mKdV hierarchy. In contrast with the non-periodic case, periodic solutions for dispersive equations do not have a (local) smoothing effect, which becomes a major obstacle to considering the Cauchy problem for dispersive equations on T. We establish global well-posedness of the fifth-order mKdV in the energy space (H²(T)), which is an improvement of the former result by the first author ([27]). The main idea to overcome the lack of the smoothing effect is introducing a suitable short-time space initially motivated by the work by Ionescu, Kenig, and Tataru ([14]). The new idea is to combine the (frequency) localized modified energy with additional weight in the spaces, which eventually handles the logarithmic divergence appearing in the energy estimates. The main contribution of the result is to lead to the possibility of studying the global dynamics of solutions in the energy spaces. Moreover, we show that the flow map of the fifth-order mKdV equation is not C³ at H^s for any s.