Research works on Camassa-Holm equation and Novikov equation for shallow water wave

We consider two models from shallow water wave.
               m_t + u m_x + 2 u_x m = 0  (Camassa-Holm),      m_t + u^2 m_x + 3 u u_x m = 0  (Novikov)
with m = u - u_{xx}. The most distinguished feature of the solutions for these two systems is the formation of peakon solutions. The low regularity of the solutions (no Lipschitz continuity) causes the major difficulty in studying the global well posedness of solutions. The existence of global Holder continuous weak solutions with exponent 1/2 for Camassa-Holm equation was resolved by two different methods: vanishing viscosity method for dissipative solutions by Zhoupin Xin and Ping Zhang 2000; characteristic methods for conservative solutions by Adrian Constantin and Alberto Bressan 2007. The uniqueness of dissipative solution was also resolved by Zhoupin Xin and Ping Zhang in 2002. After a pretty long time of waiting, the uniqueness of energy conservative solution was finally resolved by our recent paper: It is worth mentioning that the Lipschitz continuous dependence on initial data for solutions under certain construction was proved in a series of seminal works by Bressan and Fonte 2005, H. Holden and X. Raynaud 2007, and K. Grunert, H. Holden, and X. Raynaud 2011, etcs. These works can derive the uniqueness of conservative solution under certain construction. But it still cannot rule out the possibility of existence of solutions not coming from this construction. Our uniqueness result finally shows that all solutions agree with the solution considered by them.

Very recently, we proved the existence and uniqueness of global energy conservative solutions for Novikov equation, which is another model for shallow water wave including cubic nonlinearity instead of quadric nonlinearity for Camassa-Holm equation. By exploring two energy laws, one in the second order and one in the fourth order, for the Cauchy problem, we prove the global existence and uniqueness of the conservative solution, which is Holder continuous with exponent 3/4 (higher than 1/2 for Camassa-Holm equation).