Global well-posedness for nonlinear generalized Camassa-Holm equation

Abstract

We establish local and global well-posedness for the Cauchy problem of a generalized Camassa-Holm equation where orders of the momentum and the nonlinearity can be arbitrarily high. More precisely, we consider the equation \begin{equation*} m_t + m_x u^p + b m u^{p-1}u_x = -(g(u))_x + (b+1)u^p u_x, \quad m = (1-\partial_x^2)^k u, \end{equation*} where $p \geq 1$, $k \geq 1$ are arbitrary, $b$ is a real parameter, and $g(u)$ is a smooth function. %The standard Camassa-Holm equation corresponds to $k=1$, $p=1$, $b=2$, and $g(u)=0$. The local well-posedness is shown by using Kato's semigroup approach, where we treat the nonlinearity directly using commutator estimates and the fractional Leibniz rule without having to transform it in any specific differential form. This well-posedness is obtained in the phase space $H^s$ for $s > 2(k-1) + 3/2$, which is consistent with the results for the classical Camassa-Holm equation. We also prove the global existence of solutions by obtaining conserved quantity and applying the same idea from our local theory.