Global well-posedness for nonlinear generalized Camassa-Holm equation

TL;DR

This paper proves local and global well-posedness for a generalized Camassa-Holm equation with high-order momentum and nonlinearity, using Kato's approach and commutator estimates in Sobolev spaces.

Contribution

It extends well-posedness results to equations with arbitrarily high orders of momentum and nonlinearity, broadening the understanding of such nonlinear PDEs.

Findings

Established local well-posedness in Sobolev spaces for high-order equations.

Proved global existence using conserved quantities.

Results are consistent with classical Camassa-Holm equation cases.

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.