Speaker
Description
The classical Camassa-Holm (CH) equation is used to describe dynamics of shallow water waves, and features interesting behavior such as solitons or wave breaking. The study of CH has been extensively investigated in the literature. In this talk, we consider a generalized version of CH where the momentum can be of arbitrarily high order and the nonlinearity can be of any polynomial order. More precisely, the equation reads as
\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 \text{where } m = (1-\partial_x^2)^k u,
\end{equation}
where $p \geq 1$, $k \geq 1$, $b$ is a real parameter, and $g(u)$ is a smooth function. We prove local well-posedness using Kato's semigroup, where nonlinearity is treated directly using commutator estimates and the fractional Leibniz rule without having to use tricky manipulation. Furthermore, in the case where the momentum is conserved, we show that the solution is in fact global.
Affiliation
University of Graz