Related papers: Zero-filter limit issue for the Camassa-Holm equat…
This paper investigates the zero-filter limit problem associated with the Camassa-Holm equation. In the work cited as \cite{C.L.L.W.L}, it was established that, under the hypothesis of initial data $u_0\in B^s_{2,r}(\R)$ with $s>\frac32$…
In the short note, we prove that given initial data $\mathcal{u}_0 \in \pmb{H}^s(\mathbb{R})$ with $s>\frac32$ and for some $T>0$, the solution of the Camassa-Holm equation does not converges uniformly with respect to the initial data in…
The aim of this paper is to answer the question left in \cite{GL} (Math. Z. (2015) 281). We prove that the zero-filter limit of the Camassa-Holm equation is the Burgers equation in the same topology of Sobolev spaces as the initial data.
In the paper, we revisit the uniform continuity properties of the data-to-solution map of the Camassa--Holm equation on the real-line case. We show that the data-to-solution map of the Camassa--Holm equation is not uniformly continuous on…
In the paper, we show that the continuity of the solution can not be improved to the H\"{o}lder continuity. Precisely speaking, the solution of the Camassa-Holm equation belongs to $\mathcal{C}([0,T];B^s_{p,r})$ but not to…
In the paper, we gave a strengthening of our previous work in [32] (J. Differ. Equ. 269 (2020)) and proved that the data-to-solution map for the Camassa-Holm equation is nowhere uniformly continuous in $B^s_{p,r}(\R)$ with…
Whether or not the data-to-solution map of the Cauchy problem for the Camassa-Holm equation and Novikov equation in the critical Besov space $B_{2,1}^{3/2}(\R)$ is not uniformly continuous remains open. In the paper, we aim at solving the…
In this paper, we investigate the dependence on initial data of solutions to higher dimensional Camassa-Holm equations. We show that the data-to-solution map is not uniformly continuous dependence in Besov spaces…
This document provides a proof that the solutions to the convectively filtered Burgers equation, will converge to the entropy solution of the inviscid Burgers equation when certain restrictions are put on the initial conditions. It does so…
Considered in this paper is the generalized Camassa-Holm-Novikov equation with high order nonlinearity, which unifies the Camassa-Holm and Novikov equations as special cases. We show that the solution map of generalized Camassa-Holm-Novikov…
In the paper, by constructing a initial data $u_{0}\in B^{\sigma}_{p,\infty}$ with $\sigma-2>\max\{1+\frac 1 p, \frac 3 2\}$, we prove that the corresponding solution to the higher dimensional Camassa-Holm equations starting from $u_{0}$ is…
In the paper, we consider the initial value problem to the Camassa-Holm equation in the real-line case. Based on the local well-posedness result and the lifespan, we proved that the data-to-solution map of this problem is not uniformly…
In this paper, we give a construction of $u_0\in B^\sigma_{p,\infty}$ such that corresponding solution to the Camassa-Holm equation starting from $u_0$ is discontinuous at $t = 0$ in the metric of $B^\sigma_{p,\infty}$, which implies the…
In this paper, we consider the Cauchy problem for the generalized Camassa-Holm equation proposed by Hakkaev and Kirchev (2005) \cite{Hakkaev 2005}. We prove that the solution map of the generalized Camassa-Holm equation is not uniformly…
In this paper, we consider the fractional Camassa-Holm equation modelling the propagation of small-but-finite amplitude long unidirectional waves in a nonlocally and nonlinearly elastic medium. First, we establish the local well-posedness…
In this paper, we consider the Cauchy problem for the generalized Camassa-Holm equation that includes the Camassa-Holm as well as the Novikov equation on the line. We present a new and unified method to prove the sharp ill-posedness for the…
The present paper is the continuation of work [14], devoted to the study of an inviscid zero-Mach number system in the framework of \emph{endpoint} Besov spaces of type $B^s_{\infty,r}(\mathbb{R}^d)$, $r\in [1,\infty]$, $d\geq 2$, which can…
In this paper, we consider the solution map of the initial value problem to the two-component Camassa-Holm equation on the line. We prove that the solution map of this problem is not uniformly continuous in Sobolev spaces $H^s(\R)\times…
In this paper, we prove that the Cauchy problem for a generalized Camassa-Holm equation with higher-order nonlinearity is ill-posed in the critical Besov space $B^1_{\infty,1}(\R)$. It is shown in (J. Differ. Equ., 327:127-144,2022) that…
In this paper, we first establish the local well-posedness and continuous dependence for the rotation Camassa-Holm equation modelling the equatorial water waves with the weak Coriolis effect in nonhomogeneous Besov spaces $B^s_{p,r}$ with…