相关论文: Non-uniform dependence on initial data for the gen…
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…
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 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 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…
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 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 the paper, we consider the Cauchy problem for a generalized Degasperis-Procesi equation. We prove that the data-to-solution map is not uniformly continuous.
We show that the Cauchy problem for the Camassa-Holm equation has a unique, global, weak, and dissipative solution for any initial data $u_0\in H^1(\mathbb{R})$, such that $u_{0,x}$ is bounded from above almost everywhere. In particular, we…
This paper is mainly concerned with the Cauchy problem for a generalized Camassa-Holm equation with analytic initial data. The analyticity of its solutions is proved in both variables, globally in space and locally in time. Then, we present…
The Cauchy problem for the two dimensional compressible Euler equations with data in the Sobolev space $H^s(\mathbb R^2)$ is known to have a unique solution of the same Sobolev class for a short time, and the data-to-solution map is…
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…
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 study the Cauchy problem for the Camassa-Holm equation on the real line. By presenting a new construction of initial data, we show that the solution map in the smaller space $B_{p,1}^{1}\cap C^{0,1}$ with $p\in(2,\infty]$…
In this paper, we consider a generalized two component Camassa-Holm system. Based on local well-posedness results and lifespan estimates, we establish sharpness of continuity on the data-to-solution map by showing that it is not uniformly…
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…
We show the lack of uniform continuity of the flow map for the Camassa-Holm equation on the line, in the Sobolev spaces of index s > 3/2.
The failure of uniform dependence on the data is an interesting property of classical solution for a hyperbolic system. In this paper, we consider the solution map of the Cauchy problem to the 2D viscous shallow water equations which is a…
This work studies the Cauchy problem of a two-component higher order Camassa-Holm system, which is well-posed in Sobolev spaces $H^{s}(\mathbb{R})\times H^{s-2}(\mathbb{R})$, $s>\frac{7}{2}$ and its solution map is continuous. We show that…
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 this paper, we study the Cauchy problem for a generalized cross-coupled Camassa-Holm system with peakons and higher-order nonlinearities. By the transport equation theory and the classical Friedrichs regularization method, we obtain the…