相关论文: Ill-posedness for the Camassa-Holm equation in $B_…
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 prove that the Cauchy problem for the Fornberg-Whitham equation is not locally well-posed in $B^s_{p,r}(\R)$ with $(s,p,r)\in (1,1+\frac1p)\times[2,\infty)\times [1,\infty]$ or $(s,p,r)\in \{1\}\times[2,\infty)\times…
For the Fornberg-Whitham equation, the local well-posedness in the critical Besov space $B_{p, 1}^{1+\frac{1}{p}}(\mathbb{R})$ with $1\leq p <\infty$ has been studied in (Guo, Nonlinear Anal. RWA., 2023). However, for the endpoint case…
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…
The paper provides a direct proof the uniqueness of solutions to the Camassa-Holm equation, based on characteristics. Given a conservative solution $u=u(t,x)$, an equation is introduced which singles out a unique characteristic curve…
We study the Cauchy problem for the incompressible Navier-Stokes equation \begin{align} u_t -\Delta u+u\cdot \nabla u +\nabla p=0, \ \ {\rm div} u=0, \ \ u(0,x)= \delta u_0. \label{NS} \end{align} For arbitrarily small $\delta>0$, we show…
We consider the Cauchy problem for compressible Navier--Stokes equations of the ideal gas in the three-dimensional spaces. It is known that the Cauchy problem in the scaling critical spaces of the homogeneous Besov spaces $\dot…
We prove that the Cauchy problem for the three dimensional Navier-Stokes equations is ill posed in $\dot{B}^{-1,\infty}_{\infty}$ in the sense that a ``norm inflation'' happens in finite time. More precisely, we show that initial data in…
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 consider the Cauchy problem to the Euler equations in $\mathbb{R}^d$ with $d\geq2$. We construct an initial data $u_0\in B^\sigma_{p,\infty}$ showing that the corresponding solution map of the Euler equations starting from…
In this paper, we present a construction of $u_{0}\in B^{1}_{\infty,1}$ and get the local ill-posedness for the rotation Camassa-Holm equation modelling the equatorial water waves with the weak Coriolis effect by proving the norm inflation.
For the real-line case, it is shown that both the Camassa--Holm \cite{Guo} and Novikov equations \cite{Li-arx} are ill-posed in $B_{\infty,1}^{1}$. In this paper, by presenting a new construction of initial data which leads to the norm…
In this paper, we study the Cauchy problem of a higher-order $\mu$-Camassa-Holm equation. We first establish the Green's function of $(\mu-\partial_{x}^{2}+\partial_{x}^{4})^{-1}$ and local well-posedness for the equation in Sobolev spaces…
In this paper, we consider the Cauchy problem for a two-component Novikov system on the line. By specially constructed initial data $(\rho_0, u_0)$ in $B_{p, \infty}^{s-1}(\mathbb{R})\times B_{p, \infty}^s(\mathbb{R})$ with…
Considered in this paper is the modified Camassa-Holm equation with cubic nonlinearity, which is integrable and admits the single peaked solitons and multi-peakon solutions. The short-wave limit of this equation is known as the short-pulse…
It is proved in \cite[J. Funct. Anal., 2020]{AP} that the Cauchy problem for some Oldroyd-B model is well-posed in $\B^{d/p-1}_{p,1}(\R^d) \times \B^{d/p}_{p,1}(\R^d)$ with $1\leq p<2d$. In this paper, we prove that the Cauchy problem for…
In this paper we mainly study the Cauchy problem for a generalized Camassa-Holm equation. First, by using the Littlewood-Paley decomposition and transport equations theory, we establish the local well-posedness for the Cauchy problem of the…
In this paper, we study the Cauchy problem for a generalized integrable Camassa-Holm equation with both quadratic and cubic nonlinearity. By overcoming the difficulties caused by the complicated mixed nonlinear structure, we firstly…
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…
In this paper, we consider the Cauchy problem for the $b$-equation. Firstly, for $s>\frac32,$ if $u_{0}(x)\in H^{s}(\mathbb{R})$ and $m_{0}(x)=u_{0}(x)-u_{0xx}(x)\in L^{1}(\mathbb{R}),$ the global solutions of the $b$-equation is…