相关论文: Ill-posedness for the two component Degasperis-Pro…

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…

For the famous Camassa-Holm equation, the well-posedness in $B^{1+\frac{1}{p}}_{p,1}(\mathbb{R})$ with $ p\in [1,\infty)$ and the ill-posedness in $B^{1+\frac{1}{p}}_{p,r}(\mathbb{R})$ with $ p\in [1,\infty],\ r\in (1,\infty]$ had been…

In this paper, we prove the norm inflation and get the ill-posedness for the modified Camassa-Holm equation in $B_{\infty,1}^0$. Therefore we completed all well-posedness and ill-posedness problem for the modified Camassa-Holm equation in…

We study the Cauchy problem of the 2D viscous shallow water equations in some critical Besov spaces $\dot B^{\frac{2}{p}}_{p,1}(\mathbb{R}^2)\times \dot B^{\frac{2}{p}-1}_{p,q}(\mathbb{R}^2)$. As is known, this system is locally well-posed…

In this paper, we mainly consider the Cauchy problem of a weakly dissipative Camassa-Holm equation. We first establish the local well-posedness of equation in Besov spaces $B^{s}_{p,r}$ with $s>1+\frac 1 p$ and $s=1+\frac 1 p , r=1,p\in…

In this paper, we obtain the local-in-time existence and uniqueness of solution to the Degasperis-Procesi equation in $B^1_{\infty,1}(\R)$. Moreover, we prove that the data-to-solution of this equation is continuous but not uniformly…

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…

In this paper we mainly investigate the Cauchy problem of a two-component Novikov system. We first prove the local well-posedness of the system in Besov spaces $B^{s-1}_{p,r}\times B^s_{p,r}$ with…

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…

This paper is concerned with the Cauchy problem for a two-component Degasperis-Procesi system. Firstly, the local well-posedness for this system in the nonhomogeneous Besov spaces is established. Then the precise blow-up scenario for strong…

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…

It is shown that both the Camassa-Holm and Novikov equations are ill-posed in $B_{p,r}^{1+1/p}(\mathbb{R})$ with $(p,r)\in[1,\infty]\times(1,\infty]$ in \cite{Guo2019} and well-posed in $B_{p,1}^{1+1/p}(\mathbb{R})$ with $p\in[1,\infty)$ in…

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 prove norm inflation and hence ill-posedness for a class of shallow water wave equations, such as the Camassa-Holm equation, Degasperis-Procesi equation and Novikov equation etc., in the critical Sobolev space $H^{3/2}$ and even in the…

In this paper, we study the Cauchy problem for a two-component higher order Camassa-Holm systems with fractional inertia operator $A=(1-\partial_x^2)^r,r\geq1$, which was proposed by Escher and Lyons. By the transport equation theory and…

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…

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 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…

In this paper, we first establish the local well-posednesss for the Cauchy problem of a modified Camassa-Holm (MOCH) equation in critical Besov spaces $B^{\frac 1 p}_{p,1}$ with $1\leq p<+\infty.$ The obtained results improve considerably…

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…