Related papers: Non-uniform continuous dependence on initial data …

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 study the Cauchy problem of the Novikov equation in $\mathbb{R}$ for initial data belonging to the Triebel-Lizorkin spaces, i.e, $u_0\in F^{s}_{p,r}$ with $1< p, r<\infty$ and $s>\max\{\frac32,1+\frac1p\}$. We prove…

In this paper we consider the two component $b$-family of equations on $\mathbb R$. We write the equations on a Sobolev type diffeomorphism group. As an application of this formulation we show that the dependence on the initial data is…

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…

We start with the classic result that the Cauchy problem for ideal compressible gas dynamics is locally well posed in time in the sense of Hadamard; there is a unique solution that depends continuously on initial data in Sobolev space $H^s$…

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…

We construct an example showing that the solution map of the Euler equations is not continuous in the H\"older space from $C^{1,\alpha}$ to $L^\infty_tC^{1,\alpha}_x$ for any $0<\alpha<1$. On the other hand we show that it is continuous…

In this paper, we mainly investigate the Cauchy problem for the incompressible Navier-Stokes equations in homogeneous Besov spaces $\dot{B}^{\frac{d}{p}-1}_{p,r}$ with $1\leq p<\infty,\ 1\leq r\leq \infty, \ d\geq 2$. Firstly, we prove the…

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…

This work studies a two-component Fornberg-Whitham (FW) system, which can be considered as a model for the propagation of shallow water waves. It's known that its solutions depend continuously on their initial data from the local…

This paper is devoted to the study of the Cauchy problem of incompressible magneto-hydrodynamics system in framework of Besov spaces. In the case of spatial dimension $n\ge 3$ we establish the global well-posedness of the Cauchy problem of…

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.

This paper is mainly concerned with the well-posedness and exponential decay of solution for a integrable three-component Novikov system, which admits bi-Hamiltonian structure and infinitely many conserved quantities. The local…

The present paper establishes well-posedness for the two-component Fornberg-Whitham system in Besov spaces. First the existence and uniqueness of its solution is proved, then it is shown that the corresponding data-to-solution map is…

In this paper, we first establish the local well-posednesss of a Two-Component b-Family equations in nonhomogeneous Besov spaces $B^{1+\frac 1 p}_{p,1}$ with $1\leq p<+\infty.$ Then we present a new blow-up result for the Two-Component…

In this paper we consider the inviscid 2D Boussinesq equation on the Sobolev spaces $H^s(\R^2)$, $s > 2$. Using a geometric approach we show that for any $T > 0$ the corresponding solution map, $(u(0),\theta(0)) \mapsto (u(T),\theta(T))$,…

We prove that the two-component peakon solutions are orbitally stable in the energy space. The system concerned here is a two-component Novikov system, which is an integrable multicomponent extension of the integrable Novikov equation. We…

The present paper is the continuation of work \cite{XC}, devoted to extending it to a critical functional framework which is not related to the energy space. Employing the special dissipative structure of the non-conservative viscous…

In this paper, we prove the non-uniform continuity of the data-to-solution map for the incompressible magnetohydrodynamic (MHD) equations with only magnetic diffusion in Sobolev spaces $H^s(\mathbb{R}^d)$ for all $s>0$ and $d=2,3$. Our…

In this paper, we study the Cauchy problem for the following Hamilton-Jacobi equation \bbal\bca \pa_tu-\De u=|\na u|^2,\quad t>0, \ x\in \R^d,\\ u(0,x)=u_0, \quad \quad x\in \R^d. \eca\end{align*} We show that the solution map in Besov…