Related papers: Non-uniform dependence for Euler equations in Beso…
For Besov spaces $B^s_{p,r}(\rr)$ with $s>\max\{ 2 + \frac1p , \frac52\} $, $p \in (1,\infty]$ and $r \in [1 , \infty)$, it is proved that the data-to-solution map for the FORQ equation is not uniformly continuous from $B^s_{p,r}(\rr)$ to…
By constructing a series of perturbation functions through localization in the Fourier domain and translation, we show that the data-to-solution map for the Euler-Poincar\'e equations is nowhere uniformly continuous in $B^s_{p,r}(\mathbb{R}…
In this paper, we investigate the dependence on initial data of solutions to the Novikov equation. We show that the solution map is not uniformly continuous dependence on the initial data in Besov spaces $B^s_{p,r}(\R),\ s>\max\{1+\frac…
In the paper, we consider the initial value problem to the higher dimensional Euler equations in the whole space. Based on the new technical which is developed in \cite{Li2}, we proved that the data-to-solution map of this problem is not…
In this paper, we show that the solution map of the two-component Novikov system is not uniformly continuous on the initial data in Besov spaces $B_{p, r}^{s-1}(\mathbb{R})\times B_{p, r}^s(\mathbb{R})$ with $s>\max\{1+\frac{1}{p},…
We show that the incompressible Euler equations on $\mathbb{R}^2$ are not locally well-posed in the sense of Hadamard in the Besov space $B^1_{\infty,1}$. Our approach relies on the technique of Lagrangian deformations of Bourgain and Li.…
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…
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…
This paper establishes non-uniform continuity of the data-to-solution map in the periodic case, for the two-component Fornberg-Whitham system in Besov spaces $B^s_{p,r}(\mathbb{T}) \times B^{s-1}_{p,r}(\mathbb{T})$ for $s>…
In this paper, we solve an open problem left in the monographs \cite[Bahouri-Chemin-Danchin, (2011)]{BCD}. Precisely speaking, it was obtained in \cite[Theorem 7.1 on pp293, (2011)]{BCD} the existence and uniqueness of $B^s_{p,\infty}$…
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…
We give intrinsic characterisations for the uniformly localized versions of the Besov spaces $B^{s}_{p,q}({\mathbb R}^n)$, where $p,q\in [1,+\infty]$, and of the Lizorkin-Triebel spaces $F^{s}_{p,q}({\mathbb R}^n)$, where $q\in [1,+\infty]$…
In this paper we consider the incompressible Euler equation on the Sobolev space $H^s(\R^n)$, $s > n/2+1$, and show that for any $T > 0$ its solution map $u_0 \mapsto u(T)$, mapping the initial value to the value at time $T$, is nowhere…
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…
This work is the continuation of the recent paper \cite{D2} devoted to the density-dependent incompressible Euler equations. Here we concentrate on the well-posedness issue in Besov spaces of type $B^s_{\infty,r}$ embedded in the set of…
We consider the 3D incompressible Euler equations in bounded domains $\Omega$ with smooth boundary $\partial\Omega$. Based on the paper by Iwabuchi, Matsuyama and Taniguchi (2019), we define the Besov space $B^s_{p, q}(A)$ by means of the…
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…
The present paper is devoted to the study of the well-posedness issue for the density-dependent Euler equations in the whole space. We establish local-in-time results for the Cauchy problem pertaining to data in the Besov spaces embedded in…
We study the incompressible Euler equations in the $\alpha$-modulation $M^{s,\alpha}_{p,q}$ and H\"older $C^{1+\sigma}$ spaces on the plane. We show that for these spaces the associated data-to-solution map is not continuous on bounded…
In this paper, we investigate the continuity of solution to the Euler-Poincar\'{e} equations. We show that the continuity of the solution cannot be improved to the H\"{o}lder continuity. That is, the solution of the Euler-Poincar\'{e}…