Related papers: Ill-posedness for the Euler equations in Besov spa…

This work is concerned with the Cauchy problem for a Zakharov system with initial data in Sobolev spaces $H^k(\mathbb R^d)\!\times\!H^l(\mathbb R^d)\!\times\!H^{l-1}\!(\mathbb R^d)$. We recall the well-posedness and ill-posedness results…

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…

In this paper, we consider the Cauchy problem for $(abcd)$-Boussinesq system posed on one- and two-dimensional Euclidean spaces. This model, initially introduced by Bona, Chen, and Saut, describes a small-amplitude waves on the surface of…

In this note, using the ideas from our recent article \cite{EM}, we prove strong ill-posedness for the 2D Euler equations in $C^k$ spaces. This note provides a significantly shorter proof of many of the main results in \cite{BLi2}. In the…

The concern of this paper is the Cauchy problem for the Prandtl equation. This problem is known to be well-posed for analytic data, or for data with monotonicity properties. We prove here that it is linearly ill-posed in Sobolev type…

An evidence of temporal dis-continuity of the solution in $F^s_{1, \infty}(\mathbb{R}^d)$ is presented, which implies the ill-posedness of the Cauchy problem for the Euler equations. Continuity and weak-type continuity of the solutions in…

We study the Cauchy problem for the dissipative Benjamin-Ono equations $u_t+\H u_{xx}+|D|^\alpha u+uu_x=0$ with $0\leq\alpha\leq 2$. When $0\leq\alpha< 1$, we show the ill-posedness in $H^s(\R)$, $s\in\R$, in the sense that the flow map…

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 a fractional Sobolev space $H^s(\mathbb{R}^2)$ with $s\leq\frac74$, we prove the low-regularity ill-posedness for the 2D compressible Euler equations and the 2D ideal compressible MHD system. Our ill-posedness results match the…

In this paper we prove that the Cauchy problem for first-order quasi-linear systems of partial differential equations is ill-posed in Gevrey spaces, under the assumption of an initial ellipticity. The assumption bears on the principal…

We prove that the 2D Euler equations are not locally well-posed in $C^1$. Our approach relies on the technique of Lagrangian deformations and norm inflation of Bourgain and Li. We show that the assumption that the data-to-solution map is…

The Cauchy problem for the nonlinear Schr\"odinger equation is called unconditionally well posed in a data space $E$ if it is well posed in the usual sense and the solution is unique in the space $C([0,T]; E)$. In this paper, this notion of…

The Cauchy problem for the Kadomtsev-Petviashvili-II equation (u_t+u_{xxx}+uu_x)_x+u_{yy}=0 is considered. A small data global well-posedness and scattering result in the scale invariant, non-isotropic, homogeneous Sobolev space \dot…

This paper is concerned with the Cauchy problem of the quadratic nonlinear Schr\"{o}dinger equation in $\mathbb{R} \times \mathbb{R}^2$ with the nonlinearity $\eta |u|^2$ where $\eta \in \mathbb{C} \setminus \{0\}$ and low regularity…

In this work we shall show that the Cauchy problem \begin{equation} \left\{ \begin{aligned} &(u_t+u^pu_x+\mathcal H\partial_x^2u+ \alpha\mathcal H\partial_y^2u )_x - \gamma u_{yy}=0 \quad p\in{\nat} &u(0;x,y)=\phi{(x,y)} \end{aligned}…

We study a class of higher-order KdV equations. We show that the associated initial value problem is well posed in weighted Besov and Sobolev spaces for small initial data. We also prove ill-posedness results when in H^s(\R), for any real…

In this paper, we mainly investigate the Cauchy problem of the non-viscous MHD equations with magnetic diffusion. We first establish the local well-posedness (existence,~uniqueness and continuous dependence) with initial data $(u_0,b_0)$ in…

In this work we shall study the well-posedness and ill-posedness of the Cauchy problem associated to the equation \begin{equation*} u_{t}+a(u^{n})_{x}+(b\mathscr{H} u_{t}+u_{yy})_{x}=0, \end{equation*} in anisotropic weigthed Sobolev…

These notes are devoted to the notion of well-posedness of the Cauchy problem for nonlinear dispersive equations. We present recent methods for proving ill-posedness type results for dispersive PDE's. The common feature in the analysis is…

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 $p\in(2,\infty]$…