Related papers: Ill-Posedness of the Third Order NLS Equation with…

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 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…

We consider fractional Hartree and cubic nonlinear Schr\"odinger equations on Euclidean space $\mathbb R^d$ and on torus $\mathbb T^d$. We establish norm inflation (a stronger phenomena than standard ill-posedness) at every initial data in…

In this paper, we study the Cauchy problem for a generalized cross-coupled Camassa-Holm system with peakons and higher-order nonlinearities. By the transport equation theory and the classical Friedrichs regularization method, we obtain the…

We prove that the Cauchy problem for the Dirac-Klein-Gordon system of equations in 1+3 dimensions is locally well-posed in a range of Sobolev spaces for the Dirac spinor and the meson field. The result contains and extends the earlier known…

The nonlinear wave and Schrodinger equations on Euclidean space of any dimension, with general power nonlinearity and with both the focusing and defocusing signs, are proved to be ill-posed in the Sobolev space of index s whenever the…

In this paper, we study the Cauchy problem for the two component Degasperis-Procesi equation in critical Besov space $B^1_{\infty,1}(\mathbb R)$. By presenting a new construction of initial data, we proved the norm inflation of the…

The Cauchy problem for the L^2-critical boson star equation with initial data of low regularity in spatial dimension d=3 is studied. Local well-posedness in H^s for s > 1/4 is proved. Moreover, for radial initial data, local well-posedness…

We consider the Cauchy problem for one-dimensional dispersive equations with a general nonlinearity in the periodic setting. Our main hypotheses are both that the dispersive operator behaves for high frequencies as a Fourier multiplier by $…

We study the Cauchy problem for the Klein-Gordon-Zakharov system in spatial dimension $d \ge 4$ with radial or non-radial initial datum $(u, \partial_t u, n, \partial_t n)|_{t=0}\in H^{s+1}(\mathbb{R}^d) \times H^s(\mathbb{R}^d) \times…

The Cauchy problem for the Hardy-H\'enon parabolic equation is studied in the critical and subcritical regime in weighted Lebesgue spaces on the Euclidean space $\mathbb{R}^d$. Well-posedness for singular initial data and existence of…

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…

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 article, we initiate the study of the Cauchy problem for the two-dimensional relativistic Euler equations in a low-regularity setting. By introducing good variables--a rescaled velocity, logarithmic enthalpy, and an appropriately…

The sharp range of Sobolev spaces is determined in which the Cauchy problem for the classical Zakharov system is well-posed, which includes existence of solutions, uniqueness, persistence of initial regularity, and real-analytic dependence…

The Cauchy problem for the Yang-Mills system in three space dimensions with data in Fourier-Lebesgue spaces $\hat{H}^{s,r}$ , $1 < r \le 2$ , is shown to be locally well-posed, where we have to assume only almost optimal minimal regularity…

This paper studies the Cauchy problem for a helical vortex filament evolving by the 3D incompressible Navier-Stokes equations. We prove global-in-time well-posedness and smoothing of solutions with initial vorticity concentrated on a helix.…

In this paper, we are concerned with solutions to the Cauchy problem for Chern-Simons-Schr\"odinger equations in the mass supercritical case. First we establish the local well-posedness of solutions in the radial space. Then we consider…

In this article, we study the Cauchy problem to the micropolar Rayleigh-B\'{e}nard convection problem without velocity dissipation in two dimension. We first prove the local well-posedness of a smooth solution, and then establish a blow up…

Inspired by a pioneer work of Andersson-Kapitanski \cite{AK}, we prove the local well-posedness of the Cauchy problem of incompressible neo-Hookean equations if the initial deformation and velocity belong to $H^{s+1}(\mathbb{R}^n) \times…