Related papers: Global well-posedness for the Kawahara equation wi…

An ill-posed Cauchy problem for the wave equation is considered: the solution is to be determined by the Cauchy data on some part of the time-space boundary. By means of Fourier method we obtain a regularization algorithm for this problem,…

Inspired by the recent successful completion of the study of the well-posedness theory for the Cauchy problem of the Korteweg-de Vries (KdV) equation \[ u_t +uu_x +u_{xxx}=0, \quad \left. u \right |_{t=0}=u_{0} \] in the space $H^{s}…

We prove global well-posedness of the fifth-order Korteweg-de Vries equation on the real line for initial data in $H^{-1}(\mathbb{R})$. By comparison, the optimal regularity for well-posedness on the torus is known to be…

We prove new well-posedness results for dispersion-generalized Kadomtsev--Petviashvili I equations in $\mathbb{R}^2$, which family links the classical KP-I equation with the fifth order KP-I equation. For strong enough dispersion, we show…

We study global-in-time well-posedness and the behaviour and of the solution to Cauchy problem in the classical Keller-Segel system with logistic term \begin{equation*} \left. \aligned \partial_tn-\Delta n=&-\chi\nabla\cdot(n\nabla c)+\la…

In the first part of this work we study the local well-posedness of dispersive equations in the weighted spaces $H^s(\mathbb{R})\cap L^2(|x|^{2b}dx)$. We then apply our results for several dispersive models such as the Hirota-Satsuma…

This paper is dedicated to the study of the derivative nonlinear Schr\"odinger equation on the real line. The local well-posedness of this equation in the Sobolev spaces is well understood since a couple of decades, while the global…

We show the ill-posedness of the Cauchy problem for the Dirac-Klein-Gordon system in one dimension in the critical Sobolev space. From this, we finish the classification of the regularities for which this problem is well-posed or ill-posed.

We study the Cauchy problems for the Hartree-type nonlinear Dirac equations with Yukawa-type potential in two and three spatial dimensions. This paper improves our previous results \cite{chohlee,cholee}; we establish global well-posedness…

We show the global well-posedness for the two-dimensional Zakharov-Kuznetsov equation in $H^{s}({\mathbb{R}^2})$ when $\frac{11}{13}<s<1$ via the I-method. Additionally, local well-posedness for the symmetrized ZK equation in $…

We study the Cauchy problem for the Klein-Gordon-Zakharov system in 3D with low regularity data. We lower down the regularity to the critical value with respect to scaling up to the endpoint. The decisive bilinear estimates are proved by…

In this article, we address the Cauchy problem for the KP-I equation \[\partial_t u + \partial_x^3 u -\partial_x^{-1}\partial_y^2u + u\partial_x u = 0\] for functions periodic in $y$. We prove global well-posedness of this problem for any…

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 1D Cauchy problem for the Zakharov system is shown to be locally well-posed for low regularity Schr\"odinger data u_0 \in \hat{H^{k,p}} and wave data (n_0,n_1) \in \hat{H^{l,p}} \times \hat{H^{l-1,p}} under certain assumptions on the…

In this article we present local well-posedness results in the classical Sobolev space H^s(R) with s > 1/4 for the Cauchy problem of the Gardner equation, overcoming the problem of the loss of the scaling property of this equation. We also…

The initial value problem for the Korteweg-deVries equation on the line is shown to be globally well-posed for rough data. In particular, we show global well-posedness for initial data in H^s({\mathbb{R}), -3/10<s.

We study the critical dissipative quasi-geostrophic equations in $\bR^2$ with arbitrary $H^1$ initial data. After showing certain decay estimate, a global well-posedness result is proved by adapting the method in [11] with a suitable…

The Cauchy problem for the Zakharov system in four dimensions is considered. Some new well-posedness results are obtained. For small initial data, global well-posedness and scattering results are proved, including the case of initial data…

We consider the initial value problem (IVP) associated to a Boussinesq type system. After rewriting the system in an equivalent form of coupled KdV-type equations, we prove that this is locally well-posed in $(H^s(\R^2))^4$, $s>3/2$, using…

We prove that the Cauchy problem for the Chern-Simons-Higgs equations on the (2+1)-dimensional Minkowski space-time is globally well posed for initial data with finite energy. This improves a result of Chae and Choe, who proved global…