Related papers: Well-posedness for the NLS hierarchy

We prove local and global well-posedness in $H^{s,0}(\mathbb{R}^{2})$, $s > -1/2$, for the Cauchy problem associated with the Kadomotsev-Petviashvili-Burgers-I equation (KPBI) by working in Bourgain's type spaces. This result is almost…

We consider the global well-posedness for the Cauchy probelem of the Kawahara equation which is one of the fifth order KdV type equations. We first establish the local well-posedness in a more suitable function space for the global…

We show that the Cauchy problem for a class of dispersive perturbations of Burgers' equations containing the low dispersion Benjamin-Ono equation $\partial$\_t u -- D^$\alpha$\_x $\partial$\_x u = $\partial$\_x(u^2), 0 < $\alpha$ $\le$ 1,…

We study the local and global well-posedness for the coupled system of Schr\"odinger and Kawahara equations on the real line. The Sobolev space $L^{2} \times H^{-2}$ is the space where the lowest regularity local solutions are obtained. The…

We prove the local well-posedness for the two-dimensional Zakharov-Kuznetsov equation in $H^s(\mathbb{R}^2)$, for $s\in [1,2]$, on the background of an $L^\infty(\mathbb{R}^3)$-function $\Psi(t,x,y)$, with $\Psi(t,x,y)$ satisfying some…

We consider the cubic non-linear Schr\"odinger equation on general closed (compact without boundary) Riemannian surfaces. The problem is known to be locally well-posed in $H^s(M)$ for $s>1/2$. Global well-posedness for $s\geq 1$ follows…

We prove the local well-posedness for the nonlinear fourth-order Schr\"odinger equation (NL4S) in Sobolev spaces. We also studied the regularity of solutions in the sub-critical case. A direct consequence of this regularity is the global…

We consider the initial value problem (IVP) associated to a higher order nonlinear Schr\"odinger (h-NLS) equation $ \partial_{t}u+ia \partial^{2}_{x}u+ b\partial^{3}_{x}u+ic_1|u|^{2}u+c_2 |u|^{2}\partial_{x}u=0, \quad x,t \in \mathbb{R}, $…

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…

This paper is concerned with well-posedness of the Boussinesq system. We prove that the $n$ ($n\ge2$) dimensional Boussinesq system is well-psoed for small initial data $(\vec{u}_0,\theta_0)$ ($\nabla\cdot\vec{u}_0=0$) either in…

In this paper we prove a wellposedness result of the KdV equation on the space of periodic pseudo-measures, also referred to as the Fourier Lebesgue space $\mathscr{F}\ell^{\infty}(\mathbb{T},\mathbb{R})$, where…

We prove global well-posedness for low regularity data for the $L^2-critical$ defocusing nonlinear Schr\"odinger equation (NLS) in 2d. More precisely we show that a global solution exists for initial data in the Sobolev space $H^{s}(\mathbb…

We study the well-posedness of the initial-value problem for the periodic nonlinear "good" Boussinesq equation. We prove that this equation is local well-posed for initial data in Sobolev spaces \textit{$H^s(\T)$} for $s>-1/4$, the same…

Consider the $3$-d primitive equations in a layer domain $\Omega=G \times (-h,0)$, $G=(0,1)^2$, subject to mixed Dirichlet and Neumann boundary conditions at $z=-h$ and $z=0$, respectively, and the periodic lateral boundary condition. It is…

We prove the unconditional well-posedness for the fourth order nonlinear Schrodinger type equations in H^s(\mathbb{T}) when s \geq 1, which includes the non-integrable case. This regularity threshold is optimal because the nonlinear terms…

We study well-posedness of boundary value problems of Dirichlet and Neumann type for elliptic systems on the upper half-space with coefficients independent of the transversal variable, and with boundary data in fractional…

The Fourier transforms of the products of two respectively three solutions of the free Schroedinger equation in one space dimension are estimated in mixed and, in the first case weighted, L^p - norms. Inserted into an appropriate variant of…

We study the well-posedness of the generalized derivative nonlinear Schr\"odinger equation (gDNLS) $$iu_t+u_{xx}=i|u|^{2\sigma}u_x,$$ for small powers $\sigma$. We analyze this equation at both low and high regularity, and are able to…

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.…

We prove that the Yang-Mills equations in the Lorenz gauge (YM-LG) is locally well-posed for data below the energy norm, in particular, we can take data for the gauge potential $A$ and the associated curvature $F$ in $H^s\times H^{s-1}$ and…