arXiv subjects

Ademir Pastor

Publications and source records attributed to Ademir Pastor.

43 records · Page 3Linked to original sources

arXiv2013

The IVP for the Benjamin-Ono-Zakharov-Kuznetsov equation in weighted Sobolev spaces

In this paper we study the initial-value problem associated with the Benjamin-Ono-Zakharov-Kuznetsov equation. We prove that the IVP for such equation is locally well-posed in the usual Sobolev spaces $H^{s}(\R^2),$ $s>2$, and in the anisotropic spaces $H^{s_1,s_2}(\R^2)$, $s_2>2$, $s_1\geq s_2$. We also study the persistence properties of the solution and local well-posedness in the weighted Sobolev class $$ \mathcal{Z}_{s,r}=H^{s}(\R^{2})\cap L^{2}((1+x^{2} +y^{2})^rdxdy), $$ where $s>2$, $r\geq 0$, and $s\geq 2r$. Unique continuation properties of the solution are also established. These continuation principles show that our persistence properties are sharp. Most of our arguments are accomplished taking into account that ones for the Benjamin-Ono equation.

math.AP↗

arXiv2012

Global well-posedness for the k-dispersion generalized Benjamin-Ono equation

We consider the $k$-dispersion generalized Benjamin-Ono equation in the supercritical case. We establish sharp conditions on the data to show global well-posedness in the energy space for this family of nonlinear dispersive equations. We also prove similar results for the generalized Benjamin-Ono equation.

math.AP↗

arXiv2012

On well-posedness and wave operator for the gKdV equation

We consider the generalized Korteweg-de Vries (gKdV) equation $\partial_t u+\partial_x^3u+μ\partial_x(u^{k+1})=0$, where $k>4$ is an integer number and $μ=\pm1$. We give an alternative proof of the Kenig, Ponce, and Vega result in \cite{kpv1}, which asserts local and global well-posedness in $\dot{H}^{s_k}(\R)$, with $s_k=(k-4)/2k$. A blow-up alternative in suitable Strichatz-type spaces is also established. The main tool is a new linear estimate. As a consequence, we also construct a wave operator in the critical space $\dot{H}^{s_k}(\R)$, extending the results of Côte [2].

math.AP↗

arXiv2011

A note on the 2D generalized Zakharov-Kuznetsov equation: local, global, and scattering results

We consider the generalized two-dimensional Zakharov-Kuznetsov equation $u_t+\partial_x Δu+\partial_x(u^{k+1})=0$, where $k\geq3$ is an integer number. For $k\geq8$ we prove local well-posedness in the $L^2$-based Sobolev spaces $H^s(\mathbb{R}^2)$, where $s$ is greater than the critical scaling index $s_k=1-2/k$. For $k\geq 3$ we also establish a sharp criteria to obtain global $H^1(\R^2)$ solutions. A nonlinear scattering result in $H^1(\R^2)$ is also established assuming the initial data is small and belongs to a suitable Lebesgue space.

math.AP↗

arXiv2010

Local and global well-posedness for the 2D generalized Zakharov-Kuznetsov equation

This paper addresses well-posedness issues for the initial value problem (IVP) associated with the generalized Zakharov-Kuznetsov equation, namely, \{equation*} \quad \left\{\{array}{lll} {\displaystyle u_t+\partial_x Δu+u^ku_x = 0,}\qquad (x,y) \in \mathbb{R}^2, \,\,\,\, t>0, {\displaystyle u(x,y,0)=u_0(x,y)}. \{array} \right. \{equation*} For $2\leq k \leq 7$, the IVP above is shown to be locally well-posed for data in $H^s(\mathbb{R}^2)$, $s>3/4$. For $k\geq8$, local well-posedness is shown to hold for data in $H^s(\mathbb{R}^2)$, $s>s_k$, where $s_k=1-3/(2k-4)$. Furthermore, for $k\geq3$, if $u_0\in H^1(\mathbb{R}^2)$ and satisfies $\|u_0\|_{H^1}\ll1$, then the solution is shown to be global in $H^1(\mathbb{R}^2)$. For $k=2$, if $u_0\in H^s(\mathbb{R}^2)$, $s>53/63$, and satisfies $\|u_0\|_{L^2}<\sqrt3 \, \|ϕ\|_{L^2}$, where $ϕ$ is the corresponding ground state solution, then the solution is shown to be global in $H^s(\mathbb{R}^2)$.

math.AP↗

arXiv2010

The supercritical generalized KdV equation: Global well-posedness in the energy space and below

We consider the generalized Korteweg-de Vries (gKdV) equation $\partial_t u+\partial_x^3u+μ\partial_x(u^{k+1})=0$, where $k\geq5$ is an integer number and $μ=\pm1$. In the focusing case ($μ=1$), we show that if the initial data $u_0$ belongs to $H^1(\R)$ and satisfies $E(u_0)^{s_k} M(u_0)^{1-s_k} < E(Q)^{s_k} M(Q)^{1-s_k}$, $E(u_0)\geq0$, and $\|\partial_x u_0\|_{L^2}^{s_k}\|u_0\|_{L^2}^{1-s_k} < \|\partial_x Q\|_{L^2}^{s_k}\|Q\|_{L^2}^{1-s_k}$, where $M(u)$ and $E(u)$ are the mass and energy, then the corresponding solution is global in $H^1(\R)$. Here, $s_k=\frac{(k-4)}{2k}$ and $Q$ is the ground state solution corresponding to the gKdV equation. In the defocusing case ($μ=-1$), if $k$ is even, we prove that the Cauchy problem is globally well-posed in the Sobolev spaces $H^s(\mathbb{R})$, $s>\frac{4(k-1)}{5k}$.

math.AP↗

arXiv2009

On the periodic Schrödinger-Boussinesq System

We study the local and global well-posedness of the periodic boundary value problem for the nonlinear Schrödinger-Boussinesq system. The existence of periodic pulses as well as the stability of such solutions are also considered.

math.AP↗