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Luiz Gustavo Farah

Publications and source records attributed to Luiz Gustavo Farah.

34 records · Page 2Linked to original sources
arXiv2017

Instability of solitons - revisited, II: the supercritical Zakharov-Kuznetsov equation

We revisit the phenomenon of instability of solitons in the two dimensional generalization of the Korteweg-de Vries equation, the generalized Zakharov-Kuznetsov (ZK) equation, $u_t + \partial_{x_1} (Δu + u^p) = 0, (x_1,x_2) \in \mathbb R^2$. It is known that solitons are unstable in this two dimensional equation for nonlinearities $p > 3$. This was shown by Anne de Bouard in [4] generalizing the arguments of Bona-Souganidis-Strauss in [1] for the generalized KdV equation. In this paper, we use a different method to obtain the instability of solitons, namely, truncation and monotonicity properties. Not only does this approach simplify the proof, but it can also be useful for studying various other stability questions in the ZK equation as well as other generalizations of the KdV equation.

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arXiv2017

Scattering for the radial focusing INLS equation in higher dimensions

We consider the inhomogeneous nonlinear Schrödinger equation $$ i u_t +Δu+|x|^{-b}|u|^αu = 0, $$ where $\frac{4-2b}{N}<α<\frac{4-2b}{N-2}$ (when $N=2$, $\frac{4-2b}{N}<α<\infty$) and $0<b<\min\{N/3,1\}$. For a radial initial data $u_0\in H^1(\mathbb{R}^N)$ under a certain smallness condition we prove that the corresponding solution is global and scatters. The smallness condition is related to the ground state solution of $-Q+ΔQ+ |x|^{-b}|Q|^αQ=0$ and the critical Sobolev index $s_c=\frac{N}{2}-\frac{2-b}α$. This is an extension of the recent work \cite{paper2} by the same authors, where they consider the case $N=3$ and $α=2$. The proof is inspired by the concentration-compactness/rigidity method developed by Kenig-Merle \cite{KENIG} to study $H^1(\mathbb{R}^N)$-critical problem and also Holmer-Roudenko \cite{HOLROU} in the case of $H^1(\mathbb{R}^N)$-subcritical equations.

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arXiv2016

Scattering for a 3D coupled nonlinear Schrödinger system

We consider the three-dimensional cubic nonlinear Schrödinger system \begin{equation*} \begin{cases} i\partial_tu+Δu+(|u|^2+β|v|^2)u=0,\\ i\partial_tv+Δv+(|v|^2+β|u|^2)v=0. \end{cases} \end{equation*} Let $(P,Q)$ be any ground state solution of the above Schrödinger system. We show that for any initial data $(u_0,v_0)$ in $H^1(\mathbb{R}^3)\times H^1(\mathbb{R}^3)$ satisfying $M(u_0,v_0)A(u_0,v_0)<M(P,Q)A(P,Q)$ and $M(u_0,v_0)E(u_0,v_0)<M(P,Q)E(P,Q)$, where $M(u,v)$ and $E(u,v)$ are the mass and energy (invariant quantities) associated to the system, the corresponding solution is global in $H^1(\mathbb{R}^3)\times H^1(\mathbb{R}^3)$ and scatters. Our approach is in the same spirit of Duyckaerts-Holmer-Roudenko, where the authors considered the 3D cubic nonlinear Schrödinger equation.

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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.

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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].

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arXiv2010

Local well-posedness for the Sixth-Order Boussinesq Equation

This work studies the local well-posedness of the initial-value problem for the nonlinear sixth-order Boussinesq equation $u_{tt}=u_{xx}+βu_{xxxx}+u_{xxxxxx}+(u^2)_{xx}$, where $β=\pm1$. We prove local well-posedness with initial data in non-homogeneous Sobolev spaces $H^s(\R)$ for negative indices of $s \in \R$.

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arXiv2010

Global solutions for the generalized Boussinesq equation in low-order Sobolev spaces

We show that the Cauchy problem for the defocusing generalized Boussinesq equation $u_{tt}-u_{xx}+u_{xxxx}-(|u|^{2k}u)_{xx}=0$, $k\geq1$, on the real line is globally well-posed in $H^{s}(\R)$ for $s>1-({1}/{3k})$. We use the "$I$-method" to define a modification of the energy functional that is "almost conserved" in time. Our result extends the previous one obtained by Farah and Linares (2010 \textit{J. London Math. Soc.} \textbf{81} 241-254) when $k=1$.

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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 $H^s(\mathbb{R})$, $s>\frac{4(k-1)}{5k}$.

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arXiv2009

On the periodic "good" Boussinesq equation

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 range of the real case obtained in Farah \cite{LG4}.

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.

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