arXiv subjects

Andressa Gomes

Publications and source records attributed to Andressa Gomes.

3 recordsLinked to original sources

arXiv2024

Energy-critical inhomogeneous nonlinear Schrödinger equation with two power-type nonlinearities

We consider the initial value problem for the inhomogeneous nonlinear Schrödinger equation with double nonlinearities (DINLS) \begin{equation*} i \partial_t u + Δu = λ_1 |x|^{-b_1}|u|^{p_1}u + λ_2|x|^{-b_2}|u|^{\frac{4-2b_2}{N-2}}u, \end{equation*} where $λ_1,λ_2\in \mathbb{R}$, $3\leq N<6$ and $0<b_1,b_2<\min\{2,\frac{6-N}{2}\}$. In this paper, we establish global well-posedness results for certain parameter regimes and prove finite-time blow-up phenomena under specific conditions. Our analysis relies on stability theory, energy estimates, and virial identities adapted to the DINLS model.

math.AP↗

arXiv2022

Global control aspects for long waves in nonlinear dispersive media

A class of models of long waves in dispersive media with coupled quadratic nonlinearities on a periodic domain $\mathbb{T}$ are studied. We used two distributed controls, supported in $ω\subset\mathbb{T}$ and assumed to be generated by a linear feedback law conserving the "mass" (or "volume"), to prove global control results. The first result, using spectral analysis, guarantees that the system in consideration is locally controllable in $H^s(\mathbb{T})$, for $s\geq0$. After that, by certain properties of Bourgain spaces we show a property of global exponential stability. This property together with the local exact controllability ensures for the first time in the literature that long waves in nonlinear dispersive media are globally exactly controllable in large time. Precisely, our analysis relies strongly on the bilinear estimates using the Fourier restriction spaces in two different dispersions that will guarantee a global control result for coupled systems of the Korteweg-de Vries type. This result, of independent interest in the area of control of coupled dispersive systems, provides a necessary first step for the study of global control properties to the coupled dispersive systems in periodic domains.

math.AP↗

arXiv2020

Solitary wave solutions and global well-posedness for a coupled system of gKdV equations

In this work we consider the initial-value problem associated with a coupled system of generalized Korteweg-de Vries equations. We present a relationship between the best constant for a Gagliardo-Nirenberg type inequality and a criterion for the existence of global solutions in the energy space. We prove that such a constant is directly related to the existence problem of solitary-wave solutions with minimal mass, the so called ground state solutions. To guarantee the existence of ground states we use a variational method.

math.AP↗