Márcio Cavalcante - Universidade Federal de Alagoas

Horário: 10:00 - 11:00

The Stability of Solitons for the KdV equation on the Half-Line

Resumo: In this talk, we discuss recent results of stability obtained for the Korteweg-de Vries (KdV) equation on the positive half-line. First, we consider the problem of orbital stability for the Solitons concerning the initial boundary value problem (IBVP) associated with the KdV equation in the energy space with homogeneous boundary conditions on the positive half-line. We also discuss the result of asymptotic stability of the solitons. Finally, we explore some improvements to the previous results and discuss how to consider stability for more general boundary conditions. 

Isaías Pereira - Universidade Federal do Piauí

Horário: 11:00 - 12:00

Hierarchical control for the wave equation 

Resumo: This talk deals with the hierarchical control for a one-dimensional wave equation in domains with moving boundary. This equation models small vibrations of a string where an endpoint is fixed and the other is moving. As usual, we consider one main control (the leader) and an additional secondary control (the follower). We use Stackelberg strategy.

Roberto Capistrano - Universidade Federal de Pernambuco

Horário: 14:00 - 15:30

Fourier transform restriction phenomena and applications to control of dispersive equations 

Resumo: J. Bourgain discovered a subtle smoothing property of solutions of the KdV equation posed on a periodic domain. Since this celebrated article, the smoothing properties for dispersive systems are now well known. In this talk, we will see that in the last 10 years, the Bourgain spaces are fundamental to addressing the global control problems in periodic frameworks. Precisely, we will show that propagation of compactness and regularity have been observed thanks to these spaces in various control problems for dispersive systems. 

Luccas Campos - Universidade Federal de Minas Gerais 

Horário: 16:00 - 17:00

Monotonicity formulae and the long-time behavior of Schrödinger equations 

Resumo: The concentration-compactness-rigidity method, pioneered by Kenig and Merle, has become standard in the study of global well-posedness and scattering in the context of dispersive and wave equations. Albeit powerful, it requires building some heavy machinery in order to obtain the desired space-time bounds.

In this talk, we present a simpler method, based on Tao's scattering criterion and on Dodson-Murphy's Virial/Morawetz inequalities, first proved for the 3d cubic nonlinear Schrödinger (NLS) equation.

Tao's criterion is, in some sense, universal, and it is expected to work in similar ways for dispersive problems. On the other hand, the Virial/Morawetz inequalities need to be established individually for each problem, as they rely on monotonicity formulae.

This approach is versatile, as it was shown to work in the energy-subcritical setting for different nonlinearities, as well as for higher-order equations.