Estudo das soluções para algumas equações de evolução não-lineares do tipo dispersivo

Author(s)

Francisco Javier Vielma Leal

• Document type: Doctoral Thesis
• Press: Campinas, SP.
• Institution: Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática, Estatística e Computação Científica
• Defense date: 2020-07-14

Examining board members

Mahendra Prasad Panthee; Ademir Pastor Ferreira; Lucas Catão de Freitas Ferreira; Jose Felipe Linares Ramirez; Ademir Fernando Pazoto

• Advisor: Mahendra Prasad Panthee

Abstract

In this thesis, we study the controllability and stabilization of Benjamin and Intermediate Long Wave (ILW) equations on a periodic domain. In the first part of this work we consider the Benjamin equation derived by Benjamin in [12]. This model describes the unidirectional propagation of long waves in a two-fluid system where the lower fluid with greater density is infi nitely deep and the interface is subject to capillarity. First we deal with the controllability and stabilization of the nonhomogenous linear system associated to the Benjamin equation. We obtain the existence and uniqueness of solutions of this system via semigroup theory. Then, we use the classical moment method (see [79]) to show that the linear system is globally exactly controllable, and consequently to get a global exponential stabilization result with an arbitrary decay rate. Next, we derive propagation of compactness, the propagation of smoothness and the unique continuation property for the nonlinear Benjamin equation in associated Bourgain's spaces in the periodic setting. We use these properties to obtain the global exponential stability with a natural feedback law and an arbitrary decay rate. Finally, we also obtain the global controllability result for the Benjamin equation. The second part of this work we focus on the controllability and the stabilization properties of the ILW equation which models nonlinear dispersive waves of moderate amplitude on the interface between two fluids of different positive densities contained at rest in a long channel with a horizontal top and bottom, the lighter fluid forming a horizontal layer above a layer of the same depth of the heavier fluid. We prove that the ILW equation with periodic boundary conditions is exactly controllable and exponentially stabilizable. Speci cally, we incorporate a feedback law in the form of localized damping into the equation to establish a smoothing effect. Using this smoothing effect together with the propagation of regularity property and the unique continuation property we show the semi-global stabilization in L^{2}_{0}(T) of weak solutions obtained by the method of vanishing viscosity. The local-well posedness and the local exponential stability in H^{s}_{0}(T) with s>1/2 is also established using the contraction mapping theorem. Finally, the local exact controllability is derived in H^{s}_{0}(T) with s>1/2 by combining the above feedback law with some open-loop control. These results are similar to the ones obtained by Linares and Rosier [59] for the BO (AU)

• FAPESP's process: 15/06131-5 - Study of solutions to some non-linear evolution equations of dispersive type
• Grantee: Francisco Javier Vielma Leal
• Support Opportunities: Scholarships in Brazil - Doctorate