DSpace Collection:

Local and global well-posedness for some dispersive equations in modulation spaces

2026-02-07T03:00:10Z

Title: Local and global well-posedness for some dispersive equations in modulation spaces Author(s)/Inventor(s): Balvin, Fidel Cuba Advisor: Paredes, Xavier Carvajal Abstract: We consider the initial value problem (IVP) associated with a system consisting of modified Korteweg-de Vries (mKdV) type equations ∂tv + ∂3 xv + ∂x(vw2) = 0, v(x, 0) = v0(x), ∂tw + α∂3 xw + ∂x(v2w) = 0, w(x, 0) = w0(x), (3) using the theory of T. Oh and Y. Wang [31] we derive trilinear estimates and use it in the contraction argument, so we prove local well-posedness (LWP) to the IVP (3) for given data in the modulation spaces M2,p s (R) ×M2,p s (R), with s ≥ 1 4 , 2 ≤ p < ∞ and α ̸= 0. Also, at the second part of this work we consider the initial value problem (IVP) associated to the higher order nonlinear Schrödinger (h-NLS) equation with variable coefficients ∂tu + ia(t)∂2 xu + b(t)∂3 xu + ic(t)|u|2u + c1(t)|u|2∂xu = 0, u(x, 0) = u0(x), (4) using the theory of Killip, Visan, Zhang [24], Oh, Wang [30], [31], Carvajal, Gamboa and Santos [8], so we establish a priori estimates and prove global well-posedness (GWP) to the IVP (4) for given data in the modulation spaces M2,p s (R), with s ≥ 1 4 and 2 ≤ p < ∞. Publisher: Universidade Federal do Rio de Janeiro Type: Tese

Um estudo do fenômeno de dissipação anômala em equações de fluidos 2D forçadas

2026-01-30T03:00:10Z

Title: Um estudo do fenômeno de dissipação anômala em equações de fluidos 2D forçadas Author(s)/Inventor(s): Paternina Salgado, David Antonio Advisor: Lopes, Helena Judith Nussenzveig Abstract: In this work, we study the phenomenon of anomalous dissipation of potential enstrophy for the family of Generalized Camassa-Holm equations (GCH) in a two-dimensional periodic domain, which are obtained as an interpolation between second-grade fluid equations and the Camassa-Holm equations. In this context, we first prove the existence and uniqueness of solutions for the GCH equations. Existence is established using the Galerkin method, while for uniqueness, we develop a differential inequality and apply a Grönwall-type inequality.We then demonstrate that for values of β in the range 1/2 < β ≤ 1, there is no anomalous dissipation of potential enstrophy. On the other hand, in the case β = 0, the system exhibits anomalous dissipation of potential enstrophy, specifically infinite dissipation. The analysis of the phenomenon of anomalous dissipation of potential enstrophy was carried out by investigating the inviscid limit of the long-term time averages of the solutions to the GCH equations and subsequently identifying these long-term time averages with stationary statistical solutions in the phase space of potential vorticity. Publisher: Universidade Federal do Rio de Janeiro Type: Tese

Some Properties of ASH attractors

2025-11-13T03:00:12Z

Title: Some Properties of ASH attractors Author(s)/Inventor(s): Pineda Reyes, Miguel Angel Advisor: Arbieto Mendoza, Alexander Eduardo Abstract: This text refers to a doctoral thesis in dynamical systems, which focuses on continuous-time dynamics. More specifically, it advances the theory of asymptotically sectional-hyperbolic (ASH) flows through several fundamental results. First, we prove that every star ASH attractor of a C¹ vector field with positive topological entropy is necessarily sectional-hyperbolic. Second, we establish that all ASH attractors satisfy the intermediate entropy property. Third, we demonstrate that any ASH attractor in three-dimensional vector fields is entropy-expansive and admits periodic orbits. Finally, we provide a lower bound for the growth rate of periodic orbits in an ASH attractor. Publisher: Universidade Federal do Rio de Janeiro Type: Tese

On the rationality of theta elements for a Rankin-Selberg setting

2025-11-13T03:00:12Z

Title: On the rationality of theta elements for a Rankin-Selberg setting Author(s)/Inventor(s): Toso, Ricardo Silva Advisor: Pacheco, Amilcar Abstract: In 1987 Mazur and Tate used the theory of modular symbols to define theta elements that interpolate special values of the Hasse-Weil L-function of an elliptic curve. These elements are essentially finite layer analogues of the L-function, and were conjectured to satisfy many interesting properties. In this thesis, we aim to follow their steps and try to study theta elements for a different L-function. Instead of the Hasse-Weil L-function, we will consider a certain Rankin-Selberg L-function of an elliptic curves twisted by an Artin representation (under some technical conditions). We show that, despite the lack of a theory of modular symbols for our setting, it is still possible to use a theorem of Rankin-Selberg-Shimura to construct similar rational group ring elements that interpolate special values of the L-function. For the original Mazur-Tate setting, the theta elements can be quickly computed due to the many algorithms and libraries that exist for modular symbols. For the Rankin-Selberg setting however, not much is implemented. Still, we show that our theta elements can be explicitly computed and we implement them in SageMath. Publisher: Universidade Federal do Rio de Janeiro Type: Tese