Current Research Projects

Qualitative mathematical study of the solutions of a parabolic-elliptical Navier-Stokes type system.

Internal Project PII-DM-2026-01, Escuela Politécnica Nacional.

Collaborators: David LLerena (EPN), Miguel Yangari (EPN), Oscar Jarrín (University of the Americas).

Since April 2026.

First theoretical analysis of a novel Navier-Stokes-$\alpha$ model with non-linear filtering (coupled parabolic-elliptic system). The study establishes the existence and uniqueness of weak and strong solutions, proves convergence to classical equations ($\alpha \to 0$), characterizes long-term dynamics via global attractors, and derives estimates for the energy cascade. 

Previous Research Projects

Problems in partial differential equations involving local and non-local operators. 

Internal Project PII-DM-2019-05, Escuela Politécnica Nacional.

Collaborators: Miguel Yangari, (EPN), Oscar Jarrín, (University of the Americas).

From 2019-07-24 to 2020-07-23.

Abstract: The project aims to study parabolic problems involving both local and non-local operators in their spatial and temporal variables. Our research will focus on two main groups of problems: firstly, within the framework of viscous solutions, we will investigate the homogenization of equations and the regularity of solutions to problems involving non-local operators. Secondly, the second type of problems will focus on Cauchy problems for certain dispersive and dissipative equations. 

The non-existence of global solutions for certain semilinear parabolic Cauchy problems associated with arbitrarily small initial conditions in homogeneous functional spaces."

Internal Project PII-DM-08-2016, Escuela Politécnica Nacional.

Collaborators: Paul Acevedo (EPN), Lorenzo Brandolese (University Claude Bernard).

From 2016-11-01 to 2018-04-30.

Abstract: In some Cauchy problems, starting from the assumption of smallness in the norm of the initial condition in a certain homogeneous Besov space, the global existence of the solution in time is verified. An interesting and widely addressed topic is to determine the largest homogeneous Besov space (with respect to inclusion) such that the assumption of smallness of the initial condition implies the global existence of the solution. Similarly, from the limiting Besov space, it can be shown that if a homogeneous Besov space contains the limiting space, then there exists a small-sized initial condition such that its associated solution blows up in finite time. In particular cases, the limiting space is known, such as in the Cauchy problem for the incompressible Navier-Stokes equation; while in other equations, the answers are partial or scarce. Our research will focus on constructing arbitrarily small initial conditions whose associated solution does not exist globally in time for certain semilinear parabolic Cauchy problems.