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Current Research Projects
Main Collaborators:
Felipe Poblete, Universidad Austral de Chile (International Project Coordinator).
Diego Chamorro, Université de Paris-Saclay (French Coordinator).
Fernando Cortez, Escuela Politécnica Nacional (Ecuadorian Coordinator).
Claudio Muñoz, Universidad de Chile.
Oscar Jarrín, Universidad de las Americas.
Jiao He, Université de Paris Saclay.
MATH AmSud: WAFFLE: código 23-MATH-18 (Waters, Atmosphere, Fluids and Friedman Lemaître evolutions)
MATH AmSud: WAFFLE: código 23-MATH-18 (Waters, Atmosphere, Fluids and Friedman Lemaître evolutions)
Abstract : This proposal describes an emerging but genuinely cohesive collaborative network between researchers in France, Ecuador and Chile specialized in fluid dynamics from a mathematical point of view. What distinguishes us from other groups present in the region is our particular flavor involving i) deep harmonic analysis of Navier-Stokes and Euler models, ii) the comprehensive understanding of the dynamics in water wave models, and iii) a profound work on 2D fluid dispersive models such as KP, ZK, and more recently, studies in Fluids appearing in Climatology and General Relativity. We are a forming group with broad interests and high synergy looking for the advance of Science and Mathematics in Latin America through the French connection .
Sharp well-posedness in Sobolev spaces and analyticity of the solutions of dispersive equations of the Kuramoto-Sivashinsky type
Internal Project PII-DM-2024-01, , Escuela Politécnica Nacional.
Collaborators: Miguel Yangari (EPN), Oscar Jarrín (University of the Americas).
Since April 2024.
Abstract: In this research project, we focus on analyzing, firstly, the optimality of the regularity index in Sobolev spaces for the well-posedness of the Cauchy problem of the Kuramoto-Sivashinsky equation in two dimensions, as well as the regularity of the associated solutions. Secondly, we will focus on the analyticity of the solutions of the problem, by demonstrating that such solutions belong to a certain class of Gevrey functions.
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.
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