Second WAFFLE Meeting

Announcement: SECOND WAFFLE MEETING 

We are pleased to announce the organization of the second and last meeting of the WAFFLE team, which will take place from March 30 to 31, 2026, at the Center for Mathematical Modelling CMM, U. de Chile. This event will gather the members of the WAFFLE team with the aim of strengthening the collaboration networks of the WAFFLE Math-Amsud project and fostering the exchange of ideas among participants. 

VENUE

John Von Neumann Room - CMM 7th floor

SCHEDULE

ABSTRACTS

D. Chamorro (Université Paris-Saclay)   

A new pointwise inequality for rough operators and applications

Abstract:

We study in this article a new pointwise estimate for “rough” singular integral operators. From this pointwise estimate we will derive Sobolev type inequalities in a variety of functional spaces. This is a joint work with L. Marcoci and A. Marcoci. 

Fernando Cortez (Escuela Politécnica Nacional)  

Spatial Asymptotics for a Nonlocal 1D Fluid Model 

Abstract: 

We study the spatial behavior of solutions to a one-dimensional nonlocal equation arising in the modeling of equatorial oceanic flows. While the equation has been previously analyzed in the periodic setting, its behavior on the real line exhibits fundamentally different features due to the absence of compactness and the presence of spatial infinity.

Ricardo Freire (Universidad de Chile)  

On the asymptotic dynamics for the L2-supercritical gKDV equation

Abstract:

We study the L2-supercritical generalized Korteweg-de Vries equation (gKdV) with nonlinearities p > 5. The long-time dynamics in the supercritical regime remains largely unexplored beyond small data global solutions, the construction of multi-solitons for any power and self-similar blow-up near the critical power p = 5. We develop a unified description of the non-solitonic region for arbitrary H1 solutions, both global and blowing up. Our analysis shows that the asymptotic L2 and Lp  dynamics in this region is completely determined by the growth rate of the L2 norm of the gradient. In particular, we prove sharp far-field decay on both half-lines and establish normalized local vanishing along sequences of times, with improved estimates in the case of even-power nonlinearities. Joint work with Claudio Muñoz (U. Chile).

Oscar Jarrín (UDLA, Ecuador)

Mathematical study of a new Navier-Stokes-alpha model with nonlinear filter equation

Abstract: 

This project was devoted to the mathematical study of a new Navier–Stokes–alpha model with a nonlinear filter equation. Numerically, this nonlinear filter equation is applied to the nonlinear term in the Navier–Stokes equations in order to provide a precise analysis of numerical diffusion and error estimates.

From a mathematical point of view, the resulting alpha-model is described by a doubly nonlinear parabolic–elliptic coupled system. We therefore undertake the first theoretical study of this model. Specifically, we address the existence and uniqueness of weak Leray-type solutions, their rigorous convergence to weak Leray solutions of the classical Navier–Stokes equations, and their long-time dynamics through the concept of the global attractor together with upper bounds for its fractal dimension.

Finally, we present some forthcoming articles and outline several perspectives for future research. This is joint work with Manuel Fernando Cortez.

María Eugenia Martínez  (Universidad de Chile) 

Existence and interaction of solitary waves in the Zakharov Water Waves system under a slowly variable botos

Abstract: The Zakharov Water Waves system (ZWW) models the evolution of an inviscid irrotational fluid with free surface in two dimensions. It is characterized by a quasilinear system for the free surface and the fluid potential at the free boundary. In the finite flat bottom case, Amick-Kirchgässner proved the existence of small solitary waves. However, in practical situations, the bottom is always non-constant. In this work, we deal with the generalized solitary wave problem for the ZWW system with surface tension and a non-flat bottom, in a surface of one dimension, in the form of a slowly varying (in space) bottom. Our main result establishes that, under suitable conditions on the variation of the bottom, such a generalized nonlinear wave exists and interacts with the bottom in a well-defined fashion, surviving the weak long interaction and exiting the interaction region with well-defined final scaling and shift parameters. The techniques used in the proof of the main result are extensions of the construction of a multi-soliton like solution, introduced in [M. Ming, F. Rousset and N. Tzvetkov, 2013], and the interaction of solitary waves and different media, established in [C. Muñoz, 2011] for KdV, and extends our previous work on the simplified Whitham model. However, the ZWW case presents a considerable amount of new challenges, including: shape derivatives of Dirichlet-Neumann and Neumann-Neumann boundary operators, the quasilinear character of the model, and the lack of a suitable asymptotic stability theory for solitary waves.

Claudio Muñoz (Universidad de Chile)  

On the mathematical validity of the UVic system.

Abstract:

The UVic model is an Earth system model of intermediate complexity (EMIC) introduced in 2001 to model long time climate in the Earth. In this talk we will present this model, its components, and if time permits, announce a result concerning the well-posedness of this model in suitable spaces. This is joint work with Oscar Jarrín.

Felipe Poblete (Universidad Austral de Chile)  

Uniqueness of Quasimonochromatic Breathers for the Generalized Korteweg–de Vries and Zakharov–Kuznetsov Models

Abstract: In this talk, we consider the focusing modified Zakharov-Kuznetsov (mZK) equation in spatial dimension N≥1. This model can be viewed as a higher-dimensional extension of the completely integrable modified Korteweg-de Vries (mKdV) equation. We will present recent results on the existence and nonexistence of quasimonochromatic breather solutions for the mZK equation, showing that the answer strongly depends on the spatial dimension N. In addition, we will explain how the classical breather solution of the mKdV equation appears as a remarkable and unique example of a quasimonochromatic breather.

Benjamín Tardy (U. Austral de Chile and U. Chile)

Caracterización de soluciones 2-soliton para la ecuación KP-II. 

Abstract: En esta charla se presenta una caracterización completa de las soluciones 2-solitón para la ecuación de Kadomtsev-Petviashvili II (KP-II), integrando los tipos O, P y T mediante una nueva descripción basada en operadores diferenciales especiales, mapeando las fases en conos invariantes cuya dimensión refleja la complejidad y el número de términos exponenciales en la fase. Partiendo de la relación conocida entre los coeficientes de las matrices en el Grassmanniano positivo Gr(2,4) y el tipo de solitón generado, se utilizan los distintos valores de dichos coeficientes para reconstruir la solución analizando el espacio de llegada de las fases al ser evaluadas en diferentes operadores. A través de este análisis, probamos que las frecuencias y direcciones de la solución se mantienen constantes asintóticamente en las direcciones que definen la solución 2-solitón, concluyendo con la reconstrucción de la fase mediante la inversión de sistemas de Vandermonde. De este modo, logramos establecer un vínculo directo entre la geometría algebraica del Grassmanniano y la regularidad analítica de las soluciones.

Jessica Trespalacios (U. Austral de Chile) 

Long time dynamics in Einstein-Belinski-Zakharov soliton spacetimes 

Abstract: 

In this talk, we consider the vacuum Einstein field equations under the Belinski-Zakharov symmetry. We will provide a self-contained introduction to this model, describing some recent results on soliton-like solutions and their long-time behavior in this setting.  More precisely, for long-term behavior results, we introduce new energy and virial functionals to describe the energy decay of smooth global cosmological metrics inside the light cone. This is joint work with Claudio Muñoz.

Nicolás Valenzuela  (Universidad de Chile)

An artificial intelligence method to find extremizers in classical Strichartz and Maximal inequalities

Abstract: 

In this talk we present a new method where Maching Learning provides solutions to key mathematical problems. More precisely, this method unifies Artificial Intelligence ideas with classical Harmonic Analysis for the discovery of extremizers in several unsolved Strichartz and Maximal inequalities. This method is primarly based (but not bounded to) Physics Informed Neural Networks (PINNs) with a novel use of minimization procedure. We provide several examples of extremizers found by this method, expecting that some of them are proved as correct solutions to the theoretical minimization problem. This numerical work is joint with R. Freire and C. Muñoz.