FIRST WAFFLE MEETING

Announcement: FIRST WAFFLE MEETING in Ecuador 

SCHEDULE

ABSTRACTS

J. Carvalho (Universidade Federal de Pernambuco)  Quintic Defocusing Schrödinger Equation in R^{3}: Control and Stabilization 

We have gathered control and stabilization results for the nonlinear Schrödinger equation with critical exponent in the defocusing case. More precisely, we study the well-posedness of this system and present a result of local null controllability. Additionally, we discuss the exponential decay for solutions of a perturbed formulation of the aforementioned equation, considering the H^{1}-energy of the system, where we resort to a profile decomposition for both equations (linear and nonlinear) and to arguments from microlocal analysis. 

D. Chamorro (Université Paris-Saclay)   Teoremas de tipo Liouville para las ecuaciones de Navier-Stokes estacionarias.

La unicidad de las soluciones H1 de las ecuaciones de Navier-Stokes estacionarias en dimension 3 sigue siendo un problema completamente abierto.Sin embargo, si se exigen algunas condiciones adicionales a estas soluciones, entonces se puede deducir la unicidad de la solución trivial. Estas condiciones adicionales pueden expresarse en diferentes espacios funcionales, como por ejemplo pertenecer a un espacio de Lebesgue. En esta charla veremos cómo el uso de los espacios de Lebesgue de exponente variable permite dar un nuevo punto de vista a este tipo de problema. 

Frédéric Charve (Université Paris-Est - Créteil)  New asymptotics for the strong solutions of the strongly stratified Boussinesq system with ill-prepared initial data.

It is known that when the Froude number ε tends towards zero, the solutions of the strongly stratified Boussinesq system tend towards those of a two-component Navier-Stokes system (but dependent on the three space variables). Surprisingly this limit system does not depend on the thermal diffusivity ν > 0. In a previous work we obtained, for unconventional initial data, and in the framework of weak solutions, a general limit system dependent on all parameters: the two-component 3D Navier-Stokes system coupled with a heat equation in the vertical variable.In this talk we focus on the same limit for strong solutions with unconventional and ill-prepared initial data. We manage to obtain global solutions when the Froude number is sufficiently small, as well as estimates (explicit in the small parameter ε) of convergence rates. These results can also be rewritten as an asymptotic expansion around explicit particular solutions for the classical Boussinesq system.

Fernando Cortez (Escuela Politécnica Nacional)  Estudio Matemático de un Nuevo Modelo Alpha de Navier-Stokes con un filtro nolineal.

Este charla está  dedicada al estudio matemático de un nuevo modelo alphad e Navier-Stokes con una ecuación de filtro no lineal. Para una función indicadora dada, esta ecuación de filtro fue considerada por primera vez por W. Layton, G.Rebholz y C. Trenchea en [Modular Nonlinear Filter Stabilization of Methods forHigher Reynolds Numbers Flow, J. Math. Fluid Mech. 14: 325–354 (2012)] para seleccionar vórtices que deben ser amortiguados, basado en la comprensión de cómo actúa la no linealidad en problemas reales de flujo. Numéricamente, esta ecuación de filtro no lineal fue aplicada al término no lineal en las ecuaciones de Navier-Stokes para proporcionar un análisis preciso de la difusión numérica y estimaciones de error. Matemáticamente, el modelo alpha resultante se describe mediante un sistema acoplado parabólico-elíptico doblemente no lineal. Por lo tanto, llevamos a cabo el primer estudio teórico de este sistema considerando condiciones de contorno periódicas en la variable espacial. Específicamente, abordamos la existencia y unicidadde soluciones débiles tipo Leray, su convergencia rigurosa a soluciones débiles de Leray de las ecuaciones clásicas de Navier-Stokes, y su dinámica a largo plazo a través del concepto de atractor global y algunas cotas superiores para su dimensión fractal.

Ricardo Freire (Universidad de Chile)  Error Bounds for PINNS in gKdV equations placed on unbounded domains

In this talk, we consider the generalized nonlinear Korteweg–de Vries (gKdV) equation on the real line and address the challenge of approximating its solutions in unbounded domains using Physics-Informed Neural Networks (PINNs). While most prior studies have focused on bounded settings, our approach adapts PINNs to infinite domains, incorporating the equation’s dynamics directly into the learning process.We derive rigorous bounds on the approximation error in terms of the energy norm, under suitable numerical conditions. Numerical experiments demonstrate the effectiveness of our method in capturing key nonlinear structures such as traveling waves, breathers, and solitons.

Francisco Gancedo (Universidad de Sevilla & IAS Princeton) On evolution of vortex filaments

In this talk we discuss two new results about vortex filament evolution for incompressible Navier-Stokes and Euler equations. For Navier-Stokes, we prove global-in-time regularity for initial helical vortex filament. For Euler, we give existence of weak dissipative solutions with initial vorticity concentrated in a circle.

Rafael Granero (Universidad de Cantabria)  On certain dispersive equations and the behavior of their solutions

In many natural phenomena dispersive motion plays a key role. In particular, in this talk we will introduce a number of nonlocal dispersive PDE describing the motion of certain two-dimensional fluid problems.

Boris Haspot (Université Paris-Dauphine)  Vanishing viscosity limit for hyperbolic system of Temple class in 1-d with nonlinear viscosity 

We consider hyperbolic system with nonlinear viscosity such that the viscosity matrix  B(u) is commutating with A(u) the matrix associated to the convective term. The drift matrix is assumed to be Temple class. First, we prove the global existence of smooth solutions for initial data with small total variation. We show that the solution to the parabolic equation converges to a semi-group solution of the hyperbolic system as viscosity goes to zero. Furthermore, we prove that the zero diffusion limit coincides with the one obtained in [Bianchini and Bressan, Indiana Univ. Math. J. 2000]. 

Jiao He (Université Paris-Saclay) The vanishing limit of a rigid body in three-dimensional viscous incompressible fluid                                                                                                                                                       

We consider the evolution of a small rigid body in an incompressible viscous fluid filling the whole space R3 . When the small rigid body shrinks to a “massless” point in the sense that its density is constant, we prove that the solution of the fluid-rigid body system converges to a solution of the Navier–Stokes equations in the full space.Based on some L p − L q estimates of the fluid–structure semigroup and a fixed point argument, we obtain a uniform estimate of velocity of the rigid body. This allows us to construct admissible test functions which plays a key role in the procedure of passing to the limit.

Pierre-Gilles Lemarié-Rieusset (Université Paris-Saclay) Highly singular (frequentially sparse) steady solutions for the 2D Navier-Stokes equations on the torus 

We construct non-trivial steady solutions in H1 for the 2D Navier-Stokes equations on the torus. In particular, the solutions are not square integrable, so that we have to redefine the notion of solutions. 

María Eugenia Martínez  (Universidad de Chile) Global weak solutions for a variation of the Whitham equation 

We study a variation of the Whitham equation, introduced as an alternative to the KdV equation. Our analysis begins by proving the global existence of weak solutions, followed by establishing a regularity criterion that enables the deduction of their uniqueness. Additionally, we provide local-in-time criteria for regularity and uniqueness.

In the field of mathematical physics, the Whitham equation serves as a non-local model for nonlinear dispersive waves. Introduced by Whitham in 1967, it was designed to capture phenomena beyond the classical KdV equation framework, such as wave peaking and breaking. From a mathematical perspective, the Whitham equation presents a compelling model for free surface dynamics,acting as a try model for more complex nonlinear hyperbolic equations when a dispersion term is incorporated. This is a joint work with Diego Chamorro, from Université d’Evry.

Christopher Maulén (Bielefeld University) Asymptotic stability of the solitary waves solutions to the Good Boussinesq model in the energy space. 

In this talk, I will present the stability problem for the solitary waves solutions to the Good Boussinesq model. We will see a brief overview of previous results on the stability problem. Then, we will present some recent developments on the stability of solitary waves, and by using an approach based on virial identities, inspired by the work of Kowalczyk, Martel, Muñoz, and Van den Bosch, we will prove that these solutions are asymptotically stable for any perturbation in the energy space. This work was done jointly with Claudio Muñoz. 

Claudio Muñoz (Universidad de Chile)  Interaction of jamitons in second-order macroscopic traffic models (with R. Burger and S. Tapia).

Jamitons are self-sustained traveling wave solutions that arise in certain second-order macroscopic models of vehicular traffic. A necessary condition for a jamiton to appear is that the local traffic density breaks the so-called sub-characteristic condition. This condition states that the characteristic velocity of the corresponding first-order Lighthill-Whitham-Richards (LWR) model formed with the same desired speed function is enclosed by the characteristic speeds of the corresponding second-order model. The phenomenon of collision of jamitons in second-order models of traffic flow will be discussed analytically and numerically for the particular case of the second-order Aw-Rascle-Zhang (ARZ) traffic model. A compatibility condition is first defined to select jamitons that can collide with each other. The collision of jamitons produces a new jamiton with a velocity different from the initial ones. It is observed that the exit velocities smooth out the velocity of the test jamiton and the initial velocities of the jamitons that collide. Other properties such as the amplitude of the exit jamitons, lengths, and maximum density are also studied as well.

Felipe Poblete (Universidad Austral de Chile)  On uniqueness of KP soliton structures

In this talk, we explore the Kadomtsev-Petviashvili II (KP), focusing on solutions of  smooth data that are not necessarily  in a  Sobolev space, specifically those of "soliton type"  described  by a phase  and a unidimensional profile. We show that  these solutions are characterized  through a set of nonlinear differential equations and special functionals like Wronskian, Airy, and Heat types. These functional equations only depend on the new variables in terms of the phase and profile. A distinct characteristic of this set of functionals is its special and rigid structure tailored to the considered soliton. We establish the uniqueness of line-solitons, multi-solitons, and other degenerate solutions among a large class of KP solutions. Our results are also valid for other 2D dispersive models such as the quadratic and cubic Zakharov-Kuznetsov equations.

Thyago Santos (Instituto de Matemática Pura e Aplicada - IMPA)  Sharp well-posedness for the k-dispersion generalized Benjamin-Ono equations

We consider the Cauchy problem related to the family of k-dispersion generalized Benjamin-Ono (k-DGBO) equations:

    u_t + D^α_x u + μ u^ku_x = 0, (t, x) ∈ ℝ × ℝ,

    u(0, x) = u_0(x),

 where u = u(t, x) is real-valued, α ∈ [1, 2], μ ∈ {±1} and k ∈ ℤ^+. Here, D^α_x represents the 1-dimensional fractional Laplacian operator in the spatial variable x. For k ≥ 4, we establish local and global well-posedness results for this equation in both the critical: s = (k−2α) / (2k)  and subcritical: s > (k−2α) / (2k) regimes, addressing sharp regularity in homogeneous and inhomogeneous Sobolev spaces. Additionally, our method enables the formulation of a scattering criterion and a scattering theory for small data. We also investigate the case k = 3 via frequency-restricted estimates, obtaining local well-posedness results for the initial value problem associated with the 3-DGBO equation and generalizing the existing results in the literature for the whole subcritical range. For higher dispersion, these local results can be extended globally even for rough data, particularly for initial data in Sobolev spaces with negative indices. As a byproduct, we derive new nonlinear smoothing estimates.

Jéssica Trespalacios (Universidad de Chile) An Introduction to Gravitational Solitons: Belinski-Zakharov Spacetimes

In this talk we introduce the particular class of spacetimes that admit two space-like Killing vector fields, these spacetimes correspond to the also called gravisolitons or gravitational solitons. More specifically, we will focus on the Einstein vacuum model Rµν(g̃) = 0, where g̃ is the metric tensor and Rµν is the Ricci tensor, in the setting of the Belinski-Zakharov formalism. This ansatz is compatible with the well-known Gowdy symmetry. The aim is to describe rigorously the conditions for the global existence of small solutions, and their decay in the light cone, as well as the stability of a first set of solitonic solutions (gravisolitons), for the so-called reduced Einstein equation, seen as an identification of the Principal Chiral Field (PCF) model.