Noticias en español
Kink networks for scalar fields in dimension 1+1.
Abstract: Consider a real scalar wave equation in dimension 1+1 with a positive external potential having non-degenerate isolated zeros. I will speak about the problem of construction of weakly interacting pure multi-solitons, that is solutions converging exponentially in time to a superposition of Lorentz-transformed solitons (“kinks”), in the case of distinct velocities. In a joint work with Gong Chen from the University of Toronto, we prove that these solutions form a 2K-dimensional smooth manifold in the space of solutions, where K is the number of the kinks. This manifold...
Read MoreDeep Learning Schemes For Parabolic Nonlocal Integro-Differential Equations.
Abstract: In this work we consider the numerical approximation of nonlocal integro differential parabolic equations via neural networks. These equations appear in many recent applications, including finance, biology and others, and have been recently studied in great generality starting from the work of Caffarelli and Silvestre. Based in the work by Hure, Pham and Warin, we generalize their Euler scheme and consistency result for Backward Forward Stochastic Differential Equations to the nonlocal case. We rely on Lévy processes and a new neural network approximation of thenonlocal part to...
Read MoreSpatial behavior of solutions for a large class of non-local PDE’s arising from stratified flows.
Abstract: We propose a theoretical model of a non-local dipersive-dissipative equation which contains as a particular case a large class of non-local PDE’s arising from stratified flows. Within this fairly general framework, we study the spatial behavior of solutions proving some sharp pointwise and averaged decay properties as well as some pointwise grow properties.
Read MoreBoson stars and their linear stability.
Abstract: Boson stars composed of massive scalar fields are among the most promising exotic objects that may populate the universe. Even though they remain hypothetical they are frequently considered as candidates for black hole mimickers, massive compact objects, or even the core of the galactic halos in the context of dark matter. In this talk I will focus on static, spherically symmetric boson stars and first explain how they arise as solutions of a nonlinear eigenvalue problem which is obtained from the Einstein-Klein-Gordon system. This eigenvalue problem is solved by means of a...
Read MoreAsymptotic stability manifolds for solitons in the generalized Good Boussinesq equation.
Abstract: In this talk, I shall consider the generalized Good-Boussinesq model in one dimension, with power nonlinearity and data in the energy space $H^1\times L^2$. I will present in more detail the long-time behavior of zero-speed solitary waves, or standing waves. By using virial identities, in the spirit of Kowalczyk, Martel, and Muñoz, we construct and characterize a manifold of even-odd initial data around the standing wave for which there is asymptotic stability in the energy space.
Read MoreAbout infinite energy solutions to the incompressible Navier-Stokes equations.
Abstract: We study estimates for the Navier–Stokes equations, in a sufficiently robust context to be applied to the construction of : 1) Discretely self-similar solutions, for initial data satisfying the weak condition to be locally square integrable. 2) Regular axially symmetrical solutions without swirl, for initial data which together with his gradient belong to a weighted Lebesgue space.
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