Noticias en español
Spectrum of the linearized Vlasov-Poisson system.
Abstract: The Vlasov-Poisson system describes a macroscopic number of particles with their mutual gravitational attraction in a mean-field approximation. Its steady-state solutions are known as “polytropes” and a popular model for galaxies in the astronomy literature. In many cases, they are known to be neutrally stable, but the question of asymptotic stability is widely open. The goal of this talk is to present some results on the linearized equation around a steady state. This is based on joint work with Matías Moreno and Paola Rioseco.
Read MoreHopf’s lemmas and boundary point results for the fractional p-Laplacian.
Abstract: In this talk, we will discuss different versions of the classical Hopf’s boundary lemma in the setting of the fractional $p-$Laplacian, for $p \geq 2$. We will start with a Hopf’s lemma based on comparison principles and for constant-sign potentials. Afterwards, we will present a Hopf’s result for sign-changing potentials describing the behavior of the fractional normal derivative of solutions around boundary points. As we wiil see, the main contribution here is that we do not need to impose a global condition on the sign of the solution. Applications of the...
Read MoreVariational Approach for the Singular Perturbation Domain Wall Coupled System.
Abstract: In this talk, I will present results on a singular perturbation problem modeling domain walls. I will discuss the existence of solutions both when the perturbation parameter is non-zero and when it is set to zero (Thomas-Fermi approximation), demonstrating their continuous connection as the parameter approaches zero. Finally, I will show that the behavior of one of the variables can be modeled by a Painlevé II equation in the limit, by the use of an appropriate change of variables.
Read MoreSharp Fourier restriction over finite fields.
Abstract: Fourier sharp restriction theory has been a topic of interest over the last decades. On the other hand, efforts have been made in order to develop the theory of Fourier restriction over finite fields. In this talk, we will present some recently made developments (in a joint work with Diogo Oliveira e Silva) in the intersection of these two topics.
Read MoreBounds on the approximation error for Deep Neural Networks applied to dispersive models: Nonlinear waves.
Abstract: In this talk we present a comprehensive framework for deriving rigorous and efficient bounds on the approximation error of deep neural networks in PDE models characterized by branching mechanisms, such as waves, Schrödinger equations, and other dispersive models. This framework utilizes the probabilistic setting established by Henry-Labordère and Touzi. We illustrate this approach by providing rigorous bounds on the approximation error for both linear and nonlinear waves in physical dimensions d = 1, 2, 3, and analyze their respective computational costs starting from time...
Read MoreOn the nonexistence of NLS breathers.
Abstract: In this talk, we will show a proof of the nonexistence of breather solutions for NLS equations. By using a suitable virial functional, we are able to characterize the nonexistence of breather solutions by only using their inner energy and the power of the corresponding nonlinearity of the equation. We extend this result for several NLS models with different power nonlinearities and even the derivative NLS equation.
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