Noticias en español
A multiple time renewal equation for neural assemblies with elapsed time model.
Abstract: We introduce and study an extension of the classical elapsed time equation in the context of neuron populations that are described by the elapsed time since last discharge. In this extension we incorporate the elapsed since the penultimate discharge and we obtain a more complex system of integro-differential equations. For this new system we prove convergence to stationary state by means of Doeblin’s theory in the case of weak non-linearities in an appropriate functional setting, inspired by the case of the classical elapsed time equation. Moreover, we present some numerical...
Read MoreOn the fractional Zakharov-Kuznetsov equation.
Abstract: In this talk, we will present some new results related to the regularity properties of the initial value problem (IVP) for the equation” Bt u ́ Bx1 ( ́∆)α/2u + uBx1 u = 0, 0 ă α ă 2, u(x, 0) = u0(x), x = (x1, x2, . . . , xn ) P Rn, n ě 2, t P R, (0.1) where ( ́∆)α/2 denotes the n ́dimensional fractional Laplacian. In the particular case that α= 2, the equation is known as the Zakharov-Kuznetsov-(ZK) equation and it was proposed by Zakharov and Kuznetsov as amodel to describe the propagation of ion-sound waves in magnetic fields in dimen-sion n = 3. A property that enjoys the...
Read Morep-harmonic functions with Neumann conditions and measure data.
Abstract: In this talk I will discuss the problem of finding solutions to some nonlinear elliptic equations with measure data. To this end I will introduce the concept of Renormalized Solutions, which is a very useful tool to solve both Dirichlet and Neumann problems. I will present some results in this area and also discuss some open problems.
Read MoreA nonlocal isoperimetric problem: density perimeter.
Abstract: We will discuss a variant of a classical geometric minimization problem, known as the “nonlocal isoperimetric problem”, which arises from studies in Nuclear Physics by Gamow in the 1930’s. By introducing a density in the perimeter functional, we obtain features that differ substantially from existing results in the framework of the classical problem without densities. In the regime of “small” non-local contribution, we completely characterize the minimizer, in the case the density is a monomial radial weight.
Read MoreThe search of finite-time singularity solutions of the Euler equations for incompressible and inviscid fluids.
Abstract: Despite 250 years of history, the nature of solutions of the Euler equations remains an open problem. To date, it is not known if general smooth initial conditions of the Euler equations with finite energy do or do not blow-up in finite time. I will review the approach initiated by Leray of self-similar blow-up solutions. Lastly, I will show that under some conditions an axisymmetric incompressible and inviscid flow presents a finite-time singularity. This singularity appears to be generic and robust for a wide number of finite energy initial conditions.
Read MoreSobre ecuaciones tipo Cummins y convertidores de energía de olas.
Abstract: En esta charla comenzaremos abordando algunas formulaciones recientes de las ecuaciones water-waves, para luego tomar ventaja de ellas y establecer algunos problemas de transmisión explícitos que describen interacciones fluido-estructuras. En una segunda parte estudiaremos el cómo bajo ciertas restricciones es posible obtener algunas generalizaciones de la ecuación de Cummins. Por último, mostraremos métodos con los cuales la comunidad que estudia la extracción de energía a partir de las olas del mar utiliza dichas ecuaciones integro-diferenciales para obtener energías limpias e...
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