# CAPDE

## On Kac’s model, ideal Thermostats, and finite Reservoirs

Event Date: Dec 03, 2018 in CAPDE, Núcleo Modelos Estocásticos de Sistemas Complejos y Desordenados, Seminars

Abstract:   In 1956, Mark Kac introduced a stochastic model to derive a Boltzmann-like equation. Like the space-homogeneous Boltzmann’s  equation, Kac’s equation is ergodic with centered Gaussians as the  unique equilibrium state. In this talk, I will introduce Kac’s model, the thermostat used in [1] to guarantee exponentially fast convergence to equilibrium, and sketch the result in [2] how this infinite  thermostat can be approximated by a finite but large reservoir. References: [1]Bonetto, F., Loss, M.,Vaidyanathan, R.: J. Stat. Phys. 156(4), 647– 667 (2014) [2]Bonetto...

## Well-posedness and long time behavior for the Schrödinger-Korteweg-de Vries interactions on the half-Line

Event Date: Nov 19, 2018 in CAPDE, Seminars

Abstract:   In this talk we discuss about the initial-boundary value problem for the Schrödinger-Korteweg-de Vries system on the left and right half-line for a wide class of initial-boundary data, including the energy regularity H1(R±) × H1(R±) for initial data. Assuming homogeneous boundary conditions it is shown for positive coupling interactions that local solutions can be extended globally in time for initial data in the energy space; furthermore, for negative coupling interactions it was proved, for a certain class of regular initial data, the following result: if the respective...

## Resonances in deformed tubes: twisting and bending & Gross-Pitaevskii equation and integrable systems

Event Date: Nov 12, 2018 in CAPDE, Seminars

16:00 hrs.   Expositor Pablo Miranda, (USach) Título Resonances in deformed tubes: twisting and bending Abstract: In this talk we will consider  an infinite straight tube and  we will deform it  by a periodic twisting and a local bending. On the deformed tube we will define the Laplacian and will study the existence of scattering resonances created by the deformations. We will show the existence of exactly one resonance or one eigenvalue near the bottom of the essential spectrum, depending on the strength of the twisting and the bending. We will also obtain the asymptotic behavior of the...

## Uniqueness and stability of semi-wavefronts for KPP-Fisher equation with delay

Event Date: Oct 16, 2018 in CAPDE, Seminars

Abstract:   In this talk I will preset some recent results on the stability and uniqueness of semi-wavefronts of the equation  u_t(t,x)=u_{xx}(t,x)+u(t,x)(1-u(t-h,x)),    t >0,      x in \R; where the parameter h>0 is a delay. The uniqueness (up to translations) of semi-wavefronts (i.e., solutions in the form u(t,x)=\phi_c(x+ct) satisfying $\phi_c(-\infty)=0$ and $\liminf_{z\to +\infty}\phi_c(z)>0$)  is `largely open’ problem. By a simple approach we have obtained the uniqueness (up to translations) of semi-wavefronts for all speed, i.e., c >= 2, and the stability on each...

## Breathers and the dynamics of solutions to the KdV type equations

Event Date: Oct 01, 2018 in CAPDE, Seminars

Abstract: Our first aim is to identify a large class of non-linear functions f(⋅) for which the IVP for the generalized Korteweg-de Vries equation does not have breathers or “small” breathers solutions. Also we prove that all small, uniformly in time L^1 ∩ H^1 bounded solutions to KdV and related perturbations must converge to zero, as time goes to infinity, locally in an increasing-in-time region of space of order t^1/2 around any compact set in space. This set is included in the linearly dominated dispersive region x≪t. Moreover, we prove this result independently of the...