arXiv subjects

M. Escobedo

Publications and source records attributed to M. Escobedo.

8 recordsLinked to original sources

arXiv2026

On the onset of correlations in Wave Turbulence close to singularities

In this paper we describe in a formal way how the derivation of the turbulent wave equation for the Schr\"odinger equation breaks down for times close to the self similar blow up of the wave turbulence kinetic equation. To this end, we study how the derivation of the cumulants hierarchy can not be approximated using solutions of the wave turbulence kinetic equation near the blow up time. It tuns out that near the blow up time the kinetic equation has to be replaced by a hierarchy of equations which is equivalent to a random field, defined for times $t\in (-\infty, \infty)$ and satisfying a nonlinear non autonomous Schr\"odinger equation.

math.AP↗

arXiv2019

On a Boltzmann equation for Compton scattering, from non relativistic electrons at low density

A Boltzmann equation, used to describe the evolution of the density function of a gas of photons interacting by Compton scattering with electrons at low density and non relativistic equilibrium, is considered. A truncation of the very singular redistribution function is introduced and justified. The existence of weak solutions is proved for a large set of initial data. A simplified equation, where only the quadratic terms are kept and that appears at very low temperature of the electron gas, for small values of the photon's energies, is also studied. The existence of weak solutions, and also of more regular solutions that are very flat near the origin, is proved. The long time asymptotic behavior of weak solutions of the simplified equation is described.

math.AP↗

arXiv2017

A short remark on a Growth Fragmentation equation

An explicit solution for a growth fragmentation equation with constant dislocation measure is obtained. In this example the necessary condition for the general results in \cite{BW} about the existence of global solutions in the so called self similar case is not satisfied. The solution is local and blows up in finite time.

math.AP↗

arXiv2014

On the blow up and condensation of supercritical solution of the Nordheim equation for bosons

In this paper we prove that the solutions of the isotropic, spatially homogeneous Nordheim equation for bosons, with bounded initial, data blow up in finite time in the $L^\infty$ norm if the values of the energy and particle density are in the range of values where the corresponding equilibria contains a Dirac mass. We also prove that, in the weak solutions, whose initial data are measures with values of particle and energy densities satisfying the previous condition, a Dirac measure at the origin forms in finite time.

math-ph↗

arXiv2010

Classical non mass preserving solutions of coagulation equations

In this paper we construct classical solutions of a family of coagulation equations with homogeneous kernels that exhibit the behaviour known as gelation. This behaviour consists in the loss of mass due to the fact that some of the particles can become infinitely large in finite time.

math-ph↗

arXiv2009

Local well posedness for a linear coagulation equation

In this paper we derive some a priori estimates for a class of linear coagulation equations with particle fluxes towards large size particles. The derived estimates allow us to prove local well posedness for the considered equations. Some regularizing effects exhibited by the equations in the particle distributions for large particle sizes are discussed in detail.

math-ph↗

arXiv2009

On the Fundamental Solution of a Homogeneous Linearized Coagulation Equation

In this paper we study the fundamental solution of the equation obtained by the linearisation of the Smoluchowski coagulation equation with the multiplicative kernel $(x y)^{λ/2}$ with $λ\in (1, 2)$ around the steady state $f(x)=x^{-(3+λ)/2}$. An explicit representation formula as well as detailed estimates on its asymptotics are obtained. We also describe in a detailed form particle fluxes between different sizes for this linearised equation.

math-ph↗

arXiv2008

On the onset of interference effects during the formation of the Bose-Einstein condensate

In this paper we derive the equations characterizing the boundary layer which describes the transition of the distribution function of a gas of weakly interacting bosons to the distribution function of the gas in the presence of a Bose-Einstein condensate. To this end, we first rederive the classical Uehling-Uhlenbeck equation taking as a starting point the dynamics of a system of many weakly interacting quantum particles. The solutions of the Uehling-Uhlenbeck equation yield blow-up in finite time. Near the blow-up time the approximations used to derive the Uehling-Uhlenbeck equation break down. We derive the set of equations that describe the building of correlations and the onset of quantum interference effects for the many-particle hamiltonian system under the assumption that the blow-up for the Uehling-Uhlenbeck equation takes place in a self-similar form.

math-ph↗