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Miguel Escobedo

Publications and source records attributed to Miguel Escobedo.

At least 19 recordsLinked to original sources
arXiv2026

Global Well-Posedness near Rayleigh-Jeans Equilibria for the Cubic NLS Wave Kinetic Equation

We study the dynamics of the kinetic wave equation associated to the three dimensional Schrödinger equation close to Rayleigh-Jeans equilibria. We first prove that the linearised operator generates a semigroup of contractions in $L^2((0,\infty);\sqrt ω\ddω)$. Considering the family of nonsingular Rayleigh-Jeans spectra, we prove that the linearised operator possesses a spectral gap, despite the non-compactness of the integral collisional operator, and thus obtain an exponential relaxation for the linear semigroup. We then prove bilinear and trilinear estimates in the relevant norm for the nonlinear terms and deduce global well-posedness and exponential relaxation for sufficiently small relative perturbations of Rayleigh-Jeans equilibria. To our knowledge, this is the first global strong well-posedness and asymptotic stability result near a nonzero thermodynamic equilibrium for the full spatially homogeneous four-wave kinetic equation associated with the cubic nonlinear Schrödinger equation.

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arXiv2025

Local classical solutions of a kinetic equation for three waves interactions in presence of a Dirac measure at the origin

The existence of local, classical solutions is proved, for a system of two coupled equations that describe, in the framework of the wave turbulence theory, the fluctuations around an equilibrium, of a system of nonlinear waves satisfying the 3-d cubic Schrödinger equation, weakly interacting in presence of a condensate. The function that describes the density of waves behaves like a singular Rayleigh Jeans equilibria near the origin, and induces a strictly increasing behavior in time of the function describing the condensate's density.

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arXiv2024

Entropy maximizers for kinetic wave equations set on tori

We consider the kinetic wave equation, or phonon Boltzmann equation, set on the torus (physical system set on the lattice). We describe entropy maximizers for fixed mass and energy; our framework is very general, being valid in any dimension, for any dispersion relation, and even including the quantum kinetic wave equation. Of particular interest is the presence of condensation in certain regimes which we characterize.

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arXiv2024

Instability of singular equilibria of a wave kinetic equation

We consider the singular Rayleigh-Jeans equilibrium of the $4$-waves kinetic turbulence equation for the three dimensional Schrödinger equation. We first show the formation in finite time of a Dirac measure at zero frequency in the solution of the wave kinetic equation when the initial data has the form of Rayleigh-Jeans, truncated at large values of the energy. The initial value problem for the linearization around the singular Rayleigh-Jeans equilibria is then solved in several functional spaces. Then, long time convergence to a Dirac measure at the origin is described in detail for some of the solutions. This determines a basin of attraction of the Dirac measure.

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arXiv2024

Regularizing effects in a linear kinetic equation for cubic interactions

We describe regularizing effects in the linearization of a kinetic equation that arises in study of a system of nonlinear waves satisfying the Schrödinger equation in terms of weak turbulence and condensate. The problem is first considered in spaces of bounded functions with weights, where existence of solutions and some first regularity properties are proved. After a suitable change of variables the equation is written in terms of a pseudo differential operator. Homogeneity of the equation and classical arguments of freezing of coefficients may then be used to prove regularizing effect in local Sobolev type spaces.

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arXiv2023

An inverse problem: recovering the fragmentation kernel from the short-time behaviour of the fragmentation equation

The present paper provides a new representation of the solution to the fragmentation equation as a power series in the Banach space of Radon measures endowed with the total variation norm. This representation is used to justify how the fragmentation kernel, which is one of the two key parameters of the fragmentation equation, can be recovered from short-time experimental measurements of the particle size distributions when the initial condition is a delta function. A new stability result for this equation is also provided using a Wasserstein-type norm. We exploit this stability to prove the robustness of our reconstruction formula with respect to noise and initial data.

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arXiv2022

Local classical solutions of a three excitations kinetic system for a homogeneous condensed gas of bosons

Short time existence of classical solutions is proved for a system of equations that involves a three excitations kinetic operator. The system is related to the description of a gas of bosons below but close to the critical temperature, where the three excitations integral aims at describing the interaction between the particles in the condensate and the excitations in the normal gas. Some qualitative properties of the solutions are obtained.

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arXiv2022

On the linearized system of equations for the condensate-normal fluid interaction near the critical temperature

The Cauchy problem for the linearization of a system of equations arising in the kinetic theory of a condensed gas of bosons near the critical temperature around one of its equilibria is solved for radially symmetric initial data. It is proved that the linearized system has global classical solutions that satisfy the natural conservation laws for a large set of initial data. Some regularity properties of the solutions and their long time asymptotic behavior are described.

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arXiv2021

On the linearized system of equations for the condensate-normal fluid interaction at very low temperature

The linearization around one of its equilibrium of a system that describes the correlations between the superfluid component and the normal fluid part of a condensed Bose gas in the approximation of very low temperature and small condensate density, is studied. A simple and transparent argument gives a necessary and sufficient condition on the initial data for the existence of global solutions satisfying the conservation of the total number of particles and energy. Their convergence to a suitable stationary state is also shown and rates of convergence for the normal fluid and superfluid components are obtained.

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arXiv2020

Classical approximation of a linearized three waves kinetic equation

The fundamental solution of the classical approximation of a three waves kinetic equation that happens in the kinetic theory of a condensed gas of bosons near the critical temperature is obtained. It is also proved to be unique in a suitable space of distributions and several of its properties are described. The fundamental solution is used to solve the initial value problem for a general class of initial data.

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arXiv2018

Semigroup theory for the Stokes operator with Navier boundary condition on $L^p$ spaces

We consider the incompressible Navier-Stokes equations in a bounded domain with $\mathcal{C}^{1,1}$ boundary, completed with slip boundary condition. Apart from studying the general semigroup theory related to the Stokes operator with Navier boundary condition where the slip coefficient $α$ is a non-smooth scalar function, our main goal is to obtain estimate on the solutions, independent of $α$. We show that for $α$ large, the weak and strong solutions of both the linear and non-linear system are bounded uniformly with respect to $α$. This justifies mathematically that the solution of the Navier-Stokes problem with slip condition converges in the energy space to the solution of the Navier-Stokes with no-slip boundary condition as $α\to \infty$.

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arXiv2018

Estimating the division rate and kernel in the fragmentation equation

We consider the fragmentation equation $\dfrac{\partial}{\partial t}f (t, x) = --B(x)f (t, x) + \int\_{ y=x}^{ y=\infty} k(y, x)B(y)f (t, y)dy,$ and address the question of estimating the fragmentation parameters-i.e. the division rate $B(x)$ and the fragmentation kernel $k(y, x)$-from measurements of the size distribution $f (t, $\times$)$ at various times. This is a natural question for any application where the sizes of the particles are measured experimentally whereas the fragmentation rates are unknown, see for instance (Xue, Radford, Biophys. Journal, 2013) for amyloid fibril breakage. Under the assumption of a polynomial division rate $B(x) = αx^γ$ and a self-similar fragmentation kernel $k(y, x) = \frac{1}{y} k\_0 (x/ y)$, we use the asymptotic behaviour proved in (Escobedo, Mischler, Rodriguez-Ricard, Ann. IHP, 2004) to obtain uniqueness of the triplet $(α, γ, k \_0)$ and a representation formula for $k\_0$. To invert this formula, one of the delicate points is to prove that the Mellin transform of the asymptotic profile never vanishes, what we do through the use of the Cauchy integral.

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arXiv2018

On a system of equations for the normal fluid-condensate interaction in a Bose gas

The existence of global solutions for a system of differential equations is proved, and some of their properties are described. The system involves a kinetic equation for quantum particles. It is a simplified version of a mathematical description of a weakly interacting dilute gas of bosons in the presence of a condensate near the critical temperature.

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arXiv2017

On the non existence of non negative solutions to a critical Growth-Fragmentation Equation

A growth fragmentation equation with constant dislocation density measure is considered, in which growth and division rates balance each other. This leads to a simple example of equation where the so called Malthusian hypothesis $(M_+)$ of J. Bertoin and A. Watson (2016) is not necessarily satisfied. It is proved that when that happens, and as it was first suggested by these authors, no global non negative weak solution, satisfying some boundedness condition on several of its moments, exist. Non existence of local non negative solutions satisfying a similar condition, is proved to happen also. When a local non negative solution exists, the explicit expression is given.

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arXiv2016

Semi-group theory for the Stokes operator with Navier-type boundary conditions on $L^{p}$-spaces

In this article we consider the Stokes problem with Navier-type boundary conditions on a domain $Ω$, not necessarily simply connected. Since under these conditions the Stokes problem has a non trivial kernel, we also study the solutions lying in the orthogonal of that kernel. We prove the analyticity of several semigroups generated by the Stokes operator considered in different functional spaces. We obtain strong, weak and very weak solutions for the time dependent Stokes problem with the Navier-type boundary condition under different hypothesis on the initial data $\boldsymbol{u}_0$ and external force $\boldsymbol{f}$. Then, we study the fractional and pure imaginary powers of several operators related with our Stokes operators. Using the fractional powers, we prove maximal regularity results for the homogeneous Stokes problem. On the other hand, using the boundedness of the pure imaginary powers we deduce maximal $L^{p}-L^{q}$ regularity for the inhomogeneous Stokes problem.

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arXiv2015

Time Asymptotics for a Critical Case in Fragmentation and Growth-Fragmentation Equations

Fragmentation and growth-fragmentation equations is a family of problems with varied and wide applications. This paper is devoted to description of the long time time asymptotics of two critical cases of these equations, when the division rate is constant and the growth rate is linear or zero. The study of these cases may be reduced to the study of the following fragmentation equation:$$\frac{\partial}{\partial t} u(t,x) + u(t,x)=\int\limits\_x^\infty k\_0(\frac{x}{y}) u(t,y) dy.$$Using the Mellin transform of the equation, we determine the long time behavior of the solutions. Our results show in particular the strong dependence of this asymptotic behavior with respect to the initial data.

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arXiv2014

Convergence to equilibrium of a linearized quantum Boltzmann equation for bosons at very low temperature

We consider an approximation of the linearised equation of the homogeneous Boltzmann equation that describes the distribution of quasiparticles in a dilute gas of bosons at low temperature. The corresponding collision frequency is neither bounded from below nor from above. We prove the existence and uniqueness of solutions satisfying the conservation of energy. We show that these solutions converge to the corresponding stationary state, at an algebraic rate as time tends to infinity.

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