Related papers: On convergence to equilibrium distribution for Dir…
We consider the singular Rayleigh-Jeans equilibrium of the $4$-waves kinetic turbulence equation for the three dimensional Schr\"{o}dinger equation. We first show the formation in finite time of a Dirac measure at zero frequency in the…
This paper studies the large time behavior of aggregation-diffusion equations. For one spatial dimension with certain assumptions on the interaction potential, the diffusion index $m$, and the initial data, we prove the convergence to the…
We prove that the solution of the Kac analogue of Boltzmann's equation can be viewed as a probability distribution of a sum of a random number of random variables. This fact allows us to study convergence to equilibrium by means of a few…
We study spectral properties of a one-dimensional Dirac equation with various disorder. We use replicas to calculate the exact density of state and typical localization length of a Dirac particle in several cases. We show that they can be…
We study a mean field planning problem in which the initial density is a Dirac mass. We show that there exists a unique solution which converges to a self-similar profile as time tends to $0$. We proceed by studying a continuous rescaling…
We study the long time statistics of a class of semi--linear damped wave equations with polynomial nonlinearities and perturbed by additive Gaussian noise in dimensions 2 and 3. We find that if sufficiently many directions in the phase…
The purpose of this paper is to investigate several issues concerning the Dirac equation from a time-frequency analysis perspective. More precisely, we provide estimates in weighted modulation and Wiener amalgam spaces for the solutions of…
This paper concerns the kinetic limit of the Dirac equation with random electromagnetic field. We give a detailed mathematical analysis of the radiative transport limit for the phase space energy density of solutions to the Dirac equation.…
It is shown that the amplitude for reflection of a Dirac particle with arbitrarily low momentum incident on a potential of finite range is -1 and hence the transmission coefficient T=0 in general. If however the potential supports a…
We present direct and elementary commutator techniques for the Dirac equation with long-range electric and mass perturbations. The main results are absence of generalized eigenfunctions and locally uniform resolvent estimates, both in terms…
It is proved that the solutions to the singular stochastic $p$-Laplace equation, $p\in (1,2)$ and the solutions to the stochastic fast diffusion equation with nonlinearity parameter $r\in (0,1)$ on a bounded open domain $\Lambda\subset\R^d$…
The main result of the article is the rate of convergence to the Rosenblatt-type distributions in non-central limit theorems. Specifications of the main theorem are discussed for several scenarios. In particular, special attention is paid…
In this work we solve the Dirac equation by constructing the exact bound state solutions for a mixing of vector and scalar generalized Hartmann potentials. This is done provided the vector potential is equal to or minus the scalar…
This paper deals with a one--dimensional model for granular materials, which boils down to an inelastic version of the Kac kinetic equation, with inelasticity parameter $p>0$. In particular, the paper provides bounds for certain distances…
In this paper, we consider the Maxwell-Dirac system in 3 dimension under zero magnetic field. We prove the global well-posedness and modified scattering for small solutions in the weighted Sobolev class. Imposing the Lorenz gauge condition,…
We provide the first quantitative result of convergence to equilibrium in the context of the spatially homogeneous Boltzmann-Fermi-Dirac equation associated to hard potentials interactions under angular cut-off assumption, providing an…
A manifestly covariant relativistic statistical mechanics of the system of $N$ indistinguishable events with motion in space-time parametrized by an invariant ``historical time'' $\tau $ is considered. The relativistic mass distribution for…
This paper investigates solutions of hyperbolic diffusion equations in $\mathbb{R}^3$ with random initial conditions. The solutions are given as spatial-temporal random fields. Their restrictions to the unit sphere $S^2$ are studied. All…
We discuss the application of recent results on generalized solutions to the Cauchy problem for hyperbolic systems to Dirac equations with external fields. In further analysis we focus on the question of existence of associated…
We consider the drift-diffusion equation $$ u_t-\varepsilon \Delta u+\nabla\cdot(u\nabla K\star u)=0 $$ in the whole space with global-in-time bounded solutions. Mass concentration phenomena for radially symmetric solutions of this equation…