33 research outputs found
From Bloch model to the rate equations II: the case of almost degenerate energy levels
Bloch equations give a quantum description of the coupling between an atom
and a driving electric force. In this article, we address the asymptotics of
these equations for high frequency electric fields, in a weakly coupled regime.
We prove the convergence towards rate equations (i.e. linear Boltzmann
equations, describing the transitions between energy levels of the atom). We
give an explicit form for the transition rates. This has already been performed
in [BFCD03] in the case when the energy levels are fixed, and for different
classes of electric fields: quasi or almost periodic, KBM, or with continuous
spectrum. Here, we extend the study to the case when energy levels are possibly
almost degenerate. However, we need to restrict to quasiperiodic forcings. The
techniques used stem from manipulations on the density matrix and the averaging
theory for ordinary differential equations. Possibly perturbed small divisor
estimates play a key role in the analysis. In the case of a finite number of
energy levels, we also precisely analyze the initial time-layer in the rate
aquation, as well as the long-time convergence towards equilibrium. We give
hints and counterexamples in the infinite dimensional case
Uniform regularity for the Navier-Stokes equation with Navier boundary condition
We prove that there exists an interval of time which is uniform in the
vanishing viscosity limit and for which the Navier-Stokes equation with Navier
boundary condition has a strong solution. This solution is uniformly bounded in
a conormal Sobolev space and has only one normal derivative bounded in
. This allows to get the vanishing viscosity limit to the
incompressible Euler system from a strong compactness argument
An overview on the approximation of boundary Riemann problems through physical viscosity
This note aims at providing an overview of some recent results concerning the viscous approximation of so-called boundary Riemann problems for nonlinear systems of conservation laws in small total variation regimes. \ua9 2016, Sociedade Brasileira de Matem\ue1tica
A uniqueness criterion for viscous limits of boundary Riemann problems
We deal with initial-boundary value problems for systems of conservation laws in one space dimension and we focus on the boundary Riemann problem. It is known that, in general, different viscous approximations provide different limits. In this paper, we establish sufficient conditions to conclude that two different approximations lead to the same limit. As an application of this result, we show that, under reasonable assumptions, the self-similar second-order approximation and the classical viscous approximation provide the same limit. Our analysis applies to both the characteristic and the non characteristic case. We require neither genuine nonlinearity nor linear degeneracy of the characteristic fields