772 research outputs found
On the analyticity and Gevrey class regularity up to the boundary for the Euler Equations
We consider the Euler equations in a three-dimensional Gevrey-class bounded
domain. Using Lagrangian coordinates we obtain the Gevrey-class persistence of
the solution, up to the boundary, with an explicit estimate on the rate of
decay of the Gevrey-class regularity radius
Mean field limit for bosons and propagation of Wigner measures
We consider the N-body Schr\"{o}dinger dynamics of bosons in the mean field
limit with a bounded pair-interaction potential. According to the previous work
\cite{AmNi}, the mean field limit is translated into a semiclassical problem
with a small parameter , after introducing an
-dependent bosonic quantization. The limit is expressed as a
push-forward by a nonlinear flow (e.g. Hartree) of the associated Wigner
measures. These object and their basic properties were introduced in
\cite{AmNi} in the infinite dimensional setting. The additional result
presented here states that the transport by the nonlinear flow holds for rather
general class of quantum states in their mean field limit.Comment: 10 page
Mean field limit for Bosons with compact kernels interactions by Wigner measures transportation
We consider a class of many-body Hamiltonians composed of a free (kinetic)
part and a multi-particle (potential) interaction with a compactness assumption
on the latter part. We investigate the mean field limit of such quantum systems
following the Wigner measures approach. We prove the propagation of these
measures along the flow of a nonlinear (Hartree) field equation. This enhances
and complements some previous results in the subject.Comment: 27 pages. arXiv admin note: text overlap with arXiv:1111.5918 by
other author
Hilbert Expansion from the Boltzmann equation to relativistic Fluids
We study the local-in-time hydrodynamic limit of the relativistic Boltzmann
equation using a Hilbert expansion. More specifically, we prove the existence
of local solutions to the relativistic Boltzmann equation that are nearby the
local relativistic Maxwellian constructed from a class of solutions to the
relativistic Euler equations that includes a large subclass of near-constant,
non-vacuum fluid states. In particular, for small Knudsen number, these
solutions to the relativistic Boltzmann equation have dynamics that are
effectively captured by corresponding solutions to the relativistic Euler
equations.Comment: 50 page
Interplay between the Beale-Kato-Majda theorem and the analyticity-strip method to investigate numerically the incompressible Euler singularity problem
Numerical simulations of the incompressible Euler equations are performed
using the Taylor-Green vortex initial conditions and resolutions up to
. The results are analyzed in terms of the classical analyticity strip
method and Beale, Kato and Majda (BKM) theorem. A well-resolved acceleration of
the time-decay of the width of the analyticity strip is observed at
the highest resolution for while preliminary 3D visualizations
show the collision of vortex sheets. The BKM criterium on the power-law growth
of supremum of the vorticity, applied on the same time-interval, is not
inconsistent with the occurrence of a singularity around .
These new findings lead us to investigate how fast the analyticity strip
width needs to decrease to zero in order to sustain a finite-time singularity
consistent with the BKM theorem. A new simple bound of the supremum norm of
vorticity in terms of the energy spectrum is introduced and used to combine the
BKM theorem with the analyticity-strip method. It is shown that a finite-time
blowup can exist only if vanishes sufficiently fast at the
singularity time. In particular, if a power law is assumed for then
its exponent must be greater than some critical value, thus providing a new
test that is applied to our Taylor-Green numerical simulation.
Our main conclusion is that the numerical results are not inconsistent with a
singularity but that higher-resolution studies are needed to extend the
time-interval on which a well-resolved power-law behavior of takes
place, and check whether the new regime is genuine and not simply a crossover
to a faster exponential decay
Semiclassical Theory of Time-Reversal Focusing
Time reversal mirrors have been successfully implemented for various kinds of
waves propagating in complex media. In particular, acoustic waves in chaotic
cavities exhibit a refocalization that is extremely robust against external
perturbations or the partial use of the available information. We develop a
semiclassical approach in order to quantitatively describe the refocusing
signal resulting from an initially localized wave-packet. The time-dependent
reconstructed signal grows linearly with the temporal window of injection, in
agreement with the acoustic experiments, and reaches the same spatial extension
of the original wave-packet. We explain the crucial role played by the chaotic
dynamics for the reconstruction of the signal and its stability against
external perturbations.Comment: 4 pages, 1 figur
Macro stress testing with a macroeconomic credit risk model: Application to the French manufacturing sector.
The aim of this paper is to build and estimate a macroeconomic model of credit risk for the French manufacturing sector. This model is based on Wilson's CreditPortfolioView model (1997a, 1997b); it enables us to simulate loss distributions for a credit portfolio for several macroeconomic scenarios. We implement two simulation procedures based on two assumptions relative to probabilities of default (PDs): in the first procedure, firms are assumed to have identical default probabilities; in the second, individual risk is taken into account. The empirical results indicate that these simulation procedures lead to quite different loss distributions. For instance, a negative one standard deviation shock on output leads to a maximum loss of 3.07% of the financial debt of the French manufacturing sector, with a probability of 99%, under the identical default probability hypothesis versus 2.61% with individual default probabilities.macro stress test ; credit risk model ; loss distribution.
Mean-field evolution of fermions with singular interaction
We consider a system of N fermions in the mean-field regime interacting
though an inverse power law potential , for
. We prove the convergence of a solution of the many-body
Schr\"{o}dinger equation to a solution of the time-dependent Hartree-Fock
equation in the sense of reduced density matrices. We stress the dependence on
the singularity of the potential in the regularity of the initial data. The
proof is an adaptation of [22], where the case is treated.Comment: 16 page
Approach to equilibrium for the phonon Boltzmann equation
We study the asymptotics of solutions of the Boltzmann equation describing
the kinetic limit of a lattice of classical interacting anharmonic oscillators.
We prove that, if the initial condition is a small perturbation of an
equilibrium state, and vanishes at infinity, the dynamics tends diffusively to
equilibrium. The solution is the sum of a local equilibrium state, associated
to conserved quantities that diffuse to zero, and fast variables that are
slaved to the slow ones. This slaving implies the Fourier law, which relates
the induced currents to the gradients of the conserved quantities.Comment: 23 page
Inviscid Large deviation principle and the 2D Navier Stokes equations with a free boundary condition
Using a weak convergence approach, we prove a LPD for the solution of 2D
stochastic Navier Stokes equations when the viscosity converges to 0 and the
noise intensity is multiplied by the square root of the viscosity. Unlike
previous results on LDP for hydrodynamical models, the weak convergence is
proven by tightness properties of the distribution of the solution in
appropriate functional spaces
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