485 research outputs found
Global exponential convergence to variational traveling waves in cylinders
We prove, under generic assumptions, that the special variational traveling
wave that minimizes the exponentially weighted Ginzburg-Landau functional
associated with scalar reaction-diffusion equations in infinite cylinders is
the long-time attractor for the solutions of the initial value problems with
front-like initial data. The convergence to this traveling wave is
exponentially fast. The obtained result is mainly a consequence of the gradient
flow structure of the considered equation in the exponentially weighted spaces
and does not depend on the precise details of the problem. It strengthens our
earlier generic propagation and selection result for "pushed" fronts.Comment: 23 page
Fine properties of self-similar solutions of the Navier-Stokes equations
We study the solutions of the nonstationary incompressible Navier--Stokes
equations in , , of self-similar form , obtained from small and homogeneous initial
data . We construct an explicit asymptotic formula relating the
self-similar profile of the velocity field to its corresponding initial
datum
Interaction of vortices in viscous planar flows
We consider the inviscid limit for the two-dimensional incompressible
Navier-Stokes equation in the particular case where the initial flow is a
finite collection of point vortices. We suppose that the initial positions and
the circulations of the vortices do not depend on the viscosity parameter \nu,
and we choose a time T > 0 such that the Helmholtz-Kirchhoff point vortex
system is well-posed on the interval [0,T]. Under these assumptions, we prove
that the solution of the Navier-Stokes equation converges, as \nu -> 0, to a
superposition of Lamb-Oseen vortices whose centers evolve according to a
viscous regularization of the point vortex system. Convergence holds uniformly
in time, in a strong topology which allows to give an accurate description of
the asymptotic profile of each individual vortex. In particular, we compute to
leading order the deformations of the vortices due to mutual interactions. This
allows to estimate the self-interactions, which play an important role in the
convergence proof.Comment: 39 pages, 1 figur
Probability density of quantum expectation values
We consider the quantum expectation value \mathcal{A}=\ of an
observable A over the state |\psi\> . We derive the exact probability
distribution of \mathcal{A} seen as a random variable when |\psi\> varies over
the set of all pure states equipped with the Haar-induced measure. The
probability density is obtained with elementary means by computing its
characteristic function, both for non-degenerate and degenerate observables. To
illustrate our results we compare the exact predictions for few concrete
examples with the concentration bounds obtained using Levy's lemma. Finally we
comment on the relevance of the central limit theorem and draw some results on
an alternative statistical mechanics based on the uniform measure on the energy
shell.Comment: Substantial revision. References adde
Orbital stability: analysis meets geometry
We present an introduction to the orbital stability of relative equilibria of
Hamiltonian dynamical systems on (finite and infinite dimensional) Banach
spaces. A convenient formulation of the theory of Hamiltonian dynamics with
symmetry and the corresponding momentum maps is proposed that allows us to
highlight the interplay between (symplectic) geometry and (functional) analysis
in the proofs of orbital stability of relative equilibria via the so-called
energy-momentum method. The theory is illustrated with examples from finite
dimensional systems, as well as from Hamiltonian PDE's, such as solitons,
standing and plane waves for the nonlinear Schr{\"o}dinger equation, for the
wave equation, and for the Manakov system
Risk Factors for Mortality after COVID-19 in Patients with Preexisting Interstitial Lung Disease.
International audienc
The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations
In the present study we prove rigorously that in the long-wave limit, the
unidirectional solutions of a class of nonlocal wave equations to which the
improved Boussinesq equation belongs are well approximated by the solutions of
the Camassa-Holm equation over a long time scale. This general class of
nonlocal wave equations model bidirectional wave propagation in a nonlocally
and nonlinearly elastic medium whose constitutive equation is given by a
convolution integral. To justify the Camassa-Holm approximation we show that
approximation errors remain small over a long time interval. To be more
precise, we obtain error estimates in terms of two independent, small, positive
parameters and measuring the effect of nonlinearity and
dispersion, respectively. We further show that similar conclusions are also
valid for the lower order approximations: the Benjamin-Bona-Mahony
approximation and the Korteweg-de Vries approximation.Comment: 24 pages, to appear in Discrete and Continuous Dynamical System
Coherent vortex structures and 3D enstrophy cascade
Existence of 2D enstrophy cascade in a suitable mathematical setting, and
under suitable conditions compatible with 2D turbulence phenomenology, is known
both in the Fourier and in the physical scales. The goal of this paper is to
show that the same geometric condition preventing the formation of
singularities - 1/2-H\"older coherence of the vorticity direction - coupled
with a suitable condition on a modified Kraichnan scale, and under a certain
modulation assumption on evolution of the vorticity, leads to existence of 3D
enstrophy cascade in physical scales of the flow.Comment: 15 pp; final version -- to appear in CM
Renormalizing Partial Differential Equations
In this review paper, we explain how to apply Renormalization Group ideas to
the analysis of the long-time asymptotics of solutions of partial differential
equations. We illustrate the method on several examples of nonlinear parabolic
equations. We discuss many applications, including the stability of profiles
and fronts in the Ginzburg-Landau equation, anomalous scaling laws in
reaction-diffusion equations, and the shape of a solution near a blow-up point.Comment: 34 pages, Latex; [email protected]; [email protected]
- …