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

Mean-Field- and Classical Limit of Many-Body Schr\"odinger Dynamics for Bosons

We present a new proof of the convergence of the N-particle Schroedinger dynamics for bosons towards the dynamics generated by the Hartree equation in the mean-field limit. For a restricted class of two-body interactions, we obtain convergence estimates uniform in the Planck constant , up to an exponentially small remainder. For h=0, the classical dynamics in the mean-field limit is given by the Vlasov equation.Comment: Latex 2e, 18 page

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 $V(x)=1/|x|^{\alpha}$, for $\alpha\in(0,1]$. 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 $\alpha=1$ is treated.Comment: 16 page

Do investors see through mistakes in reported earnings?

This study investigates whether investors see through materially misstated earnings, and whether they anticipate earnings restatements. For firms that restate at least one annual report, we find that investors are misled by mistakes in reported earnings at the time of initial earnings announcements. Investors react positively to the component of the favorable earnings surprise that will subsequently be restated, and attach the same valuation to it as to the true earnings surprise. We also find that investors anticipate the subsequent downward restatements and start marking stock prices down several months before a restatement announcement, so that the full impact of a restatement is about three times as large as the initial announcement effect. Overall our findings indicate that although investors anticipate restatements several months before its announcement, they are misled by misstated earnings for several years and therefore would benefit from better quality of financial information

Blow-up of the hyperbolic Burgers equation

The memory effects on microscopic kinetic systems have been sometimes modelled by means of the introduction of second order time derivatives in the macroscopic hydrodynamic equations. One prototypical example is the hyperbolic modification of the Burgers equation, that has been introduced to clarify the interplay of hyperbolicity and nonlinear hydrodynamic evolution. Previous studies suggested the finite time blow-up of this equation, and here we present a rigorous proof of this fact

Some Applications of Fractional Equations

We present two observations related to theapplication of linear (LFE) and nonlinear fractional equations (NFE). First, we give the comparison and estimates of the role of the fractional derivative term to the normal diffusion term in a LFE. The transition of the solution from normal to anomalous transport is demonstrated and the dominant role of the power tails in the long time asymptotics is shown. Second, wave propagation or kinetics in a nonlinear media with fractal properties is considered. A corresponding fractional generalization of the Ginzburg-Landau and nonlinear Schrodinger equations is proposed.Comment: 11 page

The analytic structure of 2D Euler flow at short times

Using a very high precision spectral calculation applied to the incompressible and inviscid flow with initial condition $\psi_0(x_1, x_2) = \cos x_1+\cos 2x_2$, we find that the width $\delta(t)$ of its analyticity strip follows a $\ln(1/t)$ law at short times over eight decades. The asymptotic equation governing the structure of spatial complex-space singularities at short times (Frisch, Matsumoto and Bec 2003, J.Stat.Phys. 113, 761--781) is solved by a high-precision expansion method. Strong numerical evidence is obtained that singularities have infinite vorticity and lie on a complex manifold which is constructed explicitly as an envelope of analyticity disks.Comment: 19 pages, 14 figures, published versio

Semiclassical Propagation of Coherent States for the Hartree equation

In this paper we consider the nonlinear Hartree equation in presence of a given external potential, for an initial coherent state. Under suitable smoothness assumptions, we approximate the solution in terms of a time dependent coherent state, whose phase and amplitude can be determined by a classical flow. The error can be estimated in $L^2$ by C \sqrt {\var}, \var being the Planck constant. Finally we present a full formal asymptotic expansion

Energy decay for the damped wave equation under a pressure condition

We establish the presence of a spectral gap near the real axis for the damped wave equation on a manifold with negative curvature. This results holds under a dynamical condition expressed by the negativity of a topological pressure with respect to the geodesic flow. As an application, we show an exponential decay of the energy for all initial data sufficiently regular. This decay is governed by the imaginary part of a finite number of eigenvalues close to the real axis.Comment: 32 page

Rapid response to abalone virus depletion in western Victoria: information acquisition and reef code assessment

Future management of disease-affected abalone must adapt to the changing circumstances, and adopting a precautionary approach will allow maximum potential for stock recovery. This approach is mandated by the observation that no documented examples are known of abalone populations recovering from catastrophic impacts such as have occurred in the abalone fisheries of Victoria's Western and Central zones. Indeed the balance of international evidence points towards the contrary, so these fisheries are in dangerous territory. This need not mean that recovery cannot occur. However, the modelling results from this project confirm the above precautionary view and suggest that unless it is known with certainty that disease-induced mortalities have been moderate (less than 40%), then any resumption of fishing in the near term risks the future of the fishery. Acquisition of accurate mortality data is the only basis upon which fishing can recommence in the short term (within 5 years) and in many instances, such as for some among those reefs considered in our study, the opportunity has passed. The simulation results provide guidance, but their validity is conditional on myriad assumptions as well as on the accuracy of data employed. We already know that catches early in the fishery’s history were higher than reported officially, but how much higher is conjecture. Growth is highly variable over small spatial scales and feedback effects from reduced abundance together with changed size structure and persistence of habitat will play roles in determining the rate, if any, of recovery. The extent of the contemporary illegal catch is uncertain, particularly given the unprecedented closure of the fisheries. The results show that even small illegal catches can significantly degrade recovery where the viral impact is high, with clear implications for the enforcement aspects of managing these fisheries