Pulsating Fronts in a 2D Reactive Boussinesq System

We consider a reactive Boussinesq system with no stress boundary conditions in a periodic domain which is unbounded in one direction. Specifically, we couple the reaction-advection-diffusion equation for the temperature, $T$, and the linearized Navier-Stokes equation with the Boussinesq approximation for the fluid flow, $u$. We show that this system admits smooth pulsating front solutions that propagate with a positive, fixed speed.Comment: 35 pages, 3 figure

C^\infty smoothing for weak solutions of the inhomogeneous Landau equation

We consider the spatially inhomogeneous Landau equation with initial data that is bounded by a Gaussian in the velocity variable. In the case of moderately soft potentials, we show that weak solutions immediately become smooth and remain smooth as long as the mass, energy, and entropy densities remain under control. For very soft potentials, we obtain the same conclusion with the additional assumption that a sufficiently high moment of the solution in the velocity variable remains bounded. Our proof relies on the iteration of local Schauder-type estimates.Comment: 23 pages, updated with to-be-published versio

Can banks circumvent minimum capital requirements? The case of mortgage portfolios under Basel II

The recent mortgage crisis has resulted in several bank failures as the number of mortgage defaults increased. The current Basel I capital framework does not require banks to hold sufficient amounts of capital to support their mortgage lending activities. The new Basel II capital rules are intended to correct this problem. However, Basel II models could become too complex and too costly to implement, often resulting in a trade-off between complexity and model accuracy. In addition, the variation of the model, particularly how mortgage portfolios are segmented, could have a significant impact on the default and loss estimated and, thus, could affect the amount of capital that banks are required to hold. This paper finds that the calculated Basel II capital varies considerably across the default prediction model and segmentation schemes, thus providing banks with an incentive to choose an approach that results in the least required capital for them. The authors also find that a more granular segmentation model produces smaller required capital, regardless of the economic environment. In addition, while borrowers' credit risk factors are consistently superior, economic factors have also played a role in mortgage default during the financial crisis.Capital ; Banks and banking ; Basel capital accord

Super-linear spreading in local and non-local cane toads equations

In this paper, we show super-linear propagation in a nonlocal reaction-diffusion-mutation equation modeling the invasion of cane toads in Australia that has attracted attention recently from the mathematical point of view. The population of toads is structured by a phenotypical trait that governs the spatial diffusion. In this paper, we are concerned with the case when the diffusivity can take unbounded values, and we prove that the population spreads as $t^{3/2}$. We also get the sharp rate of spreading in a related local model

Super-linear propagation for a general, local cane toads model

We investigate a general, local version of the cane toads equation, which models the spread of a population structured by unbounded motility. We use the thin-front limit approach of Evans and Souganidis in [Indiana Univ. Math. J., 1989] to obtain a characterization of the propagation in terms of both the linearized equation and a geometric front equation. In particular, we reduce the task of understanding the precise location of the front for a large class of equations to analyzing a much smaller class of Hamilton-Jacobi equations. We are then able to give an explicit formula for the front location in physical space. One advantage of our approach is that we do not use the explicit trajectories along which the population spreads, which was a basis of previous work. Our result allows for large oscillations in the motility