This paper is devoted to the analysis of a numerical scheme for the coagulation and fragmentation equation with diffusion in space. A finite volume scheme is developed, based on a conservative formulation of the spatially nonhomogeneous coagulation-fragmentation model. It is shown that the scheme preserves positivity, total volume and global steady states. Finally, several numerical simulations are performed to investigate the long time behavior of the solution.

In this paper we are interested in the finite-time stability of transition solutions of the Cahn-Hilliard equation and its connection to the Willmore functional. We show that the Willmore functional locally decreases or increases in time in the linearly stable or unstable case respectively. This linear analysis explains the behavior near stationary solutions of the Cahn-Hilliard equation. We perform numerical examples in one and two dimensions and show that in the neighbourhood of transition solutions local instabilities occur in finite time. We also show convergence of solutions of the Cahn-Hilliard equation for arbitrary dimension to a stationary state by proving asymptotic decay of the Willmore functional in time.

In a time reversal experiment, a signal recorded by an array of transducers and sent back time-reversed into the same medium approximately refocuses on the original source center. The refocusing resolution is improved in an inhomogeneous medium. In this work we study the effect of changes in the medium, namely, the case when back propagation takes place in a perturbed medium.

Under the paraxial approximation assumption for a medium with weak inhomogeneities we consider a high frequency white noise regime. We show that relatively small perturbations do not affect the stable refocusing (self-averaging) for a localized source, but produce an interesting blurring of the refocused time signal. In some simple situations this effect can be explicitly quantified and related to the statistical model for the medium.

The aim of this paper is to study the local in time well posedness of the incompressible k-ε model in a 3d periodic domain. In the case when turbulent diffusion is much smaller than dissipation, asymptotic expansions are also derived, supposing that the velocity of the fluid can be neglected.

A reduced strongly nonlinear model is derived for the evolution of one-dimensional internal waves over an arbitrary bottom topography. Two layers containing inviscid, immiscible, irrotational fluids of different densities are defined. The upper layer is shallow compared with the characteristic wavelength at the interface, while the bottom region’s depth is comparable to the wavelength. The nonlinear evolution equations obtained are in terms of the internal wave elevation and the mean upper-velocity for the configuration described. The system is a generalization of the one proposed by Choi and Camassa for the flat bottom case in the same physical settings. Due to the presence of a topography a variable coefficient system of partial differential equations arises. These Boussinesq-type equations contain the Intermediate Long Wave (ILW) equation and the Benjamin- Ono (BO) equation when restricted to the unidirectional propagation regime. We intend to use this model to study the interaction of waves with the bottom profile. The dynamics include wave scattering, dispersion and attenuation, among other phenomena.

In this paper we introduce a simple continuum model for swarming of organisms in which there is a nonlocal aggregation term with an asymmetric sensing kernel countered by a nonlinear diffusion. The model can be thought of as capturing the dynamics of organisms that aggregate responding to visual cues and whose field of vision is predominantly in the direction of motion. The model has a variety of traveling solutions, including compact swarms where the speed of the swarm increases with the total number of organisms in the swarm (up to a largest swarm), traveling fronts and periodic waves. The dynamics of fully time-dependent solutions include complicated inelastic swarm interactions and the spontaneous breakup of small populations into compact swarms. We also comment on a kinetic formulation that leads to our equations and introduce a bi-directional model where the organisms are allowed to change directions.

This paper is devoted mainly to the global existence problem for the two-dimensional parabolic-parabolic Keller-Segel system in the full space. We derive a critical mass threshold below which global existence is ensured. Carefully using energy methods and ad hoc functional inequalities, we improve and extend previous results in this direction. The given threshold is thought to be the optimal criterion, but this question is still open. This global existence result is accompanied by a detailed discussion on the duality between the Onofri and the logarithmic Hardy-Littlewood-Sobolev inequalities that underlie the following approach.

High frequency solutions to partial differential equations (PDEs) are notoriously difficult to simulate numerically due to the large number of grid points required to resolve the wave oscillations. In applications, one often must rely on approximate solution methods to describe the wave field in this regime. Gaussian beams are asymptotically valid high frequency solutions concen- trated on a single curve through the domain. We show that one can form integral superpositions of such Gaussian beams to generate more general high frequency solutions to PDEs.

As a particular example, we look at high frequency solutions to the constant coefficient wave equation and construct Gaussian beam solutions with Taylor expansions of several orders. Since this PDE can be solved via a Fourier transform, we use the Fourier transform solution to gauge the error of the constructed Gaussian beam superposition solutions. Furthermore, we look at an example for which the solution exhibits a cusp caustic and investigate the order of magnitude of the wave amplitude as a function of frequency at the tip of the cusp. We show that the observed behavior is in agreement with the predictions of Maslov theory.

The goal of this paper is to give a non-local sufficient condition for generalized Poincaré inequalities which extends the well-known Bakry-Emery condition. Such generalized Poincaré inequalities have been introduced by W. Beckner in the Gaussian case and provide, along the Ornstein-Uhlenbeck flow, the exponential decay of some generalized entropies which interpolate between the L2 norm and the usual entropy. Our criterion improves on results which, for instance, can be deduced from the Bakry-Emery criterion and Holley-Stroock type perturbation results. In a second step, we apply the same strategy to non-linear equations of porous media type. This provides new interpolation inequalities and decay estimates for the solutions of the evolution problem. The criterion is again a non-local condition based on the positivity of the lowest eigenvalue of a Schrödinger operator. In both cases, we relate the Fisher information with its time derivative. Since the resulting criterion is non-local, it is better adapted to potentials with, for instance, a non-quadratic growth at infinity, or to unbounded perturbations of the potential.

Global existence is proved in the case of positive cosmological constant, and asymptotic behavior is investigated.

We generalize the Perron–Frobenius Theorem for nonnegative matrices to the class of nonnegative tensors.

We study the asymptotic behavior of the solutions of barotropic two-phase two-pressure models, with pressure relaxation, drag force and external forces. Using Chapman-Enskog expansions close to the expected equilibrium, a drift-flux model with a Darcy type closure law is obtained. Also, restricting this closure law to permanent flows (defined as steady flows in some Lagrangian frame), we can obtain a drift-flux model with an algebraic closure law, in the spirit of Zuber-Findlay models. The example of a two-phase flow in a vertical pipe is described.