A logarithmic fourth-order parabolic equation in one space dimension with periodic boundary conditions is studied. This equation arises in the context of fluctuations of a stationary nonequilibrium interface and in the modeling of quantum semiconductor devices. The existence of global-in-time non-negative weak solutions and some regularity results are shown. Furthermore, we prove that the solution converges exponentially fast to its mean value in the ``entropy norm'' and in the Fisher information, using a new optimal logarithmic Sobolev inequality for higher derivatives. In particular, the rate is independent of the solution and the constant depends only on the initial value of the entropy.

On étudie une équation non linéaire parabolique du quatrième ordre en une dimension d'espace avec des conditions au bord péeriodiques. Cette équation apparaît dans le contexte des fluctuations d'interfaces stationnaires hors d'équilibre et dans la modélisation de dispositifs semi-conducteurs. On donne un critère d'unicité des solutions faibles positives ou nulles. Pour finir, on démontre que la solution converge exponentiellement vite vers sa valeur moyenne en "norme entropique'' en utilisant une nouvelle inégalité de Sobolev logarithmique optimale pour des dérivées d'ordre élevé.

We prove a global in time existence theorem for the initial value problem for the Einstein-Boltzmann system, with positive cosmological constant and arbitrarily large initial data, in the spatially homogeneous case, in a Robertson-Walker space-time.

In this paper, we extend the model for supply chains developed in [8]. The model consists of partial differential equations governing the dynamics on each processor. Furthermore, a modelling of different types of vertices is motivated and discussed. Then, optimization problems are introduced and numerically investigated. A comparison of computing times shows the efficiency of partial differential equations for solving supply chain problems.

Two nonlinear Maxwell systems are considered: Kerr model exhibiting an instantaneous response of the medium, Kerr-Debye model which contains some delay term and is a relaxation approximation of the first one. In one space dimension, we prove that the limit of the solution to the ingoing wave condition for Kerr-Debye model is a solution to the Kerr model.

We perform stability analysis on the finite-difference time domain method (FDTD) when extended to incorporate local space-time adaptive mesh refinement (AMR). The neutrally stable Yee algorithm becomes extremely sensitive to perturbations introduced by the interpolation schemes employed at grid refinement interfaces. In this paper we investigate the stability of a range of interpolation schemes using Gustafsson-Kreiss-Sundstrom-Trefethen (GKS-T) mode and reflection/transmission coefficients analysis on the infinite domain with a single interface. This analysis allows detection of trapping instabilities, exponentially growing modes, mode resonances with the interface and mode-mode resonances. We also apply matrix stability analysis for more complicated computational domains containing multiple grid refinement interfaces.

At the equator, the Coriolis force from rotation vanishes identically so that multiple time scale dynamics for the equatorial shallow water equation naturally leads to singular limits of symmetric hyperbolic systems with fast variable coefficients. The classical strategy of using energy estimates for higher spatial derivatives has a fundamental difficulty since formally the commutator terms explode in the limit. Here this fundamental difficulty is circumvented by exploiting the special structure of the equatorial shallow water equations in suitable new variables involving the raising and lowering operators for the quantum harmonic oscillator, and obtaining uniform higher derivative estimates in a new function space based on the Hermite operator. The result is a completely new theorem characterizing the singular limit of the equatorial shallow water equations in the long wave regime, even with general unbalanced initial data, as a solution of the equatorial long wave equation. The results presented below point the way for rigorous PDE analysis of both the equatorial shallow water equations and the equatorial primitive equations in other physically relevant singular limit regimes.

Three dimensional (3D) Navier-Stokes-$\alpha$ equations are considered for uniformly rotating geophysical fluid flows (large Coriolis parameter $f=2\Omega$). The Navier-Stokes-$\alpha$ equations are a nonlinear dispersive regularization of usual Navier-Stokes equations obtained by Lagrangian averaging. The focus is on the existence and global regularity of solutions of the 3D rotating Navier-Stokes-$\alpha$ equations and the uniform convergence of these solutions to those of the original 3D rotating Navier-Stokes equations for large Coriolis parameters $f$ as $\alpha\rightarrow 0$. Methods are based on fast singular oscillating limits and results are obtained for periodic boundary conditions for all domain aspect ratios, including the case of three wave resonances which yields nonlinear resonant limit $\alpha$-equations for $f\rightarrow\infty$. The existence and global regularity of solutions of resonant limit $\alpha$-equations is established, uniformly in $\alpha$. Bootstrapping from global regularity of the resonant limit $\alpha$-equations, the existence of a regular solution of the full 3D rotating Navier-Stokes-$\alpha$ equations for large $f$ for an infinite time is established. Then we prove the existence of exponential attractors of the 3D rotating Navier-Stokes-$\alpha$ equations ($\alpha\neq 0$) and the convergence of the exponential attractors to those of the original 3D rotating Navier-Stokes equations ($\alpha =0$) for $f$ large but fixed as $\alpha\rightarrow 0$. All the estimates are uniform in $\alpha$, in contrast with previous estimates in the literature which blow up as $\alpha\rightarrow 0$.

The two-mass model of the vocal folds is a popular representation of their dynamical structure used in phonation studies. This paper presents an analysis of a recent piecewise-smooth version of the model. This version has two equilibrium positions, and in one of them (the initial prephonatory position) the system is nondifferentiable. Standard methods of stability analysis do not apply for that position, because they require smoothness of the system. A geometrical approach is applied instead, which is an extension of a method previously developed for planar systems. The analysis shows the existence of a transcritical bifurcation between the equilibrium positions, and a Hopf bifurcation related to each of them. The oscillation region of the model is next determined as the area delimited by the Hopf bifurcations. The results are illustrated by a bifurcation diagram and trajectory plots.

In this short note we study the model of subsonic detonation introduced by Sivashinsky. The model is described by the system of reaction-diffusion equations involving temperature, pressure and concentration of deficient reactant. It is shown that initial data with small support lead to quenching (decay of solution). In contrast, initial data with support large enough lead to propagation with finite velocity.