Simulation of waves in complex poroelastic media is crucial in providing important geophysical information that cannot be obtained via simple elastic or acoustic model. Thus there is a need to design an artificial boundary condition for simulation using the numerical approximation of such problem. In this paper, our aim is to derive an exact nonreflecting boundary condition for the three dimensional poroelastic wave equations based on the Grote-Keller method. The proposed boundary condition is nonlocal in space, but local in time and can be coupled easily with standard numerical approaches for the computation of numerical solutions. Numerical results computed by the finite difference method demonstrate the effectiveness of our method.

In this work we discuss a few ways to create chaotic families that are not entropically chaotic on Kac's Sphere. We present two types of examples: limiting convex combination of an en-tropically chaotic family with a particularly 'bad' non-entropic family, and two explicitly computable families that vary rapidly with N, causing loss of support on the sphere or high entropic tails.

We introduce a numerical scheme to approximate a quasi-linear hyperbolic system which models the movement of cells under the influence of chemotaxis. Since we expect to find solutions which contain vacuum parts, we propose an upwinding scheme which handles properly the presence of vacuum and, besides, which gives a good approximation of the time asymptotic states of the system. For this scheme we prove some basic analytical properties and study its stability near some of the steady states of the system. Finally, we present some numerical simulations which show the dependence of the asymptotic behavior of the solutions upon the parameters of the system.

A coupled quantum-classical model describing the transport of electrons confined in nanoscale semiconductor devices is considered. Using the subband decomposition approach allows to separate the transport directions from the condinement direction. The motion of the gas in the transport direction is assumed to be classical. Then a hierarchy of adiabatic quantum-classical model is obtained, leading to subband SHE and energy-transport models, with explicit expression of the diffusion coefficients. The energy-transport-Schrodinger-Poisson model is then used for the numerical simulation of the transport of the electron gas in an ultra-scaled Double-Gate-MOSFET.

We prove the existence and provide the asymptotics for non local fronts in homogeneous media.

The objective of quantitative photoacoustic tomography (QPAT) is to reconstruct optical and thermodynamic properties of heterogeneous media from data of absorbed energy distribution inside the media. There have been extensive theoretical and computational studies on the inverse problem in QPAT, however, mostly in the diusive regime. We present in this work some numerical reconstruction algorithms for multisource QPAT in the radiative transport regime with energy data collected at either single or multiple wavelengths. We show that when the medium to be probed is nonscattering, explicit reconstruction schemes can be derived to reconstruct the absorption and the Gruneisen coefficients. When data at multiple wavelengths are utilized, we can reconstruct simultaneously the absorption, scattering and Gruneisen coefficients. We show by numerical simulations that the reconstructions are stable.

We derive WKB approximations for a class of Airy and parabolic cylinder functions in the complex plane, including quantitative error bounds. We prove that all zeros of the Airy function lie on a ray in the complex plane, and that the parabolic cylinder functions have no zeros. We analyze a limiting case where the parabolic cylinder functions go over to the Airy function.

We present a convergence proof of the projective integration method for a class of deterministic multi-dimensional multi-scale systems which are amenable to centre manifold theory. The error is shown to contain contributions associated with the numerical accuracy of the microsolver, the numerical accuracy of the macrosolver and the distance from the centre manifold caused by the combined effect of micro- and macrosolvers, respectively. We corroborate our results by numerical simulations.

This paper is concerned with diffuse-interface approximations of the Willmore flow. We first present numerical results of standard diffuse-interface models for colliding one dimensional interfaces. In such a scenario evolutions towards interfaces with corners can occur that do not necessarily describe the adequate sharp-interface dynamics. We therefore propose and investigate alternative diffuse-interface approximations that lead to a different and more regular behavior if interfaces collide. These dynamics are derived from approximate energies that converge to the L1-lower-semicontinuous envelope of the Willmore energy, which is in general not true for the more standard Willmore approximation.

A class of approximate Lennard-Jones (LJ) potentials with a small parameter is found whose Fourier transforms have a simple asymptotic behavior as the parameter goes to zero. When the LJ potential is replaced by the approximate LJ potential, the total energy functional becomes simple and exactly the same as replacing the LJ potential by a delta function. Such a simple energy functional can be used to derive the Poisson-Nernst-Planck equations with steric effects (PNP-steric equations), a new mathematical model for the LJ interaction in ionic solutions. Using formal asymptotic analysis, stability and instability conditions for the 1D PNP-steric equations with the Dirichlet boundary conditions for one anionic and cationic species are expressed by the valences, diffusion constants, ionic radii and coupling constants. This is the first step to study the dynamics of solutions of the PNP-steric equations.

We present a result on existence of solutions for a system of highly nonlinear and singular partial dierential equations obtained by coupling the two-crystallization Allen-Cahn equations to a singular Navier-Stokes system and a nonlinear heat equation. Such system constitutes a phase field model for non-isothermal solidication/melting processes of certain metallic alloys for which two different kinds of crystallization are possible. In this model, the liquid phase and each one of the possible crystallization states are described by their own phase fields. The possibility of occurrence of fluid flow in a a priori unknown non-solid region is also considered, turning the model into a free-boundary value problem.

The present work deals with the homogenization and indepth asymptotic analysis of a nonlinear stochastic evolution equation with non Lipschtiz nonlinearities in a domain with fine grained boundaries in which the obstacles have a non periodic distribution. Under appropriate conditions on the data it is proved that a solution of the initial problem converges in suitable topologies to a solution of a limit problem which contains an additional term of capacity type. The notion of solution is that of weak probabilistic which is a system consisting of a probability space, Wiener process and a solution in the distribution sense of the problem.

We discuss a 3D model describing the time evolution of nematic liquid crystals in the framework of Landau-de Gennes theory, where the natural physical constraints are enforced by a singular free energy bulk potential proposed by J.M. Ball and A. Majumdar. The thermal effects are present through the component of the free energy that accounts for intermolecular interactions. The model is consistent with the general principles of thermodynamics and mathematically tractable. We identify the a priori estimates for the associated system of evolutionary partial dierential equations and construct global-in-time weak solutions for arbitrary physically relevant initial data.

We consider the Vlasov-Poisson system in three space dimensions in the electrostatic case. For smooth solution with compactly supported initial datum, the growth estimate of its velocity support is improved and as a consequence, we obtain a better decay estimate of the electrical field.

In this paper, a two-dimensional version of the BBM equation will be considered. The existence and scattering of global small amplitude solutions to this equation will be studied. The orbital stability of solitary wave solutions of this equation will be also investigated.

Without imposing the so-called compatibility condition on the initial data, we obtain an asymptotic expansion of the Stokes solutions at small viscosity \epsilon as the sum of the linearized Euler solution and a corrector function, which balances the discrepancy on the boundary of the Stokes and the linearized Euler solutions. Using such an expansion and smallness of the corrector, as the viscosity \epsilon tends to zero, we obtain the uniform L^2 convergence of the Stokes solutions to the linearized Euler solution with rate of order \epsilon^{1/4}.

In this work, the multiscale problem of modeling fluctuations in boundary layers in stochastic elliptic partial differential equations is solved by homogenization. A homogenized equation for the covariance of the solution of stochastic elliptic PDEs is derived. In addition to the homogenized equation, a rate for the covariance and variance as the cell size tends to zero is given. For the homogenized problem, an existence and uniqueness result and further properties are shown. The multiscale problem stems from the modeling of the electrostatics in nanoscale field-effect sensors, where the fluctuations arise from random charge concentrations in the cells of a boundary layer. Finally, numerical results and a numerical verification are presented.

We study the evolution of the interface given by two incompressible fluids with different densities in the porous strip. This problem is known as the Muskat problem and is analogous to the two phase Hele-Shaw cell. The main goal of this paper is to compare the qualitative properties between the model when the fluids move without boundaries and the model when the fluids are confined. We find that, in a precise sense, the boundaries decrease the diffusion rate and the system becomes more singular.

We present in this work a rigorous passage to the limit (in dimension 1) in a system of three reaction-diffusion equations coming out of population dynamics, towards a system of two reaction-diffusion equations, one of which includes a cross diffusion term.

In this paper, we prove a blow-up criterion for 3D compressible visco-elasticity in terms of the upper bound of the density and the deformation tensor. Due to the special structure of the equation, we get the cancelation to the derivatives of the density and transform tensor, which brings us the desired results.

We study the asymptotic emergence of the complete synchronization of Kuramoto oscillators with hierarchical leadership. We show that asymptotic complete frequency synchronization occurs for some class of initial phase congurations, when Kuramoto oscillators are scattered on the nodes of a directed graph with a hierarchical leadership topology and the coupling strength is sufficiently large. This generalizes previous studies on the complete synchronization of Kuramoto oscillators on connected and symmetric graphs. We also provide several numerical results to confirm our analytical results.

Superparameterization is a multiscale numerical method wherein solutions of prognostic equations for small scale processes on local domains embedded within the computational grid of a large scale model are computed and used to force the large scales. It was developed initially in the atmospheric sciences, but stands on its own as a nascent numerical method for the simulation of multiscale phenomena. Here we develop a stochastic version of superparameterization in a difficult one dimensional test problem involving self-similarly collapsing solitons, dispersive waves, and an intermittent inverse cascade of energy from small to large scales. We derive the nonlinear model equations by imposing a formal scale separation between resolved large scales and unresolved small scales; this allows the use of subdomains embedded within the large scale grid to describe the local small scale processes. To decrease the computational cost, we make a systematic quasi-linear stochastic approximation of the nonlinear small scale equations and use the statistical mean of the nonlinear small scale forcing (the covariance) in the large scale equations. The stochastic approximation allows the embedded domains to be formally infinite (unrealistically large scales are suppressed on the embedded domains). Further simplifications allow us to precompute the small scale forcing terms in the large scale equations as functions of the large scale variables only, which results in significant computational savings. The results are positive. The method increases the energy in overdamped simulations, decreases the energy in underdamped simulations, and improves the spatial distribution and frequency of collapsing solitons.

The existence of global non-negative weak solutions is proved for coupled one-dimensional lubrication systems that describe the evolution of nanoscopic bilayer thin polymer films that take account of Navier-slip or no-slip conditions at both liquid-liquid and liquid-solid interfaces. In addition, in the presence of attractive van der Waals and repulsive Born intermolecular interactions existence of positive smooth solutions is shown.

We consider the initial-boundary value problem for the coupled Navier-Stokes-Keller-Segel-Fisher-Kolmogorov-Petrovskii-Piskunov system in two- and three-dimensional domains. The problem describes oxytaxis and growth of Bacillus subtilis in moving water. We prove existence of global weak solutions to the problem. We distinguish between two cases determined by the cell diffusion term and the space dimension, which are referred to as the supercritical and subcritical ones. At the first case, the choice of the kinetic function enjoys wide range of possibilities: in particular, it can be zero. Our results are new even at the absence of the kinetic term. At the second case, the restrictions on the kinetic function are less relaxed: for instance, it cannot be zero but can be Fisher-like. In the case of linear cell diffusion, the solution is regular and unique provided the domain is the whole plane. In addition, we study the long-time behaviour of the problem, find dissipative estimates, and construct attractors.

The aim of this paper is to derive and analyze diffusion models for semiconductor spintronics. We begin by presenting and studying the so called "spinor" Boltzmann equation. Starting then from a rescaled version of linear Boltzmann equation with different spin-ip and non spin-ip collision operators, different continuum (drift-diffusion) models are derived. By compar- ing the strength of the spin-orbit scattering with the scaled mean free paths, we explain how some models existing in the literature (like the two-component models) can be obtained from the spinor Boltzmann equation. A new spin-vector drift-diffusion model keeping spin relaxation and spin precession effects due to the spin-orbit coupling in semiconductor structures is derived and some of its mathematical properties are checked.

We apply the penalty method to the curve straightening flow of inextensible planar open curves generated by the Kirchho bending energy. Thus we consider the curve straightening flow of extensible planar open curves generated by a combination of the Kirchho bending energy and a functional penalising deviations from unit arc-length. We start with the governing equations of the explicit parametrisation of the curve and derive an equivalent system for the two quantities indicatrix and arc-length. We prove existence and regularity of solutions and use the indicatrix/arc-length representation to compute the energy dissipation. We prove its coercivity and conclude exponential decay of the energy. Finally, by an application of the Lions-Aubin Lemma, we prove convergence of solutions to a limit curve which is the solution of an analogous gradient flow on the manifold of inextensible open curves. This procedure also allows to characterise the Lagrange multiplier in the limit model as a weak limit of force terms present in the relaxed model.

We investigate the long term behavior in terms of global attractors, as time goes to infinity, of solutions to a continuum model for biological aggregations in which individuals experience long-range social attraction and short range dispersal. We consider the aggregation equation with both degenerate and non-degenerate diffusion in a bounded domain subject to various boundary conditions. In the degenerate case, we prove the existence of the global attractor and derive some optimal regularity results. Furthermore, in the non-degenerate case we give a complete structural characterization of the global attractor, and also discuss the convergence of any bounded solutions to steady states. In particular, under suitable assumptions on the parameters of the problem, we establish the convergence of the bounded solution u(t) to a single steady state and the rate of convergence. Finally, the existence of an exponential attractor is also demonstrated for suciently smooth kernels in the case of non-degenerate diffusion.

The low Mach number limit of global smooth solutions to the compressible magnetohydrodynamic equations in a bounded smooth domain in R^2 with perfectly conducting boundary is verified for all time, provided that the initial data are well-prepared.

We consider an elliptic equation with a divergence-free drift b. We prove that an inequality of Harnack type holds under the assumption b \in L^{n/2+\delta} where \delta>0. As an application we provide a one-sided Liouville's theorem provided that b\in L^{n/2+\delta}(R^n).