Showing papers in "Quarterly of Applied Mathematics in 2015"

Moment growth bounds on continuous time Markov processes on non-negative integer lattices

Abstract: We consider Markov processes in continuous time with state space $\posint^N$ and provide two sufficient conditions and one necessary condition for the existence of moments $E(\|X(t)\|^r)$ of all orders $r \in at$ for all $t \geq 0$. The sufficient conditions also guarantee an exponential in time growth bound for the moments. The class of processes studied have finitely many state independent jumpsize vectors $ u_1,\dots, u_M$. This class of processes arise naturally in many applications such as stochastic models of chemical kinetics, population dynamics and queueing theory for example. We also provide a necessary and sufficient condition for stochiometric boundedness of species in terms of $ u_j$.

Approximation of dynamic and quasi-static evolution problems in elasto-plasticity by cap models

Abstract: This work is devoted to the analysis of elasto-plasticity models arising in soil mechanics. Contrary to the typical models mainly used for metals, it is required here to take into account plastic dilatancy due to the sensitivity of granular materials to hydrostatic pressure. The yield criterion thus depends on the mean stress and the elasticity domain is unbounded and not invariant in the direction of hydrostatic matrices. In the mechanical literature, so-called cap models have been introduced, where the elasticity domain is cut in the direction of hydrostatic stresses by means of a strain-hardening yield surface, called a cap. The purpose of this article is to study the well-posedness of plasticity models with unbounded elasticity sets in dynamical and quasi-static regimes. An asymptotic analysis as the cap is moved to in nity is also performed, which enables one to recover solutions to the uncapped model of perfect elasto-plasticity.

Conservative formulation for compressible multiphase flows

Abstract: Derivation of governing equations for multiphase flow on the base of thermodynamically compatible systems theory is presented. The mixture is considered as a continuum in which the multiphase character of the flow is taken into account. The resulting governing equations of the formulated model belong to the class of hyperbolic systems of conservation laws. In order to examine the reliability of the model, the one-dimensional Riemann problem for the four phase flow is studied numerically with the use of the MUSCL-Hancock method in conjunction with the GFORCE flux.

Nonlinear surface waves on the plasma-vacuum interface

Abstract: In this paper we study the propagation of weakly nonlinear surface waves on a plasma-vacuum interface. In the plasma region we consider the equations of incompressible magnetohydrodynamics, while in vacuum the magnetic and electric fields are governed by the Maxwell equations. A surface wave propagate along the plasma-vacuum interface, when it is linearly weakly stable. Following the approach of Ali and Hunter, we measure the amplitude of the surface wave by the normalized displacement of the interface in a reference frame moving with the linearized phase velocity of the wave, and obtain that it satisfies an asymptotic nonlocal, Hamiltonian evolution equation. We show the local-in-time existence of smooth solutions to the Cauchy problem for the amplitude equation in noncanonical variables, and we derive a blow up criterion.

The role of initial curvature in solutions to the generalized inviscid Proudman-Johnson equation

Abstract: In [20], we derived representation formulae for spatially periodic solutions to the generalized, inviscid Proudman-Johnson equation and studied their regularity for several classes of initial data. The purpose of this paper is to extend these results to larger classes of functions including those having arbitrary local curvature near particular points in the domain. Mathematics Subject Classification (2010). 35B44, 35B10, 35B65, 35Q35.

Well-posedness of the conductivity reconstruction from an interior current density in terms of Schauder theory

Abstract: We show the well-posedness of the conductivity image reconstruction problem with a single set of interior electrical current data and boundary conductivity data. Isotropic conductivity is considered in two space dimensions. Uniqueness for similar conductivity reconstruction problems has been known for several cases. However, the existence and the stability are obtained in this paper for the first time. The main tool of the proof is the method of characteristics of a related curl equation.

Correctors for the Neumann problem in thin domains with locally periodic oscillatory structure

Abstract: In this paper we are concerned with convergence of solutions of the Poisson equation with Neumann boundary conditions in a two-dimensional thin domain exhibiting highly oscillatory behavior in part of its boundary. We deal with the resonant case in which the height, amplitude and period of the oscillations are all of the same order which is given by a small parameter $\epsilon > 0$. Applying an appropriate corrector approach we get strong convergence when we replace the original solutions by a kind of first-order expansion through the Multiple-Scale Method.

A new two-component system modelling shallow-water waves

Abstract: For propagation of surface shallow-water waves on irrotational flows, we derive a new two-component system. The system is obtained by a variational approach in the Lagrangian formalism. The system has a non-canonical Hamiltonian formulation. We also find its exact solitary-wave solutions.

Asymptotic estimates of viscoelastic Green’s functions near the wavefront

Abstract: Asymptotic behavior of viscoelastic Green's functions near the wavefront is expressed in terms of a causal function $g(t)$ defined in \cite{SerHanJMP} in connection with the Kramers-Kronig dispersion relations. Viscoelastic Green's functions exhibit a discontinuity at the wavefront if $g(0) < \infty$. Estimates of continuous and discontinuous viscoelastic Green's functions near the wavefront are obtained.

A high–order perturbation of surfaces (HOPS) approach to Fokas integral equations: Three–dimensional layered–media scattering

Abstract: In this paper we discuss a novel High–Order Perturbation of Surfaces (HOPS) method for the simulation of linear acoustic waves in a three–dimensional layered, periodic structure. The model we consider is that of linear time–harmonic wave propagation which generates three–dimensional, quasiperiodic, outgoing solutions of Helmholtz equations, coupled across irregular layer interfaces. We significantly enhance and stabilize the approach of the author and D. Ambrose, which is a formulation of the problem utilizing the integral equation formalism of Fokas and collaborators. The method is phrased in terms of interfacial variables resulting in a method which has an order of magnitude fewer unknowns than a volumetric approach to this problem. In contrast to classical integral equation formulations, the current contribution does not require specialized quadratures or periodized fundamental solutions. Additionally, as a result of the HOPS philosophy, our new approach is not only faster and better conditioned than the algorithm of the author and Ambrose, it also delivers the scattering returns of an entire family of solutions with a single simulation rather than requiring a new approximation for each profile of interest. Detailed numerical simulations are presented which demonstrate the efficiency, fidelity, and spectral accuracy which can be realized with this new methodology.

Run-up amplification of transient long waves

Abstract: The extreme characteristics of long wave run-up are studied in this paper. First we give a brief overview of the existing theory which is mainly based on the hodograph transformation (Carrier & Greenspan, 1958). Then, using numerical simulations, we build on the work of Stefanakis et al. (2011) for an infinite sloping beach and we find that resonant run-up amplification of monochromatic waves is robust to spectral perturbations of the incoming wave and resonant regimes do exist for certain values of the frequency. In the setting of a finite beach attached to a constant depth region, resonance can only be observed when the incoming wavelength is larger than the distance from the undisturbed shoreline to the seaward boundary. Wavefront steepness is also found to play a role in wave run-up, with steeper waves reaching higher run-up values.

A fundamental improvement to Ericksen-Leslie kinematics

Abstract: We demonstrate theory and computations for finite-energy line defect solutions in an improvement of Ericksen-Leslie liquid crystal theory. Planar director fields are considered in two and three space dimensions, and we demonstrate straight as well as loop disclination solutions. The possibility of static balance of forces in the presence of a disclination and in the absence of flow and body forces is discussed. The work exploits an implicit conceptual connection between the Weingarten-Volterra characterization of possible jumps in certain potential fields and the Stokes-Helmholtz resolution of vector fields. The theoretical basis of our work is compared and contrasted with the theory of Volterra disclinations in elasticity. Physical reasoning precluding a gauge-invariant structure for the model is also presented.

Why condensation by compression in pure water vapor cannot occur in an approach based on Euler equations

Abstract: Phase transitions are in the focus of the modeling of multiphase flows. A large number of models is available to describe such processes. We consider several different two phase models that are based on the Euler equations of compressible fluid flows and which take into account phase transitions between a liquid phase and its vapor. Especially we consider the flow of liquid water and water vapor. We give a mathematical proof, that all these models are not able to describe the process of condensation by compression. This behavior is in agreement with observations in experiments, that simulate adiabatic flows, and shows that the Euler equations give a fairly good description of the process. The mathematical proof is valid for the official standard {\em IAPWS-IF97} for water and for any other good equation of state. Also the opposite case of expanding the liquid phase will be discussed.

Hyperbolic relaxation of reaction-diffusion equations with dynamic boundary conditions

Abstract: Under consideration is the hyperbolic relaxation of a semilinear reaction-diffusion equation on a bounded domain, subject to a dynamic boundary condition. We also consider the limit parabolic problem with the same dynamic boundary condition. Each problem is well-posed in a suitable phase space where the global weak solutions generate a Lipschitz continuous semiflow which admits a bounded absorbing set. We prove the existence of a family of global attractors of optimal regularity. After fitting both problems into a common framework, a proof of the upper-semicontinuity of the family of global attractors is given as the relaxation parameter goes to zero. Finally, we also establish the existence of exponential attractors.

On the lifespan and the blowup mechanism of smooth solutions to a class of 2-D nonlinear wave equations with small initial data

Abstract: This paper is concerned with the lifespan and the blowup mechanism for smooth solutions to the 2-D nonlinear wave equation $\\p_t^2u-\\ds\\sum_{i=1}^2\\p_i(c_i^2(u)\\p_iu)$ $=0$, where $c_i(u)\\in C^{\\infty}(\\Bbb R^n)$, $c_i(0)\ eq 0$, and $(c_1'(0))^2+(c_2'(0))^2\ eq 0$. This equation has an interesting physics background as it arises from the pressure-gradient model in compressible fluid dynamics and also in nonlinear variational wave equations. Under the initial condition $(u(0,x), \\p_tu(0,x))=(\\ve u_0(x), \\ve u_1(x))$ with $u_0(x), u_1(x)\\in C_0^{\\infty}(\\Bbb R^2)$, and $\\ve>0$ is small, we will show that the classical solution $u(t,x)$ stops to be smooth at some finite time $T_{\\ve}$. Moreover, blowup occurs due to the formation of a singularity of the first-order derivatives $\ a_{t,x}u(t,x)$, while $u(t,x)$ itself is continuous up to the blowup time $T_{\\ve}$.

The minimum and other free energies for non-linear materials with memory

Abstract: Expressions are obtained for free energies of materials with a certain type of non-linear constitutive relation. In particular, the minimum and related free energies are considered in some detail. Minimal states are defined for these materials, and it is shown that any free energy yielding a linear constitutive equation that is a functional of the minimal state has a counterpart in the non-linear case which is also a minimal state functional in this more general context. These results are explored for simple examples, including discrete spectrum materials.

The fundamental solution of a conservation law without convexity

Abstract: The signed fundamental solution of a scalar conservation law is constructed explicitly or implicitly when its flux is nonconvex. The flux is assumed to have finite number of inflection points. The fundamental solution constructed consists of a series of rarefaction waves, contact discontinuities and a shock. These analytically constructed fundamental solutions are also compared with numerical approximations, which possess the structure of the analytically constructed fundamental solution.

Persistence property in weighted Sobolev spaces for nonlinear dispersive equations

Abstract: We generalize here the Abstract Interpolation Lemma proved by the authors in [1]. Using this extension, we show in a more general con- text, the persistence property for the generalized Korteweg-de Vries equation, in the weighted Sobolev space with low regularity in the weight. The method used can be applied for other nonlinear disper- sive models, for instance the multidimensional nonlinear Schrodinger equation. References [1] X. Carvajal and W. Neves, Persistence of solutions to higher order non- linear Schrodinger equation, J. Diff. Equations, v. 249, p. 2214-2236, 2010. Instituto de Matematica, Universidade Federal do Rio de Janeiro, C.P. 68530, Cidade Universitaria 21945-970, Rio de Janeiro, Brazil. E-mail: carvajal@im.ufrj.br, wladimir@im.ufrj.br.

On the quenching behavior of the MEMS with fringing field

Abstract: The singular parabolic problem $u_t-\triangle u=\lambda{\frac{1+\delta| abla u|^2}{(1-u)^2}}$ on a bounded domain $\Omega$ of $\mathbb{R}^n$ with Dirichlet boundary condition, models the Microelectromechanical systems (MEMS) device with fringing field. In this paper, we focus on the quenching behavior of the solution to this equation. We first show that there exists a critical value $\lambda_\delta^*>0$ such that if $0 \lambda_\delta^*$, all the solution will quench in finite time. The estimate of the quenching time in terms of large voltage $\lambda$ is investigated. Furthermore, the quenching set is a compact subset of $\Omega$, provided $\Omega$ is a convex bounded domain in $\mathbb{R}^n$. In particular, if the domain $\Omega$ is radially symmetric, then the origin is the only quenching point. We not only derive the one-side estimate of the quenching rate, but also further study the refined asymptotic behavior of the finite quenching solution.