Showing papers in "Quarterly of Applied Mathematics in 2016"

On the high frequency limit of the LLL equation

Abstract: We derive heuristically an integro-differential equation, as well as a shell model, governing the dynamics of the Lowest Landau Level equation in a high frequency regime.

Hydrodynamic limit of granular gases to pressureless Euler in dimension 1

Abstract: We investigate the behavior of granular gases in the limit of small Knudsen number, that is very frequent collisions. We deal with the strongly inelastic case, in one dimension of space and velocity. We are able to prove the convergence toward the pressureless Euler system. The proof relies on dispersive relations at the kinetic level, which leads to the so-called Oleinik property at the limit.

Sensitivity of anomalous localized resonance phenomena with respect to dissipation

Abstract: We analyze cloaking due to anomalous localized resonance in the quasistatic regime in the case when a general charge density distribution is brought near a slab superlens. If the charge density distribution is within a critical distance of the slab, then the power dissipation within the slab blows up as certain electrical dissipation parameters go to zero. The potential remains bounded far away from the slab in this limit, which leads to cloaking due to anomalous localized resonance. On the other hand, if the charge density distribution is further than this critical distance from the slab, then the power dissipation within the slab remains bounded and cloaking due to anomalous localized resonance does not occur. The critical distance is shown to strongly depend on the the rate at which the dissipation outside of the slab goes to zero.

A Caginalp phase-field system based on type III heat conduction with two temperatures

Abstract: Our aim in this paper is to study a generalization of the Caginalp phasefield system based on the theory of type III thermomechanics with two temperatures for the heat conduction. In particular, we obtain well-posedness results and study the dissipativity of the associated solution operators. We consider here both regular and singular nonlinear terms. Furthermore, we endow the equations with two types of boundary conditions, namely, Dirichlet and Neumann. Finally, we study the spatial behavior of the solutions in a semi-infinite cylinder, when such solutions exist.

A non-traditional view on the modeling of nematic disclination dynamics

Abstract: Nonsingular disclination dynamics in a uniaxial nematic liquid crystal is modeled within a mathematical framework where the kinematics is a direct extension of the classical way of identifying these line defects with singularities of a unit vector field representing the nematic director. It is well known that the universally accepted Oseen-Frank energy is infinite for configurations that contain disclination line defects. We devise a natural augmentation of the Oseen-Frank energy to account for physical situations where, under certain conditions, infinite director gradients have zero associated energy cost, as would be necessary for modeling half-integer strength disclinations within the framework of the director theory. Equilibria and dynamics (in the absence of flow) of line defects are studied within the proposed model. Using appropriate initial/boundary data, the gradient-flow dynamics of this energy leads to non-singular, line defect equilibrium solutions, including those of half-integer strength. However, we demonstrate that the gradient flow dynamics for this energy is not able to adequately describe defect evolution. Motivated by similarity with dislocation dynamics in solids, a novel 2D-model of disclination dynamics in nematics is proposed. The model is based on the extended Oseen-Frank energy and takes into account thermodynamics and the kinematics of conservation of defect topological charge. We validate this model through computations of disclination equilibria, annihilation, repulsion, and splitting. We show that the energy function we devise, suitably interpreted, can serve as well for the modeling of equilibria and dynamics of dislocation line defects in solids making the conclusions of this paper relevant to mechanics of both solids and liquid crystals.

Quasineutral limit of the pressureless Euler-Poisson equation for ions

Abstract: In this paper, we consider the quasineutral limit of the Euler-Poisson equation for a clod, ion-acoustic plasma when the Debye length tends to zero. When the ion-acoustic plasma is cold, the Euler-Poisson equation is pressureless and hence fails to be Friedrich symmetrisable, which excludes the application of the classical energy estimates method. This brings new difficulties in proving uniform estimates independent of $\varepsilon$. The main novelty in this article is to introduce new $\varepsilon$-weighted norms of the unknowns and to combine energy estimates in different levels with weights depending on $\varepsilon$. Finally, that the quasineutral regimes are the incompressible Euler equations is proven for well prepared initial data.

Sharp local well-posedness of KdV type equations with dissipative perturbations

Abstract: In this work, we study the initial value problems associated to some linear perturbations of KdV equations. Our focus is in the well-posedness issues for initial data given in the $L^2$-based Sobolev spaces. We derive bilinear estimate in a space with weight in the time variable and obtain sharp local well-posedness results.

Global Riemann solver and front tracking approximation of three-component gas floods

Abstract: We study a 2× 2 system of non-strictly hyperbolic conservation laws arising in three– component gas flooding for enhanced oil recovery. The system is not strictly hyperbolic. In fact, along a curve in the domain one family is linearly degenerate, and along two other curves the system is parabolic degenerate. We construct global solutions for the Riemann problem, utilizing the splitting property of thermo-dynamics from the hydro-dynamics. Front tracking simulations are presented, using the global Riemann Solver. 1 The Three-Component Gas Flooding Model We consider a simplified compositional displacement model for a three-component system at constant temperature and pressure [11], (1.1) (C1)t + (F1(C1, C2))x = 0 , (C2)t + (F2(C1, C2))x = 0 , associated with initial data (1.2) C1(0, x) = C1(x) , C2(0, x) = C2(x) . The independent variables (t, x) are normalized such that the overall velocity is 1. Here Ci is the overall ith component volume fraction, and Fi is the overall i th component flux. For the third component, we trivially have C3 = 1− C1 − C2 , F3 = 1− F1 − F2 . The couplet (C1, C2) takes values in a triangular domain D = {(C1, C2) | C1 ≥ 0, C2 ≥ 0, 1− C1 − C2 > 0} . For the phase behaviors that are considered in this paper, there exists a subset D2 ⊂ D, referred to as the two-phase region, where the fluid splits into two phases, the liquid and the gaseous phases. In the single phase region D1 = D \D2, we trivially have F1(C1, C2) = C1, F2(C1, C2) = C2.

Instabilities and oscillations in coagulation equations with kernels of homogeneity one

Abstract: We discuss the long-time behaviour of solutions to Smoluchowski's coagulation equation with kernels of homogeneity one, combining formal asymptotics, heuristic arguments based on linearization, and numerical simulations. The case of what we call diagonally dominant kernels is particularly interesting. Here one expects that the long-time behaviour is, after a suitable change of variables, the same as for the Burgers equation. However, for kernels that are close to the diagonal one we obtain instability of both, constant solutions and traveling waves and in general no convergence to N-waves for integrable data. On the other hand, for kernels not close to the diagonal one these structures are stable, but the traveling waves have strong oscillations. This has implications on the approach towards an N-wave for integrable data, which is also characterized by strong oscillations near the shock front.

Sensitivity via the complex-step method for delay differential equations with non-smooth initial data

Abstract: In this report, we use the complex-step derivative approximation technique to compute sensitivities for delay differential equations (DDEs) with non-smooth (discontinuous and even distributional) history functions. We compare the results with exact derivatives and with those computed using the classical sensitivity equations whenever possible. Our results demonstrate that the implementation of the complex-step method using the method of steps and the Matlab solver dde23 provides a very good approximation of sensitivities as long as discontinuities in the initial data do not cause loss of smoothness in the solution: that is, even when the underlying smoothness with respect to the initial data for the Cauchy-Riemann derivation of the the method does not hold. We conclude with remarks on our findings regarding the complex-step method for computing sensitivities for simpler ordinary differential equation systems in the event of lack of smoothness with respect to parameters.

A bistable belief dynamics model for radicalization within sectarian conflict

Abstract: We introduce a two-variable model to describe spatial polarization, radicalization, and conflict. Individuals in the model harbor a continuous belief variable as well as a discrete radicalization level expressing their tolerance to neighbors with different beliefs. A novel feature of our model is that it incorporates a bistable radicalization process to address memory-dependent social behavior. We demonstrate how bistable radicalization may explain contradicting observations regarding whether social segregation exacerbates or alleviates conflicts. We also extend our model by introducing a mechanism to include institutional influence, such as propaganda or education, and examine its effectiveness. In some parameter regimes, institutional influence may suppress the progression of radicalization and allow a population to achieve social conformity over time. In other cases, institutional intervention may exacerbate the spread of radicalization through a population of mixed beliefs. In such instances, our analysis implies that social segregation may be a viable option against sectarian conflict.

Evolution of the Boson gas at zero temperature: Mean-field limit and second-order correction

Abstract: A large system of N integer-spin atoms, called Bosons, manifests one of the most coherent macroscopic quantum states known to date, the “Bose-Einstein condensate”, at extremely low temperatures. As N → ∞, this system is usually described by a mean-field limit: a single-particle wave function, the condensate wave function, that satisfies a nonlinear Schrödinger-type equation. In this expository paper, we review kinetic aspects of the mean-field Boson evolution. Furthermore, we discuss recent advances in the rigorous study of second-order corrections to this mean-field limit. These corrections originate from the quantum-kinetic mechanism of pair excitation, which lies at the core of pioneering works in theoretical physics including ideas of Bogoliubov, Lee, Huang, Yang and Wu. In the course of our exposition, we revisit the formalism of Fock space, which is indispensable for the analysis of pair excitation.

Towards monitoring critical microscopic parameters for electropermeabilization

Abstract: Electropermeabilization is a clinical technique in cancer treatment to locally stimulate the cell metabolism. It is based on electrical fields that change the properties of the cell membrane. With that, cancer treatment can reach the cell more easily. Electropermeabilization occurs only with accurate dosage of the electrical field. For applications, a monitoring for the amount of electropermeabilization is needed. It is a first step to image the macroscopic electrical field during the process. Nevertheless, this is not complete, because electropermeabilization depends on critical individual properties of the cells such as their curvature. From the macroscopic field, one cannot directly infer that microscopic state. In this article, we study effective parameters in a homogenization model as the next step to monitor the microscopic properties in clinical practice. We start from a physiological cell model for electropermeabilization and analyze its well-posedness. For a dynamical homogenization scheme, we prove convergence and then analyze the effective parameters, which can be found by macroscopic imaging methods. We demonstrate numerically the sensitivity of these effective parameters to critical microscopic parameters governing electropermeabilization. This opens the door to solve the inverse problem of rreconstructing these parameters.

Scalar Green function bounds for instantaneous shock location and one-dimensional stability of viscous shock waves

Abstract: In this paper, we investigate and prove the nonlinear stability of viscous shock wave solutions of a scalar viscous conservation law, using the methods developed for general systems of conservation laws by Howard, Mascia, Zumbrun and others, based on instantaneous tracking of the location of the perturbed viscous shock wave. In some sense, this paper extends the treatment in a previous expository work of Zumbrun ["Instantaneous shock location ..."] on Burgers equation to the general case, giving an exposition of these methods in the simplest setting of scalar equations. In particular we give by a rescaling argument a simple treatment of nonlinear stability in the small-amplitude case.

Spectral representations, and approximations, of divergence-free vector fields

Abstract: Special solutions of the equation for a solenoidal vector field subject to prescribed flux boundary conditions are described. A unique gradient solution is found and proved to be the least energy solution of the problem. This solution has a representation in terms of certain Σ− Steklov−eigenvalues and eigenfunctions. Error estimates for finite approximations of these solutions are obtained. Some results of computational simulations for two dimensional and axisymmetrical problems are presented.

Euler-Lagrange equation for a minimization problem over monotone transport maps

Abstract: A variational time discretization for the compressible Euler equations has been introduced recently. It involves a minimization problem over the cone of monotone transport maps in each timestep. A matrix-valued measure field appears in the minimization as a Lagrange multiplier for the monotonicity constraint. We show that the absolutely continuous part of this measure field vanishes in the support of the density.

Global smooth solution to a hyperbolic system on an interval with dynamic boundary conditions

Abstract: We consider a hyperbolic two component system of partial differential equations in one space dimension with ODE boundary conditions describing the flow of an incompressible fluid in an elastic tube that is connected to a tank at each end. Using the local-existence theory together with entropy methods, the existence and uniqueness of a global-in-time smooth solution is established for smooth initial data sufficiently close to the equilibrium state. Energy estimates are derived using the relative entropy method for zero order estimates while constructing entropy-entropy flux pairs for the corresponding diagonal system of the shifted Riemann invariants to deal with higher order estimates. Finally, using the linear theory and interpolation estimates, we show that the solution converges exponentially to the equilibrium state.

Bifurcation analysis of a single-group asset flow model

Abstract: We study the stability and Hopf bifurcation analysis of an asset pricing model that based on the model introduced by Caginalp and Balenovich in 1999, under the assumption of a …xed amount of cash and stock in the system. First, we study stability analysis of equilibrium points. Choosing the momentum coe¢ cient as a bifurcation parameter, we also show that Hopf bifurcation occurs when the bifurcation parameter passes through a critical value. Analytical results are supported by numerical simulations. A key conclusion for economics and …nance is the existence of periodic solutions for an interval of the bifurcation parameter, which is the trend-based (or momentum) coe¢ cient. Key words. Asset price dynamics, stability of price dynamics, Hopf bifurcation, price trend, momentum, market dynamics, liquidity, periodic solutions. AMS subject classi…cation. : 91B25, 91B26, 91B50, 91G80, 91G99, 34D20, 34C60, 37G15, 37N40 1. Introduction. A central theme in classical …nance is that market participants all have access to the same information, and all seek to optimize their returns so that a unique equilibrium price is established (see, for example, [3], [18], [20]). The approach to equilibrium is often assumed to be a process involving some randomness or noise, but otherwise smooth and rapid. Aside from noise, one can expect that prices will not overshoot the equilibrium price since the equation governing the change in price, P , is a …rst order in time, i.e., P 0 = F (D=S) where S and D are supply and demand that depend on price but not on the recent price derivative history. As such, there is no mechanism for oscillations or cyclic behavior within this setting. A well known example of cyclic behavior in economics is called the "cobweb theorem" whereby prices oscillate periodically due to the time lag between supply and demand decisions. Agricultural commodities provide a simple example with a delay between planting and harvesting (see [21] (pages 133-134 gives two agricultural examples: rubber and corn) and [36]). In …nancial markets, however, the prevailing theory (at least during latter part of the 20th century), called the e¢ cient market hypothesis (EMH), maintains the existence of in…nite arbitrage capital that would quickly exploit any deviations between the trading price and the intrinsic or fundamental value of the asset, which are necessarily unique since the participants have the same information and calculation of future returns. The absence of any delay in information or trading excludes, mathematically, the possibilities of overshooting the equilibrium price or oscillating about it. While policy makers often discuss instabilities in asset prices, classical …nance tends to treat these as rare occurrences within a stochastic setting. In particular, much of classical …nance is based on the concept that an asset’s price, P (t), is governed by Current address: Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA. Permanent address: Department of Mathematics, TOBB University of Economics and Technology, 06560-Ankara, TURKEY. H. Merdan was supported by TUBITAK (The Scienti…c and Technological Research Council of Turkey) yDepartment of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA zDepartment of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA

Analysis of integro-differential equations modeling the vertical decomposition of soil organic matter

Abstract: Analysis of integro-differential equations modeling the vertical decomposition of soil organic matter