Parabolic Limit Research Articles

A closed-form structural stress-strain relation and implications on plastic design limits in codes and standards

The deformation of elastic-perfectly plastic beam or plate under combined through-thickness membrane and bending stresses (i.e., structural stress) is analyzed in this paper. Closed-form solutions of “plane-remains-plane” strain (i.e., structural strain) distribution as a function of applied stress and yield strength is derived. The relationship between structural stress and structural strain is also revealed for gaining insights on not only the plastic collapse conditions, but also through-thickness deformation behaviors leading to limit state. Based on the closed-form strain solutions, four deformation modes are identified: (1) elastic, (2) one-side yielding, (3) two-side yielding, and (4) plastic collapse. A membrane-bending stress interaction diagram is constructed to depict four zones that are associated with the four deformation modes in the normalized membrane and bending stress space. Closed-form solutions of residual stress, residual strain, and structural stress range-strain range relation after a loading-unloading cycle are also derived. Based on the parabolic plastic collapse stress limit curve, parabolic design stress limit curves with coherent safety margins for any combinations of membrane and bending stresses are constructed. The new insights gained in this study offer a more consistent definition of design stress limit than those used in the current codes and standards, e.g., ASME BP&V Code.

Boundary behavior of positive solutions of the heat equation on a stratified Lie group

In this article, we are concerned with a certain type of boundary behavior of positive solutions of the heat equation on a stratified Lie group at a given boundary point. We prove that a necessary and sufficient condition for the existence of the parabolic limit of a positive solution u at a point on the boundary is the existence of the strong derivative of the boundary measure of u at that point. Moreover, the parabolic limit and the strong derivative are equal. We also construct an example of a positive measure on the Heisenberg group to show that the set of all points where strong derivative exists is strictly larger than the set of Lebesgue points of the measure.

A stochastic hierarchical model for low grade glioma evolution

A stochastic hierarchical model for the evolution of low grade gliomas is proposed. Starting with the description of cell motion using a piecewise diffusion Markov process (PDifMP) at the cellular level, we derive an equation for the density of the transition probability of this Markov process based on the generalised Fokker–Planck equation. Then, a macroscopic model is derived via parabolic limit and Hilbert expansions in the moment equations. After setting up the model, we perform several numerical tests to study the role of the local characteristics and the extended generator of the PDifMP in the process of tumour progression. The main aim focuses on understanding how the variations of the jump rate function of this process at the microscopic scale and the diffusion coefficient at the macroscopic scale are related to the diffusive behaviour of the glioma cells and to the onset of malignancy, i.e., the transition from low-grade to high-grade gliomas.

Navier–Stokes–Fourier equations as a parabolic limit of a general hyperbolic system of rational extended thermodynamics

This paper aims to prove that a general system of 14 balance laws for a compressible, possibly dense, gas that satisfies the universal principles of Rational Extended Thermodynamics (RET) converges to the Navier–Stokes–Fourier equations in the first step of the Maxwellian iteration. Moreover, in a theory not far from equilibrium, we show that the production terms of the hyperbolic system are uniquely determined as soon as the heat conductivity, the shear viscosity, and the bulk viscosity are assigned. The obtained results are tested on the RET theories for rarefied monatomic and polyatomic gases.

Time periodic solution to chemotaxis-shallow water system in a periodic domain

<p style='text-indent:20px;'>In this paper, we analyse the existence and uniqueness of a time-periodic solution to chemotaxis-shallow water system in a periodic domain. Under the assumption of some smallness and symmetric external force, the existence of periodic solution is established using the method of parabolic regularization and limit process. The uniqueness of the periodic solution is proved by energy estimates.</p>

Relativistic Rational Extended Thermodynamics of Polyatomic Gases with a New Hierarchy of Moments.

A relativistic version of the rational extended thermodynamics of polyatomic gases based on a new hierarchy of moments that takes into account the total energy composed by the rest energy and the energy of the molecular internal mode is proposed. The moment equations associated with the Boltzmann–Chernikov equation are derived, and the system for the first 15 equations is closed by the procedure of the maximum entropy principle and by using an appropriate BGK model for the collisional term. The entropy principle with a convex entropy density is proved in a neighborhood of equilibrium state, and, as a consequence, the system is symmetric hyperbolic and the Cauchy problem is well-posed. The ultra-relativistic and classical limits are also studied. The theories with 14 and 6 moments are deduced as principal subsystems. Particularly interesting is the subsystem with 6 fields in which the dissipation is only due to the dynamical pressure. This simplified model can be very useful when bulk viscosity is dominant and might be important in cosmological problems. Using the Maxwellian iteration, we obtain the parabolic limit, and the heat conductivity, shear viscosity, and bulk viscosity are deduced and plotted.

End-exclusion zones in strongly stretched, molten polymer brushes of arbitrary shape.

Theories of strongly stretched polymer brushes, particularly the parabolic brush theory, are valuable for providing analytically tractable predictions for the thermodynamic behavior of surface-grafted polymers in a wide range of settings. However, the parabolic brush limit fails to describe polymers grafted to convex curved substrates, such as the surfaces of spherical nanoparticles or the interfaces of strongly segregated block copolymers. It has previously been shown that strongly stretched curved brushes require a boundary layer devoid of free chain ends, requiring modifications of the theoretical analysis. While this "end-exclusion zone" has been successfully incorporated into the descriptions of brushes grafted onto the outer surfaces of cylinders and spheres, the behavior of brushes on surfaces of arbitrary curvature has not yet been studied. We present a formulation of the strong-stretching theory for molten brushes on the surfaces of arbitrary curvature and identify four distinct regimes of interest for which brushes are predicted to possess end-exclusion zones, notably including regimes of positive mean curvature but negative Gaussian curvature. Through numerical solutions of the strong-stretching brush equations, we report predicted scaling of the size of the end-exclusion zone, the chain end distribution, the chain polarization, and the free energy of stretching with mean and Gaussian surface curvatures. Through these results, we present a comprehensive picture of how the brush geometry influences the end-exclusion zones and exact strong-stretching free energies, which can be applied, for example, to model the full spectrum of brush geometries encountered in block copolymer melt assembly.

A unified asymptotic preserving and well-balanced scheme for the Euler system with multiscale relaxation

The design and analysis of a unified asymptotic preserving (AP) and well-balanced scheme for the Euler Equations with gravitational and frictional source terms is presented in this paper. The asymptotic behaviour of the Euler system in the limit of zero Mach and Froude numbers, and large friction is characterised by an additional scaling parameter. Depending on the values of this parameter, the Euler system relaxes towards a hyperbolic or a parabolic limit equation. Standard Implicit–Explicit Runge–Kutta schemes are incapable of switching between these asymptotic regimes. We propose a time semi-discretisation to obtain a unified scheme which is AP for the two different limits. A further reformulation of the semi-implicit scheme can be recast as a fully-explicit method in which the mass update contains both hyperbolic and parabolic fluxes. A space–time fully-discrete scheme is derived using a finite volume framework. A hydrostatic reconstruction strategy, an upwinding of the sources at the interfaces, and a careful choice of the central discretisation of the parabolic fluxes are used to achieve the well-balancing property for hydrostatic steady states. Results of several numerical case studies are presented to substantiate the theoretical claims and to verify the robustness of the scheme.

Diffusive limits of 2D well-balanced schemes for kinetic models of neutron transport

Two-dimensional dissipative and isotropic kinetic models, like the ones used in neutron transport theory, are considered. Especially, steady-states are expressed for constant opacity and damping, allowing to derive a scattering S-matrix and corresponding "truly 2D well-balanced" numerical schemes. A first scheme is obtained by directly implementing truncated Fourier–Bessel series, whereas another proceeds by applying an exponential modulation to a former, conservative, one. Consistency with the asymptotic damped parabolic approximation is checked for both algorithms. A striking property of some of these schemes is that they can be proved to be both 2D well-balanced and asymptotic-preserving in the parabolic limit, even when setting up IMEX time-integrators: see Corollaries 3.4 and A.1. These findings are further confirmed by means of practical benchmarks carried out on coarse Cartesian computational grids.

Maxwellian iteration of a causal relativistic model of polyatomic gases and evaluation of bulk, shear viscosity and heat conductivity

In the present paper, we give the parabolic limit of the field equations of a recent hyperbolic model of relativistic polyatomic gas in the framework of Rational Extended Thermodynamics (RET) theory. We obtain in this way the corresponding constitutive equations of the Thermodynamics of Irreversible Processes (TIP) obtained first in the context of relativity by Eckart. The limit is reached via the Maxwellian iteration procedure to make a connection between the phenomenological coefficients (shear, bulk viscosity, and heat conductivity) and the relaxation times. The classical and ultrarelativistic limit of these coefficients are also obtained finding that, in the case of classical limit, they coincide with the ones known in the literature. As a particular case, we study the monatomic gas and we can plot the dimensionless coefficients associated with bulk, shear, and heat conductivity. In contrast to a monatomic gas in which the bulk viscosity is very small and tends to the classical regime of the order of magnitude of O(1∕c4), the bulk viscosity and the dynamical pressure for polyatomic gases, due to the rotation and vibration of internal modes, are of the order of unit (except in the ultrarelativistic limit), and this might open new perspectives in cosmology.

On parabolic convergence of positive solutions of the heat equation

ABSTRACT In this article, we study a certain type of boundary behaviour of positive solutions of the heat equation on the Euclidean upper half-space of . We prove that the existence of the parabolic limit of a positive solution of the heat equation at a point in the boundary is equivalent to the existence of the strong derivative of the boundary measure of the solution at that point. Moreover, the parabolic limit and the strong derivative are equal.

Correlated random walks in heterogeneous landscapes: Derivation, homogenization, and invasion fronts

Many models for the movement of particles and individuals are based on the diffusion equation, which, in turn, can be derived from an uncorrelated random walk or a position-jump process. In those models, individuals have a location but no well-defined velocity. An alternative, and sometimes more accurate, model is based on a correlated random walk or a velocity-jump process, where individuals have a well defined location and velocity. The latter approach leads to hyperbolic equations for the density of individuals, rather than parabolic equations that result from the diffusion process. Almost all previous work on correlated random walks considers a homogeneous landscape, whereas diffusion models for uncorrelated walks have been extended to spatially varying environments. In this work, we first derive the equations for a correlated random walk in a one-dimensional spatially varying environment with either smooth variation or piecewise constant variation. Then we show how to derive the so-called parabolic limit from the resulting hyperbolic equations. We develop homogenization theory for the hyperbolic equations, and show that taking the parabolic limit and homogenization are commuting actions. We illustrate our results with two examples from ecology: the persistence and spread of a population in a patchy heterogeneous landscape.

Heterogeneous discrete kinetic model and its diffusion limit

<p style='text-indent:20px;'>A revertible discrete velocity kinetic model is introduced when the environment is spatially heterogeneous. It is proved that the parabolic scale singular limit of the model exists and satisfies a new heterogeneous diffusion equation that depends on the diffusivity and the turning frequency together. An energy functional is introduced which takes into account spatial heterogeneity in the velocity field. The monotonicity of the energy functional is the key to obtain uniform estimates needed for the weak convergence proof. The Div-Curl lemma completes the strong convergence proof.</p>

A note on the gravitational wave energy spectrum of parabolic and hyperbolic encounters

The first calculation of the frequency spectrum of gravitational wave mass quadrupole radiation for binaries on hyperbolic orbits was performed in De Vittori et al (2012 Phys. Rev. D 86 044017). Some shortcomings of their derivation were pointed out, but there are still inaccuracies and supplements that we believe are worthwhile to communicate. In this note we provide a consistent and straightforward exposition of the frequency spectrum in the case of hyperbolic encounters and explicitly determine the parabolic limit, which was not possible with the previous treatments.

Compressible multicomponent flow in porous media with Maxwell‐Stefan diffusion

We introduce a Darcy-scale model to describe compressible multi-component flow in a fully saturated porous medium. In order to capture cross-diffusive effects between the different species correctly, we make use of the Maxwell--Stefan theory in a thermodynamically consistent way. For inviscid flow the model turns out to be a nonlinear system of hyperbolic balance laws. We show that the dissipative structure of the Maxwell-Stefan operator permits to guarantee the existence of global classical solutions for initial data close to equilibria. Furthermore it is proven by relative entropy techniques that solutions of the Darcy-scale model tend in a certain long-time regime to solutions of a parabolic limit system.

Glioma invasion and its interplay with nervous tissue and therapy: A multiscale model

A multiscale mathematical model for glioma cell migration and proliferation is proposed, taking into account a possible therapeutic approach. Starting with the description of processes occurring at the subcellular level, the equation for the mesoscopic level is formulated and a macroscopic model is derived, via parabolic limit and Hilbert expansions in the moment equations. After the model set up and the study of the well-posedness of this macroscopic setting, we investigate the role of the fibers in the tumor dynamics. In particular, we focus on the fiber density function, with the aim of comparing some common choices present in the literature and understanding which differences arise in the description of the actual fiber density and orientation. Finally, some numerical simulations, based on real data, highlight the role of each modelled process in the evolution of the solution of the macroscopic equation.

Convergence rate from hyperbolic systems of balance laws to parabolic systems

It is proved recently that partially dissipative hyperbolic systems converge globally-in-time to parabolic systems in a slow time scaling, when initial data are smooth and sufficiently close to constant equilibrium states. Based on this result, we establish error estimates between the smooth solutions of the hyperbolic systems of balance laws and those of the parabolic limit systems in one space dimension. The proof of the error estimates uses a stream function technique together with energy estimates. As applications of the results, we give five examples arising from physical models.

Derivation of a bacterial nutrient-taxis system with doubly degenerate cross-diffusion as the parabolic limit of a velocity-jump process.

This paper is devoted to the justification of the macroscopic, mean-field nutrient taxis system with doubly degenerate cross-diffusion proposed by Leyva et al. (Phys A 392:5644-5662, 2013) to model the complex spatio-temporal dynamics exhibited by the bacterium Bacillus subtilis during experiments run in vitro. This justification is based on a microscopic description of the movement of individual cells whose changes in velocity (in both speed and orientation) obey a velocity jump process governed by a transport equation of Boltzmann type. For that purpose, the asymptotic method introduced by Hillen and Othmer (SIAM J Appl Math 61:751-775, 2000; SIAM J Appl Math 62:1222-1250, 2002) is applied, which consists of the computation of the leading order term in a regular Hilbert expansion for the solution to the transport equation, under an appropriate parabolic scaling and a first order perturbation of the turning rate of Schnitzer type (Schnitzer in Phys Rev E 48:2553-2568, 1993). The resulting parabolic limit equation at leading order for the bacterial cell density recovers the degenerate nonlinear cross diffusion term and the associated chemotactic drift appearing in the original system of equations. Although the bacterium B. subtilis is used as a prototype, the method and results apply in more generality.

Inertial manifolds for the hyperbolic relaxation of semilinear parabolic equations

The paper gives a comprehensive study of Inertial Manifolds for hyperbolic relaxations of an abstract semilinear parabolic equation in a Hilbert space. A new scheme of constructing Inertial Manifolds for such type of problems is suggested and optimal spectral gap conditions which guarantee their existence are established. Moreover, the dependence of the constructed manifolds on the relaxation parameter in the case of the parabolic singular limit is also studied.Bibliography: 38 titles.

Space-time fractional diffusion in cell movement models with delay

The movement of organisms and cells can be governed by occasional long distance runs, according to an approximate Lévy walk. For T cells migrating through chronically-infected brain tissue, runs are further interrupted by long pauses and the aim here is to clarify the form of continuous model equations that describe such movements. Starting from a microscopic velocity-jump model based on experimental observations, we include power-law distributions of run and waiting times and investigate the relevant parabolic limit from a kinetic equation for resting and moving individuals. In biologically relevant regimes we derive nonlocal diffusion equations, including fractional Laplacians in space and fractional time derivatives. Its analysis and numerical experiments shed light on how the searching strategy, and the impact from chemokinesis responses to chemokines, shorten the average time taken to find rare targets in the absence of direct guidance information such as chemotaxis.