1,721,041 research outputs found
Exact representation of the asymptotic drift speed and diffusion matrix for a class of velocity-jump processes
Full text linkThis paper examines a class of linear hyperbolic systems which generalizes the Goldstein–Kac model to an arbitrary finite number of speeds viwith transition rates μij. Under the basic assumptions that the transition matrix is symmetric and irreducible, and the differences vi−vjgenerate all the space, the system exhibits a large-time behavior described by a parabolic advection–diffusion equation. The main contribu-tion is to determine explicit formulas for the asymptotic drift speed and diffusion matrix in term of the kinetic parameters viand μij, establishinga complete connection between microscopic and macroscopic coefficients. It is shown that the drift speed is the arithmetic mean of the velocities vi. The diffusion matrix has a more complicate representation, based on the graph with vertices the velocities viand arcs weighted by the transition rates μij. The approach is based on an exhaustive analysis of the dispersion relation and on the application of a variant of the Kirchoff’s matrix tree Theorem from graph theory
Twenty-eight years with “Hyperbolic Conservation Laws with Relaxation”
Full text linkThis paper is a review on the results inspired by the publication “Hyperbolic conservation laws with relaxation” by Tai-Ping Liu [1], with emphasis on the topic of nonlinear waves (specifically, rarefaction and shock waves). The aim is twofold: firstly, to report in details the impact of the article on the subsequent research in the area; secondly, to detect research trends which merit attention in the (near) future
A dive into shallow water
No full textIn this tutorial, we attempt to furnish a basic introduction on shallow water modeling with specific attention to Saint--Venant equations.
We propose a selection of results, including derivation of the model, well-posedness of the Cauchy problem, existence and stability of roll-waves, kinetic formulation and the corresponding hydrodynamical limit, presented, whenever possible, in a simplified way and designed mainly for readers that are not expert in the field
Stabilità Asintotica di Onde di Shock per Leggi di Conservazione Perturbate Singolarmente
No full textL'obiettivo di questa comunicazione e' presentare un risultato di esistenza e stabilita' asintotica di soluzioni in forma di onda viaggiante per un sistema semilineare iperbolico di tipo rilassamento.The goal is to present a result on the existence and asymptotic stability of solutions in the form of a travelling wave for a hyperbolic semilinear system with relaxation
Small, medium and large shock waves for radiative Euler equations
No full textWe examine the existence of shock profiles for a hyperbolic-elliptic system arising in radiation hydrodynamics. The algebraic-differential system for the wave profile is reduced to a standard two-dimensional form that is analyzed in detail showing the existence of a heteroclinic connection between the two singular points of the system for any distance between the corresponding asymptotic states of the original model. Depending on the location of these asymptotic states, the profile can be either continuous or possesses at most one point of discontinuity. Moreover, a sharp threshold relative to the presence of an internal absolute maximum in the temperature profile-also called a Zel'dovich spike-is rigorously derived. © 2012 Elsevier B.V. All rights reserved
EDO equazioni differenziali ordinarie
No full textQuesto testo, nato da varie iterazioni del corso “Equazioni differenziali” della laurea triennale in Matematica, è indirizzato ad un lettore interessato ai primi rudimenti matematici e alla fenomenologia prodotta dalle equazioni differenziali. Nei sei Capitoli in cui è diviso, il libro presenta le metodologie di base per la risoluzione esplicita, esempi di applicazione nell’ambito della fisica e della geometria, la teoria generale delle equazioni in forma normale, la struttura delle soluzioni nel caso dei sistemi lineari, i sistemi a coefficienti costanti, i sistemi autonomi nonlineari in prossimità dei punti regolari e singolari
Numerical evidences of almost convergence of wave speeds for the Burridge– Knopoff model
No full textThis paper deals with the numerical approximation of a stick–slip system, known in the literature as Burridge–Knopoff model, proposed as a simplified description of the mechanisms generating earthquakes. Modelling of friction is crucialand we consider here the so-called velocity-weakening form. The aim of the article is twofold. Firstly, we establish the effectiveness of the classical Predictor–Corrector strategy. To our knowledge, such approach has never been applied to the model under investigation. In the first part, we determine the reliability of the proposed strategy by comparing the results with a collection of significant computational tests, starting from the simplest configuration to the more complicated (and more realistic) ones, with the numerical outputs obtained by different algorithms. Particular emphasis is laid on the Gutenberg–Richter statistical law, a classical empirical benchmark for seismic events. The second part is inspired by the result by Muratov (Phys Rev 59:3847–3857, 1999) providing evidence for the existence of traveling solutions for a corresponding continuum version of the Burridge–Knopoff model. In this direction, we aim to find some appropriate estimate for the crucial object describing the wave, namely its propagation speed. To this aim, motivated by LeVeque and Yee (J Comput Phys 86:187–210, 1990) (a paper dealing with the different topic of conservation laws), we apply a space-averaged quantity (which depends on time) for determining asymptotically an explicit numerical estimate for the velocity, which we decide to name LeVeque–Yee formula after the authors’ name of the original paper. As expected, for the Burridge–Knopoff, due to its inherent discontinuity of the process, it is not possible to attach to a single seismic event any specific propagation speed. More regularity is expected by performing some temporal averaging in the spirit of the Cesàro mean. In this direction, we observe the numerical evidence of the almost convergence of the wave speeds for the Burridge–Knopoff model of earthquakes
Spectral Stability of Weak Relaxation Shock Profiles
No full textUsing a combination of Kawashima- and Goodman-type energy estimates, we establish spectral stability of general small-amplitude relaxation shocks of symmetric dissipative systems. This extends previous results obtained by Plaza and Zumbrun [8] by singular perturbation techniques under an additional technical assumption, namely, that the background equation be noncharacteristic with respect to the shock
On Relaxation Hyperbolic Systems Violating the Shizuta-Kawashima Condition
No full textIn this paper, we start a general study on relaxation hyperbolic systems which violate the Shizuta–Kawashima coupling condition ([SK]). This investigation is motivated by the fact that this condition is in general not satisfied by various physical systems, and almost all the time in several space dimensions. First, we explore the role of entropy functionals around equilibrium solutions, which may be not constant, proposing a stability condition for such solutions. Then we find strictly dissipative entropy functions for one dimensional 2 × 2 systems which violate [SK] condition. Finally, we prove the existence of global smooth
solutions for a class of systems such that condition [SK] does not hold, but which are linearly degenerated in the non dissipative directions
The Perturbed Riemann Problem for a Balance Law
No full textWe investigate the asymptotic behavior of the bounded global entropy solutions to the hyperbolic scalar balance law ut+f(u)x=g(u), for a convex flux function f and a source term g with simple zeros. For initial data coinciding outside a compact interval with Riemann data, we describe the generic asymptotic behavior of the solutions. These may converge, as the time goes to infinity, to a sequence of traveling waves bounded by two shock waves. Such traveling waves can be smooth if they connect two consecutive zeros of the source term; otherwise they can be discontinuous and oscillating around an unstable zero. However, we are able to prove that, for generic initial data, the solutions converge to the waves of the first type
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