Weak entropy solutions to a model in induction hardening, existence and weak-strong uniqueness

In this paper, we investigate a model describing induction hardening of steel. The related system consists of an energy balance, an ordinary differential equation (ODE) for the different phases of steel, and Maxwell’s equations in a potential formulation. The existence of weak entropy solutions is shown by a suitable regularization and discretization technique. Moreover, we prove the weak-strong uniqueness of these solutions, i.e. that a weak entropy solutions coincides with a classical solution emanating form the same initial data as long as the classical one exists. The weak entropy solution concept has advantages in comparison to the previously introduced weak solutions, e.g. it allows to include free energy functions with low regularity properties corresponding to phase transitions.

Download Full-text

Global generalized solutions to a forager–exploiter model with superlinear degradation and their eventual regularity properties

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202520400072 ◽

2020 ◽

Vol 30 (06) ◽

pp. 1075-1117 ◽

Cited By ~ 1

Author(s):

Tobias Black

Keyword(s):

Solution Concept ◽

Initial Boundary ◽

Initial Boundary Value Problem ◽

Generalized Solution ◽

Global Solvability ◽

Generalized Solutions ◽

Food Density ◽

Nutrient Density ◽

Regularity Properties ◽

Eventual Regularity

In this paper, we consider a cascaded taxis model for two proliferating and degrading species which thrive on the same nutrient but orient their movement according to different schemes. In particular, we assume the first group, the foragers, to orient their movement directly along an increasing gradient of the food density, while the second group, the exploiters, instead track higher densities of the forager group. Specifically, we will investigate an initial boundary-value problem for a prototypical forager–exploiter model of the form [Formula: see text] in a smoothly bounded domain [Formula: see text], where [Formula: see text], [Formula: see text] is nonnegative and the functions [Formula: see text] are assumed to satisfy [Formula: see text], [Formula: see text] as well as [Formula: see text] respectively, with constants [Formula: see text], [Formula: see text] and [Formula: see text] and [Formula: see text]. Assuming that [Formula: see text], [Formula: see text] and that [Formula: see text] satisfies certain structural conditions, we establish the global solvability of this system with respect to a suitable generalized solution concept and then, for the more restrictive case of [Formula: see text] and [Formula: see text], investigate an eventual regularity effect driven by the decay of the nutrient density [Formula: see text].

Download Full-text

Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving Subquadratic Growth Restrictions

Advanced Nonlinear Studies ◽

10.1515/ans-2020-2107 ◽

2020 ◽

Vol 20 (4) ◽

pp. 795-817

Author(s):

Michael Winkler

Keyword(s):

Large Time ◽

Blow Up ◽

Solution Concept ◽

Neumann Boundary ◽

Generalized Solution ◽

Homogeneous Steady State ◽

Regularity Properties ◽

Growth System ◽

Growth Restrictions ◽

Large Time Limit

AbstractThe chemotaxis-growth system(⋆)\left\{\begin{aligned} \displaystyle u_{t}&\displaystyle=D\Delta u-\chi\nabla\cdot(u\nabla v)+\rho u-\mu u^{\alpha},\\ \displaystyle v_{t}&\displaystyle=d\Delta v-\kappa v+\lambda u\end{aligned}\right.is considered under homogeneous Neumann boundary conditions in smoothly bounded domains \Omega\subset\mathbb{R}^{n}, n\geq 1. For any choice of \alpha>1, the literature provides a comprehensive result on global existence for widely arbitrary initial data within a suitably generalized solution concept, but the regularity properties of such solutions may be rather poor, as indicated by precedent results on the occurrence of finite-time blow-up in corresponding parabolic-elliptic simplifications. Based on the analysis of a certain eventual Lyapunov-type feature of (⋆), the present work shows that, whenever \alpha\geq 2-\frac{2}{n}, under an appropriate smallness assumption on 𝜒, any such solution at least asymptotically exhibits relaxation by approaching the nontrivial spatially homogeneous steady state \bigl{(}\bigl{(}\frac{\rho}{\mu}\bigr{)}^{\frac{1}{\alpha-1}},\frac{\lambda}{\kappa}\bigl{(}\frac{\rho}{\mu}\bigr{)}^{\frac{1}{\alpha-1}}\bigr{)} in the large time limit.

Download Full-text

Interactive Graphics for the Analysis of Tires

Tire Science and Technology ◽

10.2346/1.2150991 ◽

1985 ◽

Vol 13 (3) ◽

pp. 127-146 ◽

Cited By ~ 1

Author(s):

R. Prabhakaran

Keyword(s):

Finite Element Analysis ◽

Finite Element ◽

Approximate Solutions ◽

Interactive Graphics ◽

Element Analysis ◽

Analysis Process ◽

Physical Problems ◽

The Finite Element Analysis ◽

Analysis Computer ◽

Discretization Technique

Abstract The finite element method, which is a numerical discretization technique for obtaining approximate solutions to complex physical problems, is accepted in many industries as the primary tool for structural analysis. Computer graphics is an essential ingredient of the finite element analysis process. The use of interactive graphics techniques for analysis of tires is discussed in this presentation. The features and capabilities of the program used for pre- and post-processing for finite element analysis at GenCorp are included.

Download Full-text

Degenerate Diffusion Operators Arising in Population Biology (AM-185)

10.23943/princeton/9780691157122.001.0001 ◽

2017 ◽

Author(s):

Charles L. Epstein ◽

Rafe Mazzeo

Keyword(s):

Adjoint Operator ◽

Integral Kernel ◽

Mathematical Finance ◽

Elliptic Operators ◽

Complete Analysis ◽

Time Behavior ◽

Degenerate Elliptic Operators ◽

Regularity Properties ◽

Degenerate Elliptic ◽

Mathematical Foundations

This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the martingale problem and therefore the existence of the associated Markov process. The book uses an “integral kernel method” to develop mathematical foundations for the study of such degenerate elliptic operators and the stochastic processes they define. The precise nature of the degeneracies of the principal symbol for these operators leads to solutions of the parabolic and elliptic problems that display novel regularity properties. Dually, the adjoint operator allows for rather dramatic singularities, such as measures supported on high codimensional strata of the boundary. The book establishes the uniqueness, existence, and sharp regularity properties for solutions to the homogeneous and inhomogeneous heat equations, as well as a complete analysis of the resolvent operator acting on Hölder spaces. It shows that the semigroups defined by these operators have holomorphic extensions to the right half plane. The book also demonstrates precise asymptotic results for the long-time behavior of solutions to both the forward and backward Kolmogorov equations.

Download Full-text

Local Regularity Properties of Almost and Quasiminimal sets with a Sliding Boundary Condition

Astérisque ◽

10.24033/ast.1077 ◽

2019 ◽

Vol 411 ◽

pp. 1-380

Author(s):

Guy DAVID

Keyword(s):

Boundary Condition ◽

Local Regularity ◽

Regularity Properties

Download Full-text

Data-Driven Many-Body Models with Chemical Accuracy for CH4/H2O Mixtures

10.26434/chemrxiv.13013057 ◽

2020 ◽

Author(s):

Marc Riera ◽

Alan Hirales ◽

Raja Ghosh ◽

Francesco Paesani

Keyword(s):

Liquid Phase ◽

Quantitative Description ◽

Liquid Mixtures ◽

Many Body ◽

Energy Functions ◽

Potential Energy Functions ◽

Dynamics Simulations ◽

Body Potential ◽

Small Clusters ◽

Many Body Interactions

<div> <div> <div> <p>Many-body potential energy functions (PEFs) based on the TTM-nrg and MB-nrg theoretical/computational frameworks are developed from coupled cluster reference data for neat methane and mixed methane/water systems. It is shown that that the MB-nrg PEFs achieve subchemical accuracy in the representation of individual many-body effects in small clusters and enables predictive simulations from the gas to the liquid phase. Analysis of structural properties calculated from molecular dynamics simulations of liquid methane and methane/water mixtures using both TTM-nrg and MB-nrg PEFs indicates that, while accounting for polarization effects is important for a correct description of many-body interactions in the liquid phase, an accurate representation of short-range interactions, as provided by the MB-nrg PEFs, is necessary for a quantitative description of the local solvation structure in liquid mixtures. </p> </div> </div> </div>

Download Full-text