Remarks about a Constructive Method in Nonlinear Elasticity

2019 ◽  
Author(s):  
Hans-Peter Gittel ◽  
Matthias Guenther
Keyword(s):  
Total Energy ◽  
Nonlinear Elasticity ◽  
Elastic Energy ◽  
Existence Result ◽  
Approximation Scheme ◽  
Local Existence ◽  
Constructive Method ◽  
Equilibrium Conditions ◽  
Incremental Method ◽  
Minimizing Movements

The paper deals with an incremental method for solving the equilibrium conditions in nonlinear elasticity which was introduced by H. Beckert in 1975. Here, the alteration rate of some stress tensor is prescribed by supplementary stresses. This yields an expression for a locally defined elastic energy and the total energy can be minimized. Hence, the considered method is in the sense of minimizing movements. The authors analyze some of its properties, derive a local existence result in a simplified way, and prove the convergence of an approximation scheme.

scholarly journals An unbalanced optimal transport splitting scheme for general advection-reaction-diffusion problems

2019 ◽  
Vol 25 ◽  
pp. 8 ◽  
Author(s):  
Thomas Gallouët ◽  
Maxime Laborde ◽  
Léonard Monsaingeon
Keyword(s):  
Optimal Transport ◽  
Diffusion Processes ◽  
Reaction Diffusion ◽  
Wasserstein Distance ◽  
Diffusion Problem ◽  
Constructive Method ◽  
Interacting Species ◽  
General Scalar ◽  
Minimizing Movements ◽  
Rao Distance

In this paper, we show that unbalanced optimal transport provides a convenient framework to handle reaction and diffusion processes in a unified metric setting. We use a constructive method, alternating minimizing movements for the Wasserstein distance and for the Fisher-Rao distance, and prove existence of weak solutions for general scalar reaction-diffusion-advection equations. We extend the approach to systems of multiple interacting species, and also consider an application to a very degenerate diffusion problem involving a Gamma-limit. Moreover, some numerical simulations are included.

scholarly journals A local existence result for Poincaré-Einstein metrics

2020 ◽  
Vol 361 ◽  
pp. 106912 ◽  
Author(s):  
Matthew J. Gursky ◽  
Gábor Székelyhidi
Keyword(s):  
Existence Result ◽  
Einstein Metrics ◽  
Local Existence

scholarly journals LOCAL EXISTENCE AND CONTINUATION CRITERIA FOR SOLUTIONS OF THE EINSTEIN–VLASOV-SCALAR FIELD SYSTEM WITH SURFACE SYMMETRY

2004 ◽  
Vol 01 (04) ◽  
pp. 691-724 ◽  
Author(s):  
D. TEGANKONG ◽  
N. NOUTCHEGUEME ◽  
A. D. RENDALL
Keyword(s):  
General Relativity ◽  
Scalar Field ◽  
Wave Equations ◽  
Existence Result ◽  
General Theorem ◽  
Local Existence ◽  
Field System ◽  
Cosmological Solutions ◽  
Time Existence ◽  
Global Existence Result

We prove in the cases of spherical, plane and hyperbolic symmetry a local in time existence theorem and continuation criteria for cosmological solutions of the Einstein–Vlasov-scalar field system, with the sources generated by a distribution function and a scalar field, subject to the Vlasov and wave equations respectively. This system describes the evolution of self-gravitating collisionless matter and scalar waves within the context of general relativity. In the case where the only source is a scalar field it is shown that a global existence result can be deduced from the general theorem.

scholarly journals Cosmological solutions of the Vlasov-Einstein system with spherical, plane, and hyperbolic symmetry

1996 ◽  
Vol 119 (4) ◽  
pp. 739-762 ◽  
Author(s):  
Gerhard Rein
Keyword(s):  
General Relativity ◽  
Phase Space ◽  
Existence Result ◽  
Local Existence ◽  
Space Distribution ◽  
Initial Value ◽  
Small Initial Data ◽  
Phase Space Distribution ◽  
Spacetime Singularity ◽  
Cosmological Solutions

AbstractThe Vlasov-Einstein system describes a self-gravitating, collisionless gas within the framework of general relativity. We investigate the initial value problem in a cosmological setting with spherical, plane, or hyperbolic symmetry and prove that for small initial data solutions exist up to a spacetime singularity which is a curvature and a crushing singularity. An important tool in the analysis is a local existence result with a continuation criterion saying that solutions can be extended as long as the momenta in the support of the phase-space distribution of the matter remain bounded.

Global well-posedness of the Cauchy problem for the 3D Jordan–Moore–Gibson–Thompson equation

2020 ◽  
pp. 2050069
Author(s):  
Reinhard Racke ◽  
Belkacem Said-Houari
Keyword(s):  
Cauchy Problem ◽  
Existence Result ◽  
Local Existence ◽  
Decay Rates ◽  
Polynomial Decay ◽  
Small Data ◽  
Third Order ◽  
Bootstrap Argument ◽  
The Cauchy Problem ◽  
Linear Decay

We consider the Cauchy problem of a third order in time nonlinear equation known as the Jordan–Moore–Gibson–Thompson (JMGT) equation arising in acoustics as an alternative model to the well-known Kuznetsov equation. We show a local existence result in appropriate function spaces, and, using the energy method together with a bootstrap argument, we prove a global existence result for small data, without using the linear decay. Finally, polynomial decay rates in time for a norm related to the solution will be obtained.

AN EFFICIENT NUMERICAL METHOD FOR CAVITATION IN NONLINEAR ELASTICITY

2011 ◽  
Vol 21 (08) ◽  
pp. 1733-1760 ◽  
Author(s):  
XIANMIN XU ◽  
DUVAN HENAO
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Method ◽  
Nonlinear Elasticity ◽  
Elastic Energy ◽  
Finite Element Approximation ◽  
Void Coalescence ◽  
Element Approximation ◽  
Nonconforming Finite Element ◽  
First Time

This paper is concerned with the numerical computation of cavitation in nonlinear elasticity. The Crouzeix–Raviart nonconforming finite element method is shown to prevent the degeneration of the mesh provoked by the conventional finite element approximation of this problem. Upon the addition of a suitable stabilizing term to the elastic energy, the method is used to solve cavitation problems in both radially symmetric and non-radially symmetric settings. While the radially symmetric examples serve to illustrate the efficiency of the method, and for validation purposes, the experiments with non-centered and multiple cavities (carried out for the first time) yield novel observations of situations potentially leading to void coalescence.

scholarly journals General decay and well-posedness of the Cauchy problem for the Jordan-Moore-Gibson-Thompson equation with memory

Filomat ◽  
2021 ◽  
Vol 35 (5) ◽  
pp. 1745-1773
Author(s):  
Salah Boulaaras ◽  
Abdelbaki Choucha ◽  
Djamel Ouchenane
Keyword(s):  
Cauchy Problem ◽  
Existence Result ◽  
Local Existence ◽  
Generalized Solution ◽  
General Decay ◽  
Small Data ◽  
Well Posedness ◽  
Third Order ◽  
The Cauchy Problem ◽  
With Memory

In this paper, we consider the Cauchy problem of a third order in time nonlinear equation known as the Jordan-Moore-Gibson-Thompson (JMGT) equation with the presence of both memory. Using the well known energy method combined with Lyapunov functionals approach, we prove a general decay result, and we show a local existence result in appropriate function spaces. Finally, we prove a global existence result for small data, and we prove the uniqueness of the generalized solution.

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