On the stability of a class of self-similar solutions to the filtration-absorption equation

2002 ◽  
Vol 13 (2) ◽  
pp. 179-194 ◽  
Author(s):  
ALINA CHERTOCK
Keyword(s):  
The Self ◽  
Initial Condition ◽  
Cauchy Problems ◽  
One Dimensional ◽  
Self Similar ◽  
Analytical Result ◽  
Axisymmetric Solutions ◽  
The Stability ◽  
The One ◽  
Similar Solutions

We consider the one-dimensional and two-dimensional filtration-absorption equation ut = uΔu−(c−1)(∇u)2. The one-dimensional case was considered previously by Barenblatt et al. [4], where a special class of self-similar solutions was introduced. By the analogy with the 1D case we construct a family of axisymmetric solutions in 2D. We demonstrate numerically that the self-similar solutions obtained attract the solutions of non-self-similar Cauchy problems having the initial condition of compact support. The main analytical result we provide is the linear stability of the above self-similar solutions both in the 1D case and in the 2D case.

A self-similar solution for expansion into a vacuum of a collisionless plasma bunch

1998 ◽  
Vol 59 (1) ◽  
pp. 83-90 ◽  
Author(s):  
A. V. BAITIN ◽  
K. M. KUZANYAN
Keyword(s):  
Electron Distribution Function ◽  
Collisionless Plasma ◽  
Electron Component ◽  
One Dimensional ◽  
Ultrarelativistic Electron ◽  
Plasma Bunch ◽  
Self Similar Solution ◽  
Self Similar ◽  
The One ◽  
Similar Solutions

The process of expansion into a vacuum of a collisionless plasma bunch with relativistic electron temperature is investigated for the one-dimensional case. Self-similar solutions for the evolution of the electron distribution function and ion acceleration are obtained, taking account of cooling of the electron component of plasma for the cases of non-relativistic and ultrarelativistic electron energies.

On the correctness of a model for phase transitions in binary alloys

1990 ◽  
Vol 1 (4) ◽  
pp. 327-338 ◽  
Author(s):  
I. G. Götz
Keyword(s):  
Phase Transitions ◽  
Heat Conduction ◽  
Stability Criterion ◽  
Binary Alloys ◽  
The Self ◽  
Self Similar Solution ◽  
Self Similar ◽  
The Stability ◽  
Similar Solutions ◽  
And Diffusion

The main result of this paper is a non-uniqueness theorem for the self-similar solutions of a model for phase transitions in binary alloys. The reason for this non-uniqueness is the discontinuity in the coefficients of heat conduction and diffusion at the inter-phase. Also the existence of a self-similar solution and the stability criterion are discussed.

Theoretical modeling of the carbon dioxide injection into the porous medium saturated with methane and water taking into account the CO2 hydrate formation

2018 ◽  
Author(s):  
Gubaidullin A. A. ◽  
Musakaev N. G. ◽  
Duong Ngoc Hai ◽  
Borodin S. L. ◽  
Nguyen Quang Thai ◽  
...  
Keyword(s):  
Carbon Dioxide ◽  
Hydrate Formation ◽  
Carbon Dioxide Injection ◽  
Reservoir Temperature ◽  
One Dimensional ◽  
Porous Reservoir ◽  
Self Similar ◽  
The One ◽  
Similar Solutions ◽  
The Mathematical Model

In this work the mathematical model is constructed and the features of the injection of warm carbon dioxide (with the temperature higher than the initial reservoir temperature) into the porous reservoir initially saturated with methane gas and water are investigated. Self-similar solutions of the one-dimensional problem describing the distributions of the main parameters in the reservoir are constructed. The effect of the parameters of the injected carbon dioxide and the reservoir on the intensity of the CO2 hydrate formation is analyzed

ASYMPTOTICALLY SELF-SIMILAR GLOBAL SOLUTIONS OF A DAMPED WAVE EQUATION

2007 ◽  
Vol 09 (02) ◽  
pp. 253-277 ◽  
Author(s):  
SEIFEDDINE SNOUSSI ◽  
SLIM TAYACHI
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Global Solutions ◽  
The Self ◽  
Initial Condition ◽  
Damped Wave Equation ◽  
Small Initial Data ◽  
Additional Hypothesis ◽  
Self Similar ◽  
Similar Solutions

We study the existence and the asymptotic behavior of global solutions of the damped wave equation [Formula: see text] where a ∈ ℝ, α >1, t > 0, x ∈ ℝn, n = 1,2,3, with initial condition (u (0), ut (0)) = (φ,ψ). For α > 2 and α > 1+2 / n, we prove the existence of mild global solutions for small initial data with low regularity and which are not in L1(ℝn). Under the additional hypothesis, (2 < α < 5, when n = 3), we prove that some of these solutions are asymptotic to the self-similar solutions of the associated semi-linear heat equation [Formula: see text] with homogeneous slowly decreasing initial data behaving like c|x|-2 / (α-1) as |x|→ ∞.

scholarly journals Impact of dust in the decay of blast waves produced by a nuclear explosion

2020 ◽  
Vol 476 (2238) ◽  
pp. 20200105 ◽  
Author(s):  
Meera Chadha ◽  
J. Jena
Keyword(s):  
Radiation Effects ◽  
Nuclear Explosion ◽  
Blast Waves ◽  
Dust Particles ◽  
One Dimensional ◽  
Self Similar ◽  
The One ◽  
Similar Solutions ◽  
The Impact ◽  
Relaxing Gas

In this paper, we have studied the impact created by the introduction of up to 5% dust particles in enhancing the decay of blast waves produced by a nuclear explosion. A mathematical model is designed and modified using appropriate assumptions, the most important being treating a nuclear explosion as a point source of energy. A system of partial differential equations describing the one-dimensional, adiabatic, unsteady flow of a relaxing gas with dust particles and radiation effects is considered. The symmetric nature of an explosion is captured using the Lie group invariance and self-similar solutions obtained for the gas undergoing strong shocks. The enhancements in decay caused by varying the quantity of dust are studied. The energy released and the damage radius are found to decrease with time with an increase in the dust parameters.

Comments on the paper ‘Self-similar state of a weakly turbulent plasma’

1982 ◽  
Vol 27 (1) ◽  
pp. 189-190 ◽  
Author(s):  
G. E. Vekstein ◽  
D. D. Ryutov ◽  
R. Z. Sagdeev
Keyword(s):  
Electric Field ◽  
External Electric Field ◽  
Collisionless Plasma ◽  
Turbulent Plasma ◽  
The Self ◽  
Similar Solution ◽  
One Dimensional ◽  
Self Similar Solution ◽  
Self Similar ◽  
The One

In a recent paper, Balescu (1980) criticizes the self-similar solution in the problem of anomalous plasma resistivity (Vekstein, Ryutov & Sagdeev 1970) and comes to the conclusion that this solution is not correct. The aim of this comment is to show why Balescu's arguments are erroneous.We consider here the simplest case of the one-dimensional collisionless plasma in the presence of an external electric field.

scholarly journals Self-similar solutions of the one-dimensional Landau–Lifshitz–Gilbert equation

Nonlinearity ◽  
2015 ◽  
Vol 28 (5) ◽  
pp. 1307-1350 ◽  
Author(s):  
Susana Gutiérrez ◽  
André de Laire
Keyword(s):  
One Dimensional ◽  
Self Similar ◽  
The One ◽  
Similar Solutions

Self-similar solutions for Vlasov and water-bag models

1978 ◽  
Vol 19 (1) ◽  
pp. 135-146 ◽  
Author(s):  
J. R. Burgan ◽  
J. Gutierrez ◽  
E. Fijalkow ◽  
M. Navet ◽  
M. R. Feix
Keyword(s):  
Single Species ◽  
Complete Agreement ◽  
Self Similarity ◽  
One Dimensional ◽  
Virtual Particle ◽  
Electron Plasmas ◽  
Finite Group Theory ◽  
Self Similar ◽  
The One ◽  
Similar Solutions

Using finite group theory, self-similar solutions for the one-dimensional Vlasov– Poisson system are considered, both for electron plasmas, with a fixed ion background, and for a single species beam. Difficulties arising from the Poisson equation are pointed out and handled by using an exponential group of transformations. Introducing, for the beam, a water-bag model, we find that the boundary condition problem, imposed by self-similarity, can be solved through the introduction of a ‘virtual particle’ concept. In order to test the analytical solution obtained, we perform a numerical simulation using a particle code. Complete agreement is found.

scholarly journals Stability of a one-dimensional morphoelastic model for post-burn contraction

2021 ◽  
Vol 83 (3) ◽  
Author(s):  
Ginger Egberts ◽  
Fred Vermolen ◽  
Paul van Zuijlen
Keyword(s):  
Wound Healing ◽  
Residual Stresses ◽  
Truncation Error ◽  
Signaling Molecules ◽  
Discrete Problem ◽  
One Dimensional ◽  
Stability Constraint ◽  
Biological Interpretation ◽  
The Stability ◽  
The One

AbstractTo deal with permanent deformations and residual stresses, we consider a morphoelastic model for the scar formation as the result of wound healing after a skin trauma. Next to the mechanical components such as strain and displacements, the model accounts for biological constituents such as the concentration of signaling molecules, the cellular densities of fibroblasts and myofibroblasts, and the density of collagen. Here we present stability constraints for the one-dimensional counterpart of this morphoelastic model, for both the continuous and (semi-) discrete problem. We show that the truncation error between these eigenvalues associated with the continuous and semi-discrete problem is of order $${{\mathcal {O}}}(h^2)$$ O ( h 2 ) . Next we perform numerical validation to these constraints and provide a biological interpretation of the (in)stability. For the mechanical part of the model, the results show the components reach equilibria in a (non) monotonic way, depending on the value of the viscosity. The results show that the parameters of the chemical part of the model need to meet the stability constraint, depending on the decay rate of the signaling molecules, to avoid unrealistic results.

Sign in / Sign up

Export Citation Format

Share Document