Singularly Perturbed Evolution Equations with Applications to Kinetic Theory

10.1142/2621 ◽  
1995 ◽  
Author(s):  
J R Mika ◽  
J Banasiak
Keyword(s):  
Kinetic Theory ◽  
Evolution Equations ◽  
Singularly Perturbed

scholarly journals A Class of Shock Wave Solution to Singularly Perturbed Nonlinear Time-Delay Evolution Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Yi-Hu Feng ◽  
Lei Hou
Keyword(s):  
Shock Wave ◽  
Time Delay ◽  
Wave Solution ◽  
Evolution Equations ◽  
Multiple Scales ◽  
Expansion Method ◽  
Initial Boundary ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Shock Wave Solution

Nonlinear singularly perturbed problem for time-delay evolution equation with two parameters is studied. Using the variables of the multiple scales method, homogeneous equilibrium method, and approximation expansion method from the singularly perturbed theories, the structure of the solution to the time-delay problem with two small parameters is discussed. Under suitable conditions, first, the outer solution to the time-delay initial boundary value problem is given. Second, the multiple scales variables are introduced to obtain the shock wave solution and boundary layer corrective terms for the solution. Then, the stretched variable is applied to get the initial layer correction terms. Finally, using the singularly perturbed theory and the fixed point theorem from functional analysis, the uniform validity of asymptotic expansion solution to the problem is proved. In addition, the proposed method possesses the advantages of being very convenient to use.

scholarly journals Lifts of Symmetric Tensors: Fluids, Plasma, and Grad Hierarchy

Entropy ◽  
2019 ◽  
Vol 21 (9) ◽  
pp. 907 ◽  
Author(s):  
Oğul Esen ◽  
Miroslav Grmela ◽  
Hasan Gümral ◽  
Michal Pavelka
Keyword(s):  
Kinetic Theory ◽  
Lie Algebra ◽  
Particle Motion ◽  
Evolution Equations ◽  
Poisson Brackets ◽  
Hamiltonian Approach ◽  
Symmetric Tensors ◽  
Field Equations ◽  
Algebra Homomorphisms

Geometrical and algebraic aspects of the Hamiltonian realizations of the Euler’s fluid and the Vlasov’s plasma are investigated. A purely geometric pathway (involving complete lifts and vertical representatives) is proposed, which establishes a link from particle motion to evolution of the field variables. This pathway is free from Poisson brackets and Hamiltonian functionals. Momentum realizations (sections on T * T * Q ) of (both compressible and incompressible) Euler’s fluid and Vlasov’s plasma are derived. Poisson mappings relating the momentum realizations with the usual field equations are constructed as duals of injective Lie algebra homomorphisms. The geometric pathway is then used to construct the evolution equations for 10-moments kinetic theory. This way the entire Grad hierarchy (including entropic fields) can be constructed in a purely geometric way. This geometric way is an alternative to the usual Hamiltonian approach to mechanics based on Poisson brackets.

scholarly journals A Comment on ‘‘On the corotating frame and evolution equations in kinetic theory’’

1986 ◽  
Vol 85 (4) ◽  
pp. 2341-2342 ◽  
Author(s):  
A. S. Lodge ◽  
R. B. Bird ◽  
C. F. Curtiss ◽  
M. W. Johnson
Keyword(s):  
Kinetic Theory ◽  
Evolution Equations ◽  
Corotating Frame

Continuous families of exponential attractors for singularly perturbed equations with memory

2010 ◽  
Vol 140 (2) ◽  
pp. 329-366 ◽  
Author(s):  
Stefania Gatti ◽  
Alain Miranville ◽  
Vittorino Pata ◽  
Sergey Zelik
Keyword(s):  
Evolution Equations ◽  
General Theorem ◽  
Perturbation Parameter ◽  
Singular Limit ◽  
Singularly Perturbed ◽  
Exponential Attractors ◽  
Abstract Evolution Equations ◽  
Singularly Perturbed Equations ◽  
Image Position ◽  
With Memory

For a family of semigroups Sε(t) : ℌε → ℌε depending on a perturbation parameter ε ∈ [0, 1], where the perturbation is allowed to become singular at ε = 0, we establish a general theorem on the existence of exponential attractors εε satisfying a suitable Hölder continuity property with respect to the symmetric Hausdorff distance at every ε ∈ [0, 1]. The result is applied to the abstract evolution equations with memorywhere kε(s) = (1/ε)k(s/ε) is the rescaling of a convex summable kernel k with unit mass. Such a family can be viewed as a memory perturbation of the equationformally obtained in the singular limit ε → 0.

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