Stable and not too unstable solutions on 𝑅ⁿ for small diffusion

10.1090/fic/048/04 ◽  
2006 ◽  
pp. 67-93 ◽  
Author(s):  
Norman Dancer
Keyword(s):  
Small Diffusion ◽  
Unstable Solutions

Evolution of unstable system.

2015 ◽  
Vol 9 (3) ◽  
pp. 2487-2502 ◽  
Author(s):  
Igor V. Lebed
Keyword(s):  
Reynolds Number ◽  
Vortex Shedding ◽  
Stokes Flow ◽  
Stable Solution ◽  
The State ◽  
Unstable System ◽  
Unstable Flow ◽  
Unstable Solutions ◽  
Vortex Shedding Modes ◽  
Hydrodynamics Equations

Scenario of appearance and development of instability in problem of a flow around a solid sphere at rest is discussed. The scenario was created by solutions to the multimoment hydrodynamics equations, which were applied to investigate the unstable phenomena. These solutions allow interpreting Stokes flow, periodic pulsations of the recirculating zone in the wake behind the sphere, the phenomenon of vortex shedding observed experimentally. In accordance with the scenario, system loses its stability when entropy outflow through surface confining the system cannot be compensated by entropy produced within the system. The system does not find a new stable position after losing its stability, that is, the system remains further unstable. As Reynolds number grows, one unstable flow regime is replaced by another. The replacement is governed tendency of the system to discover fastest path to depart from the state of statistical equilibrium. This striving, however, does not lead the system to disintegration. Periodically, reverse solutions to the multimoment hydrodynamics equations change the nature of evolution and guide the unstable system in a highly unlikely direction. In case of unstable system, unlikely path meets the direction of approaching the state of statistical equilibrium. Such behavior of the system contradicts the scenario created by solutions to the classic hydrodynamics equations. Unstable solutions to the classic hydrodynamics equations are not fairly prolonged along time to interpret experiment. Stable solutions satisfactorily reproduce all observed stable medium states. As Reynolds number grows one stable solution is replaced by another. They are, however, incapable of reproducing any of unstable regimes recorded experimentally. In particular, stable solutions to the classic hydrodynamics equations cannot put anything in correspondence to any of observed vortex shedding modes. In accordance with our interpretation, the reason for this isthe classic hydrodynamics equations themselves.

Formation of High Resistivity Phases of Nickel Silicide at Small Diffusion Region

2007 ◽  
Author(s):  
Ryuji Tomita ◽  
Hiroshi Kimura ◽  
Makoto Yasuda ◽  
Tomowo Nakayama ◽  
Kazutaka Maeda ◽  
...  
Keyword(s):  
Nickel Silicide ◽  
High Resistivity ◽  
Diffusion Region ◽  
Small Diffusion

The first boundary exit point of a diffusion process with small diffusion

Mathematical Notes ◽  
10.1007/bf02412502 ◽  
1977 ◽  
Vol 22 (3) ◽  
pp. 720-725
Author(s):  
A. D. Venttsel'
Keyword(s):  
Diffusion Process ◽  
Exit Point ◽  
Small Diffusion

Metastability for a generalized Burgers equation with applications to propagating flame fronts

1999 ◽  
Vol 10 (1) ◽  
pp. 27-53 ◽  
Author(s):  
X. SUN ◽  
M. J. WARD
Keyword(s):  
Flame Front ◽  
Principal Eigenvalue ◽  
Vertical Channel ◽  
Front Propagation ◽  
Slow Motion ◽  
Diffusion Limit ◽  
Asymptotic Results ◽  
Small Diffusion ◽  
Generalized Burgers Equation ◽  
Exponentially Small

In the small diffusion limit ε→0, metastable dynamics is studied for the generalized Burgers problemformula hereHere u=u(x, t) and f(u) is smooth, convex, and satisfies f(0)=f′(0)=0. The choice f(u)=u2/2 has been shown previously to arise in connection with the physical problem of upward flame-front propagation in a vertical channel in a particular parameter regime. In this context, the shape y=y(x, t) of the flame-front interface satisfies u=−yx. For this problem, it is shown that the principal eigenvalue associated with the linearization around an equilibrium solution corresponding to a parabolic-shaped flame-front interface is exponentially small. This exponentially small eigenvalue then leads to a metastable behaviour for the time- dependent problem. This behaviour is studied quantitatively by deriving an asymptotic ordinary differential equation characterizing the slow motion of the tip location of a parabolic-shaped interface. Similar metastability results are obtained for more general f(u). These asymptotic results are shown to compare very favourably with full numerical computations.

scholarly journals Arbitrary nonlinearities in convective population dynamics with small diffusion

Physical Review E ◽  
2005 ◽  
Vol 72 (4) ◽  
Author(s):  
I. D. Peixoto ◽  
L. Giuggioli ◽  
V. M. Kenkre
Keyword(s):  
Population Dynamics ◽  
Small Diffusion
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