Theory of Laminar Flames

Buckmaster, J.; Ludford, G. S. S.; Emmons, Howard W.

doi:10.1115/1.3167129

Cited by 137 publications

(57 citation statements)

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“…Note that the obtained reduced model in (2.4) shares many common features with the classical models in combustion theory and population biology (see e.g. [4,23,30]). In particular, from dimensional analysis the diffusion length L = D s /k s naturally emerges.…”

Section: Modelmentioning

confidence: 68%

Waves of Autocrine Signaling in Patterned Epithelia

Muratov

Shvartsman

2010

Math. Model. Nat. Phenom.

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Abstract. A biophysical model describing long-range cell-to-cell communication by a diffusible signal mediated by autocrine loops in developing epithelia in the presence of a morphogenetic prepattern is introduced. Under a number of approximations, the model reduces to a particular kind of bistable reaction-diffusion equation with strong heterogeneity. In the case of the heterogeneity in the form of a long strip a detailed analysis of signal propagation is possible, using a variational approach. It is shown that under a number of assumptions which can be easily verified for particular sets of model parameters, the equation admits a unique (up to translations) variational traveling wave solution. A global bifurcation structure of these solutions is investigated in a number of particular cases. It is demonstrated that the considered setting may provide a robust developmental regulatory mechanism for delivering chemical signals across large distances in developing epithelia.

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Section: Modelmentioning

confidence: 68%

Waves of Autocrine Signaling in Patterned Epithelia

Muratov

Shvartsman

2010

Math. Model. Nat. Phenom.

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“…(1.2) Equation (1.1) models heat transfer in a medium where the thermal conductivity, convective transport, and sources or sinks of thermal energy may depend on the temperature but not on the place or time [4,5,38,109,113,146,147]. The equation also arises in various guises in numerous other fields [4,5,37,38,50,113] soil physics [19,33,74,128,137], population genetics [44,53,56,83,84,110,111,114], fluid dynamics [25, 28 30], neurology [53,112,132], combustion theory [20,26,27,144] and reaction chemistry [7,8,53], to name but a few. In these situations, the second-order term on the right-hand side of (1.1) describes a diffusive process, the first-order term corresponds to a convective or advective process, and the zero-order term is associated with reaction, sorption, sources or sinks.…”

Section: Finite Speed Of Propagationmentioning

confidence: 98%

The Characterization of Reaction-Convection-Diffusion Processes by Travelling Waves

Gilding

Kersner

1996

Journal of Differential Equations

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It has long been known that the heat equation displays infinite speed of propagation. This is to say that if the initial data are nonnegative and have nonempty compact support, the solution of an initial-value problem is positive everywhere after any infinitesimal time. However, since the nineteen-fifties it has also been known that certain nonlinear diffusion equations of degenerate parabolic type do not display this phenomenon. For these equations, the (generalized) solution of an initialvalue problem with compactly-supported initial data will have bounded support with respect to the spatial variable at all times. In this paper the necessary and sufficient criterion for finite speed of propagation for the general nonlinear reactionconvection-diffusion equationis determined. The assumptions on the coefficients a, b and c are such that the classification unifies and generalizes previously-known results. The technique employed is comparision of an arbitrary solution of the equation with suitably-constructed travelling-wave solutions and subsolutions. Basically the central conclusion is that the equation exhibits finite speed of propagation if and only if it admits a travelling-wave solution with bounded support. Concurrently, the search for a travelling-wave solution with bounded support can be reduced to the study of a singular nonlinear integral equation whose solution must satisfy a certain constraint.

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“…In combustion theory, Problem (1.1) is the high activation limit as → 0 of a scalar reaction-diffusion equation, which is of the form Problem (1.1) is a particular case of the model for premixed Near-Equidiffusional Flames (NEF's) which goes back to [19], see also [8,9], and reads…”

Section: Introductionmentioning

confidence: 99%

A critical case of stability in a free boundary problem

Brauner

Hulshof²,

Lunardi

2001

J.evol.equ.

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We study the stability of the planar travelling wave solution to a free boundary problem for the heat equation in the whole R 2 . We turn the problem into a fully nonlinear parabolic system and establish a stability result which is the proper generalization of the one-dimensional case. The curvature terms contribute a gradient squared corresponding to critical growth. The latter is eliminated by means of the Hopf-Cole transformation.

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Theory of Laminar Flames

Cited by 137 publications

References 0 publications

Waves of Autocrine Signaling in Patterned Epithelia

Waves of Autocrine Signaling in Patterned Epithelia

The Characterization of Reaction-Convection-Diffusion Processes by Travelling Waves

A critical case of stability in a free boundary problem

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