Propagation of a combustion front in a striated solid medium: a homogenization analysis

Brauner, CM; Fife, Paul C.; Namah, Gawtum; Schmidt-Lainé, Claudine

doi:10.1090/qam/1233525

Cited by 11 publications

(16 citation statements)

References 2 publications

(1 reference statement)

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“…Similar existence results have been obtained for other classes of equations arising in different models [23,31,105].…”

Section: Flows With An Array Of Vortical Cells and More General Flowssupporting

confidence: 84%

“…Note here that the Laplace operator has been replaced with a general nonhomogeneous diffusion operator div(A∇u). Such diffusion operators have also been considered in the one-dimensional case or in the case of the whole space (see [85,102,103,104,106]), and also in similar problems modeling the propagation of fronts in periodic solid media; see [23,84].…”

Section: General Periodic Domains and Periodic Excitable Mediamentioning

confidence: 99%

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Front propagation in periodic excitable media

Berestycki

Hamel

2002

Comm Pure Appl Math

346

546

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This paper is devoted to the study of pulsating traveling fronts for reactiondiffusion-advection equations in a general class of periodic domains with underlying periodic diffusion and velocity fields. Such fronts move in some arbitrarily given direction with an unknown effective speed. The notion of pulsating traveling fronts generalizes that of traveling fronts for planar or shear flows.Various existence, uniqueness, and monotonicity results are proved for two classes of reaction terms. First, for a combustion-type nonlinearity, it is proved that the pulsating traveling front exists and that its speed is unique. Moreover, the front is increasing with respect to the time variable and unique up to translation in time. We also consider one class of monostable nonlinearity that arises either in combustion or biological models. Then, the set of possible speeds is a semiinfinite interval, closed and bounded from below. For each possible speed, there exists a pulsating traveling front that is increasing in time. This result extends the classical Kolmogorov-Petrovsky-Piskunov case. Our study covers in particular the case of flows in all of space with periodic advections such as periodic shear flows or a periodic array of vortical cells. These results are also obtained for cylinders with oscillating boundaries or domains with a periodic array of holes.

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“…Similar existence results have been obtained for other classes of equations arising in different models [23,31,105].…”

Section: Flows With An Array Of Vortical Cells and More General Flowssupporting

confidence: 84%

Section: General Periodic Domains and Periodic Excitable Mediamentioning

confidence: 99%

Front propagation in periodic excitable media

Berestycki

Hamel

2002

Comm Pure Appl Math

346

546

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“…with R smooth and 1-periodic, we look for solutions of the form 6) where u → v(u) is a C 1 function of the variable u, satisfying condition (2.4) and such that v > 0. The function z → φ α (z) is assumed to be periodic.…”

Section: Model Level Set Formulation Main Resultsmentioning

confidence: 99%

The `hump' effect in solid propellant combustion

Namah¹,

Roquejoffre²

2000

Interfaces Free Bound.

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An eikonal equation modelling the propagation of combustion fronts in striated media is studied via a level set formulation. Travelling fronts are obtained, and their speeds turn out to be monotone with respect to the angle of the striations. An effective equation for thin meniscus striations is derived from a homogenization process, thus explaining the so-called 'hump' effect. Finally, the time-asymptotic stabilization of unsteady fronts propagating in straight striations is proved.

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“…However, most works concerning viscosity solutions of first order PDE deal with continuous Hamiltonians. In many applications ( [1,2,6,11] for example), spatial inhomogeneity often leads to a discontinuous Hamiltonian, and it is not clear at all whether the uniqueness is still valid in this case. We will show, under very general assumptions on v, that (1.1) admits a unique viscosity solution.…”

Section: Introductionmentioning

confidence: 99%

Viscosity solutions of discontinuous Hamilton–Jacobi equations

Chen¹,

Hu²

2008

Interfaces Free Bound.

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We define viscosity solutions for the Hamilton-Jacobi equation ϕ t = v(x, t)H (∇ϕ) in R N × (0, ∞) where v is positive and bounded measurable and H is non-negative and Lipschitz continuous. Under certain assumptions, we establish the existence and uniqueness of Lipschitz continuous viscosity solutions. The uniqueness result holds in particular for those v which are independent of t and piecewise continuous with discontinuity sets consisting of finitely many smooth lower dimensional surfaces not tangent to each other at any point of their intersection.

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Propagation of a combustion front in a striated solid medium: a homogenization analysis

Cited by 11 publications

References 2 publications

Front propagation in periodic excitable media

Front propagation in periodic excitable media

The `hump' effect in solid propellant combustion

Viscosity solutions of discontinuous Hamilton–Jacobi equations

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