×

zbMATH — the first resource for mathematics

Multi-dimensional travelling-wave solutions of a flame propagation model. (English) Zbl 0711.35066
The existence of a travelling wave solution \(u(x_ 1+ct,y)\) of the equation \[ \partial u/\partial t+\alpha (y)\partial u/\partial x_ 1=\Delta u+g(u)\text{ in } {\mathbb{R}}\times \omega \] is shown, where \(\omega\) is a bounded and smooth open domain. The function u satisfies an elliptic equation with parameter c where both u and c are unknown. In contrast to previous papers \(c+\alpha (y)\) may in general change the sign in the domain \(\omega\). Conditions for the occurrence of this phenomenon which may be interpreted as an inversion of the velocity field are given. The proofs are based on a priori estimates which are derived from methods typical for the elliptic case as the maximum principle, energy estimates, and elliptic regularity.
Reviewer: P.Kröger

MSC:
35K57 Reaction-diffusion equations
80A25 Combustion
35J60 Nonlinear elliptic equations
PDF BibTeX Cite
Full Text: DOI
References:
[1] S. Agmon, A. Douglis & L. Nirenberg, ?Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I?, Comm. Pure Appl. Math. 21, pp. 623-727 (1959). · Zbl 0093.10401
[2] F. Benkhaldoun & B. Larrouturou, ?A finite-element adaptive investigation of curved stable and unstable flame fronts?, 76, pp. 119-134 (1989). · Zbl 0687.76071
[3] H. Berestycki & B. Larrouturou, ?A semilinear elliptic equation in a strip arising in a two-dimensional flame propagation model?, J. für Reine und Angewandte Mathematik, 396, pp. 14-40 (1989). · Zbl 0658.35036
[4] H. Berestycki & B. Larrouturou, ?Quelques aspects mathématiques de la propagation des flammes prémélangées?, Nonlinear partial differential equations and their applications, Collège de France seminar, Brezis & Lions, eds., Research Notes in Mathematics, Pitman-Longman, London, to appear.
[5] H. Berestycki, B. Nicolaenko & B. Scheurer, ?Travelling wave solutions to combustion models and their singular limits?, SIAM J. Math. Anal. 16 (6), pp. 1207-1242 (1985). · Zbl 0596.76096
[6] H. Berestycki & L. Nirenberg, ?Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations?, J. Geom. and Phys. (special issue dedicated to I. M. Gelfand), to appear. · Zbl 0698.35031
[7] H. Berestycki & L. Nirenberg, ?Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains, to appear. · Zbl 0705.35004
[8] R. Gardner, ?Existence of multidimensional travelling wave solutions of an initialboundary value problem?, J. Diff. Equ. 61, pp. 335-379 (1986). · Zbl 0577.35064
[9] W. E. Johnson, ?On a first-order boundary value problem for laminar flame theory?, Arch. Rational Mech. Anal. 13, pp. 46-54 (1963). · Zbl 0114.42402
[10] W. E. Johnson & W. Nachbar, ?Laminar flame theory and the steady linear burning of a monopropellant?, Arch. Rational Mech. Anal. 12, pp. 58-91 (1963). · Zbl 0111.40502
[11] Ja. I. Kanel’, ?Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory?, Mat. Sbornik 59, pp. 245-288 (1962).
[12] Ja. I. Kanel’, ?On steady state solutions to systems of equations arising in combustion theory?, Dokl. Akad. Nauk USSR 149 (2), pp. 367-369 (1963).
[13] B. Larrouturou, ?Introduction to combustion modelling?, Springer Series in Computational Physics, to appear, (1990).
[14] C. M. Li, Thesis, Courant Institute of Mathematical Sciences, New York-University, in preparation.
[15] G. I. Sivashinsky, ?Instabilities, pattern formation and turbulence in flames?, Ann. Rev. Fluid Mech. 15, pp. 179-199 (1983). · Zbl 0538.76053
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.