Abstract
We consider the explosion problem in an incompressible flow introduced in [5]. We use a novel L p − L ∞ estimate for elliptic advection-diffusion problems to show that the explosion threshold obeys a positive lower bound which is uniform in the advecting flow. We also identify the flows for which the explosion threshold tends to infinity as their amplitude grows and obtain an effective description of the explosion threshold in the strong flow asymptotics in two-dimensional cellular flows.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. Audoly, H. Berestycki and Y. Pomeau, Réaction-diffusion en écoulement stationnaire rapide, C. R. Acad. Sci. Paris, Sér. II 328 (2000), 255–262.
M. Belk and V. Volpert, Modeling of heat explosion with convection, Chaos 14 (2004), 263–273.
H. Berestycki, X. Cabre and L. Ryzhik, in preparation.
H. Berestycki, F. Hamel and N. Nadirashvili, Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena, Comm. Math. Phys. 253 (2005), 451–480.
H. Berestycki, L. Kagan, G. Joulin and G. Sivashinsky, The effect of stirring on the limits of thermal explosion, Combustion Theory and Modelling 1 (1997), 97–112.
H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for u t − Δu = g(u) revisited, Adv. Differential Equations 1 (1996), 73–90.
H. Brezis and J. L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 (1997), 443–469.
X. Cabré and A. Capella, Regularity of radial minimizers and extremal solutions of semiliniar elliptic equations, J. Funct. Anal. 238 (2006), 709–733.
S. Childress, Alpha-effect in flux ropes and sheets, Phys. Earth Planet Int. 20 (1979), 172–180.
P. Constantin, A. Kiselev, L. Ryzhik and A. Zlatos, Diffusion and mixing in a fluid flow, Ann. of Math. (2) 168 (2008), 643–674.
P. Constantin, A. Novikov and L. Ryzhik, Relaxation in reactive flows, Geom. Funct. Anal. 18 (2008), 1145–1167.
M. Crandall and P. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rational Mech. Anal. 58 (1975), 207–218.
A. Fannjiang and G. Papaniclaou, Convection enhanced diffusion for periodic flows, SIAM J. Appl. Math. 54 (1994), 333–408.
W. Feller, On second order differential operators, Ann. of Math. (2) 61 (1955), 90–105.
D. A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, Plenum Press, New York, 1969.
M. Freidlin, Reaction-diffusion in incompressible fluid: Asymptotic problems, J. Differential Equations 179 (2002), 44–96.
M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, second edn., Springer, New York, 1998.
A. Friedman, The asymptotic behavior of the first real eigenvalue of a second order elliptic operator with a small parameter in the highest derivatives, Indiana Univ. Math. J. 22 (1973), 1005–1015.
N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices II: Dynamic case, NoDEA Nonlinear Differential Equations Appl. 15 (2008), 115–145.
N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: stationary case, SIAM J. Math. Anal. 38 (2006/07), 1423–1449.
S. Heinze, Diffusion-advection in cellular flows with large Péclet numbers, Arch. Rational Mech. Anal. 168 (2003), 329–342.
D. Joseph and T. Lundgren, Quasilinear Dirichlet problem driven by positive sources, Arch. Rational Mech. Anal. 49 (1972/73), 241–269.
G. Joulin, A. Mikishev and G. Sivashinsky, A Semenov-Rayleigh-Bernard problem, preprint, 1996.
J. Keener and H. Keller, Positive solutions of convex nonlinear eigenvalue problems, J. Differential Equations 16 (1974), 103–125.
Y. Kifer, Random Perturbations of Dynamical Systems, Birkhäuser Boston, Boston, 1988.
A. Kiselev and L. Ryzhik, Enhancement of the traveling front speeds in reaction-diffusion equations with advection, Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001), 309–358
L. Koralov, Randomperturbations of two-dimensionalHamiltonian flows, Probab. Theory Related Fields 129 (2004), 37–62.
P. Mandl, Analytical Treatment of One-Dimensional Markov Processes, Springer, Prague, Academia, 1968.
A. S. Merzhanov and E. A. Shtessel, Free convection and thermal explosion in reactive systems, Astronaut. Acta 18 (1973), 191–199.
G. Nedev, Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris, Sér. I 330 (2000), 997–1002.
A. Novikov, G. Papanicolaou and L. Ryzhik, Boundary layers for cellular flows at high Péclet numbers, Comm. Pure Appl. Math. 58 (2005), 867–922.
A. Novikov and L. Ryzhik, Bounds on the speed of propagation of the KPP fronts in a cellular flow, Arch. Rational Mech. Anal. 184 (2007), 23–48.
M. N. Rosenbluth, H. L. Berk, I. Doxas and W. Horton, Effective diffusion in laminar convective flows, Phys. Fluids 30 (1987), 2636–2647.
L. Ryzhik and A. Zlatos, KPP pulsating front speed-up by flows, Commun. Math. Sci. 5 (2007), 575–593.
N. Semenov, Chemical Kinetics and Chain Reaction, Clarendon Press, Oxford, 1935.
B. Shraiman, Diffusive transport in a Rayleigh-Bernard convection cell, Phys. Rev. A 36 (1987), 261–267.
A. M. Soward, Fast dynamo action in fluid flow, J. Fluid Mech. 180 (1987), 267–295.
Ya. B. Zeldovich, G. I. Barenblatt, V. B. Librovich and G. M. Makhviladze, The Mathematical Theory of Combustion and Explosions (translated from the Russian by Donald H. McNeill), Consultants Bureau [Plenum], New York, 1985.
A. Zlatos, Pulsating front speed-up and quenching of reaction by fast advection, Nonlinearity 20 (2007), 2907–2921.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Berestycki, H., Kiselev, A., Novikov, A. et al. The explosion problem in a flow. JAMA 110, 31–65 (2010). https://doi.org/10.1007/s11854-010-0002-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11854-010-0002-7