Article

Communications in Mathematical Physics

, Volume 259, Issue 2, pp 451-474

First online:

Traveling Fronts in a Reactive Boussinesq System: Bounds and Stability

  • Brandy WinnAffiliated withDepartment of Mathematics, University of Chicago Email author 

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Abstract

This paper considers a simplified model of active combustion in a fluid flow, with the reaction influencing the flow. The model consists of a reaction-diffusion-advection equation coupled with an incompressible Navier-Stokes system under the Boussinesq approximation in an infinite vertical strip. We prove that for certain ignition nonlinearities, including all that are C2, and for any domain width, planar traveling front solutions are nonlinearly and exponentially stable within certain weighted H2 spaces, provided that the Rayleigh number ρ is small enough. The same result holds for bistable nonlinearities in unweighted H2 spaces. We also obtain uniform bounds on the Nusselt number, the bulk burning rate, and the average maximum vertical velocity for chemistries that include bistable and ignition nonlinearities.