Relaxation in Reactive Flows

Abstract

We consider convective systems in a bounded domain, in which viscous fluids described by the Stokes system are coupled using the Boussinesq approximation to a reaction-advection-diffusion equation for the temperature. We show that the resulting flows possess relaxation-enhancing properties in the sense of [12]. In particular, we show that solutions of the nonlinear problems become small when the gravity is sufficiently strong due to the improved interaction with the cold boundary. As an application, we deduce that the explosion threshold for power-like nonlinearities tends to infinity in the large Rayleigh number limit. We also discuss the behavior of the principal eigenvalues of the corresponding advection-diffusion problem and the quenching phenomenon for reaction-diffusion equations.

On the Euler equations of incompressible fluids

“…Classical methods of dynamical systems are relevant for the long-time effects ( [118]). Strong rapid effects are studied using PDE methods ( [17], [50], [55]). …”

The explosion problem in a flow

“…The fully nonlinear problem when the flow itself satisfies a Navier-Stokes type equation coupled to the explosion problem for temperature has been studied in [2,23,29] using numerics and formal asymptotics. Recently, some rigorous results for the behavior of the solutions to the coupled system in the regime of a strong gravity have been obtained in [11]. Here we derive several qualitative properties of the explosion threshold λ * (u) in terms of the geometry and the amplitude of the flow u.…”

“…11) r A = 0 on ∂Q 2h . Now, as in the proof of Lemma 1.3 we conclude that there exists a constant C(h) such thatr A L ∞ (Q 2h ) ≤ C(h) for all A > 0.The same proof shows that C(h) → 0 as h → 0 -this happens because the principal eigenvalue of the Dirichlet Laplacian in Q 2h tends to infinity as h → 0 while the constants K p (h) in the Poincaré inequality…”

A stability result for the Stokes-Boussinesq equations in infinite 3d channels

“…Existence of traveling fronts for the Navier-Stokes-Boussinesq system in infinite channels has been the subject of research during recent years [3,2,11,5,8,10]. Of interest has been also an understanding of the regularizing and mixing effect of convection [6,4]. In [3] and [2], the solutions of the system with front-like datum in a 2d strip have been considered and uniform estimates for the full Navier-Stokes-Boussinesq system have been obtained for the stress-free boundary conditions on u.…”