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

We consider the Stokes-Boussinesq (and the stationary Navier-Stokes-Boussinesq) equations in a slanted, i.e. not aligned with the gravity's direction, 3d channel and with an arbitrary Rayleigh number. For the front-like initial data and under the no-slip boundary condition for the flow and no-flux boundary condition for the reactant temperature, we derive uniform estimates on the burning rate and the flow velocity, which can be interpreted as stability results for the laminar front.


Introduction and the main results
The Boussinesq system for a reactive flow is a model describing flame propagation in a gravitationally stratified medium [13]. It consists of the reaction-advection-diffusion equation for the reactant temperature θ (normalised so that 0 ≤ θ ≤ 1) and the Navier-Stokes equations for the evolution of the incompressible flow u. The variables u and θ are coupled through the advection velocity in the reaction equation, and through the force term in the fluid equation. After passing to nondimensional variables [2], the simplified Stokes-Boussinesq system takes the form: Here, ν > 0 is the Prandtl number (inversely proportional to the Reynolds number), while the reaction rate is given by a nonnegative, nonlinear function f (θ) of ignition type. That is, f is Lipschitz continuous, and there is a threshold temperature 0 < ϑ 0 < 1 such that: (1.4) f (θ) = 0 for θ ≤ ϑ 0 and θ ≥ 1, f (θ) > 0 on (ϑ 0 , 1), f ′ (1) < 0.
The vector ρ = ρ g corresponds to the non-dimensional gravity g scaled by the Rayleigh number ρ, and we assume that ρ is non-parallel to the unbounded direction of the 3d channel D, which amounts to studying the system (1.1) -(1.3) in: with: ρ · e 3 = 0.
The crossection Ω is a sufficiently regular, connected and bounded domain in R 2 . For the solutions to (1.1) -(1.3), we impose the Neumann condition in θ, and the no-slip (Dirichlet) boundary condition in u: (1.5) ∂θ ∂ n = 0 and u = 0 on ∂D.
The classical result by Kanel [7,1] states that there exists the unique speed c 0 (necessarily positive) and a (necessarily decreasing) traveling wave profile Φ : R −→ [0, 1], satisfying: We call c 0 and Φ the laminar speed and laminar front, and to fix the ideas, we let Φ(0) = ϑ 0 . Then θ(x, t) = Φ(x−c 0 t) is the unique (up to translations) traveling-wave solution to: joining the two equilibria 1 and 0. Moreover, if the initial data θ 0 (x) differs from the Heaviside function H(x) by a compactly supported error, then there exists a shift x 0 such that, for the solution θ to (1.6) there holds: Our purpose is to reproduce this result for the problem (1.1) -(1.3), (1.5). Following [3] and [2], we define the bulk burning rateB(t) and the Nusselt numberN (t) by: (1.7) and the average flowŪ (t) by is compactly supported in D. Let u 0 ∈ W 1,2 (D) and let (θ, u, p) be a global solution of (1.1) -(1.3), (1.5) with: θ(·, 0) = θ 0 and u(·, 0) = u 0 . Then, there exists a constant C Ω , depending only on Ω, such that as t → +∞: The above result shows that the solution of the initial-boundary problem for (1.1) -(1.3), with small ρ/ν propagates with finite speed close to the laminar front speed. Also, note that if we replace θ with the laminar front Φ, then a simple integration of (1.6) gives:B(t) = c 0 . This corresponds with the estimates in Theorem 1.1 when ρ = 0, i.e. the system (1.1) -(1.3) turns out to be a regular perturbation of the reaction-diffusion equation.
Our next result states that with front-like initial data, the solution to the studied system stays front-like: With the same assumptions as in Theorem 1.1, we have: with appropriate x 0 > 0 and C Ω,0 depending on Ω, f and the initial data θ 0 .
Finally, we have: −ν∆u + u · ∇u + ∇p = θ ρ, and the crossection Ω is sufficiently "thin", i.e it satisfies the same type of condition as in [10]: 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. In this paper we generalize these results to dimension 3 and the more physically relevant no-slip boundary conditions. The analysis follows [2] closely; it first seeks bounds forN andB using the parabolic equation (1.1). Where our argument diverges from that of [2] is in finding an estimate for u ∞ . In [2], this has been done using Poincaré's inequality for vorticity, based on the assumption that the vorticity vanishes at the boundary, and using the rather simple form of the vorticity equation in dimension two. In the present case, we rely on the almost-uniform Xie bound in Lemma 2.2.
Acknowledgments. M.L. was partially supported by the NSF grants DMS-0707275 and DMS-0846996, and by the Polish MN grant N N201 547438. M.R. was partially supported by the IPM grant no 90350025.

Auxiliary results
Our first result is a technical lemma, which extends Lemma 4.3 in [2], proved there for 2d channels (see also [4]). Here we prove a similar result in dimension 3, showing that diffusion in a tube behaves like 1d heat equation.
with div u = 0 be a given solenoidal flow satisfying u = 0 on ∂D. Let φ be the solution to the advection-diffusion equation: Then there exists a constant C Ω depending only on Ω (in particular independent of u and φ 0 ) such that, for all sufficiently large t, there holds: Proof. 1. We first prove a Nash-type inequality, valid for solutions of (2.1): ∀t.
To simplify the notation, in what follows we suppress the dependence on t and write φ instead of φ(·, t), etc.
Notice that, by Cauchy-Schwarz inequality: Similarly: Letψ : R −→ C be the Fourier transform of ψ, i.e.ψ(ω) =´R ψ(s)e −2πiωs ds. By Plancherel's identity, we have: . For a given positive m, we now write: where the estimate of the first term follows by: |ψ(ω)| ≤ ψ L 1 (D) , while to estimate the second term we used (2.5).

2.
Recall now that: Indeed, the L 1 norm of a solution to (2.1) is conserved when the initial data is positive. In the general case one can write φ 0 = φ + 0 − φ − 0 where φ + 0 and φ − 0 are positive, with disjoint supports. Solving (2.1) for each, one obtains the inequality (2.6). Further, integrating (2.1) against φ and using incompressibility of u and the boundary condition, it follows that: In view of (2.2), (2.6), (2.7) we now obtain: which, after integrating in time, gives: is increasing and it converges to 0 as α → 0. Let β be the unique solution to β 4 β 3 +1 = C Ω t , so that α ≤ β. Now, for t → ∞ clearly C Ω t → 0, hence also β → 0 and β 4 2 ≤ β 4 β 3 +1 = C Ω t . Consequently, β ≤ C Ω t 1/4 and we arrive at: 3. We now argue as in [2]. Let P t be the solution operator for (2.1). Thus far, we have showed that: Now if P * t is the adjoint operator, then P * t is the solution operator to (2.1), with u replaced by −u. Therefore the above argument works again: Finally, we conclude the lemma: We now present a lemma taken from [10]: Lemma 2.2. Let g ∈ L 2 (D). There exists a constant C Ω , depending only on the crossection Ω, such that for any solenoidal flow u ∈ W 2,2 ∩ W 1,2 0 (D) satisfying: −ν∆u + ∇p = g, div u = 0 in D, there holds: We remark that the proof of (2.9) relies on Xie's estimate [12]: valid for u ∈ W 2,2 (D) ∩ W 1,2 0 (D), and on a commutator estimate [9]: where P is the Helmholz projection onto the space of solenoidal vector fields. The interest in the inequality (2.9) lies in the independence of the constant √ 2 √ πν at the term involving g. Indeed, this was the key argument allowing to prove [10] existence of traveling waves for the full Navier-Stokes-Boussinesq system in 3d channels, satisfying appropriate "thinness" condition on the crossection Ω. The same argument is needed to obtain the uniform bounds for the stationary Navier-Stokes-Boussinesq system in Theorem 1.3 (ii).
On the other hand, by elliptic estimates and the Sobolev imbedding, it follows directly that: In fact, already this inequality is sufficient for the estimates in case of the Stokes-Boussinesq system.
We will also need the following result, in the line of Lemma 3.6 from [10]: with C Ω depending only on Ω. In fact: where C P W stands for the Poincare-Wirtinger constant of Ω.
Proof. Let e 1 , e 2 , e 3 be the standard basis for R 3 . Suppressing the time variable t, we define: Therefore, the following identity concludes the proof of lemma: by the Poincare-Wirtinger inequality: θ − ffl θ L 2 (Ω) ≤ C P W ∇θ L 2 (Ω) .

Proofs of the main result
The following lemma has been proven in [2] for the case of 2d channels and vorticity of the flow u vanishing at ∂D. Exactly the same proof, relying on the construction of super and sub-solutions to (1.1) is valid also in the present 3d case. Since the argument uses the estimate in Lemma 2.1, we partially reproduce it below for the sake of completeness.
Lemma 3.1. There exists a constant C Ω,0 depending on Ω, f , and on the initial data θ 0 , such that: Moreover, there exists x 0 > 0 and q ∈ L 1 (R), such that for all sufficiently large t >> 1:

2)
where Q is the solution to: Proof. We only prove (3.2), since it implies (3.1) as in [2], Lemma 4.2. Define: where x 0 , C > 0 are to be determined later, and Q is as in (3.3) with q appropriately chosen. To prove that ψ is a subsolution, we first need to show the non-positivity of the following expression:
Proof of Theorem 1.1.
Multiplying the fluid equation (1.2) by u and integrating over D, one obtains: Integrating (1.2) against u t gives, in turn: where in both inequalities above we used Lemma 2.3 to "replace" the term θ ρ by ρ∇θ, Taking averages in time, we get: (3.13) By Lemma 2.2 or (2.10), and Lemma 2.3 it now follows that: Using (3.13) we obtain: (3.14) By (3.1) and (3.7) we hence get, for large t >> 1: where C all depends on Ω, f, ν, θ 0 and u 0 . Consequently: and, returning to (3.14): In view of Lemma 3.2, Theorem 1.1 is hence proven.
Proof of Theorem 1.3. We only prove the assertion (ii), because the 2d case in (i) follows with exactly the same calculations as in Theorem 1.1 and Theorem 1.2 .
For the stationary Navier-Stokes-Boussinesq system, using the Poincaré inequality and (2.11), we obtain the following counterpart of (3.12): (3.16) ∇u L 2 (D) ≤ ρ ν C P C P W + | g · (0,x)| 2 dx where C P and C P W are, respectively, the Poincaré and the Poincaré-Wirtinger constants of Ω. By Lemma 2.1 and argueing as (3.14), we arrive at: while the counterpart of (3.15) in the present case is: It is therefore clear that when the constant in front ofN (t) in the last term of the right hand side above is smaller than 1, the results of Theorem 1.1 and Theorem 1.2 follow as in the case of the Stokes-Boussinesq system.