Non-Planar Fronts in Boussinesq Reactive Flows

Henri Berestycki^{∗}  Peter Constantin ^{†}  Lenya Ryzhik^{‡}

February 16, 2005

Abstract

We consider the reactive Boussinesq equations in a slanted cylinder, with zero stress boundary conditions and arbitrary Rayleigh number. We show that the equations have non-planar traveling front solutions that propagate at a constant speed. We also establish uniform upper bounds on the burning rate and the flow velocity for general front-like initial data for the Cauchy problem.

1 Introduction

The existence of traveling fronts for reaction-diffusion equations and their stability has been extensively studied since the pioneering work of Kolmogorov, Petrovskii and Piskunov [26] and Fisher [17]. A large number of results have been obtained during the last decade on the generalization of the notion of a traveling front to reaction-diffusion-advection equations in a prescribed flow. These include non-planar traveling fronts in shear flows [9, 10, 12], and pulsating traveling fronts in periodic flows [6, 39, 40], as well as results for monotonic systems in a unidirectional flow [35, 36, 37]. One of the main qualitative effects of a flow is the speed-up of front propagation due to front stretching. Various bounds have been obtained for the speed of propagation of fronts in prescribed flows [1, 2, 3, 7, 13, 22, 25, 23, 24, 31], including variational principles for the front speed [7, 8, 18, 19, 21, 22]. The homogenization limit in a periodic flow has also been studied [27]. Extensive recent overviews can be found in [5, 30, 41].

However, those results have been obtained under the assumption that the flow is imposed from outside, and that it is not affected by the evolution of the solution of the reaction-diffusion-advection equation, that is, by the temperature or concentration of the reactant. This is known as the constant density approximation in the combustion literature. A first step in the coupling of the temperature and fluid flow evolution is via the Boussinesq approximation: the density mismatch is so small that the density difference is accounted by a buoyancy force in the equation for an incompressible flow. Recently a number of works considered systems of a reaction-diffusion-advection equation coupled to a flow equation of the Boussinesq type. Global existence and regularity of solutions in two dimensions was studied in [28]. It has been shown that non-planar convective traveling fronts may not exist in a vertical cylinder if the Rayleigh number is too small while for large Rayleigh numbers the planar fronts become unstable [14, 32, 33]. Moreover, there exists a bifurcation at a critical value ρ_{c} > 0 – non-trivial convective fronts may exist for the Rayleigh numbers close to ρ_{c} [32, 33]. Numerical computations [34] show that non-planar convective fronts exist and are stable for a large range of Rayleigh numbers ρ > ρ_{c}. The fingering instability in this regime was investigated in [15].

∗EHESS, CAMS, 54 Boulevard Raspail, F - 75006 Paris, France; hb@ehess.fr

†Department of Mathematics, University of Chicago, Chicago, IL 60637, USA; ryzhik@math.uchicago.edu

‡Department of Mathematics, University of Chicago, Chicago, IL 60637, USA; const@math.uchicago.edu

One of the difficulties in the analysis of the Boussinesq problem at large Rayleigh numbers in a vertical cylinder is the presence of unstable planar fronts that make uniform lower bounds on the front speed quite difficult. However, it has been observed in [4] that such planar fronts cannot exist in a horizontal cylinder. One of the main results of [4] is that non-planar fronts in a horizontal cylinder exist for small Rayleigh numbers. A purpose of the present paper is to extend this result to all positive Rayleigh numbers; we use an approach that is different from [4] and is based on the a priori bounds developed in [14].

The reactive Boussinesq equations for the temperature T and flow u have the dimensional form

T_{t} + u · ∇T = κΔT + \frac{v_{0}^{2}}{κ} f(T)  (1.1)
u_{t} + u · ∇u − νΔu + ∇p = gT e_{z}
∇ · u = 0.

Here e_{z} is the unit vector in the vertical direction, g is the strength of gravity, the speed v_{0} is proportional to the traveling front speed in the absence of gravity, κ is the thermal diffusivity and ν is the fluid viscosity. The temperature is normalized so that 0 ≤ T ≤ 1. The nonlinearity f(T) is assumed to be a Lipschitz function of the ignition type

f(T) = 0 for 0 ≤ T ≤ θ_{0} with θ_{0} > 0, f(T) > 0 for T ∈ (θ_{0}, 1) and f(1) = 0.  (1.2)

We consider the equations (1.1) in a slanted two-dimensional cylinder x ∈ ℝ, αx ≤ z ≤ αx + H with a finite slope α < ∞. It is convenient to rotate the cylinder in order to make it horizontal to simplify the notation. Then (1.1) becomes

T_{t} + u · ∇T = κΔT + \frac{v_{0}^{2}}{κ} f(T)  (1.3)
u_{t} + u · ∇u − νΔu + ∇p = gT ê
∇ · u = 0,

where u is the flow velocity measured relative to the new coordinate system. The gravity on the right points in a direction ê that is non-parallel to the x-axis, as the original cylinder was assumed to be non-vertical (α < ∞). The new rotated problem is posed in a cylinder D = ℝ_{x} × [0, L]_{z}, L = H/\sqrt{1 + α^{2}}. The boundary conditions for the temperature T are set to be front-like:

T → 1 as x → −∞, T → 0 as x → +∞, \frac{∂T}{∂z} = 0 at z = 0, L.  (1.4)

The flow u = (v, w) satisfies the no stress boundary conditions:

u, ω → 0 as x → ±∞ and w, ω = 0 at z = 0, L.  (1.5)

Here ω = w_{x} − v_{z} is the flow vorticity so that

Δv = −ω_{z}, Δw = ω_{x}.

In order to pass to the non-dimensional variables we introduce the laminar front width δ = κ/v_{0} and reaction time t_{c} = κ/v_{0}^{2} and rescale the space and time variables: x_{new} = x_{old}/δ and t_{new} = t_{old}/t_{c}. We also rescale the flow u_{new} = u_{old}/v_{0}. Then the Boussinesq equations become

T_{t} + u · ∇T = ΔT + f(T)  (1.6)
u_{t} + u · ∇u − σΔu + ∇p = ρT ê
∇ · u = 0,

where σ = ν/κ is the Prandtl number and ρ = gδ^{3}/κ^{2} is the Rayleigh number. The problem is now posed in the strip D = ℝ_{x} × [0, λ]_{z}, λ = L/δ, with the boundary conditions that come from (1.4) and (1.5).

The traveling front solutions of (1.6) are solutions of the form T (x − ct, z), u(x − ct, z) with the speed c to be determined. They satisfy

−cT_{x} + u · ∇T = ΔT + f(T)  (1.7)
−cu_{x} + u · ∇u − σΔu + ∇p = ρT ê
∇ · u = 0,

with the boundary conditions

T → T_{−} as x → −∞, T → 0 as x → +∞, \frac{∂T}{∂z} = 0 at z = 0, λ  (1.8)

and

w, ω = 0 at z = 0, λ.  (1.9)

Here T_{−} is a constant that is not a priori prescribed. We recall that, as has been observed in [4], if the direction of gravity ê is not parallel to the x-axis, any traveling front solution of (1.7) must be non-planar, that is, it must depend on both variables x and z. This is the main difference between the cases of a vertical and slanted cylinder: planar fronts exist in the former case but not in the latter. Our main result is the following theorem.

Theorem 1.1 Let the nonlinearity f(T) be of the ignition type (1.2). Then a traveling front solution (c, T, u) of (1.7) exists such that it is non-planar: T_{z} ≢ 0, the flow u ≢ 0 and the reaction rate f(T) ≢ 0. Moreover, the solution satisfies the following properties: c > 0, T ∈ C^{2,α}(D), ∇T ∈ L^{2}(D), u ∈ H^{1}(D) ∩ C^{2,α}(D). If we assume in addition that

f(T) ≤ (T − θ_{0})_{+}^{2}/λ^{2},  (1.10)

then the left limit T_{−} = 1.

The assumption (1.10) is of technical nature. It does not involve the Rayleigh number ρ, it is rather a restriction on the channel width λ. We do not address the question of the uniqueness of the traveling front speed or profile in this paper – this problem requires an additional study. Our results can be generalized to the no-slip boundary conditions u = 0 on ∂D at the expense of a more technical proof – we leave this problem for a future publication.

The general idea of the proof is as follows. We first consider the problem (1.7) on a finite domain D_{a} = [−a, a]_{x} × [0, λ]. Solutions (T_{a}^{c}, u_{a}^{c}) of the restricted problem exist for all c ∈ ℝ. We normalize them by the requirement that

max_{x≥0} T_{a}^{c}(x, z) = θ_{0}.  (1.11)

This imposes a restriction on the speed c. In order to show that there exists a speed c_{a} so that (1.11) holds we first obtain some a priori bounds on c, T and u under the condition (1.11). Then we use the Leray-Schauder topological degree theory and the above a priori bounds to show that c_{a} exists. The a priori bounds allow us to pass to the limit a → ∞. Finally we show that the right limit of T as x → +∞ is equal to zero, and that the left limit is equal to one under the additional assumption on f(T) in Theorem 1.1. This general strategy is similar to that in the proof of existence of traveling fronts in a prescribed decoupled flow, as in, for example, [9, 12]. The main difficulty and novelty are in the a priori bounds for the solution of the coupled problem in a bounded domain.

Our second result shows that the solution of the Cauchy problem for (1.6) propagate with a finite speed and that this speed is close to the speed of the laminar front c_{0} when the Rayleigh number is small. Recall that there exists a unique speed c_{0} so that a traveling front solution of

−c_{0}Φ_{x} = Φ_{xx} + Φ(U), Φ(−∞) = 1, Φ(+∞) = 0

exists.

In order to make this precise we define the bulk burning rate V̄ (t), the Nusselt number N̄(t) and the average horizontal flow Ū(t) by

V̄ (t) = \frac{1}{t} ∫_{0}^{t} ∫ V (s)ds, V (t) = ∫ f(T)\frac{dxdz}{λ} ,  (1.12)
N̄(t) = \frac{1}{t} ∫_{0}^{t} N(s)ds, N(t) = ∫ |∇T |^{2} \frac{dxdz}{λ} ,  (1.13)
Ū(t) = \frac{1}{t} ∫_{0}^{t} ∥v(s)∥_{∞}ds.  (1.14)

The following theorem provides uniform bounds on these bulk quantities. It also shows that the coupled problem (1.6) is in a sense a ”regular perturbation” of the single reaction-diffusion equation with ρ = 0.

Theorem 1.2 Assume that there exists R so that T_{0}(x, z) = 0 for x > R and T_{0}(x, z) = 1 for x < −R and that the initial vorticity ω_{0} ∈ L^{2}(D). There exists a constant C > 0 so that under the above assumptions on the initial data T_{0}, u_{0} we have the following bounds

c_{0} − C[ρ + ρ^{2}] + o(1) ≤ V̄ (t) ≤ c_{0} + C[ρ + ρ^{2}] + o(1)  (1.15)

N̄(t) ≤ [Cρ + \sqrt{\frac{c_{0}}{2} + C^{2}ρ^{2}}]^{2} + o(1)

Ū(t) ≤ Cρ[1 + ρ] + o(1)

as t → +∞.

This theorem may be interpreted as a stability result for a perturbation of a homogeneous reactiondiffusion equation by the buoyancy coupling. The proof is based on the construction of super- and sub- solutions, and a bound on the decay of the solutions of advection-diffusion equations that is uniform in the advection flow.

The third result of this paper deals with the Boussinesq system in a narrow domain. It has been shown in [14] that if a vertical strip is sufficiently narrow and gravity is sufficiently weak then solutions of the Cauchy data become planar as t → +∞. The following theorem generalizes this result to inclined cylinders.

Theorem 1.3 Let ê = (e_{1}, e_{2}) be the unit vector in the direction of gravity and let ρ_{j} = ρe_{j}, j = 1, 2 and let the initial data (T_{0}, u_{0}) be as in Theorem 1.2. There exist two constants λ_{0} and ρ_{0} so that if the domain is sufficiently narrow: λ ≤ λ_{0} and gravity is sufficiently small: ρ ≤ ρ_{0} then the burning rate is bounded by

V̄ (t) ≤ c_{0} + Cρ_{2} + o(1) as t → +∞.  (1.16)

Moreover, the front is nearly planar in the sense that

N̄_{z}(t) = \frac{1}{t} ∫_{0}^{t} ∥T_{z}(s)∥_{2}^{2}ds ≤ Cρ_{2}^{2} + o(1) as t → +∞.  (1.17)

The main observation of this theorem is that only the gravity strength in the direction perpendicular to the strip enters in the upper bounds (4.35) and (4.36).

The paper is organized as follows: Theorem 1.1 is proved in Sections 2 and 3. Theorems 1.2 and 1.3 are proved in Section 4.

Acknowledgment. We thank Vitaly Volpert for explaining to us the results of [4] prior to its publication. We also thank Marta Lewicka for a careful reading of the preliminary version of the manuscript. This research was supported in part by the ASCI Flash center at the University of Chicago under DOE contract B341495. PC was partially supported by the NSF grant DMS-0202531, LR by NSF grant DMS-0203537, ONR grant N00014-02-1-0089 and an Alfred P. Sloan Fellowship.

2 The finite domain problem

We consider in this section the approximating problem

−cT_{x} + u · ∇T = ΔT + f(T)  (2.1)
−cu_{x} + u · ∇u − σΔu + ∇p = ρT ê
∇ · u = 0,

in a finite domain D_{a} = [−a, a]_{x} × [0, λ]_{y}, a > 0, with the boundary conditions

T (−a, z) = 1, T (a, z) = 0, \frac{∂T}{∂z} = 0 at z = 0, λ  (2.2)

and

w = 0, ω = 0 at z = 0, λ and v(±a, z) = ω(±a, z) = 0 at x = ±a.  (2.3)

One can show with the techniques of the present section that a solution T_{a}, u_{a} of (2.1) in D_{a} with the boundary conditions (2.2) and (2.3) exists for all c ∈ ℝ. However, given an arbitrary c there is no way to control the limit of T_{a} and u_{a} as a → ∞. Hence, following the standard procedure, we impose an additional constraint (1.11). This ensures that the non-trivial part of the solution does not escape to infinity when we pass to the limit a → ∞.

Proposition 2.1 There exists a speed c_{a} ∈ ℝ so that there exists a solution (T_{a}, u_{a}) of (2.1) in D_{a} with the boundary conditions (2.2) and (2.3) such that

max_{x≥0,z∈[0,λ]} T_{a}(x, z) = θ_{0}.  (2.4)

We denote the corresponding solution as (c_{a}, T_{a}, u_{a}). Moreover, there exists a_{0} > 0 and a constant C > 0 that is independent of a, so that we have for all a > a_{0}

|c_{a}| ≤ C,  (2.5)

and

∫_{D_{a}} |∇T_{a}|^{2}dxdz + ∫_{D_{a}} |∇u_{a}|^{2}dxdz + ∥u_{a}∥_{∞} ≤ C.  (2.6)

Moreover, the uniform Hölder estimates hold: there exists a_{0} > 0 and a constant C > 0 independent of a so that we have for all a > a_{0}

∥ω_{a}∥_{C^{1,α}(D_{a})} + ∥u_{a}∥_{C^{1,α}(D_{a})} + ∥T_{a}∥_{C^{1,α}(D_{a})} ≤ C  (2.7)

provided that 0 < α < 1.

Proof. The proof consists of two parts. First, we introduce a family of problems depending on a parameter τ ∈ [0, 1] so that at τ = 0 we have a simple linear problem without advection or coupling and at τ = 1 we have the full problem (2.1) with the correct boundary conditions. The normalization condition (2.4) is imposed for all τ ∈ [0, 1]. We obtain the a priori bounds as in (2.5), (2.6) and (2.7) for such solutions that are uniform in τ ∈ [0, 1]. In the second step we use the a priori bounds, the Leray-Schauder topological degree argument and the information on the linear problem at τ = 0 to show that solutions of the nonlinear coupled problem at τ = 1 exist. We drop the subscript a throughout the proof to make the notation less cumbersome.

Step 1. A priori bounds for solutions. Let us first define a one-parameter (homotopy) family of finite domain Boussinesq problems in the vorticity formulation

−c^{τ} T_{x}^{τ} + τu^{τ} · ∇T^{τ} = ΔT^{τ} + τf(T^{τ})  (2.8)
−c^{τ} ω_{x}^{τ} + u^{τ} · ∇ω^{τ} − σΔω^{τ} = τρê · ∇^{⊥}T := ρτ[e_{2}T_{x}^{τ} − e_{1}T_{z}^{τ}]
ω^{τ} = w_{x}^{τ} − v_{z}^{τ} , ∇ · u^{τ} = 0.

As mentioned above, τ is the homotopy parameter: τ ∈ [0, 1], with τ = 0 corresponding to the linear problem, and τ = 1 to the full problem (2.1)-(2.3). The problem (2.8) is posed in D_{a} with the same boundary conditions

\frac{∂T^{τ}}{∂z} = 0, w^{τ} = ω^{τ} = 0 for z = 0, λ  (2.9)

and

T^{τ} (−a, z) = 1, T^{τ} (a, z) = 0, v^{τ} (±a, z) = ω^{τ} = (±a, z) = 0 for x = ±a,  (2.10)

as (2.1). We also require that

max_{x≥0,z} T^{τ} (x, z) = θ_{0}  (2.11)

and obtain a priori bounds on c^{τ}, T^{τ} and ω^{τ}. We drop the superscript τ below wherever it causes no confusion. The general plan is as follows. First, we bound the speed c above and below by a linear function of ∥v∥_{∞} in Lemma 2.2. Next we bound ∥u∥_{∞} from above by a linear function of ∥∇T ∥_{2} in Lemma 2.4. The other direction, a bound on ∥∇T ∥_{2}^{2} in terms of a linear function of ∥u∥_{∞} is established in Lemmas 2.5 and 2.6. Since the latter bound is quadratic in ∥∇T ∥_{2}, the last estimates allow to obtain a uniform bound on this quantity, from which all other a priori bounds follow in a fairly straightforward manner: see Corollary 2.7 and Lemma 2.8.

We begin with a lemma that bounds the speed c in terms of the horizontal flow velocity ∥v∥_{L^{∞}(D_{a})}.

Lemma 2.2 Let (c, T, u) satisfy (2.8)-(2.10) with the normalization (2.11) and let u = (v, w). Then there exists a_{0} > 0 so that for all a ≥ a_{0} we have

−1 − τ∥v∥_{∞} ≤ c ≤ 1 + Mτ + τ∥v∥_{∞}.  (2.12)

Proof. First, we observe that the function ψ_{A}(x) = Ae^{−α(x+a)} is a super-solution for the reactiondiffusion-advection equation with the flow u fixed if A > 1 and

c ≥ α + \frac{Mτ}{α} + τ∥v∥_{∞},  (2.13)

that is,

−c\frac{∂ψ_{A}}{∂x} + τu · ∇ψ_{A} ≥ Δψ_{A} + τf(ψ_{A}),  (2.14)

provided that (2.13) holds with

M = sup_{0≤T ≤1} \frac{f(T)}{T}.

Furthermore, we have

T (−a, z) = 1 < A = ψ_{A}(−a), T (a, z) = 0 < ψ_{A}(a)  (2.15)

at the two ends of the domain D_{a}. We now show that this together with (2.14) implies that

T (x, z) ≤ ψ_{A}(x)  (2.16)

for all (x, z) ∈ D_{a} and A > 1. Indeed, consider the family of functions ψ_{A}(x). Then all ψ_{A} are supersolutions in the sense that the inequality (2.14) holds. Moreover, as the maximum principle implies that 0 ≤ T ≤ 1, for A > 5e^{2αa} sufficiently large we have ψ_{A}(x) > 5 > T (x, z) for all (x, z) ∈ D_{a}. We define

A_{0} = inf {A ∈ ℝ : ψ_{A}(x) ≥ T (x, z) for all (x, z) ∈ D_{a}} .

The previous argument implies that A_{0} is finite, A_{0} ≤ 5e^{2αa} and, moreover, clearly A_{0} > 0. Observe that since the domain D_{a} is compact, we should have ψ_{A_{0}}(x) ≥ T (x, z) – otherwise this inequality would be violated for A slightly larger than A_{0} at some point in D_{a}. Moreover, the equation ψ_{A_{0}}(x) = T (x, z) should have a solution. We claim that A_{0} = 1. Indeed, otherwise the point (x_{0}, z_{0}) that solves ψ_{A_{0}}(x_{0}) = T (x_{0}, z_{0}) cannot be at the boundary of D_{a} because of the boundary conditions on the function T. Hence this point has to lie in the interior of D_{a}. The continuity of ψ_{A}(x) with respect to A implies that the graphs of ψ_{A_{0}}(x) and T (x, z) are tangent at (x_{0}, z_{0}). Then the strong maximum principle implies that ψ_{A_{0}}(x) ≡ T (x, z) which is a contradiction, as they differ on the boundary. Hence we conclude that A_{0} = 1 and thus (2.16) holds for all A > 1 and thus for A = 1, so that

T (x, z) ≤ e^{−α(x+a)}.  (2.17)

However, the existence of such a super-solution contradicts the normalization condition (2.11) if α ≥ ln(θ_{0}^{−1})/a because (2.11) implies that there exists z_{0} so that T (0, z_{0}) = θ_{0}. Therefore, the existence of a solution T that satisfies (2.11) implies

c ≤ inf_{α≥ln(θ_{0}^{−1})/a} (α + \frac{Mτ}{α} ) + τ∥v∥_{∞} ≤ 1 + Mτ + τ∥v∥_{∞}  (2.18)

provided that a ≥ ln(1/θ_{0}). This proves the upper bound in (2.12). In order to prove the lower bound we observe that the function ϕ = 1 − e^{α(x−a)} is a sub-solution for T with the flow u fixed if

c ≤ −α − τ∥v∥_{∞}.  (2.19)

That is, if (2.19) holds, then T (x, z) ≥ 1 − e^{α(x−a)}. This is shown in a way similar to the proof of (2.17) under the assumption (2.13) above. However, ϕ(0) = 1 − e^{−αa} > θ_{0} for

a > \frac{ln((1 − θ_{0})^{−1})}{α}.  (2.20)

This implies that max_{x≥0} T (x, z) ≥ ϕ(0) > θ_{0} provided that both (2.19) and (2.20) hold. Hence in order for (2.11) to be possible we need

c ≥ sup_{α> \frac{ln((1−θ_{0})^{−1})}{a}} [−α − τ∥v∥_{∞}] ≥ −1 − τ∥v∥_{∞}  (2.21)

provided that a ≥ ln((1 − θ_{0})^{−1}). This is the lower bound in (2.12) and the proof of Lemma 2.2 is complete. □

Next we establish a bound on ∥u∥_{L^{∞}(D_{a})} and ∥ω∥_{L^{∞}(D_{a})} in terms of ∥∇T ∥_{L^{2}(D_{a})}. These bounds are all obtained from the following type of estimates.

Lemma 2.3 Let S_{a} = [−a, a]_{x} ×Ω_{y} be a finite cylinder with a smooth bounded cross-section Ω ∈ ℝ^{d}, d = 1, 2. Let ϕ be a function that satisfies either of the following three conditions: (i) ϕ(x, y) = 0 on the whole boundary ∂S_{a}, (ii) ϕ(x, y) = 0 for y ∈ ∂Ω and \frac{∂ϕ(x,y)}{∂x} = 0 for x = −a, a, or (iii) \frac{∂ϕ(x,y)}{∂n} = 0 for y ∈ ∂Ω, and ϕ(x, y) = 0 for x = −a, a. Then there exists a constant C that depends only on the domain Ω, but not on the cylinder length a, so that we have

∥ϕ∥_{L^{∞}(S_{a})} ≤ C [∥Δϕ∥_{L^{2}(S_{a})} + ∥ϕ∥_{L^{2}(S_{a})}] .  (2.22)

Proof. Let Q be any cylinder of the form [x_{0}, x_{0} + 1] × Ω ⊂ S_{a} with −a ≤ x_{0} ≤ a − 1. The standard interior elliptic estimates up to the boundary [20] can be applied to Q in all the three cases (i)-(iii). The corners at x = ±a are not an obstacle. Indeed, both in the case of the Dirichlet and Neumann boundary conditions prescribed on the lines x = ±a, one can extend the solution to a larger cylinder [−a − 1, a + 1] × Ω by reflecting the solution across the line x = ±a, either in the even or odd way, respectively. Hence the usual elliptic estimates up to the boundary can be applied to all such cylinders Q to obtain

∥ϕ∥_{H^{2}(Q)} ≤ C [∥Δϕ∥_{L^{2}(S_{a})} + ∥ϕ∥_{L^{2}(S_{a})}]  (2.23)

in all three cases (i)-(iii). Then the Sobolev embedding theorem in dimensions d = 2, 3 implies that

∥ϕ∥_{L^{∞}(Q)} ≤ C∥ϕ∥_{H^{2}(Q)} ≤ C [∥Δϕ∥_{L^{2}(Q)} + ∥ϕ∥_{L^{2}(Q)}] ≤ C [∥Δϕ∥_{L^{2}(S_{a})} + ∥ϕ∥_{L^{2}(S_{a})}]

with the constant C that depends only on the domain Ω. □

This lemma can be easily extended to higher dimensions using the appropriate Sobolev embeddings. It implies immediately the following bounds on ∥u∥_{∞} and ∥ω∥_{∞} in terms of ∥∇T ∥_{L^{2}(D_{a})}.

Lemma 2.4 Let (c, T, u) satisfy (2.8)-(2.10) with the normalization (2.11). There exists a_{0} > 0 and a constant C > 0 so that, for all a > a_{0}, it holds that

∥u∥_{L^{∞}(D_{a})} ≤ C∥∇T ∥_{L^{2}(D_{a})}.  (2.24)

and

∥ω∥_{L^{∞}(D_{a})} ≤ C∥∇T ∥_{L^{2}(D_{a})} [1 + ∥∇T ∥_{L^{2}(D_{a})}] .  (2.25)

Moreover, ∇u satisfies the same bound:

∥∇u∥_{L^{∞}(D_{a})} ≤ C∥∇T ∥_{L^{2}(D_{a})} [1 + ∥∇T ∥_{L^{2}(D_{a})}] .  (2.26)

Proof. We use the vorticity equation

−cω_{x} + u · ∇ω − σΔω = ρτ(ê · ∇^{⊥}T), ω = 0 on ∂D_{a}.  (2.27)

Case (i) of Lemma 2.3 implies that

∥ω∥_{L^{∞}(D_{a})} ≤ C [∥∇T ∥_{L^{2}(D_{a})} + (|c| + ∥u∥_{∞})∥∇ω∥_{L^{2}(D_{a})} + ∥ω∥_{L^{2}(D_{a})}] .  (2.28)

Here the constant C depends only on ρ and λ. Note that multiplying the vorticity equation by ω and integrating by parts, using the boundary conditions we obtain

∫_{D_{a}} |∇ω|^{2}dxdz = τρ ∫ (ê · ∇^{⊥}T ) ωdxdz ≤ τρ∥∇T ∥_{2}∥ω∥_{2}.

The Dirichlet boundary conditions for ω imply that the Poincaré inequality applies to ω so that ∥ω∥_{L^{2}(D_{a})} ≤ (λ/π)∥∇ω∥_{L^{2}(D_{a})}. Hence we obtain

∥∇ω∥_{2} ≤ \frac{λ}{π}τρ∥∇T ∥_{2}  (2.29)

and thus

∥ω∥_{L^{2}(D_{a})} ≤ C∥∇T ∥_{L^{2}(D_{a})},  (2.30)

with the constant C independent of the cylinder length a. This, together with (2.28) and the bound (2.12) on the speed c implies (2.25), provided that we show (2.24).

We now prove (2.24). The horizontal flow component v satisfies the Poisson equation

Δv = −ω_{z}, v(±a, z) = 0, \frac{∂v}{∂z} = 0, at z = 0, λ.  (2.31)

The boundary conditions at z = 0, λ are obtained from v_{z} = w_{x} − ω = 0 as follows from (2.9). The third case (iii) of Lemma 2.3 implies that

∥v∥_{L^{∞}(D_{a})} ≤ C[∥∇ω∥_{L^{2}(D_{a})} + ∥v∥_{L^{2}(D_{a})}].  (2.32)

The first term in the right side is bounded by (2.29). In order to bound the second we multiply (2.31) by v and integrate to obtain, using the boundary conditions and (2.29)

∫_{D_{a}} |∇v|^{2}dxdz = ∫_{D_{a}} ω_{z}(x, z)v(x, z)dxdz ≤ ∥ω_{z}∥_{2}∥v∥_{2} ≤ C∥∇T ∥_{2}∥v∥_{2}.  (2.33)

Now, observe that (2.31), the Neumann boundary conditions for v and the Dirichlet boundary condition for ω at z = 0, λ imply that

\frac{d^{2}}{dx^{2}} ∫ v(x, z)dz = 0.

It follows then from the Dirichlet boundary conditions for v at x = ±a that

∫_{0}^{λ} v(x, z)dz = 0  (2.34)

for all x. One may alternatively deduce (2.34) from incompressibility of the flow u and the boundary conditions. Therefore, it follows from the Poincaré inequality that ∥v∥_{L^{2}(D_{a})} ≤ (λ/2π)∥∇v∥_{L^{2}(D_{a})}. Thus, (2.33) implies that both ∥∇v∥_{L^{2}(D_{a})} ≤ C∥∇T ∥_{L^{2}(D_{a})} and ∥v∥_{L^{2}(D_{a})} ≤ C∥∇T ∥_{L^{2}(D_{a})} with a constant independent of a. Hence, (2.32) implies (2.24) for the horizontal flow component.

The vertical flow component satisfies

Δw = ω_{x}, w(x, 0) = w(x, λ) = 0, \frac{∂w}{∂x} (±a, z) = 0.  (2.35)

The Neumann boundary condition at x = ±a is deduced from the relation w_{x} = ω + v_{z} and the Dirichlet boundary conditions for v and ω at x = ±a. The case (ii) in Lemma 2.3 implies that

∥w∥_{L^{∞}(D_{a})} ≤ C[∥∇ω∥_{L^{2}(D_{a})} + ∥w∥_{L^{2}(D_{a})}].  (2.36)

As before, we use (2.29) to bound the first term in the right side. In order to bound the second we multiply (2.35) by w and integrate, using the boundary conditions and (2.29) again, to obtain that

∫_{D_{a}} |∇w|^{2}dxdz = − ∫_{D_{a}} ω_{x}(x, z)w(x, z)dxdz ≤ ∥ω_{x}∥_{2}∥w∥_{2} ≤ C∥∇T ∥_{2}∥w∥_{2}.  (2.37)

The Dirichlet boundary conditions for w at z = 0, λ imply that ∥w∥_{L^{2}(D_{a})} ≤ λ/π∥∇w∥_{L^{2}(D_{a})}. Thus, (2.37) implies that

∥∇w∥_{L^{2}(D_{a})} ≤ C∥∇T ∥_{L^{2}(D_{a})},  (2.38)

and hence ∥w∥_{L^{2}(D_{a})} ≤ C∥∇T ∥_{L^{2}(D_{a})} with a constant independent of a. Therefore, now (2.36) implies (2.24) for the vertical flow component. Thus, the proof of (2.24) is complete. We recall that then (2.25) follows as well, as explained in the paragraph below (2.30).

In order to complete the proof of Lemma 2.4 it remains to bound the derivatives of u. First, we observe that the function ψ = v_{z} satisfies the boundary value problem

−Δψ = ω_{zz}, ψ = 0 on ∂D_{a}.  (2.39)

Hence, case (i) of Lemma 2.3 applies to the function ψ. Moreover, the elliptic estimates for ω, as in (2.23) imply that ∥ω_{zz}∥_{L^{2}(D_{a})} ≤ ∥Δω∥_{L^{2}(D_{a})} ≤ C∥∇T ∥_{L^{2}(D_{a})}(1 + ∥∇T ∥_{L^{2}(D_{a})}). Hence, the same proof as in the derivation of the bound (2.25) applies to ψ and we obtain that

∥v_{z}∥_{L^{∞}(D_{a})} ≤ C∥∇T ∥_{L^{2}(D_{a})}(1 + ∥∇T ∥_{L^{2}(D_{a})}).

This, together with (2.25) implies that

∥w_{x}∥_{L^{∞}(D_{a})} ≤ C∥∇T ∥_{L^{2}(D_{a})}(1 + ∥∇T ∥_{L^{2}(D_{a})}).

The other pair of derivatives, v_{x} and w_{z}, do not satisfy a homogeneous boundary condition on the lines x = ±a. Therefore, one cannot apply the standard elliptic estimates up to the boundary to the function η = w_{z} = −v_{x} (the second equality follows from the incompressibility of the flow). In order to circumvent this difficulty, we extend the function w to a larger cylinder D_{a+1} = [−a−1, a+1]×[0, λ] by setting w(−a − x, z) = w(−a + x, z) and w(a + x, z) = w(a − x, z) for 0 ≤ x ≤ 1. The resulting function is of a class C^{2}(D_{a+1}) since w(x, z) satisfies the Neumann boundary condition at x = ±a. This also extends the function η = w_{z} to the larger cylinder. Moreover, η satisfies the Neumann boundary condition along the horizontal lines z = 0, λ:

η_{z} = w_{zz} = −v_{zx} = 0 on z = 0, λ,

and

Δη = ω_{xz},

with the function ω extended to the larger cylinder by the same reflection. Hence, the interior elliptic estimates up to the boundary for solutions of the Neumann problem imply that

∥η∥_{H^{2}(Q)} ≤ ∥Δη∥_{L^{2}(D_{a})} + ∥η∥_{L^{2}(D_{a})}

for any rectangle Q = [x_{0}, x_{0} + 1] × [0, λ] that is strictly contained inside the larger cylinder D_{a+1}. Therefore, the Sobolev embedding theorem together with the above estimates imply that

∥η∥_{L^{∞}(D_{a})} ≤ C[∥Δη∥_{L^{2}(D_{a})} + ∥η∥_{L^{2}(D_{a})}] = C[∥ω_{xz}∥_{L^{2}(D_{a})} + ∥η∥_{L^{2}(D_{a})}].  (2.40)

However, as the function ω satisfies the Dirichlet boundary conditions in D_{a}, we can apply the estimate (2.23) to the function ω up to boundary, to obtain

∥ω∥_{H^{2}(D_{a})} ≤ C [∥Δω∥_{L^{2}(D_{a})} + ∥ω∥_{L^{2}(D_{a})}] .

We now use the vorticity equation (2.27) to bound ∥Δω∥_{L^{2}(D_{a})} and the estimate (2.30) to estimate ∥ω∥_{L^{2}(D_{a})}, and conclude that

∥ω_{xz}∥_{L^{2}(D_{a})} ≤ C∥∇T ∥_{L^{2}(D_{a})}(1 + ∥∇T ∥_{L^{2}(D_{a})}).  (2.41)

Furthermore, the estimate (2.38) for ∥∇w∥_{L^{2}(D_{a})} implies that

∥η∥_{L^{2}(D_{a})} = ∥w_{z}∥_{L^{2}(D_{a})} ≤ ∥∇w∥_{L^{2}(D_{a})} ≤ C∥∇T ∥_{L^{2}(D_{a})}.  (2.42)

We infer from the bounds (2.40), (2.41) and (2.42) that

∥η∥_{L^{∞}(D_{a})} ≤ C∥∇T ∥_{L^{2}(D_{a})}(1 + ∥∇T ∥_{L^{2}(D_{a})}).

This proves the uniform bound on w_{z} and hence the proof of Lemma 2.4 is complete. □

Let us now proceed to estimate ∥∇T ∥_{L^{2}(D_{a})} in terms of ∥v∥_{L^{∞}(D_{a})}, a bound in the direction opposite to that in Lemma 2.4. Most importantly, we will bound the square ∥∇T ∥_{2}^{2} in terms of a linear function of ∥v∥_{∞}. As we are unable to obtain such bound by the standard ellitpic estimates, we have to proceed with an explicit calculation. As a preliminary step we show the following.

Lemma 2.5 Let (c, T, u) satisfy (2.8)-(2.10) with the normalization (2.11). Then there exists a constant C > 0 and a constant a_{0} > 0 so that we have for all a > a_{0} and 0 ≤ τ ≤ 1

∫_{D_{a}} |∇T |^{2}dxdz + ∫_{0}^{λ} T_{x}(a, z)dz ≤ C [1 + ∥v∥_{∞}] .  (2.43)

Proof. Recall that the function T satisfies

−cT_{x} + τu · ∇T = ΔT + τf(T)  (2.44)

with the boundary conditions

T (−a, z) = 1, T (a, z) = 0, \frac{∂T}{∂z} = 0 at z = 0, λ.  (2.45)

We multiply (2.44) by (1 − T) and use the boundary conditions and incompressibility of the flow u to obtain

\frac{cλ}{2} = ∫_{0}^{λ} T_{x}(a, z)dz + ∫_{D_{a}} |∇T |^{2}dxdz + τ ∫ (1 − T)f(T)dxdz.  (2.46)

Hence Lemma 2.2, and the fact that (1 − T)f(T) ≥ 0 imply that

∫_{D_{a}} |∇T |^{2}dxdz + ∫_{0}^{λ} T_{x}(a, z)dz ≤ \frac{cλ}{2} ≤ C [1 + ∥v∥_{∞}]  (2.47)

and Lemma 2.5 is proved. □

In order to close the bounds (2.12), (2.24) and (2.43) we need to bound the integral of T_{x} in (2.43). This is done in the next Lemma.

Lemma 2.6 Let (c, T, u) satisfy (2.8)-(2.10) with the normalization (2.11). There exists a constant C > 0 and a constant a_{0} so that we have for all a ≥ a_{0} and 0 ≤ τ ≤ 1

0 ≤ − ∫_{0}^{λ} T_{x}(a, z)dz ≤ C [1 + ∥∇T ∥_{2}] .  (2.48)

Proof. In order to find a bound for ∫_{0}^{λ} T_{x}(x = ±a, z)dz we introduce

I(x) = \frac{1}{λ} ∫_{0}^{λ} T (x, z)dz

and integrate equation (2.8) for T in z. Using the boundary conditions we obtain

−I_{xx} = G(x), I(−a) = 1, I(a) = 0, G(x) = \frac{τ}{λ} ∫ f(T (x, z))dz − ∫ (τu · ∇T − cT_{x})\frac{dz}{λ} . (2.49)

This equation can be solved explicitly:

I(x) = − ∫_{−a}^{x}(x − s)G(s)ds + Ax + B

with constants

A = −\frac{1}{2a} + \frac{1}{2a} ∫_{−a}^{a}(a − s)G(s)ds, B = \frac{1}{2} + \frac{1}{2} ∫_{−a}^{a}(a − s)G(s)ds

that are determined from the boundary conditions. Thus, we have

I_{x}(−a) = A, I_{x}(a) = A − ∫_{−a}^{a} G(s)ds.

Using the expression for the function G(x) in (2.49), we now infer that

0 ≤ −I_{x}(a) = \frac{1}{2a} + \frac{1}{2a} ∫_{−a}^{a}(a + s)G(s)ds = \frac{1}{2a} + \frac{τ}{2a} ∫_{−a}^{a} ∫_{0}^{λ}(a + x)f(T (x, z))\frac{dzdx}{λ}

−\frac{τ}{2a} ∫_{−a}^{a} ∫_{0}^{λ}(a + x)u · ∇T (x, z)\frac{dzdx}{λ} + \frac{c}{2a} ∫_{−a}^{a} ∫_{0}^{λ}(a + x)T_{x} \frac{dzdx}{λ}.

Integrating by parts, using the boundary conditions and incompressibility of u we obtain

0 ≤ −I_{x}(a) = \frac{1}{2a} + \frac{τ}{2} ∫_{−a}^{a} ∫_{0}^{λ} f(T (x, z))\frac{dzdx}{λ} + \frac{τ}{2} ∫_{−a}^{a} ∫_{0}^{λ} \frac{x}{a}f(T (x, z))\frac{dzdx}{λ}

+\frac{τ}{2a} ∫_{−a}^{a} ∫_{0}^{λ} v(x, z)T (x, z)\frac{dzdx}{λ} − \frac{c}{2a} ∫_{−a}^{a} ∫_{0}^{λ} T \frac{dzdx}{λ}.

However, the normalization condition (2.11) implies that f(T (x, z)) = 0 for x ≥ 0 since there is no reaction to the right of x = 0. Therefore, we can drop the third term above. This is one of the crucial points in the proof of the current lemma. Hence, we conclude that

0 ≤ −I_{x}(a) ≤ \frac{1}{2a} + \frac{τ}{2} ∫_{−a}^{a} ∫_{0}^{λ} f(T (x, z))\frac{dzdx}{λ} + \frac{τ}{2a} ∫_{−a}^{a} ∫_{0}^{λ} v(x, z)T (x, z)\frac{dzdx}{λ}

−\frac{c}{2a} ∫_{−a}^{a} ∫_{0}^{λ} T \frac{dzdx}{λ} ≤ \frac{1}{2a} + \frac{τ}{2} ∫_{−a}^{a} ∫_{0}^{λ} f(T (x, z))\frac{dzdx}{λ} + τ∥v∥_{∞} + |c|.  (2.50)

We used the fact that 0 ≤ T ≤ 1 to bound the last term above. Next, we look at I_{x}(−a):

0 ≤ −I_{x}(−a) = \frac{1}{2a} − \frac{1}{2a} ∫_{−a}^{a}(a − s)G(s)ds = \frac{1}{2a} − \frac{τ}{2a} ∫_{−a}^{a} ∫_{0}^{λ}(a − x)f(T (x, z))\frac{dzdx}{λ}

+\frac{τ}{2a} ∫_{−a}^{a} ∫_{0}^{1}(a − x)u · ∇T (x, z)dzdx − \frac{c}{2a} ∫_{−a}^{a} ∫_{0}^{λ}(a − x)T_{x} \frac{dzdx}{λ}.

We can drop the second term above, as (a − x)f(T) ≥ 0, so that, after integration by parts, we get

0 ≤ −I_{x}(−a) ≤ \frac{1}{2a} + \frac{τ}{2a} ∫_{−a}^{a} ∫_{0}^{λ} v(x, z)T (x, z)\frac{dzdx}{λ} − \frac{c}{2a} ∫_{−a}^{a} ∫_{0}^{λ} T \frac{dzdx}{λ} + |c|

≤ \frac{1}{2a} + τ∥v∥_{∞} + |c|.  (2.51)

Let us now put together (2.50), (2.51) and (2.46) We observe that, with F = τ ∫ f(T)\frac{dxdz}{λ}, we

have the following three inequalities:

c = I_{x}(a) − I_{x}(−a) + F

0 ≤ −I_{x}(a) ≤ \frac{1}{2a} + \frac{F}{2} + τ∥v∥_{∞} + |c|

0 ≤ −I_{x}(−a) ≤ \frac{1}{2a} + τ∥v∥_{∞} + |c|.

This implies that

F ≤ \frac{1}{a} + 2τ∥v∥_{∞} + 4|c|  (2.52)

and thus

−I_{x}(a) ≤ 3|c| + \frac{1}{a} + 2τ∥v∥_{∞} for a ≥ a_{0}

Lemma 2.2 implies then that

−I_{x}(a) ≤ C [1 + τ∥v∥_{∞} + \frac{1}{a}] .

Finally, Lemma 2.4 implies that

−I_{x}(a) ≤ C [1 + ∥∇T ∥_{2} + \frac{1}{a}] for a ≥ a_{0}.

Thus, Lemma 2.6 is proved. □

The previous lemmas imply uniform bounds that we summarize as follows.

Corollary 2.7 Let (c, T, u) satisfy (2.8)-(2.10) with the normalization (2.11). There exists a constant C > 0 and a_{0} > 0 so that we have for all a ≥ a_{0} and 0 ≤ τ ≤ 1

∥u∥_{L^{∞}(D_{a})} + ∥∇u∥_{L^{∞}(D_{a})} + ∥ω∥_{L^{2}(D_{a})} + |c| + ∥∇T ∥_{L^{2}(D_{a})} + τ ∫_{D_{a}} f(T)dxdz ≤ C.  (2.53)

In particular, as a consequence we also have

∥∇u∥_{L^{2}(D_{a})} + ∥ω∥_{L^{∞}(D_{a})} ≤ C.  (2.54)

Proof. Lemmas 2.6 and 2.5 imply that

∫ |∇T |^{2}dxdz ≤ C [1 + τ∥v∥_{∞} + ∥∇T ∥_{L^{2}(D_{a})}] .

Then Lemma 2.4 implies that

∥∇T ∥_{L^{2}(D_{a})}^{2} ≤ C(1 + ∥∇T ∥_{L^{2}(D_{a})})

and thus the estimate on ∥∇T ∥_{L^{2}(D_{a})} in (2.53) holds. Then Lemma 2.4 implies the bounds on ∥u∥_{L^{∞}(D_{a})}, ∥∇u∥_{L^{∞}(D_{a})} and ∥ω∥_{L^{2}(D_{a})}. The bound on |c| in (2.53) now follows from Lemma 2.2. Finally, the estimate on the total reaction rate follows from the above bounds and (2.52). One can elliminate the factor τ in front of the total reaction rate in (2.53): actually, one can show that it remains bounded as τ → 0. However, unlike the other estimates in (2.53), we will use the bound on the total reaction rate only at τ = 1. □

It remains to prove the uniform Hölder C^{1,α}-estimates for T (x, z), ω and u in order to finish the proof of Proposition 2.1.

Lemma 2.8 There exist two constants C > 0 and a_{0} > 0 so that the following bound holds for all a ≥ a_{0}:

∥ω∥_{C^{1,α}(D_{a})} + ∥u∥_{C^{1,α}(D_{a})} + ∥T ∥_{C^{1,α}(D_{a})} ≤ C  (2.55)

provided that 0 ≤ α < 1.

Proof. The bound for T follows from the standard elliptic local regularity estimates up to the boundary [20], the C^{1}-bound on the flow u and the uniform bound on the speed c in Corollary 2.7. The Hölder estimate for ω follows then from the vorticity equation (2.27) with the Dirichlet boundary conditions, the above mentioned C^{1,α}-bound on T, the same uniform estimates in Corollary 2.7 and the same results of [20]. Finally, the Hölder bounds on u follow from the Poisson equations (2.31) and (2.35) on the horizontal and vertical flow components, respectively, and the Hölder estimates for ω. □

This completes the proof of the a priori bounds in Proposition 2.1. We now turn to the proof of the existence part of this proposition.

Step 2. The degree argument. The a priori bounds proved in the first step of the proof allow us to use the Leray-Schauder topological degree argument to establish existence of solutions to the problem (2.8)-(2.10) with the normalization (2.11) in the bounded domain D_{a}. This method of construction of traveling wave solutions goes back to [11]. We introduce a map

K_{τ} : (c, ω, T) → (θ^{τ} , Ω^{τ} , Z^{τ})

as the solution operator of the linear system

−cZ_{x}^{τ} + τu · ∇Z^{τ} = ΔZ^{τ} + τf(T)  (2.56)
−cΩ_{x}^{τ} + u · ∇Ω^{τ} − σΔΩ^{τ} = τρ[e_{2}T_{x} − e_{1}T_{z}]

in D_{a} with the no stress boundary conditions

\frac{∂Z^{τ}}{∂z} = 0, w̃^{τ} = Ω^{τ} = 0 at z = 0, λ  (2.57)

and

Z^{τ} (−a, y) = 1, Z^{τ} (a, y) = 0, ũ^{τ} = Ω^{τ} = 0 at x = ±a.  (2.58)

Here the unknown flow ũ^{τ} = (ũ^{τ} , w̃^{τ}) and the given flow u are the incompressible flows corresponding to the vorticities Ω^{τ} and ω, respectively, and satisfying the no-stress boundary conditions. The number θ^{τ} is defined by

θ^{τ} = θ_{0} − max_{x≥0} T (x, z) + c.

The operator K_{τ} is a mapping of the Banach space X = ℝ × C^{1,α}(D_{a}) × C^{1,α}(D_{a}), equipped with the norm ∥(c, ω, T)∥_{X} = max(|c|, ∥ω∥_{C^{1,α}(D_{a})}, ∥T ∥_{C^{1,α}(D_{a})}), onto itself. A solution q^{τ} = (c^{τ} , ω^{τ} , T^{τ}) of (2.8)-(2.10) is a fixed point of K_{τ} and satisfies K_{τ} q^{τ} = q^{τ}, and vice versa: a fixed point of K_{τ} provides a solution to (2.8)-(2.10). Hence, in order to show that (2.8)-(2.10) has a traveling front solution it suffices to show that the kernel of the operator F_{τ} = Id − K_{τ} is not trivial. The standard elliptic regularity results in [20] imply that the operator K_{τ} is compact and depends continuously on the parameter τ ∈ [0, 1]. Thus the Leray-Schauder topological degree theory can be applied. Let us introduce a ball B_{M} = {∥(c, ω, T)∥_{X} ≤ M}. Then Lemma 2.8 and Lemma 2.2 show that the operator F_{τ} does not vanish on the boundary ∂B_{M} with M sufficiently large for any τ ∈ [0, 1]. It remains only to show that the degree deg(F_{1}, B_{M}, 0) in B̄_{M} is not zero. However, the homotopy invariance property of the degree implies that deg(F_{τ} , B_{M}, 0) = deg(F_{0}, B_{M}, 0) for all τ ∈ [0, 1]. Moreover, the degree at τ = 0 can be computed explicitly as the operator F_{0} is given by

F_{0}(c, ω, T) = (max_{x≥0} T (x, y) − θ_{0}, ω, T − T_{0}^{c}).

Here the function T_{0}(x) solves

\frac{d^{2}T_{0}^{c}}{dx^{2}} + c\frac{dT_{0}^{c}}{dx} = 0, T_{0}^{c}(−a) = 1, T_{0}^{c}(a) = 0

and is given by

T_{0}^{c}(x) = \frac{e^{−cx} − e^{−ca}}{e^{ca} − e^{−ca}} .

The mapping F_{0} is homotopic to

Φ(c, ω, T) = (max_{x≥0} T_{0}^{c}(x, y) − θ_{0}, ω, T − T_{0}^{c})

that in turn is homotopic to

Φ̃(c, ω, T) = (T_{0}^{c}(0) − θ_{0}, ω, T − T_{0}^{c_{∗}^{0}}),

where c_{∗}^{0} is the unique number so that T_{0}^{c_{∗}}(0) = θ_{0}. The degree of the mapping Φ̃ is the product of the degrees of each component. The last two have degree equal to one, and the first to −1, as the function T_{0}^{c}(0) is decreasing in c. Thus degF_{0} = −1 and hence degF_{1} = −1 so that the kernel of Id − K_{1} is not empty. This finishes the proof of Proposition 2.1. □

Remark 2.9 Observe that the C^{1,α}-regularity of T, u and ω can be bootstrapped to C^{2,α}-regularity: we have

∥ω∥_{C^{2,α}(D_{a})} + ∥u∥_{C^{2,α}(D_{a})} + ∥T ∥_{C^{2,α}(D_{a})} ≤ C  (2.59)

provided that 0 ≤ α < 1.

3 Identification of the limit

In order to finish the proof of Theorem 1.1 we consider the solutions (c^{a}, T^{a}, u^{a}) constructed in Proposition 2.1 and pass to the limit a → +∞. The a priori estimates in the same proposition imply that we can choose a subsequence a_{n} → ∞ so that T_{n}(x, z) = T_{a_{n}}(x, z) converges uniformly on compact sets to a function T (x, z), while the flow u_{n}(x, z) = u_{a_{n}}(x, z) converges to a flow u(x, z) = (v, w) and the front speeds also converge: c_{n} = c_{a_{n}} → c. The vorticity functions ω_{n}(x, z) = ω_{a_{n}}(x, z) converge to the limit ω = w_{x} − v_{z}. The limits satisfy the uniform bounds

|c| + ∥u∥_{∞} + ∥ω∥_{∞} + ∫ |∇T |^{2}dxdz + ∫ f(T)dxdz + ∫ |∇u|^{2}dxdz ≤ C  (3.1)

that follow from Corollary 2.7 and the Hölder estimates (2.7) and (2.59). The regularity estimates on (T^{a}, u^{a}) imply that the limit functions T and u satisfy the Boussinesq system

−cT_{x} + u · ∇T = ΔT + f(T)
−cω_{x} + u · ∇ω − σΔω = ρ(ê · ∇^{⊥}T)
ω = w_{x} − v_{z}.  (3.2)

Moreover, the boundary conditions on the lateral boundaries hold for T and u:

\frac{∂T}{∂z} = 0, w = ω = 0 on z = 0, λ.  (3.3)

The normalization condition

max_{x≥0} T (x, z) = θ_{0}  (3.4)

is also satisfied.

Therefore, to finish the proof of Theorem 1.1, it remains only to show that (i) T converges to a constant θ_{−} as x → −∞ and T → 0 as x → +∞, (ii) u → 0 as x → +∞, and (iii) θ_{−} = 1 if the reaction rate satisfies f(T) ≤ (T − θ_{0})_{+}^{2}/λ^{2}. First, we note that the uniform L^{2}-bound on ∇T in (3.1) implies that T converges to two constants θ_{−} and θ_{+} as x → ±∞, possibly passing to a subsequence x_{n} → ±∞. The elliptic regularity results imply that actually T converges to these

constants as x → ±∞. Moreover, the bound for the total reaction rate ∫ f(T)dxdz in (3.1) implies

that f(θ_{−}) = f(θ_{+}) = 0. Furthermore, integrating (3.2) we obtain

c(θ_{−} − θ_{+}) = ∫ f(T)\frac{dxdz}{λ} .  (3.5)

In order to identify the limits θ_{±} we will make use of the following lemmas that provide some additional information on solutions on a finite domain before the passage to the limit. The first result describes the behavior near the right end x = a_{n}.

Lemma 3.1 There exists a sequence a_{n} → ∞ so that

|\frac{∂T_{n}(a_{n}, z)}{∂x}| → 0  (3.6)

as n → ∞, uniformly in z. Moreover, we have lim_{n→∞} T_{n}(a_{n} − x_{0}, z) = 0 for all x_{0} ∈ ℝ.

Proof. We introduce a shifted solution Φ_{n}(x, z) = T_{n}(x + a_{n}, z) , v_{n} = u_{n}(x + a_{n}, z) defined in the domain −2a_{n} ≤ x ≤ 0. The functions Φ_{n} and v_{n} satisfy the same a priori bounds (3.1) as T_{n} and u_{n} and hence they converge as n → ∞ to some limits Φ and v that satisfy

−cΦ_{x} + v · ∇Φ = ΔΦ, Φ(0, z) = 0, x ≤ 0  (3.7)

as f(Φ_{n}) = 0 for x > −a_{n} and thus in the limit f(Φ) = 0. The function Φ satisfies the Neumann boundary conditions at z = 0, λ. The uniform upper bound on ∥∇Φ_{n}∥_{2} together with the elliptic regularity results imply that Φ has to converge to a constant Φ_{−} as x → −∞ along a subsequence. We note that, as 0 ≤ Φ(x, z) ≤ θ_{0}, the constant Φ_{−} satisfies the same bounds:

0 ≤ Φ_{−} ≤ θ_{0}.

Integrating (3.7) we obtain

cλΦ_{−} = ∫ Φ_{x}(0, z)dz ≤ 0.  (3.8)

Hence, either Φ_{−} = 0 or c ≤ 0. In the former case Φ ≡ 0 and hence Φ_{x}(0, z) = 0 for all z. That implies that both T_{x}^{n}(a_{n}, z) → 0 as n → ∞ and T_{x}^{n}(a_{n} − x_{0}, z) = Φ_{n}(−x_{0}, z) → 0 as n → ∞, as claimed in Lemma 3.1. It remains to rule out the second case, c ≤ 0. This is done in the next lemma that provides a crucial lower bound on the speed c_{n}. In particular it shows that c > 0 – this will conclude the proof of Lemma 3.1.

Lemma 3.2 The front speed is positive, c > 0.

Proof. Integrating the temperature equation in (3.2) for T_{n}, we obtain

c_{n}λ = ∫_{0}^{λ} \frac{∂T_{n}}{∂x} (a_{n}, z)dz − ∫_{0}^{λ} \frac{∂T_{n}}{∂x} (−a_{n}, z)dz + ∫ f(T_{n})dxdz ≥ ∫_{0}^{λ} \frac{∂T_{n}}{∂x} (a_{n}, z)dz + ∫ f(T_{n})dxdz.

(3.9)

Observe also that we have a uniform bound

(∫ f(T_{n})dxdz) (∫ |∇T_{n}|^{2}dxdz) ≥ C  (3.10)

that follows from the fact that T_{n}(0, z) ≤ θ_{0} and T_{n}(−a_{n}, z) = 1. The proof of (3.10) is as in [13]: there exists z_{0} ∈ (0, λ) such that both

∫_{−a_{n}}^{0} |∇T_{n}(x, z_{0})|^{2}dx ≤ 3 ∫_{D_{n}} |∇T_{n}|^{2} \frac{dxdz}{λ} , D_{n} = [−a_{n}, a_{n}]_{x} × [0, λ]_{z},

and

∫_{−a_{n}}^{0} f(T_{n}(x, z_{0}))dx ≤ 3 ∫_{D_{n}} f(T_{n}(x, z))\frac{dxdz}{λ} .

Let x_{1} be the left-most point so that T_{n}(x_{1}, z_{0}) = 1 − \frac{1 − θ_{0}}{4} :

x_{1} = inf \{x ∈ (−a_{n}, 0) : T_{n}(x, z_{0}) = 1 − \frac{1 − θ_{0}}{4} \}

and x_{2} > x_{1} be the left-most point so that T_{n}(x_{2}, z_{0}) = θ_{0} + \frac{1 − θ_{0}}{4} :

x_{2} = inf \{x ∈ (−a_{n}, 0) : T_{n}(x, z_{0}) = θ_{0} + \frac{1 − θ_{0}}{4} \} .

Existence of x_{1} and x_{2} is guaranteed by the fact that T_{n}(−a_{n}, z) = 1 and T_{n}(0, z) ≤ θ_{0} for all z ∈ [0, λ]. Then the reaction rate f(T_{n}(x, z_{0})) > C for x_{1} ≤ x ≤ x_{2} so that

C|x_{1} − x_{2}| ≤ ∫_{x_{1}}^{x_{2}} f(T_{n}(x, z_{0}))dx ≤ 3 ∫_{D_{n}} f(T_{n}(x, z))\frac{dxdz}{λ}

and

\frac{(1 − θ_{0})^{2}}{4|x_{1} − x_{2}|} ≤ ∫_{x_{1}}^{x_{2}} |∇T_{n}(x, z_{0})|^{2}dx ≤ 3 ∫_{D_{n}} |∇T_{n}|^{2} \frac{dxdz}{λ} .

Multiplying these two inequalities, we arrive at (3.10).

The estimate (3.10) and the uniform upper bound on ∥∇T_{n}∥_{2} in Corollary 2.7 imply that

∫_{D_{n}} f(T_{n})dxdz ≥ C.  (3.11)

Then, passing to the limit in (3.9), and using (3.8) we obtain

cλ(1 − Φ_{−}) ≥ C > 0,

as

∫ \frac{∂T_{n}}{∂x} (a_{n}, z)dz → ∫ Φ_{x}(0, z)dz,

with the function Φ as in the proof of Lemma 3.1. Now, we recall that Φ_{−} ≤ θ_{0} < 1 and thus the front speed c > 0. This finishes the proof of Lemma 3.2 and hence also that of Lemma 3.1. □

Lemma 3.2 and (3.5) imply that θ_{−} ≥ θ_{+}. However, if θ_{−} = θ_{+} we have f(T) = 0 everywhere and hence (3.2) is a linear equation. The maximum principle implies that T ≡ const in this case. The last condition in (3.1) implies that this constant has to be equal to θ_{0}. Hence, either θ_{−} > θ_{+} or T ≡ θ_{0}.

Let us now rule out the special case that θ_{−} = θ_{+} = θ_{0}.

Lemma 3.3 The left and right limits θ_{−} and θ_{+} satisfy θ_{−} > θ_{+}.

Proof. We have already shown that θ_{−} ≥ θ_{+} and, moreover, if θ_{−} = θ_{+} then

θ_{−} = θ_{+} = θ_{0}.  (3.12)

Hence, it suffices to show that the latter is impossible. Let us assume that (3.12) holds. As we have explained above, then

T_{n} → θ_{0}, and ∂T_{n}/∂x → 0 uniformly on compact sets.  (3.13)

Then, integrating the equation

−c_{n} \frac{∂T_{n}}{∂x} + u_{n} · ∇T_{n} = ΔT_{n} + f(T_{n})

between x = 0 and x = a_{n} we obtain, as f(T_{n}) = 0 in this region,

c_{n} ∫_{0}^{λ} T_{n}(0, z)dz − ∫_{0}^{λ} v_{n}(0, z)T_{n}(0, z)dz = ∫_{0}^{λ} \frac{∂T_{n}}{∂x} (a_{n}, z)dz − ∫_{0}^{λ} \frac{∂T_{n}}{∂x} (0, z)dz.  (3.14)

We now pass to the limit n → ∞ in (3.14). The first term on the left converges to cθ_{0}λ, as we have assumed that T converges uniformly to θ_{0} on compact intervals. The second term on the left converges to

∫_{0}^{λ} v(0, z)θ_{0}dz = 0,

as incompressibility of the flow u_{n} and the boundary conditions at x = ±a imply that

∫ v_{n}(0, z)dz = 0.

The limit (3.6) in Lemma 3.1 implies that the first term on the right side of (3.14) converges to zero. Finally, the last term on the right side of (3.14) converges to zero because of (3.13). Therefore, we obtain

cλθ_{0} = 0.

However, this implies that c = 0 which contradicts Lemma 3.2. Hence, the case θ_{−} = θ_{+} = θ_{0} is ruled out and thus θ_{−} > θ_{+}. □

We continue the analysis of the behavior of the solution at the right end of the domain.

Lemma 3.4 The gradient ∇T_{n} converges to zero “as x → +∞” uniformly in n, that is, for every ε > 0 there exists N ∈ ℕ and R so that |∇T_{n}(x, z)| < ε for all n ≥ N and all R ≤ x ≤ a_{n}.

Proof. Let us assume that this is not the case. Then there exists ε_{0} > 0 and a sequence b_{n} → +∞ so that |∇T_{n}(b_{n}, z_{n})| ≥ ε_{0} for some z_{n} ∈ [0, λ]. Note that Lemma 3.1 implies that

|a_{n} − b_{n}| → ∞.  (3.15)

Let us define the shifted solution Ψ_{n}(x, z) = T_{n}(x − b_{n}, z), v_{n}(x, z) = u_{n}(x − b_{n}, z) on the domain x ∈ [−a_{n}−b_{n}, a_{n}−b_{n}]. Then Ψ_{n} and v_{n} satisfy the same uniform bounds as T_{n} and u_{n} and thus they converge to a pair of functions Ψ, v uniformly on compact intervals, together with their derivatives. The functions Ψ and v are defined on the whole real line because of (3.15). Moreover, the function Ψ has left and right limits Ψ_{±} as x → ±∞. Hence, the same argument as in the proof of Lemma 3.1 shows that Ψ must be equal a constant, as it has left and right limits and satisfies

−cΨ_{x} + v · ∇Ψ = ΔΨ.

However, this contradicts the fact that max_{z} |∇Ψ(0, z)| ≥ ε_{0}. □

The decay of the gradient of T_{n} implies that the flow ahead of the front goes to zero for large x, uniformly in n.

Lemma 3.5 The flow u(x, z) converges to zero on the right uniformly in n, that is, for any ε > 0 there exists R > 0 and N ∈ ℕ so that |u(x, z)| < ε for all R < x ≤ a_{n}.

Proof. We choose N and R so that |∇T_{n}| < ε for all n ≥ N and x ∈ [R, a_{n}]. Then, we decompose T_{n} = T_{n}^{in} + T_{n}^{out} with supp T_{n}^{in} ⊂ {x ≤ R + 1} and supp T_{n}^{out} ⊂ {x ≥ R}. We also require that both T_{n}^{in} and T_{n}^{out} satisfy the same uniform gradient bounds as T_{n}. Moreover, we have |∇T_{n}^{out}| < ε. We also split ω_{n} = ω_{n}^{in} + ω_{n}^{out} and u_{n} = u_{n}^{in} + u_{n}^{out} accordingly:

−c_{n}ω_{n}^{in} + u_{n} · ∇ω_{n}^{in} − σΔω_{n}^{in} = ρ (e_{2} \frac{∂T_{n}^{in}}{∂x} − e_{1} \frac{∂T_{n}^{in}}{∂z} ) , ω_{n}^{in} = 0 on ∂D_{a_{n}},

and similarly for ω_{n}^{out}.

We now bound |u_{n}^{in}| and |u_{n}^{out}| separately for sufficiently large x. First, we look at u_{n}^{in} = (v_{n}^{in}, w_{n}^{in}). The function ω_{n}^{in} satisfies a homogeneous equation

−c_{n} \frac{∂ω_{n}^{in}}{∂x} + u_{n} · ∇ω_{n}^{in} − σΔω_{n}^{in} = 0  (3.16)

in the rectangle D_{R+2,a_{n}} = {R + 2 ≤ x ≤ a_{n}, 0 ≤ z ≤ λ}, as T_{n}^{in} vanishes in D_{R+2,a_{n}}. The function ω_{n}^{in} satisfies a uniform C^{2,α}-bound – this is shown in the same way as the C^{2,α}-bound for the full vorticity function ω in (2.59). This in turn implies that the function ψ(z) = ω_{n}^{in}(R+2, z) is uniformly bounded in C^{2}[0, λ]. Let g(x) be a smooth monotonic and positive cut-off function so that

g(x) = 1 for R + 2 ≤ x ≤ R + 3 and g(x) = 0 for x ≥ R + 4.  (3.17)

Then the function ω_{n}^{in} can be decomposed as

ω_{n}^{in}(x, z) = ψ(z)g(x) + ζ_{n}.

The function ζ_{n} satisfies

−c_{n} \frac{∂ζ_{n}}{∂x} + u_{n} · ∇ζ_{n} − σΔζ_{n} = f_{n} in D_{R+2,a_{n}}, ζ_{n} = 0 on ∂D_{R+2,a_{n}}.  (3.18)

The right side f_{n} is given by

f_{n} := σψ′′(z)g(x) + σψ(z)g′′(x) − c_{n}ψ(z)g′(x) − v_{n}ψ(z)g′(x) − w_{n}ψ′(z)g(x).

It is supported in R + 2 ≤ x ≤ R + 4 and is uniformly bounded since ∥ω_{n}^{in}∥_{C^{2,α}(D_{a})} ≤ C. Let us choose α > 0 sufficiently small, then the function ξ_{n}(x, z) = ζ_{n}(x, z)e^{αx} satisfies

−c_{n} \frac{∂ξ_{n}}{∂x} + αcξ_{n} + u_{n} · ∇ξ_{n} − αv_{n}ξ_{n} − σΔξ_{n} + 2σα\frac{∂ξ_{n}}{∂x} − σα^{2}ξ_{n} = g_{n} in D_{R+2,a_{n}}, ξ_{n} = 0 on ∂D_{R+2,a_{n}}

(3.19)

with g_{n} = f_{n}(x)e^{αx}. Multiplying (3.19) by ξ_{n} and integrating by parts, using the boundary conditions, we obtain

σ ∫_{D_{R+2,an}} |∇ξ_{n}|^{2}dxdz + (cα − α∥v∥_{∞} − σα^{2}) ∫_{D_{R+2,an}} |ξ_{n}|^{2}dxdz ≤ ∥g_{n}∥_{2}∥ξ_{n}∥_{2}.  (3.20)

However, as the function ξ_{n} vanishes at z = 0, λ, the Poincaré inequality implies that

∫_{D_{R+2,an}} |∇ξ_{n}|^{2}dxdz ≥ \frac{π^{2}}{λ^{2}} ∫_{D_{R+2,an}} |ξ_{n}|^{2}dxdz.

Hence, the following upper bound holds

∫_{D_{R+2,an}} |ξ_{n}|^{2}dxdz ≤ ∥g_{n}∥_{2}^{2} ≤ C,

provided that α is sufficiently small, since ∥v∥_{∞} ≤ C. Using (3.20) once again we conclude that

∫_{D_{R+2,an}} |∇ξ_{n}|^{2}dxdz ≤ C.

Therefore, the function ζ_{n} satisfies

∫_{D_{R+2,an}} [|∇ζ_{n}|^{2} + |ζ_{n}|^{2}] e^{2αx}dxdz ≤ C.

This, in turn implies the same bound for the function ω_{n}^{in}:

∫_{R+2}^{a_{n}} ∫_{0}^{λ} |ω_{n}^{in}|^{2}e^{2αx}dxdz + ∫_{R+2}^{a_{n}} ∫_{0}^{λ} |∇ω_{n}^{in}|^{2}e^{2αx}dxdz ≤ C.  (3.21)

It follows that the L^{2}-norm of ω_{n}^{in} decays uniformly in n:

∫_{r_{0}}^{a_{n}} ∫_{0}^{λ} |ω_{n}^{in}|^{2}dxdz ≤ e^{−αr_{0}} ∫_{r_{0}}^{a_{n}} ∫_{0}^{λ} |ω_{n}^{in}|^{2}e^{2αx}dxdz ≤ Ce^{−αr_{0}}.  (3.22)

for r_{0} > R + 5, and the same bound holds for ∇ω_{n}^{in}:

∫_{r_{0}}^{a_{n}} ∫_{0}^{λ} |∇ω_{n}^{in}|^{2}dxdz ≤ e^{−αr_{0}} ∫_{r_{0}}^{a_{n}} ∫_{0}^{λ} |∇ω_{n}^{in}|^{2}e^{2αx}dxdz ≤ Ce^{−αr_{0}}.  (3.23)

As the function ω_{n}^{in} satisfies the homogeneous equation (3.16) for x ≥ R + 1 with a bounded flow u, the standard local elliptic estimates now imply that

|ω_{n}^{in}(x, z)| ≤ Ce^{−αx} for x ≥ R + 5.  (3.24)

The W^{2,p} elliptic estimates imply then the uniform decay of the gradient of ω_{n}^{in}:

|∇ω_{n}^{in}(x, z)| ≤ Ce^{−αx} for x ≥ R + 5.  (3.25)

Now we can bound the flow u_{n}^{in} = (v_{n}^{in}, w_{n}^{in}) itself. First, we look at the horizontal component v_{n}^{in}. It satisfies the following Poisson equation in D_{R+2,a_{n}}:

−Δv_{n}^{in} = \frac{∂ω_{n}^{in}}{∂z} in D_{R+2,a_{n}}, \frac{∂v_{n}^{in}}{∂z} = 0 on z = 0, λ, v_{n}^{in} = 0 on x = a_{n}.

Moreover, the C^{2,α}-regularity of u_{n} implies that the boundary value ϕ(z) = v_{n}^{in}(R + 2, z) is bounded in C^{2}[0, λ]. Therefore, as we did with ω_{n}^{in}, we represent v_{n}^{in}(x, z) = ϕ(z)g(x) + v̄_{n}^{in}(x, z) with the cut-off function g(x) as in (3.17). The function v̄_{n}^{in} satisfies

−Δv̄_{n}^{in} = f̄_{n} := −ϕ_{zz}(z)g(x) − ϕ(z)g′′(x) + \frac{∂ω_{n}^{in}}{∂z} in D_{R+2,a_{n}},

with an exponentially decaying function f̄_{n}, as follows from (3.25). The boundary conditions are

\frac{∂v̄_{n}^{in}}{∂z} = 0 on z = 0, λ, v̄_{n}^{in} = 0 on x = R + 2, a_{n}.

The same argument as we used to obtain (3.21) implies that

∫_{R+2}^{a_{n}} ∫_{0}^{λ} |v_{n}^{in}|^{2}e^{2βx}dxdz ≤ C ∫ |f̄_{n}(x, z)|^{2}e^{2βx}dxdz ≤ C  (3.26)

with a sufficiently small 0 < β < α. Therefore, in the same vein as we have obtained (3.24) and (3.25), we conclude that

|v_{n}^{in}(x, z)| ≤ Ce^{−αx} for x ≥ R + 5  (3.27)

and

|∇v_{n}^{in}(x, z)| ≤ Ce^{−αx} for x ≥ R + 5.  (3.28)

The uniform bound on w_{n}^{in} now follows, as it satisfies the Dirichlet boundary condition w_{n}^{in}(x, 0) =

w_{n}^{in}(x, λ) = 0 and the derivative \frac{∂w_{n}^{in}}{∂z} = −\frac{∂v_{n}^{in}}{∂x} is exponentially decaying (3.28). We infer that

|w_{n}^{in}(x, z)| ≤ Ce^{−αx} for x ≥ R + 5.  (3.29)

Now we bound u_{n}^{out}. The corresponding vorticity satisfies

−c_{n} \frac{∂ω_{n}^{out}}{∂x} + u_{n} · ∇ω_{n}^{out} − σΔω_{n}^{out} = ρ(ê · ∇^{⊥}T_{n}^{out}) in D_{a_{n}}, ω = 0 on D_{a_{n}}.

However,

|∇T_{n}^{out}| ≤ ε  (3.30)

by construction, hence the maximum principle implies that

|ω_{n}^{out}(x, z)| ≤ ερq(z) ≤ Cε.  (3.31)

Here the non-negative function q(z) satisfies the boundary value problem

−σq′′(z) + w_{n}(z)q′(z) = 1, q(0) = q(λ) = 0.

We infer from the standard local elliptic estimates up to the boundary, (3.31) and (3.30) that

|∇ω_{n}^{out}(x, z)| ≤ Cε in D_{a_{n}}  (3.32)

as well. The vertical flow component satisfies

Δw_{n}^{out} = \frac{∂ω_{n}^{out}}{∂x} in D_{a_{n}}, w_{n}^{out}(x, 0) = w_{n}^{out}(x, λ) = 0, \frac{∂w_{n}^{out}(±a_{n}, z)}{∂x} = 0.

Therefore, the maximum principle implies once again that

|w_{n}^{out}(x, z)| ≤ \frac{Cρε}{2}z(λ − z) ≤ Cε in D_{a_{n}}.

Hence, the same local elliptic regularity results allow us to conclude that

|∇w_{n}^{out}(x, z)| ≤ Cε in D_{a_{n}}.

In order to bound the horizontal flow component v_{n}^{out} and conclude the proof of Lemma 3.5 we

observe that \frac{∂v_{n}^{out}}{∂z} = \frac{∂w_{n}^{out}}{∂z} − ω_{n}^{out} so that |\frac{∂v_{n}^{out}}{∂z} | ≤ Cε in D_{a_{n}}. However, v_{n}^{out} also satisfies the

mean-zero condition

∫_{0}^{λ} v_{n}^{out}(x, z) = 0 for all −a_{n} ≤ x ≤ a_{n}.

Hence, we have |v_{n}^{out}(x, z)| ≤ Cε in D_{a_{n}}, and the proof of Lemma 3.5 is now complete. □ The next lemma implies that the right limit θ_{+} = 0.

Lemma 3.6 The right limit θ_{+} = 0.

Proof. Let us choose R independent of n so that c_{a_{n}} > sup_{x>R} |v_{n}(x, z)| for all n. Lemma 3.2 implies that the speeds c_{n} are uniformly bounded below by a positive constant, thus it follows from Lemma 3.5 that we can find such R > 0. Then the function ϕ(x) = Ae^{−αx}, with a sufficiently small α > 0, satisfies

−c_{n}ϕ_{x} + u_{n} · ∇ϕ ≥ Δϕ.

An argument as in the proof of Lemma 2.2 shows that if A is chosen so that Ae^{−αR} > 1 then T_{n}(x, z) ≤ Ae^{−αx} on the domain x ∈ [R, a_{n}]. Therefore, the limit T (x, z) obeys the same bound, which in turn implies that θ_{+} = 0. □

Finally, we show that under the additional assumption (1.10) the left limit θ_{−} = 1. This is the only place in the proof where assumption (1.10) is used.

Lemma 3.7 Let us assume that f(T) ≤ (T − θ_{0})_{+}^{2}/λ^{2}. Then the left limit θ_{−} = 1.

Proof. We note that we have for each x ∈ ℝ

∫ |∇T (x, z)|^{2}dz ≥ \frac{(M(x) − m(x))^{2}}{λ}

with M(x) = max_{z} T (x, z) and m(x) = min_{z} T (x, z). It follows from the maximum principle that the function m(x) is non-increasing. Let us assume that θ_{−} ≤ θ_{0}, then monotonicity of m(x) implies that m(x) < θ_{0} for all x ∈ ℝ. Then we have

∫ |∇T (x, z)|^{2}dxdz ≥ ∫ (M(x) − m(x))^{2} \frac{dx}{λ} ≥ ∫ (T (x, z) − θ_{0})_{+}^{2} \frac{dxdz}{λ^{2}} .

We also observe that

cθ_{−} = ∫ f(T)\frac{dxdz}{λ} , \frac{cθ_{−}^{2}}{2} + ∫ |∇T |^{2} \frac{dxdz}{λ} = ∫ T f(T)\frac{dxdz}{λ}

so that

∫ |∇T |^{2}dxdz = ∫ (T − \frac{θ_{−}}{2} ) f(T)dxdz.

Hence we obtain using (1.10)

∫ (T − \frac{θ_{−}}{2} ) (T − θ_{0})_{+}^{2} \frac{dxdz}{λ^{2}} ≥ ∫ (T − \frac{θ_{−}}{2} ) f(T)dxdz ≥ ∫ (T (x, z) − θ_{0})_{+}^{2} \frac{dxdz}{λ^{2}} .

However, the left side is smaller than the right side unless T ≡ θ_{0}, the case that we have already ruled out. □

This finishes the proof of Theorem 1.1.

4 Bounds for the initial value problem

We consider in this section the solutions of the Cauchy problem with general front-like initial data and obtain the uniform bounds on the bulk burning rate and other average quantities stated in Theorems 1.2 and Theorem 1.3. We prove the first result, and the proof of the second result is presented in Section 4.2.

4.1 Bounds in an arbitrary strip

We prove in this section Theorem 1.2. Let T (t, x, z), u(t, x, z) be the solution of the Cauchy problem

T_{t} + u · ∇T = ΔT + f(T)  (4.1)
u_{t} + u · ∇u − σΔu + ∇p = ρT ê
∇ · u = 0,

(4.2)

with initial data T_{0}(x, z), u_{0}(x). We assume that there exists R > 0 so that T_{0}(x, z) = 0 for x > R and T_{0}(x, z) = 1 for x < −R, and that the initial vorticity is bounded in L^{2}:

∫ |ω_{0}(x, z)|^{2}dxdz < +∞.

The assumptions on the initial temperature T_{0} can be relaxed – it simply has to approach one and zero at the two ends of the domain sufficiently fast.

We recall that the bulk burning rate V̄ (t), the Nusselt number N̄(t) and the average horizontal flow Ū(t) are defined by

V̄ (t) = \frac{1}{t} ∫_{0}^{t} ∫ V (s)ds, V (t) = ∫ f(T)\frac{dxdz}{λ} ,  (4.3)
N̄(t) = \frac{1}{t} ∫_{0}^{t} N(s)ds, N(t) = ∫ |∇T |^{2} \frac{dxdz}{λ} ,  (4.4)
Ū(t) = \frac{1}{t} ∫_{0}^{t} ∥v(s)∥_{∞}ds.  (4.5)

The laminar front speed c_{0} is defined as the unique c so that equation

−cΦ′ = Φ′′ + f(Φ), Φ(−∞) = 1, Φ(+∞) = 0

has a solution 0 < Φ < 1. We recall the statement of Theorem 1.2.

Theorem 4.1 There exists a constant C > 0 so that under the above assumptions on the initial data T_{0}, u_{0}, the following bounds hold

c_{0} − C[ρ + ρ^{2}] + o(1) ≤ V̄ (t) ≤ c_{0} + C[ρ + ρ^{2}] + o(1)  (4.6)

N̄(t) ≤ [Cρ + \sqrt{\frac{c_{0}}{2} + C^{2}ρ^{2}}]^{2} + o(1)

Ū(t) ≤ Cρ[1 + ρ] + o(1)

as t → +∞.

This theorem shows that the coupled problem (4.1) is in a sense a regular perturbation of the single reaction-diffusion equation with ρ = 0. The lower bound in (4.6) is of interest only for small ρ when the left side is positive.

Proof. First, we prove the following bounds on N̄(t) and V̄ (t) in terms of Ū(t).

Lemma 4.2 There exists a constant C_{0} that depends only on the initial data T_{0} so that

N̄(t) ≤ \frac{1}{2}V̄ (t) + Ū(t) + C_{0} [\frac{1}{t} + \frac{1}{\sqrt{t}}]  (4.7)

and

V̄ (t) ≤ c_{0} + Ū(t) + C_{0} [\frac{1}{t} + \frac{1}{\sqrt{t}}] .  (4.8)

Proof. Define g(T) = T (1 − T) and its integral

R(t) = ∫ g(T)\frac{dxdz}{λ} .

The idea of using a concave function g(T) in a related context is due to B. Winn [38]. We observe that

\frac{dR}{dt} = ∫ g′(T)ΔT \frac{dxdz}{λ} + ∫ g′(T)f(T)\frac{dxdz}{λ} ≥ − ∫ g′′(T)|∇T |^{2} \frac{dxdz}{λ} − V (t)  (4.9)

with the burning rate V (t) defined in (4.3). Thus

\frac{dR}{dt} + V (t) ≥ 2 ∫ |∇T |^{2} \frac{dxdz}{λ} = 2N(t),

which after averaging in time becomes

\frac{R(t)}{t} + V̄ (t) ≥ 2N̄(t).  (4.10)

In order to obtain an upper bound for the potentially small term R(t)/t in (4.10) we construct sub- and super-solutions for T (t, x, z). This construction follows [40]. We look for a sub-solution for T of the form

ψ_{l}(t, x, z) = Φ_{0}(x − c_{0}t + x_{1} + ξ_{1}(t)) − q_{1}(t, x, z).

Here Φ_{0} is the traveling wave in the absence of convection, at ρ = 0, normalized so that Φ_{0}(0) = θ_{0}. It is the unique solution of

−c_{0}Φ′_{0} = Φ′′_{0} + f(Φ_{0}), Φ_{0}(0) = θ_{0}, Φ_{0}(−∞) = 1, Φ_{0}(+∞) = 0.

The functions ξ_{1}(t) and q_{1}(t, x, z) are to be chosen. In order for ψ_{l} to be a sub-solution we need

G[ψ_{l}] = \frac{∂ψ_{l}}{∂t} + u · ∇ψ_{l} − Δψ_{l} − f(ψ_{l}) ≤ 0.

We have

G[ψ_{l}] = ξ̇_{1}Φ′_{0} + uΦ′_{0} − \frac{∂q_{1}}{∂t} − u · ∇q_{1} + Δq_{1} + f(Φ_{0}) − f(Φ_{0} − q_{1}).

With an appropriate choice of x_{1}, that is, by shifting Φ_{0} sufficiently to the left we can ensure that T_{0}(x, z) ≥ Φ_{0}(x)−q_{10}(x) with 0 ≤ q_{10}(x) ≤ (1−θ_{0})/2 and q_{10}(x) ∈ L^{1}(ℝ). Then we choose q_{1}(t, x, z) to be the solution of

\frac{∂q_{1}}{∂t} + u · ∇q_{1} = Δq_{1}, q_{1}(0, x, z) = q_{10}(x), \frac{∂q_{1}}{∂z} = 0 at z = 0, λ.  (4.11)

The following lemma first proved in [16] provides a uniform L^{1} − L^{∞} decay estimate for q_{1} that is independent of the advection term.

Lemma 4.3 There exists a constant C > 0 that is independent of the (incompressible) flow u so that

∥q_{1}(t)∥_{∞} ≤ \frac{C}{λ\sqrt{t}}∥q_{10}∥_{L^{1}(D)}  (4.12)

for t ≥ 1.

As mentioned above, the main point of the above result is the independence of the constant in (4.12) from the flow u. We also note that this L^{1} −L^{∞} estimate behaves in a one-dimensional way for large times, as one would expect for a strip. The factor of λ in the denominator is compensated by the fact that the L^{1}-norm is taken over the strip and not only in x. We postpone the proof of Lemma 4.3 till the end of this section.

We can find δ > 0 so that if Φ_{0} ∈ (1 − δ, 1) and q_{1} ∈ (0, (1 − θ_{0})/2) then f(Φ_{0}) ≤ f(Φ_{0} − δ). Hence we have in this range of Φ_{0}:

G[ψ_{l}] ≤ ξ̇_{1}Φ′_{0} + vΦ′_{0}.  (4.13)

Furthermore, if Φ_{0} ∈ (0, δ) then f(Φ_{0}) = f(Φ_{0} − δ) = 0 and hence in this range of Φ_{0} we have (4.13) with the equality sign. Finally, if Φ_{0} ∈ (δ, 1 − δ) then |f(Φ_{0}) − f(Φ_{0} − q)| ≤ K|q| and Φ′_{0} ≤ −β. Hence G[ψ_{l}] ≤ 0 everywhere provided that

ξ̇_{1}(t) ≥ ∥v(t)∥_{∞} + \frac{K∥q(t)∥_{∞}}{β} .  (4.14)

Thus choose

ξ_{1}(t) = Ū(t)t + C\sqrt{t}.  (4.15)

Therefore we obtain a lower bound for T:

T (t, x, z) ≥ Φ_{0}(x − c_{0}t + Ū(t)t + C\sqrt{t}) − q_{1}(t, x, z).  (4.16)

In order to obtain an upper bound we set ψ_{u} = Φ_{0}(x − c_{0}t − x_{2} − ξ_{2}(t)) + q_{2}(t, x, z) and look for ξ_{2}(t) and q_{2}(t, x, z) so that G[ψ_{u}] ≥ 0. The constant x_{2} is chosen so that

T_{0}(x, z) ≤ Φ_{0}(x − x_{2}) + q_{2}(0, x, z)

with q_{2}(0, x, z) ∈ L^{1}(D) and 0 ≤ q_{2}(0, x, z) ≤ θ_{0}/2, as with q_{1}(0, x, z). The function q_{2}(t, x, z) is then chosen to satisfy the same advection-diffusion equation (4.11) similarly to q_{1}. Hence it obeys the same time decay bounds as q_{1}. With the above choice of q_{2} we have

G(ψ_{u}) = −ξ̇_{2}Φ′_{0} + vΦ′_{0} + f(Φ_{0}) − f(Φ_{0} + q_{2}).

Once again, we consider three regions of values for Φ_{0}. First, if 1 − δ ≤ Φ_{0} ≤ 1 with a sufficiently small δ > 0 then f(Φ_{0}) − f(Φ_{0} + q_{2}) ≥ 0, as q_{2} ≥ 0. Hence G[ψ_{u}] ≥ 0 in this region provided that ξ̇_{2} ≥ 0. Second, as q_{2} ≤ θ_{0}/2 we have f(Φ_{0}) = f(Φ_{0} + q_{2}) = 0 if 0 ≤ Φ_{0} ≤ δ with a sufficiently small δ > 0. Hence G[ψ_{u}] ≥ 0 in that region under the same condition ξ̇_{2} ≥ 0. Finally, if Φ_{0} ∈ (δ, 1 − δ) then Φ′_{0} ≤ −β with β > 0 and |f(Φ_{0}) − f(Φ_{0} + q_{2})| ≤ K∥q_{2}∥_{∞}. That means that G[ψ_{u}] ≥ 0 if we choose ξ_{2} so that

ξ̇_{2} ≥ ∥v(t)∥_{∞} + \frac{K∥q_{2}∥_{∞}}{β} .

Therefore we choose

ξ_{2}(t) = Ū(t)t + C\sqrt{t},

as with ξ_{1}(t). Therefore we obtain upper and lower bounds

Φ_{0}(x − c_{0}t + ξ_{1}(t) + x_{1}) − q_{1}(t, x, z) ≤ T (t, x, z) ≤ Φ_{0}(x − c_{0}t − ξ_{2}(t) − x_{2}) + q_{2}(t, x, z) (4.17)

that imply in particular that

Φ_{0}(x − c_{0}t + Ū(t)t + C_{0}[1 + \sqrt{t}]) − \frac{C_{0}}{\sqrt{t}} ≤ T (t, x, z) ≤ Φ_{0}(x − c_{0}t − Ū(t)t − C_{0}[1 + \sqrt{t}]) + \frac{C_{0}}{\sqrt{t}} (4.18)

with a constant C_{0} determined by the initial conditions. Hence, using (4.17)-(4.18) and the L^{1}- bounds ∥q_{j}(t)∥_{L^{1}(D)} ≤ C_{0}, j = 1, 2, we obtain

R(t) = ∫ T (1 − T)\frac{dxdz}{λ} = ∫_{−∞}^{c_{0}t−ξ_{2}(t)−x_{2}} ∫ T (1 − T)\frac{dzdx}{λ} + ∫_{c_{0}t−ξ_{2}(t)−x_{2}}^{c_{0}t+ξ_{1}(t)+x_{1}} ∫ T (1 − T)\frac{dzdx}{λ}

+ ∫_{c_{0}t+ξ_{1}(t)+x_{1}}^{∞} ∫ T (1 − T)\frac{dzdx}{λ} ≤ C_{0} + ∫_{−∞}^{0}(1 − Φ_{0})dx + (ξ_{1}(t) + ξ_{2}(t)) + ∫_{0}^{∞} Φ_{0}(x)dx

≤ C_{0}(1 + \sqrt{t}) + 2tŪ(t).

This together with (4.10) implies that

V̄ (t) + 2Ū(t) + C_{0} [\frac{1}{t} + \frac{1}{\sqrt{t}}] ≥ 2N̄(t)  (4.19)

so that (4.7) holds.

Moreover, we have

V̄ (t) = \frac{1}{t} ∫_{0}^{t} (∫ f(T (s, x, z))\frac{dxdz}{λ} ) ds = \frac{1}{t} ∫_{0}^{t} (∫ T_{t}(s, x, z)) \frac{dxdz}{λ} ds  (4.20)

= \frac{1}{t} ∫ [T (t, x, z) − T_{0}(x, z)] \frac{dxdz}{λ}

≤ \frac{1}{t} ∫ [Φ_{0}(x − c_{0}t − x_{2} − ξ_{2}(t)) + q_{2}(t, x, z) − Φ_{0}(x + x_{1}) + q_{1}(0, x, z)] \frac{dxdz}{λ}

≤ \frac{C_{0}}{t} + \frac{1}{t} ∫_{−∞}^{c_{0}t+ξ_{2}(t)+x_{2}}(1 − Φ_{0}(x))dx + \frac{1}{t} ∫_{0}^{∞} Φ_{0}(x)dx ≤ c_{0} + Ū(t) + C_{0} [\frac{1}{\sqrt{t}} + \frac{1}{t}]

as follows from (4.17)-(4.18). This proves (4.8) and finishes the proof of Lemma 4.2. □ On the other hand we have the following upper bound for Ū(t) in terms of N̄(t).

Lemma 4.4 There exists a constant C > 0 so that for all t > 0 the following inequality holds

Ū(t) ≤ C [ρ\sqrt{N̄(t)} + \frac{1}{\sqrt{t}}∥ω_{0}∥_{L^{2}}] .  (4.21)

Proof. We multiply the vorticity equation

\frac{∂ω}{∂t} + u · ∇ω − σω = ρ(ê · ∇^{⊥}T)

by ω and integrate:

\frac{1}{2} \frac{d}{dt} ∫ |ω(t, x, z)|^{2}dxdz + σ ∫ |∇ω(t, x, z)|^{2}dxdz = ρ ∫ ω(t, x, z)(ê^{⊥} · ∇T)dxdz,  (4.22)

with ê^{⊥} = (e_{2}, −e_{1}).

The Poincaré inequality for ω(t, x, z) implies then that

\frac{1}{2} \frac{d}{dt} ∫ |ω(t, x, z)|^{2}dxdz + \frac{σ}{2} ∫ |∇ω(t, x, z)|^{2}dxdz ≤ Cρ^{2} ∫ |∇T (t, x, z)|^{2}dxdz.

Integrating this equation in time we conclude that

\frac{1}{t} ∫_{0}^{t} ∫ |∇ω(s, x, z)|^{2}dxdzds ≤ C [ρ^{2}N̄(t) + \frac{1}{t}∥ω_{0}∥_{L^{2}}^{2}] .  (4.23)

However, as in the proof of Lemma 2.4, we have ∥v(t)∥_{L^{∞}(D)} ≤ C∥∇ω(t)∥_{L^{2}(D)}. This, together with (4.23) implies (4.21). □

Putting the bounds (4.21) and (4.7)-(4.8) together and using the Cauchy-Schwartz inequality we arrive at

2N̄(t) ≤ c_{0} + C_{0} [\frac{1}{t} + \frac{1}{\sqrt{t}}] + \frac{Cρ}{t} ∫_{0}^{t} \sqrt{N(s)}ds ≤ c_{0} + C_{0} [\frac{1}{t} + \frac{1}{\sqrt{t}}] + Cρ\sqrt{N̄(t)}.

Hence we obtain an upper bound

N̄(t) ≤ [Cρ + \sqrt{\frac{c_{0}}{2} + C^{2}ρ^{2}}]^{2} + o(1).  (4.24)

This, together with (4.21) implies that

Ū(t) ≤ Cρ [1 + ρ] + o(1).

It follows then from (4.20) that

V̄ (t) ≤ c_{0} + Cρ [1 + ρ] + o(1).

The lower bound on V̄ (t) in (4.6) is proved similarly. This finishes the proof of Theorem 4.1. It remains only to prove Lemma 4.3. □

Proof of Lemma 4.3. We will show that there exists a universal constant C > 0 so that the solution of

\frac{∂ψ}{∂t} + u · ∇ψ = σΔψ  (4.25)
ψ(0, x, z) = ψ_{0}(x, z) ≥ 0,

with the Neumann boundary conditions at z = 0 and z = λ, and u sufficiently regular, satisfies

∥ψ(t)∥_{L^{∞}(D)} ≤ Cn^{2}(t)∥ψ_{0}∥_{L^{1}(D)}.  (4.26)

Here n(t) is the unique solution of

\frac{n^{4}(t)}{1 + n^{3}(t)λ^{3}} = \frac{C}{σλ^{2}t}.  (4.27)

We multiply (4.25) by ψ and integrate over the domain D to obtain

\frac{1}{2} \frac{d}{dt}∥ψ∥_{2}^{2} = −σ∥∇ψ∥_{2}^{2}.  (4.28)

We now prove the following version of the Nash inequality [29] for a strip of width λ in two dimensions:

∥∇ψ∥_{2}^{2} ≥ C \frac{λ^{2}∥ψ∥_{2}^{6}}{∥ψ∥_{1}^{4} + λ^{3}∥ψ∥_{1}∥ψ∥_{2}^{3}} .  (4.29)

The proof of (4.29) is similar to that of the usual Nash inequality. We represent ψ in terms of its Fourier series-integral:

ψ(x, z) = ∑_{n≥0} ∫_{ℝ} e^{ikx} cos (\frac{πnz}{λ} ) ψ̂_{n}(k)\frac{dk}{2π} ,

where

ψ̂_{n}(k) = \frac{2}{λ} ∫_{D} e^{−ikx} cos (\frac{πnz}{λ} ) ψ(x, z)dxdz.

Therefore we have

|ψ̂_{n}(k)| ≤ \frac{2}{λ}∥ψ∥_{L^{1}}.  (4.30)

The Plancherel formula becomes

∫_{D} |ψ(x, z)|^{2}dxdz = ∑_{n,m≥0} ∫_{ℝ^{2}} e^{ikx−ipx} cos (\frac{πnz}{λ} ) cos (\frac{πmz}{λ} ) ψ̂_{n}(k)ψ̂_{m}^{∗}(p)\frac{dkdpdxdz}{(2π)^{2}}

= \frac{λ}{2} ∑_{n≥0} ∫ |ψ̂_{n}(k)|^{2} \frac{dk}{2π}

and similarly

∫_{D} |∇ψ(x, z)|^{2}dxdz = \frac{λ}{2} ∑_{n≥0} ∫_{ℝ} (k^{2} + \frac{π^{2}n^{2}}{λ^{2}} ) |ψ̂_{n}(k)|^{2} \frac{dk}{2π} .

Let ρ > 0 be a positive number to be chosen later. Then using the above Plancherel formula we write

∥ψ∥_{2}^{2} = I + II,

with the first term that is bounded using (4.30)

I = \frac{λ}{2} ∑_{0≤n≤ρλ} ∫_{|k|≤ρ} |ψ̂_{n}(k)|^{2} \frac{dk}{2π} ≤ \frac{Cλρ([λρ] + 1)}{λ^{2}}∥ψ∥_{1}^{2} ≤ \frac{Cρ(λρ + 1)}{λ}∥ψ∥_{1}^{2}.

The rest is bounded by

II ≤ \frac{Cλ}{ρ^{2}} ∑_{n≥0} ∫_{k∈ℝ} (k^{2} + \frac{4π^{2}n^{2}}{λ^{2}} ) |ψ̂_{n}(k)|^{2}dk ≤ \frac{C}{ρ^{2}} ∥∇ψ∥_{2}^{2}.

Therefore we have for all ρ > 0:

∥ψ∥_{2}^{2} ≤ \frac{Cρ(λρ + 1)}{λ}∥ψ∥_{1}^{2} + \frac{C}{ρ^{2}} ∥∇ψ∥_{2}^{2}.

We choose ρ so that

ρ^{3} = \frac{λ∥∇ψ∥_{2}^{2}}{∥ψ∥_{1}^{2}}

and obtain

∥ψ∥_{2}^{2} ≤ \frac{C∥∇ψ∥_{2}^{2/3}}{λ^{2/3}∥ψ∥_{1}^{2/3}} (\frac{λ^{4/3}∥∇ψ∥_{2}^{2/3}}{∥ψ∥_{1}^{2/3}} + 1) ∥ψ∥_{1}^{2} + \frac{C∥∇ψ∥_{2}^{2}∥ψ∥_{1}^{4/3}}{λ^{2/3}∥∇ψ∥_{2}^{4/3}}

= \frac{C}{λ^{2/3}} ∥ψ∥_{1}^{4/3}∥∇ψ∥_{2}^{2/3} + Cλ^{2/3}∥∇ψ∥_{2}^{4/3}∥ψ∥_{1}^{2/3}.

This is a quadratic inequality ax^{2} +bx−c ≥ 0 with x = ∥∇ψ∥_{2}^{2/3}, a = Cλ^{2/3}∥ψ∥_{1}^{2/3}, b = \frac{C}{λ^{2/3}} ∥ψ∥_{1}^{4/3},

and c = ∥ψ∥_{2}^{2} and hence

x ≥ \frac{−b + \sqrt{b^{2} + 4ac}}{2a} = \frac{2c}{b + \sqrt{b^{2} + 4ac}} ≥ \frac{c}{\sqrt{b^{2} + 4ac}}.

This implies that

∥∇ψ∥_{2}^{2/3} ≥ C∥ψ∥_{2}^{2} (\frac{∥ψ∥_{1}^{8/3}}{λ^{4/3}} + λ^{2/3}∥ψ∥_{1}^{2/3}∥ψ∥_{2}^{2})^{−1/2}

and therefore

∥∇ψ∥_{2}^{2} ≥ C∥ψ∥_{2}^{6} (\frac{4∥ψ∥_{1}^{8/3}}{λ^{4/3}} + 4λ^{2/3}∥ψ∥_{1}^{2/3}∥ψ∥_{2}^{2})^{−3/2} ≥ C∥ψ∥_{2}^{6} (\frac{∥ψ∥_{1}^{4}}{λ^{2}} + λ∥ψ∥_{1}∥ψ∥_{2}^{3})^{−1}

≥ \frac{Cλ^{2}∥ψ∥_{2}^{6}}{∥ψ∥_{1}^{4} + λ^{3}∥ψ∥_{1}∥ψ∥_{2}^{3}} .

Hence (4.29) indeed holds.

We insert (4.29) into (4.28) and use the conservation of the L^{1}-norm of ψ (recall that the initial data is non-negative) obtain

\frac{d∥ψ∥_{2}}{dt} ≤ −\frac{Cσλ^{2}∥ψ∥_{2}^{5}}{∥ψ_{0}∥_{1}^{4} + λ^{3}∥ψ_{0}∥_{1}∥ψ∥_{2}^{3}} .  (4.31)

Integrating (4.31) in time we have

Cσλ^{2}t ≤ \frac{∥ψ_{0}∥_{1}^{4}}{∥ψ∥_{2}^{4}} + \frac{λ^{3}∥ψ_{0}∥_{1}}{∥ψ∥_{2}} ≤ \frac{1}{z(t)} [λ^{3} + \frac{1}{z^{3}(t)}] ,

where z(t) = ∥ψ(t)∥_{2}/∥ψ_{0}∥_{1}, and thus

\frac{z^{4}(t)}{1 + λ^{3}z^{3}(t)} ≤ \frac{1}{Cσλ^{2}t}.  (4.32)

The function on the left side of (4.32) is monotonically increasing and hence we have

∥ψ(t)∥_{2} ≤ n(t)∥ψ_{0}∥_{1},  (4.33)

where n(t) is the solution of (4.27).

Let us denote by P_{t} the solution operator for (4.25): ψ(t) = P_{t}ψ_{0}. Then (4.33) implies that ∥P_{t}∥_{L^{1}→L^{2}} ≤ n(t). The adjoint operator P_{t}^{∗} is the solution operator for

\frac{∂ψ̃}{∂t} − u · ∇ψ̃ = σΔψ̃  (4.34)
ψ̃(0, x) = ψ̃_{0}(x), x ∈ D

with the Neumann boundary conditions at z = 0, λ. Note that the preceding estimates rely only on the anti-symmetry of the convection operator u·∇. Therefore we have the bound ∥P_{t}^{∗}∥_{L^{1}→L^{2}} ≤ n(t) and hence ∥P_{t}∥_{L^{2}→L^{∞}} ≤ n(t) so that

∥ψ(t)∥_{L^{∞}} ≤ n(t/2)∥ψ(t/2)∥_{L^{2}} ≤ n^{2}(t/2)∥ψ_{0}∥_{L^{1}}

and thus (4.26) indeed holds.

The estimate (4.12) follows from the observation that for large t ≫ 1, when n(t) is small, we have the bound

n^{2}(t) ≤ \frac{C}{(σt)^{1/2}λ}.

This finishes the proof of Lemma 4.3. □

4.2 Bounds on the burning rate in a narrow domain

We recall that no non-planar traveling fronts do exist in the reactive Boussinesq problem in a narrow vertical strip when gravity is sufficiently small [32, 33, 14]. Moreover, solutions with general frontlike initial data become asymptotically planar in the long time limit [14]. We extend now this result to the inclined cylinders. More precisely, we have the following result (this is a re-statement of Theorem 1.3).

Theorem 4.5 Let ê = (e_{1}, e_{2}) be the unit vector in the direction of gravity and let ρ_{j} = ρe_{j}, j = 1, 2. There exist two constants λ_{0} and ρ_{0} so that if the domain is sufficiently narrow: λ ≤ λ_{0} and gravity is sufficiently small: ρ ≤ ρ_{0} then the burning rate is bounded by

V̄ (t) ≤ c_{0} + Cρ_{2} + o(1) as t → +∞.  (4.35)

Moreover, the front is nearly planar in the sense that

N̄_{z}(t) = \frac{1}{t} ∫_{0}^{t} ∥T_{z}(s)∥_{2}^{2}ds ≤ Cρ_{2}^{2} + o(1) as t → +∞.  (4.36)

The key point in Theorem 4.5 is that the bounds in (4.35) and (4.36) are independent of the gravity strength ρ_{1} in the direction parallel to the cylinder.

Proof. Multiplying the vorticity equation by ω and integrating by parts we obtain

\frac{1}{2} \frac{d}{dt} ∫ |ω|^{2}dxdz + σ ∫ |∇ω|^{2}dxdz = ρ_{2} ∫ T_{x}ωdxdz − ρ_{1} ∫ T_{z}ωdxdz.  (4.37)

The Poincaré inequality applies to ω(x, z) with the Poincaré constant proportional to 1/λ. Hence, if λ < λ_{0} and ρ_{j} < ρ_{0}, (4.37) implies that

\frac{1}{2} \frac{d}{dt} ∫ |ω|^{2}dxdz + \frac{σ}{2} ∫ |∇ω|^{2}dxdz ≤ Cρ_{2}^{2} ∫ |T_{x}|^{2}dxdz + Cρ_{1}^{2} ∫ |T_{z}|^{2}dxdz.  (4.38)

We now differentiate the equation for T (t, x, z) in z to get

\frac{∂T_{z}}{∂t} + u · ∇T_{z} + u_{z} · ∇T = ΔT_{z} + f′(T)T_{z}.

Multiplying this equation by T_{z} we obtain

\frac{1}{2} \frac{d}{dt} ∫ |T_{z}|^{2}dxdz + ∫ |∇T_{z}|^{2}dxdz + ∫ T_{z}u_{z} · ∇T dxdz = ∫ f′(T)T_{z}^{2} ≤ M ∫ T_{z}^{2}dxdz. (4.39)

The last integral on the left side is bounded by

|∫ T_{z}u_{z} · ∇T dxdz| = |∫ T u_{z} · ∇T_{z}dxdz| ≤ 2 ∫ |u_{z}|^{2}dxdz + \frac{1}{2}|∇T_{z}|^{2}dxdz.

This, together with incompressibility of u, the Poincaré inequality for T_{z} and (4.39) imply that

\frac{1}{2} \frac{d}{dt} ∫ |T_{z}|^{2}dxdz + \frac{1}{4} ∫ |∇T_{z}|^{2}dxdz ≤ 4 ∫ |ω|^{2}dxdz,  (4.40)

provided that λ < λ_{0}. Combining (4.38) and (4.40) and using the Poincaré inequality for ω and T_{z} once again, we obtain the following inequalities for Ω(t) = ∥ω(t)∥_{2}^{2} and N_{z}(t) = ∥T_{z}∥_{2}^{2}:

\frac{1}{2} \frac{dΩ}{dt} + \frac{C}{λ^{2}} Ω ≤ Cρ_{2}^{2}N_{x}(t) + Cρ_{1}^{2}N_{z}(t)

and

\frac{1}{2} \frac{dN_{z}}{dt} + \frac{C}{λ^{2}} N_{z} ≤ 4Ω.

Hence, the function Q = N_{z} + Ω satisfies

\frac{1}{2} \frac{dQ}{dt} + [\frac{C}{λ^{2}} − 4 − Cρ_{1}^{2}] Q ≤ Cρ_{2}^{2}N_{x}(t).

Therefore, we have

Q(t) ≤ Q_{0}e^{−γt} + Cρ_{2}^{2} ∫_{0}^{t} e^{−γ(t−s)}N_{x}(s)ds

with γ > 0 provided that C/λ^{2} > 5 and Cρ_{1}^{2} < 1. We conclude that

Q̄(t) := \frac{1}{t} ∫_{0}^{t} Q(τ)dτ ≤ \frac{Q_{0}}{t} ∫_{0}^{t} e^{−γτ} dτ + \frac{Cρ_{2}}{t} ∫_{0}^{t} ∫_{0}^{τ} e^{−γ(τ−s)}N_{x}(s)dsdτ

≤ \frac{C_{0}}{t} + \frac{Cρ_{2}^{2}}{t} ∫_{0}^{t} N_{x}(s)e^{γs} ∫_{s}^{t} e^{−γτ} dτds ≤ \frac{C_{0}}{t} + \frac{Cρ_{2}^{2}}{γ} N̄_{x}(t) ≤ Cρ_{2}^{2} + \frac{C_{0}}{t} .

The last inequality above follows from the bound on N̄(t) in Theorem 4.1. Now, the bound (4.36) in Theorem 4.5 follows. Then, (4.38) together with (4.36) and the same uniform bound on N̄(t) in Theorem 4.1 imply that

\frac{1}{t} ∫_{0}^{t} ∥∇ω(s)∥_{2}^{2}ds ≤ Cρ_{2}^{2} + \frac{C_{0}}{t} .

We recall that ∥v(t)∥_{L^{∞}(D)} ≤ C∥∇ω∥_{L^{2}(D)} – this, together with the above, imply that

Ū(t) ≤ Cρ_{2} + \frac{C_{0}}{\sqrt{t}}.  (4.41)

Finally, using (4.20) and (4.41) we obtain (4.35). □

References

[1] M. Abel, A. Celani, D. Vergni and A. Vulpiani, Front propagation in laminar flows, Physical Review E, 64 6307 (2001).

[2] M. Abel, M. Cencini, D. Vergni and A. Vulpiani, Front speed enhancement in cellular flows, Chaos 12, p. 481.

[3] B. Audoly, H. Berestycki and Y. Pomeau, Rèaction diffusion en ècoulement stationnaire rapide, C.R.Acad. Sci., Ser. IIB, 328, 255-262.

[4] M. Belk, B. Kazmierczak, V. Volpert Existence of reaction-diffusion-convection waves in unbounded cylinders, Preprint, 2004.

[5] H. Berestycki, The influence of advection on the propagation of fronts in reaction-diffusion equations, in Nonlinear PDEs in Condensed Matter and Reactive Flows, NATO Science Series C, 569, H. Berestycki and Y. Pomeau eds, Kluwer, Doordrecht, 2003.

[6] H. Berestycki and F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math. 55, 2002, 949–1032.

[7] H. Berestycki, F. Hamel and N. Nadirashvili, The speed of propagation for KPP type problems in periodic and general domains, Preprint, 2003.

[8] H. Berestycki, F. Hamel and N. Nadirashvili, Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena, Preprint, 2003.

[9] H. Berestycki, B. Larrouturou and L. Nirenberg, A nonlinear elliptic problem describing the propagation of a curved premixed flame, in Mathematical Modeling in Combustion and Related Topics, C.-M. Brauner and C. Schmidt-Lainé, eds., NATO ASI Series, Kluwer, 1988.

[10] H. Berestycki, B. Larrouturou and P. L. Lions, Multi-dimensional traveling wave solutions of a flame propagation model, Arch. Rational Mech. Anal., 111, 1990, 33-49.

[11] H. Berestycki, B. Nicolaenco and B. Scheurer, Traveling wave solutions to combustion models and their singular limits, SIAM Jour. math. Anal., 16, 1983, 1207-1242.

[12] H. Berestycki and L. Nirenberg, Traveling fronts in cylinders, Annales de l’IHP, Analyse non linéare, 9, 1992, 497-572.

[13] P. Constantin, A. Kiselev, A. Oberman, L. Ryzhik, Bulk burning rate in passive-reactive diffusion, Arch. Rat. Mech. Anal. 154, 2000, 53-91.

[14] P. Constantin, A. Kiselev and L. Ryzhik, Fronts in reactive convection: bounds, stability and instability, Comm. Pure Appl. Math., 56, 2003, 1781-1803.

[15] A. de Wit, Fingering of chemical fronts in porous media, Phys. Rev. Let., 87, 054502.

[16] A. Fannjiang, A. Kiselev and L. Ryzhik, Unpublished notes, 2002.

[17] R. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7, 1937, 355–369.

[18] M. Freidlin and J. Gärtner, On the propagation of concentration waves in periodic and random media, Soviet Math. Dokl., 20, 1979, 1282-1286.

[19] M. Freidlin, Geometric optics approach to reaction-diffusion equations, SIAM J. Appl. Math., 46, 1986, 222-232.

[20] D. Gilbarg and. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983.

[21] F. Hamel, Formules min-max pour les vitesses d’ondes progressives multidimensionnelles, Ann. Fac. Sci. Toulouse Math., Série 6, 8, 1999, 259–280.

[22] S. Heinze, G. Papanicolau and A. Stevens, Variational principles for propagation speeds in inhomogeneous media, SIAM J. Appl. Math. 62, 2001, 129-148.

[23] L. Kagan and G. Sivashinsky, Flame propagation and extinction in large-scale vortical flows, Combust. Flame 120, 2000, 222-232.

[24] L. Kagan, P.D. Ronney and G. Sivashinsky, Activation energy effect on flame propagation in large-scale vortical flows, Combust. Theory Modelling 6, 2002, 479-485.

[25] A. Kiselev and L. Ryzhik, Enhancement of the travelling front speeds in reaction-diffusion equations with advection, Ann. Inst. H. Poincaré Anal. Non Linéaire 18, 2001, 309-358.

[26] A.N. Kolmogorov, I.G. Petrovskii and N.S. Piskunov, Étude de l’équation de la chaleurde matière et son application à un problème biologique, Bull. Moskov. Gos. Univ. Mat. Mekh. 1 (1937), 1-25. (see [30] pp. 105-130 for an English transl.)

[27] A. Majda and P. Souganidis, Large scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales, Nonlinearity, 7, 1994, 1-30.

[28] S. Malham and J. Xin, Global solutions to a reactive Boussinesq system with front data on an infinite domain, Comm. Math. Phys., 193, 1998, 287-316.

[29] J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. Jour. Math., 80, 1958, 931-954.

[30] Dynamics of curved fronts, P. Pelcé, Ed., Academic Press, 1988.

[31] G. Papanicolaou and X. Xin, Reaction diffusion fronts in periodically layered media, Jour. Stat. Phys., 63, 1991, 915-932.

[32] R. Texier-Picard and V. Volpert, Problèmes de réaction-diffusion-convection dans des cylindres non bornés, C. R. Acad. Sci. Paris Sr. I Math. 333, 2001, 1077-1082

[33] R. Texier-Picard and V. Volpert, Reaction-diffusion-convection problems in unbounded cylinders, Revista Matematica Complutense, 16, 2003, ...

[34] N. Vladimirova, R. Rosner, Model flames in the Boussinesq limit: the effects of feedback, Phys. Rev. E., 67, 2003, 066305.

[35] V.A. Volpert and A.I. Volpert, Location of spectrum and stability of solutions for monotone parabolic system, Adv. Diff. Eq., 2, 1997, 811-830.

[36] V.A. Volpert and A.I. Volpert, Existence and stability of multidimensional travelling waves in the monostable case, Israel Jour. Math., 110, 1999, 269-292.

[37] V.A. Volpert and A.I. Volpert, Spectrum of elliptic operators and stability of travelling waves, Asymptotic Anal., 23, 2000, 111-134.

[38] B. Win, Ph.D. thesis, University of Chicago, 2004.

[39] J. Xin, Existence of planar flame fronts in convective-diffusive periodic media, Arch. Rat. Mech. Anal., 121, 1992, 205-233.

[40] J. Xin, Existence and nonexistence of travelling waves and reaction-diffusion front propagation in periodic media, Jour. Stat. Phys., 73, 1993, 893-926.

[41] J. Xin, Analysis and modelling of front propagation in heterogeneous media, SIAM Rev., 42, 2000, 161-230.

[42] 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.