Global Solutions to a Reactive Boussinesq System with Front Data on an Infinite Domain

Mathematical Physics

© Springer-Verlag 1998

Global Solutions to a Reactive Boussinesq System with Front Data on an Infinite Domain

Simon Malham¹, Jack X. Xin²

1 Department of Theoretical Mechanics, University of Nottingham, Nottingham, UK.

E-mail: [email protected]

2 Department of Mathematics, University of Arizona, Tucson AZ 85721, USA.

E-mail: [email protected]

Received: 15 January 1996 / Accepted: 2 September 1997

Abstract: We prove the existence of global solutions to a coupled system of Navier–

Stokes, and reaction-diffusion equations (for temperature and mass fraction) with pre- scribed front data on an infinite vertical strip or tube. This system models a one-step exothermic chemical reaction. The heat release induced volume expansion is accounted for via the Boussinesq approximation. The solutions are time dependent moving fronts in the presence of fluid convection. In the general setting, the fronts are subject to intensive Rayleigh-Taylor and thermal-diffusive instabilities. Various physical quantities, such as fluid velocity, temperature, and front speed, can grow in time. We show that the growth is at moste^O(t)for large timetby constructing a nonlinear functional on the temperature and mass fraction components. These results hold for arbitrary order reactions in two space dimensions and for quadratic and cubic reactions in three space dimensions. In the absence of any thermal-diffusive instability (unit Lewis number), and with weak fluid coupling, we construct a class of fronts moving through shear flows. Although the front speeds may oscillate in time, we show that they are uniformly bounded for large t. The front equation shows the generic time-dependent nature of the front speeds and the straining effect of the flow field.

1. Introduction

We study the existence of global solutions to the following Boussinesq combustion system on the infinite tube := {(x, y) ∈ 6×R}, where6 ⊂ R^d−1 is an open, bounded, simply connected domain with smooth boundary∂6=6/6, outward normal nˆ, andd= 2,3 is our spatial dimension:

∂_tψ+u·∇ψ=1ψ−ψf(θ), (1.1a)

∂_tθ+u·∇θ=`1θ+ψf(θ), (1.1b)

∂_tu+u·∇u=ν1u−∇p+σθe, (1.1c)

∇·u= 0. (1.1d)

Physically we interpret:u(x, y, t) :×R→R^das the fluid velocity;p(x, y, t) :× R → Rthe pressure;ψ(x, y, t) :×R → Rthe concentration of the reactant in a one-step irreversible exothermic reaction;θ(x, y, t) : ×R → Rthe temperature of the reactant-product mixture;νthe normalised fluid viscosity or Prandtl number;`the Lewis number;σthe Rayleigh number;edenotes the unit vertical direction opposite to the propagation direction of flame (aligned with they direction). For convenience we shall writeθ:= (ψ , θ) :×R→R², i.e. the 2-tuple of reactant concentration and temperature. We assume non-homogeneous boundary conditions which allow front type initial data to be prescribed (for our main resultsb^∗≡0):

∂nˆψ = 0, ∂nˆθ= 0, u= 0, on∂6×R×R+, ψ →0, θ→1, u→b^∗, asy→+∞, ψ →1, θ→0, u→b^∗, asy→ −∞.

(1.2) We will suppose for somem∈N,

f(θ) = (

θ^m, θ >0,

0, θ≤0. (1.3)

System (1.1) models the vertical movement of flame fronts. Thermal volume expansion of the fluid due to the irreversible exothermic combustion reaction is accounted for by the Overbeck-Boussinesq approximation [10, 28]. The nonlinear chemical reaction term ψf(θ) usually takes the Arrhenius formψf(θ) exp{−E/Rθ}, or its normalised version ψf(θ) exp{(θ−1)/(1 +χ(θ−1))}, wheref(θ) is usually of the form (1.3) thoughmis not always a positive integer (in general). The constantEis the activation energy,Ris the universal gas constant,χ∈(0,1) is the thermal expansion coefficient and >0 is the reciprocal of the Zel’dovich number (see Buckmaster and Ludford [8] or Berestycki and Larrouturou [2]). Since the supremum of exp{−E/Rθ}over×Ris always bounded, the inclusion of this factor in the chemical nonlinearity would not affect any of our proofs. Consequently we neglect this exponential factor and for simplicity, choosefto be a power nonlinearity.

This simple chemical nonlinearity also arises in isothermal autocatalytic chemical reactions of the formA+mB→(m+ 1)B, wheremis the order of reaction, andψ,θ are the concentrations of reactantAand autocatalystBrespectively. The rate of reaction is thus given bykψθ^m, where the constant enthalpykcan be scaled out of the system. In this case the temperature remains fixed, yet the density of theAandBmixture increases with the reaction resulting in a change of fluid velocity. Thus chemical feedback plays the role of thermal feedback and system (1.1) then governs the dynamics of the moving concentration fronts in the presence of fluid convection.

Billingham, King, Merkin, Metcalf, Needham, and Scott [6, 39, 40] studied the au- tocatalytic reaction-diffusion system (1.1a), (1.1b), neglecting hydrodynamical effects.

They proved existence and uniqueness results for an associated boundary value problem and studied the development of travelling fronts of chemical reaction. See also Focant and Gallay [18] for a recent study of existence of traveling fronts in the quadratic-cubic

case and their stability when`is near one. The passively convected version of the non- isothermal system was considered by Berestycki, Larrouturou and Roquejoffre [3, 48]

in an infinite tube, and they proved the linear and nonlinear stability of travelling front solutions. Manley, Marion and Temam [34, 36] examined system (1.1) on a finite tube in the case of a multi-component reaction and with slightly different boundary condi- tions. They proved the global existence (d= 2,3) of suitable weak solutions uniformly bounded (d= 2) in time. Further, their estimates for the Hausdorff dimension of the uni- versal attractor indicated that for long tubes, hydrodynamical effects make a significant contribution to the complexity of the flow. Crucial to their proofs was the assumption of a bounded nonlinear reaction term.

We are interested in studying the full system (1.1) on unbounded domains while al- lowing for unbounded chemical nonlinearities. The attractor dimension results of Man- ley, Marion and Temam [34, 36] indicate the importance of studying this system when the vertical domain size is much larger than the typical length scale associated with the front width. The infinite cylindrical domain is the natural setting for examining the long time behaviour of travelling front solutions and especially for the irreversible reactions.

Numerical simulations (Patnaik and Kailasanath [45], Zhu and Xin [62]) have shown that moving fronts of system (1.1) are subject to both Rayleigh-Taylor (upward fronts) and thermal diffusive instabilities. The Rayleigh-Taylor instability from theσθeterm is due to heavier (cold) fluid lying above the lighter (hot) fluid. It leads to bubble formation on the front and growth of fluid energy and vorticity. The thermal-diffusive instability due to`6= 1 can cause chaotic front oscillation (` >1) or formations of cellular front structures (` < 1) [51, 39]. As a result, the maximum temperature can grow in time.

Last, but not least, for high Reynolds numbers (smallν) the fluid flow can become highly irregular, which in turn wrinkles the front and may induce front acceleration. In [62], power growth in time of maximum vorticity and temperature is numerically observed (d= 2,ν = 0.005,`= 0.1 or 10). Majda and Souganidis [33] studied front acceleration (front speed ofO(t^p),p >0), in a prescribed (passive) random shear flow of H¨older regularity. All this evidence suggests that in general, one should not expect the front speed to be uniformly bounded in time, instead a power growth may well happen.

Our first result implies an exponential bound e^O(t) on the front speeds for fronts in a two dimensional infinite vertical strip (for all orders of reactionm) or in a three dimensional infinite vertical tube (form∈ {1,2,3}). We treat only the one-step reaction case, as the analysis of the multi-component case is practically identical. We prove the existence of global weak solutions (d = 2,3) to (1.1). In the two dimensional case we prove uniqueness for a class of slightly more regular weak solutions as well as the existence of strong, smooth solutions for smooth initial data. Our sharpest norm upper bounds of solutions grow with time, so we are unable to discuss attractors. The growing bounds may be interpreted as the enhancement of instabilities in the system due to the unbounded chemical nonlinearities and unbounded domains. If=6×3,|3|<∞, i.e. a bounded strip or tube, we can considerably improve our growth estimates. The details however will be presented elsewhere.

Our second result concerns fronts in a reduced system when`= 1,σis small,ν >2π.

The fluid flows are laminar, the Rayleigh-Taylor effect is minimal, and the thermal- diffusive effect is absent. We construct a class of front solutions near the known passive fronts in smooth shear flows. The time dependent front speeds are proved to be uniformly bounded in time. Passive fronts in shear flows on infinite cylindrical domains have been studied at length by Berestycki, Larrouturou, Lions, Nirenberg and Roquejoffre [2, 3, 48, 4, 5], regarding the existence and stability of travelling front solutions. Similar issues on passive fronts in periodic flow fields have also been well studied by Papanicolaou,

Xin, and Zhu, [43, 56–59]. The passive fronts in these cases all propagate with constant speeds. However, with fluid coupling turned on, front speed is no longer constant as we will see from the front equation arising in the course of the proof.

2. Global Weak Solutions and Growth Estimates

2.1. Notation and Statement of Main Result. We denotehϕi:=R

ϕdxdyfor Lebesgue measurable functionsϕ:→R. Forq ∈[1,∞),n∈N,L^q(;R^d) andHⁿ(;R^d) are the usual Lebesgue and Sobolev spaces ofR^d-valued functions, equipped with the norm and inner product

kϕk^q_Lq(;R^d) :=

Xd

i=1

h|ϕ_i|^qi,

(ϕ,φ)_Hn(;R^d) :=

Xd i=1

X

|α|≤n

hD^αϕ_iD^αφ_ii.

Sincehas a smooth boundary, an equivalent norm onHⁿ(;R^d) is kϕk²_Hn(;R^d)=kϕk²_L2(;R^d)+ X

|α|=n

kD^αϕk²_L2(;R^d).

The non-reflexive spaceL^∞(;R^d) is equipped with the usual sup-norm. We adopt the notationL^q := L^q(;R²) forR²-valued functions defined on ⊂R^d. We will often use the Gagliardo–Nirenberg inequality: for allϕ∈H¹(),q∈[2,∞) whend= 2 and q∈[2,6] whend= 3

kϕkL^q ≤ck∇ϕk_L^d2−2(^dq;R^d)kϕk¹_L⁻2^d2+dq +c()kϕkL². (2.1) The last term on the right-hand side of (2.1) is zero whenϕ∈H₀¹(). Also, we shall use the interpolation inequality [19, 31]: forϕ∈L^r()∩L^p(), 1≤p≤q≤r≤ ∞, µ∈[0,1] and 1/q=µ/p+ (1−µ)/r

kϕk_L^q ≤ kϕk^µ_Lpkϕk¹_L^−µr . (2.2) The Poincar´e inequality establishes the equivalence of the normkϕk_H¹()and semi-norm k∇ϕk_L²(;R^d)onH₀¹(): forϕ∈H₀¹()

kϕk_L²≤c(6)k∇ϕk_L²(;R^d). (2.3) For a given Hilbert spaceXwith inner product (·,·)_X, we shall useh·,·iX×X⁰ to denote the bilinear form establishing the duality betweenXand its dualX⁰. For two topological vector spacesXandY, the notationX,→Yshall indicate that a continuous embedding exists fromXinto Yand we shall useX ,→,→ Ywhen the embedding is compact.

Given a Banach spaceY, we shall useL^p_loc([0,∞);Y) to denote the space of measurable functions from [0,∞) toYsuch thatk·kY∈L^p_loc([0,∞)). The notations w-Yand w^∗-Y are used to denoteYendowed with its weak and weak-star topologies respectively. By C([0,∞); w-X) we indicate the space of functions continuous from [0,∞) into w-X.

With D := {v ∈ C₀^∞(;R^d) : ∇ ·v = 0}, we specify H to be the closure of D in L²(;R^d) and V to be the closure of D in H¹(;R^d). Since there exists TC : {v ∈ L²(;R^d) : ∇·v ∈ L²()} → {v|∂ ∈ H⁻¹²(∂;R^d)}, a continu- ous linear trace operator such thatTC(v) =v·n|ˆ _∂for smoothv, we have the standard characterisations [14, 32]

H={v∈L²(;R^d) :∇·v= 0, TC(v) = 0}, V={v∈H₀¹(;R^d) :∇·v= 0}.

By the Riesz representation theorem we can identifyH≡H⁰which is our pivot space and then, using the inequalities above, we establish the Gelfand triple [56]V,→H,→V⁰, where each space is dense in the one which follows. We usePto represent the Leray- Hodge orthogonal projection onto divergence free functionsP:L²(;R^d)→H. In the standard fashion we define the linear Stokes operatorA= −P1:D(A)→H, where D(A) =H²(;R^d)∩V.

Further, with E := {ϕ ∈ C₀^∞(R^d) restricted to  : ∂nˆϕ = 0 on ∂6×R} and W :={ϕ ∈H¹() :kϕk²_H1()+k1ϕk²_L2(), we specifyWto be the closure ofEin W. Since there existTD :H¹() →H¹²(∂)⊂L²(∂) andTN : W →H⁻¹²(∂), continuous linear trace operators such thatT_D(ϕ) = ϕ|_∂ andT_N(ϕ) = ∂_nˆϕ|_∂ for smoothϕ, we can characterise [15, 56]

W={ϕ∈H¹();1ϕ∈L²() :∂nˆϕ= 0 on∂6×R},

We will also need Green’s theorem, which by density arguments, holds forϕ∈Wand v∈H¹():

h∇v·∇ϕi+hv1ϕi=hTD(v),TN(ϕ)i

H¹²(∂)×H⁻¹²(∂). (2.4) Supposeφ ∈ C₀^∞(R; [0,∞)) satisfiesR

Rφ(y) dy = 1. Set ˜ψ(y) = R_∞

y−ysφ(s)ds and θ(y) =˜ R_y

−∞φ(s)ds, wherey_sis a finite constant which we can choose to be zero. Thus θ˜ = ( ˜ψ,θ)˜ ∈ [0,1]² is smooth, satisfies the boundary conditions (1.2) and ˜ψ·θ˜has compact support inR. As in Heywood [22], we assume thatb^∗ can be continued as a function into,b ∈ H_loc² (;R^d), for which there exists a scalar distributionq(x, y) such thatf =ν1b−b·∇b−∇q∈L²(;R^d). This is trivial whenb^∗ ≡0, which we assume throughout the rest of this section. (However, with slight modifications to our proofs equivalent to those in Heywood [22], our results can include the case when f has finite Dirichlet integral – for example, when6is a disk of radiusr0, a natural choice forb^∗would be a Hagan-Poiseuille flow,b^∗(x) =∂_yp˜·(|x|²−r²₀)e/4νfor some prescribed pressure gradient∂_yp.)˜

We linearly decompose our solutions into θ = ˜θ + ˆθ. For initial data (θⁱⁿ,uⁱⁿ) satisfying the boundary conditions (1.2), our initial boundary value problem now takes the form:

∂_tψˆ+u·∇ψ=1ψ−ψf(θ), in×R+, (2.5a)

∂_tθˆ+u·∇θ=`1θ+ψf(θ), in×R+, (2.5b)

∂_tu+u·∇u=ν1u−∇p+σθˆe, in×R+, (2.5c)

∇·u= 0, in×R+, (2.5d)

(∂_nˆθˆ,u) = 0, on∂6×R×R+, (2.5e) ( ˆθ,u)→0, as|y| → ∞, (2.5f) θˆ(x, y,0) = ˆθⁱⁿ(x, y) =θⁱⁿ−θ˜, in, (2.5g) u(x, y,0) =uⁱⁿ(x, y). (2.5h) The term ˜θeis the gradient of a scalar function sopis now the modified pressure.

Definition 2.1. For given initial data ( ˆθⁱⁿ,uⁱⁿ)∈L^q()×H, for allq∈[1,∞), which satisfy ˆψⁱⁿ∈[−1,1] and ˆθⁱⁿ ∈[−1,∞) for a.e. (x, y)∈, a global weak solution of (2.5) indicates measurable functions

θˆ ∈L^∞_loc([0,∞);L^q)∩L²_loc([0,∞);H¹(;R²))∩C([0,∞);L²), (2.6a) u∈L^∞_loc([0,∞);H)∩L²_loc([0,∞);V)∩C([0,∞); w-H), (2.6b) for everyq∈[1,∞) whend= 2 and everyq∈[1,6] whend= 3, such that ˆψ∈[−1,1]

and ˆθ≥ −1, a.e. in, for everyt∈[0,∞), and which for all (ϕ,v)∈C^∞([0,∞);E× D) and [t₀, t₁]⊂[0,∞) satisfy

Z _t1

t0

hψ∂ˆ _tϕ+ψ1ϕ+ψu·∇ϕ−ψf(θ)ϕidt=hψ(tˆ ₁)ϕ(t₁)i − hψ(tˆ ₀)ϕ(t₀)i, (2.7a) Z _t1

t0

hθ∂ˆ _tϕ+`θ1ϕ+θu·∇ϕ+ψf(θ)ϕidt=hθ(tˆ ₁)ϕ(t₁)i − hθ(tˆ ₀)ϕ(t₀)i, (2.7b) Z _t1

t0

hu·∂_tv+νu·1v+u⊗u:∇v+σθvˆ eidt=hu(t₁)v(t₁)i − hu(t₀)v(t₀)i, (2.7c) and

( ˆθ(0),u(0)) = ( ˆθⁱⁿ,uⁱⁿ). (2.8) Remark 2.1. Since for these weak solutions, ( ˆθ,u)∈C([0,∞);L²×w-H), the initial condition (2.8) is satisfied in this sense.

Theorem 2.1. Supposeψⁱⁿ ∈ [0,1],θⁱⁿ ∈ [0,∞) for a.e. (x, y) ∈ . If ( ˆθⁱⁿ,uⁱⁿ) ∈ L^q×Hfor everyq∈ [1,∞), then there exists a global weak solution to (2.5) for all m∈Nwhend= 2 and form= 1,2 or 3 whend= 3. Moreover, ast→ ∞there is a positive constantcsuch that

kθˆ(t)k_L^q,ku(t)k_H=O(e^ct) (2.9) for allq∈[1,∞) ifd= 2 andq∈[1,6] ifd= 3.

Whend= 2, if ( ˆθⁱⁿ,uⁱⁿ) ∈Hⁿ(;R²)×(V∩Hⁿ(;R²)) for everyn∈N, then there exists a unique global classical solution to (2.5).

2.2. Existence of Weak Solutions (d= 2,3). We shall prove this result through a series of lemmas. We provide a-priori estimates for an associated system – the Leray regularised form of (2.5), with renormalised chemical nonlinearities (θδ= ˜θ+ ˆθδ)

∂_tψˆ_δ+uδ·∇ψ_δ =1ψ_δ−ψ_δf(θ_δ)e^−δθδ, in×R+, (2.10a)

∂_tθˆ_δ+uδ·∇θ_δ =`1θ_δ+ψ_δf(θ_δ)e^−δθδ, in×R+, (2.10b)

∂_tuδ+wδ·∇uδ =ν1uδ−∇p_δ+σθˆ_δe, in×R+, (2.10c)

∇·uδ = 0, in×R+, (2.10d)

(∂nˆθˆδ,uδ) = 0, on∂6×R×R+, (2.10e) ( ˆθδ,uδ)→0, as|y| → ∞, (2.10f) θˆδ(0) = ˆθⁱⁿ_δ =J_δ∗θˆⁱⁿ, in, (2.10g) uδ(0) =uⁱⁿδ =J_δ∗uⁱⁿ, in. (2.10h) Forδ > 0, letJ_δ ∈ C₀^∞(R^d) be a Friedrichs mollifier [1] with support onB_δ(x, y), the ball of radiusδ, centered at (x, y). We definewδ(ξ) = (J_δ ∗uδ)(ξ) =R

R^dJ_δ(ξ− η) ¯uδ(η) dη, i.e. the mollification ofuδ, where ¯uδ is the zero extension ofuδ outside

. We recall the usual mollifiers properties: ifv∈L^q(;R^d),q∈[1,∞), thenkJ_δ ∗ vkL^q ≤ kvkL^q and lim_δ→0+kJ_δ∗v−vkL^q = 0.

For everyδ >0, we know that a unique classical solution exists to (2.10) for the given initial data, at least on some interval [0, T_δ],T_δ >0. We remark that since the polynomial functionf(·) is Lipschitz continuous and the nonlinear reaction terms are also bounded for this approximate system, then such an existence result on a finite domain follows classically via a Faedo-Galerkin method, projecting initially onto the firstN eigen-functions of the appropriate elliptic operators on a smooth bounded boundary say of vertical diameterN (see for example Heywood [22]). Norm estimates can be shown to be independent of the domain size considered and the result follows by considering the limitN → ∞. Our a-priori Lebesgue and Sobolev norm estimates on ( ˆθδ,uδ) will verify that such a solution exists on [0,∞), i.e. the interval of existence is independent of δ. We will then eventually consider the limitδ→0⁺. The following preliminary lemma is an extension of the usual parabolic maximum principles (see Smoller [52] and Marion [36]).

Lemma 2.1. Ifψⁱⁿ∈[0,1],θⁱⁿ∈[0,∞) a.e. in, thenψ_δ ∈[0,1],θ_δ≥0, everywhere in×[0, T_δ].

Proof. Consider the inner product of (2.10a) withψ⁻:= max{0,−ψ_δ}inL²(), kψ⁻(t)k²_L²+ 2

Z _t

0

k∇ψ⁻k²_L²(;R^d)ds≤ kψ⁻(0)k²_L².

Analogous estimates follow forθ⁻:= max{0,−θ_δ}andψ⁺:= max{0, ψ_δ−1}. We now establish the main estimates we require to prove the existence of weak solutions ford= 2,3. The phrase “uniformly bounded” is considered to be with respect to our regularising parameterδ >0. We shall usecandc(·) to denote a generic finite positive constant which might depend on the argument indicated, but which does not depend on the regularisation parameter.

Lemma 2.2. If ( ˆθⁱⁿ,uⁱⁿ) ∈ L² ×H, then the set {θˆδ,uδ} is uniformly bounded in L^∞_loc([0,∞);L²×H)∩L²_loc([0,∞);H¹(;R²)×V), and in fact

kθˆδ(t)k_L², kuδ(t)kH, Z _t

0

kθˆδ(s)k_H¹(;R²)ds=O(e^ct).

Proof. Motivated by the work of Masuda [37], Haraux and Youkana [20], Collet and Xin [12] and Bricmont, Kupiainen and Xin [7] on reaction-diffusion systems, we consider a simple nonlinear functional that allows us to take advantage of the interaction of the chemical nonlinearities. As in Collet and Xin [12], forF ∈ C²(R²; [0,∞)), we use (2.5a)–(2.5b) to derive

d

dthF( ˆψ_δ,θˆ_δ)i+hQ(∇θˆδ)i

=h∇·(F1∇ψˆ_δ+`F2∇θˆ_δ−uδF)i

+hF₁1ψ˜+`F₂1θ˜i − hF₁uδ·∇ψ˜+F₂uδ·∇θ˜i

− h(F1−F2)ψ_δf(θ_δ)e^−δθδi, (2.11) where

Q(∇θˆδ) =F11|∇ψˆ_δ|²+ (1 +`)F12∇ψˆ<sub>δ</sub>·∇θˆ<sub>δ</sub>+`F22|∇θˆ_δ|². (2.12) Here F₁ andF₂ are the partial derivatives of F with respect to its first and second arguments. We would like to choose anF( ˆψ_δ,θˆ_δ) which satisfies:

F11F22 >(1 +`)<sup>2</sup>F<sub>12</sub><sup>2</sup>/4`, for all ˆψ_δ∈[−1,1], θˆ_δ ∈[−1,∞),

(2.13a) F₁−F₂>0, for all ˆψ_δ∈[0,1], θˆ_δ∈[0,∞). (2.13b) We impose (2.13a) to ensure the quadratic form (2.12) is non-negative and condi- tion (2.13b) to partially help control the nonlinear terms.

Step 1. We show that the following inequality holds for the mean-square reactant con- centration and temperature:

d

dthF( ˆψ_δ,θˆ_δ)i+c(`) k∇ψˆ_δk²_L²(;R^d)+k∇θˆ_δk²_L²(;R^d)

≤c kθˆδk²_L2+kuδk²_H+`²+ 1

. (2.14) We assumeF to be the quadratic formF( ˆψ_δ,θˆ_δ) :=αψˆ²_δ+βψˆ_δθˆ_δ+γθˆ²_δ. Note that for arbitrary (α, β, γ)∈R³+ and (ε₁, ε₂)∈R²+we can estimate

F( ˆθδ)≥

α−βε₁ 2

ψˆ_δ²+

γ− β 2ε1

θˆ²_δ,

Q(∇θˆδ)≥

2α−(1 +`)βε₂ 2

|∇ψˆ_δ|²+

2γ`−(1 +`)β 2ε2

|∇θˆ_δ|².

Condition (2.13b) is satisfied provided we choose

2α−β=k1>0 and β−2γ=k2>0. (2.15)

For a givenγ >0, we can always chooseβlarge enough so thatk₂>0 and thenε₁and ε₂large enough so thatγ > β/2ε₁and 2γ` > (1 +`)β/2ε₂. Finally we can chooseα sufficiently large (but finite) such thatk1 >0,α > βε1/2 and 2α >(1 +`)βε2/2 and thus we guarantee (2.15) and hence (2.13b) as well as (2.13a). In (2.11) we bound the linear convection and diffusion terms using the H¨older and Young inequalities:

|hF11ψ˜+`F21θ˜i|+|hF1uδ·∇ψ˜+F2uδ·∇θ˜i|

≤c Z

6×2|u_δ·e|²ρ(y) dxdy+ Z

6×2|θˆδ|²ρ(y) dxdy+`²+ 1

, where2 = supp{∂yψ˜} = supp{∂yθ˜},ρ(y) = max{|∂_yψ˜|,|∂_yθ˜|,|∂_y²ψ˜|,|∂_y²θ˜|}. Using (2.13b) we estimate the nonlinear term

h k₁ψˆ_δ+k₂θˆ_δ

ψ_δf(θ_δ)e^−δθδi ≥ − Z

K

|−k₁|ψˆ_δ|+k₂θˆ_δ|ψ_δf(θ_δ) dxdy

− Z

−

|k1ψˆ_δ−k2|θˆ_δ||ψ_δf(θ_δ) dxdy−c

≥ −c kθˆδk²_L²+ 1

, (2.16)

whereK = {(x, y)∈: ˆψ_δ ∈[−1,0], θˆ_δ ∈[0, K = 2k1/k2]}and−= {(x, y)∈

: ˆψ_δ ∈[0,1],θˆ_δ ∈[−1,0]}and we have made use of the fact that ˜ψ·θ˜^mhas compact support. Combining these last two inequalities yields (2.14).

Step 2. We establish the following energy inequality:

d

dtkuδk²_H+ 2νkuδk²_V≤ kuδk²_H+σ²kθˆ_δk²_L². (2.17) This inequality follows by considering the inner product of (2.10c) withuδinH,

1 2

d

dtkuδk²_H+νkuδk²_V=σ(uδ,θˆ_δe)_H, and then using the H¨older and Young inequalities to estimate

σ(uδ,θˆ_δe)_H≤σku_δk_Hkθˆ_δk_L² ≤1

2ku_δk²_H+σ² 2 kθˆ_δk²_L2.

Combine the inequalities in Steps 1 and 2 and apply the Gr¨onwall inequality, noting that by the usual mollifier properties kuⁱⁿ_δk_H ≤ kuⁱⁿk_H andkθˆⁱⁿ_δk_L² ≤ kθˆⁱⁿk_L², to establish the norm growth estimates.

We are now in a position to estimate theL^q()-norms of the mass-fraction and temper- ature fields and in particularL¹estimates which indicate the growth rates of the reacting fronts.

Lemma 2.3. If, in addition to the assumptions of Lemma 2.2, we assume ˆθⁱⁿ_δ ∈L^q, for allq∈[1,∞), then{θˆδ}is uniformly bounded inL^∞_loc([0,∞);L^q) for everyq∈[1,∞) whend = 2 andq ∈ [2,6] whend = 3, and:kθˆ(t)kL^q = O(e^ct). We can extend this estimate to include q ∈ [1,2) whend = 3, provided we additionally assumem ∈ {1, . . . ,6}.

Proof. Step 1. We establish that if ˆθⁱⁿ_δ ∈ L^q for someq ∈ [2,∞) when d = 2 or someq ∈[2,6] whend = 3, then ˆθδ lies inL^∞_loc([0,∞);L^q), which follows from the inequality: for anyn∈N,

kθˆδ(t)k_L²ⁿ≤ kθˆδ(0)k_L²ⁿ+ct+c Z _t

0

kθˆδ(s)k¹_L^/n2 +kuδ(s)k_L²ⁿ(;R^d)ds. (2.18) Forn ∈ N, letF be the polynomialF( ˆψ_δ,θˆ_δ) := P_2n

k=0α_kψˆ²_δ^n−kθˆ^k_δ, whereα_k, k = 0,1, . . . ,2nare positive constants. We can choose theα_k so thatF satisfies the conditions (2.13) and also thatF ≥cψˆ²_δⁿ+cθˆ²_δⁿ, for all ˆψ_δ ∈[−1,1], θˆ_δ ∈[−1,∞), to ensurehFiis a positive definite functional – we provide a proof in the appendix (analogous to the quadratic case). Using (2.11), since there existsA∈Rsuch that for all ˆθ_δ> A,F₁−F₂ >0, then with_A={(x, y)∈: ˆψ_δ ∈[−1,1], θˆ_δ∈[−1, A]}:

d dthFi ≤c

Z

A

|θˆδ|²ⁿ⁻¹

ψ_δf(θ_δ)dxdy+c ku_δk_L²ⁿ(;R^d)+ 1 X

i=1,2

kF_ik

L²ⁿ⁻¹²ⁿ ()

≤c Z

A

|θˆδ|²ⁿ⁻¹ψ˜·θ˜^mdxdy+c(A) Z

A

|θˆδ|²dxdy _2n¹

hFi¹⁻²ⁿ¹ +c kuδk_L²ⁿ(;R^d)+ 1

hFi¹⁻²ⁿ¹.

We have used thatk∇θk˜ L^∞ ≤c,k∇θk˜ _L²ⁿ(;R^d+2) ≤candk1θk˜ _L²ⁿ ≤c. Now using that ˜ψ·θ˜^mhas compact support the last inequality becomes

d

dt hFi²ⁿ¹

≤c(A) Z

A

|θˆδ|²dxdy ¹

2n

+c ku_δk_L²ⁿ(;R^d)+ 1 ,

which yields (2.18) after integration in time. If we use the Gagliardo–Nirenberg inequal- ity (2.1) to estimate theL²ⁿ-norm on velocity field in (2.18) we must restrict ourselves ton= 1,2 or 3 whend= 3. IntermediateL^q-estimates follow from (2.2).

Step 2. If ˆθⁱⁿ_δ ∈L¹and providedψ_δf(θ_δ)∈L¹(), then{θˆδ}is uniformly bounded in L^∞_loc([0,∞);L^q()) for everyq∈[1,2). In particular, by Step 1, the latter assumption here is true for allm∈Nwhend= 2, but whend= 3, we are restricted to reactions for whichm∈ {1, . . . ,6}.

As in Constantin and Fefferman [13], considerF(ξ) = R_ξ

0(ξ−η)φ(η)dη ∈ C²(R), where:φ(ξ) ≥ 0, for allξ ∈ R; φ → 0 asξ → 0;φ(ξ) = 0, for allξ > ρ₀ and R_ρ0

0 φ(ξ) dξ = 1. HenceF⁰(ξ) = R_ξ

0 φ(η) dη ∈ [0,1] andF⁰⁰(ξ) = φ(ξ)≥0. We form the estimate (9:={(x, y)∈:|ψˆ_δ|> ρ0})

d dt

Z

9

F(|ψˆ_δ|)dxdy+ Z

9

F⁰⁰(|ψˆ_δ|)|∇|ψˆ_δ||²dxdy

≤ Z

|ψ_δf(θ_δ)|+|uδ·∇ψ˜|+|1ψ˜|dxdy.

Taking the limit asρ₀→0, we get d

dtkψˆ_δk_L¹()≤ kψ_δf(θ_δ)k_L¹()+ckuδkH+c.

Now use thatkψ_δf(θ_δ)kL¹() ≤c 1 +kψˆ_δkL¹()+kθˆ_δkL¹()+P_m

k=2kθˆ_δk^k_Lk()

, where, by step 1, the last term on the right-hand side here is uniformly bounded for everym∈N whend = 2, but only form ∈ {1, . . . ,6}whend = 3. We can derive an analogous estimate forkθˆ_δk_L¹(). Adding the two estimates together, using Lemma 2.2 and the interpolation inequality (2.2), the result follows.

Consider the following weak form of our regularised system (2.10), for all (ϕ,v) ∈ C^∞([0,∞);E×D) and [t₀, t₁]⊂[0,∞),

R_t1

t0hψˆ_δ∂_tϕ+ψ_δ1ϕ+ψ_δuδ·∇ϕ−ψ_δf(θ_δ)e^−δθδϕidt=hψˆ_δ(t)ϕ(t)i|^t_t¹₀, (2.19a) R_t1

t0hθˆ_δ∂_tϕ+`θ_δ1ϕ+θ_δuδ·∇ϕ+ψ_δf(θ_δ)e^−δθδϕidt=hθˆ_δ(t)ϕ(t)i|^t_t¹₀, (2.19b) R_t1

t0hu_δ·∂_tv+νuδ·1v+wδ⊗uδ: ∇v+σθˆ_δveidt=hu_δ(t)v(t)i|^t_t¹₀. (2.19c) From Lemmas 2.2 and 2.3, there exists a subsequence{θˆδ,uδ}(which we label by the same subscript) such that for allq∈[1,∞) whend= 2 andq∈[2,6] whend= 3, as δ→0⁺,

θˆδ →θˆ in w^∗-L^∞_loc([0,∞); w-L^q)∩w-L²_loc([0,∞); w-H¹(;R²)),

uδ →uin w^∗-L^∞_loc([0,∞); w-H)∩w-L²_loc([0,∞); w-V). (2.20) Convergence of the linear terms in (2.19) to the appropriate terms in (2.7) follows from (2.20). By standard results we can deduce from (2.19c) that{∂_tuδ}is uniformly bounded in L⁴_loc^/d([0,∞);V⁰). The Aubin-Lions theorem [14, 54] for Bochner spaces (based on the Rellich-Kondrachov compact imbedding) implies

{φ∈L²_loc([0,∞);H¹()) :∂_tφ∈L⁴_loc^/d([0,∞);H¹()⁰)}

,→,→L²_loc([0,∞);L²_loc()), (2.21) where the target space is equipped with the strong topology. Hence{uδ}is relatively compact inL²_loc([0,∞);L²(B_R)) for anyR ∈ (0,∞). Souδ → ustrongly in both L²_loc([0,∞);L²(supp{v})) andL²_loc([0,∞);L²(supp{ϕ})), which together with ˆθδ →θˆ weakly inL²_loc([0,∞);L²(supp{ϕ})), is sufficient to establish the convergence of the nonlinear convective terms in our weak formulation since

Z _t1

t0

h∇ϕ·(uδθˆδ−uθˆ)idt≤ Z _t1

t0

kuδ−ukHkθˆδk_L²k∇ϕkL^∞(;R^d)dt +

Z _t1

t0

hu·∇ϕ( ˆθδ−θˆ)idt . By applying Green’s theorem we can establish that

k`1θˆ_δ+uδ·∇θˆ_δk_H¹()⁰≤ sup

kϕk_H1=1

n

`|hϕ1θˆ_δi|+|hϕuδ·∇θˆ_δi|o

= sup

kϕk_H1=1

n

`|h∇ϕ·∇θˆ_δi|+|h∇ϕ·uδθˆ_δi|o

≤`k∇θˆ_δk_L²()+kuδk_L⁴(;R^d)kθˆ_δk_L⁴().

Using the conclusions of Lemma 2.3 and recalling that ˜ψ·θ˜has compact support, we know that{ψ_δf(θ_δ)e^−δθδ}is uniformly bounded inL^∞_loc([0,∞);L²()) provided we assumem= 1,2 or 3 whend= 3 (no assumption whend= 2). Hence∂_tθˆ_δ =`1θ_δ− uδ·∇θ_δ+ψ_δf(θ_δ)e^−δθδ ∈L²_loc([0,∞);H¹()⁰) ford= 2,3. Analogous estimates hold for∂_tψˆ_δ. By employing the compact embedding (2.21) we can establish that ˆθδ →θˆ strongly inL²_loc([0,∞);L²(supp{ϕ})). We can now guarantee the convergence of the nonlinear reaction terms in (2.19) to the appropriate terms in (2.7) by, for example, using the identity

Z _t1

t0

D

ϕ ψ_δf(θ_δ)e^−δθδ−ψf(θ)E dt

≡ Z _t1

t0

D ϕ

(ψ_δ−ψ)f(θ_δ)e^−δθδ+ψ f(θ_δ)e^−δθδ−f(θ)E dt, and noting that strong convergence of ˆψ_δ →ψˆinL²_loc([0,∞);L²(supp{ϕ})) and weak convergence off(θ_δ)→ f(θ) inL²_loc([0,∞);L²(supp{ϕ})) is sufficient to ensure the right-hand side converges to zero.

By choosingv ∈Dindependent of time, thatu ∈ C([0,∞); w-H) follows from (2.7) and thatDis dense inH. Further, from the embedding

{φ∈L²_loc([0,∞);H¹()) :∂_tφ∈L²_loc([0,∞);H¹()⁰)},→C([0,∞);L²()), we deduce that ˆθ ∈C([0,∞);L²) after taking the limitδ→ 0⁺. This last embedding also implies that whend= 2, we haveu∈C([0,∞);H).

2.3. Stronger Solutions (d= 2). We prove uniqueness for slightly more regular solutions and then show that classical solutions exist provided we assume our initial data is smooth.

Lemma 2.4. Whend = 2, for initial data ˆθⁱⁿ ∈ H¹(;R²) anduⁱⁿ ∈ H, weak solu- tions also satisfy ˆθ ∈L^∞_loc([0,∞);H¹(;R²))∩L²_loc([0,∞);W²) and this additional regularity is sufficient to establish uniqueness.

Proof. Consider the inner product of (2.10b) with−1θˆ_δ inL²(), d

dtk∇θˆ_δk²_L2+ 2`k1θˆ<sub>δ</sub>k<sup>2</sup><sub>L</sub>2 ≤ ck1θˆ<sub>δ</sub>k<sub>L</sub><sup>2</sup> k∇θˆ<sub>δ</sub>k<sub>L</sub><sup>4</sup>kuδk<sub>L</sub><sup>4</sup>+kψ<sub>δ</sub>f(θ<sub>δ</sub>)e<sup>−δθδ</sup>k<sub>L</sub><sup>2</sup> +kuk<sub>H</sub>+k∇θˆ<sub>δ</sub>k<sub>L</sub><sup>2</sup>+`+ 1

. Using the Gagliardo–Nirenberg and Young inequalities, for arbitraryσ >0,

k1θˆ_δk_L²k∇θˆ_δk_L⁴kuδk_L⁴ ≤σk1θˆ_δk²_L²+ c

σk∇θˆ_δk²_L²kuδk_L^4−d4⁸ .

Also note from the Gagliardo–Nirenberg inequality and Lemma 2.2 thatkuδk_L^4−d84 lies inL¹_loc([0,∞)) when d = 2. Hence noting regularity already established, forming an analogous estimate fork∇ψ_δ(t)k_L², integrating with respect to time and considering the limitδ→0⁺, the regularity result follows.

We know [54] that forϕ∈C([0,∞);L²()),_d^d_tkϕk²_L2= 2(ϕ, ∂_tϕ)_L2, in the scalar distribution sense on compact subsets of [0,∞). Let (θ1,u1) and (θ2,u2) be two weak solutions to our system (2.7) with the same initial data ( ˆθⁱⁿ,uⁱⁿ)∈H¹(;R²)×H. Set