Chemically Reacting Fluid Flows: Strong Solutions and Global Attractors

Journal of Dynamics and Differential Equations, Vol. 12, No. 2, 2000

Chemically Reacting Fluid Flows: Strong Solutionsand Global Attractors

David Norman1, 2

Received May 26, 1998

In this paper we consider the existence and properties of strong solutions for amodel of incompressible chemically reacting flows where reactants enter thedomain, react, and then leave the domain. We show results which exactlyparallel those of the Navier�Stokes equations, i.e., in two dimensions strongsolutions exist for all time, and in three dimensions we show existence only forsmall times. In two dimensions, we also show the existence of global attractorswhich are compact in L2. Rather than considering a specific set of boundaryconditions, we instead state our results based on a series of assumptions, whichwould be proved using the boundary conditions. This allows our results to beapplied directly to the two sets of boundary conditions which appear in theliterature.

KEY WORDS: chemically reacting flow; combustion; Arrhenius kinetics;incompressible Navier�Stokes equations; reaction diffusion equations; Boussinesqmodels; strong solution.AMS (MOS) Classification Numbers: Primary 80A25; Secondary 35Q99,35B40, 58F39.

1. INTRODUCTION

In this paper we study the following model for chemically reacting fluidflows in nondimensional form

�t u&Pr 2u+(u } {) u+{p=f0(T ) (1.1a)

{ } u=0 (1.1b)

273

1040-7294�00�0400-0273�18.00�0 � 2000 Plenum Publishing Corporation

1 Department of Mathematics, 127 Vincent Hall, University of Minnesota, Minneapolis,Minnesota 55455.

2 To whom correspondence should be addressed at 655 Medical Drive, Bountiful, Utah 84010.E-mail: norman�math.umn.edu.

�t T&2T+(u } {) T=& :N

i=1

hi Wi (Y1 ,..., YN , T ) (1.1c)

�t Yi&1

Le2Yi+(u } {) Yi =W i (Y1 ,..., YN , T ) (1.1d)

where Pr is the Prandl number, Le is the Lewis number, the terms with(u } {) are fluid transport terms, f0(T ) is the forcing term from buoyancy,Wi=(Y1 ,..., YN , T ) describes the change in mass fractions due to the reac-tion, and hi is the enthalpy of species i divided by its molecular weight, i.e.,a measure of the amount of heat contained in species i. The first two equa-tions are the usual Navier�Stokes equations, and the second two are reac-tion�diffusion equations with a transport term added. We assume that theequations hold in a C2 bounded domain 0/Rd, d=2 or 3. We assumethat the Wi are bounded, Lipschitz functions and that f0(T ) is the usualBoussinesq model for buoyancy

f0(T )=&c0 g� (T&tl) (1.2)

where c0>0 is a constant and g� is a unit vector pointing in the directionof gravity.

In this paper we show the existence of strong solutions to this modelwhich parallel the existence results for the Navier�Stokes equations. Inparticular, we show that in two space dimensions strong solutions exist forall time, while in three dimensions we show only that strong solutions existfor a short time. The key step in the proof is to show that if the Navier�Stokes portion of the system has a strong solution, then the chemistry andtemperature equations also have strong solutions.

Our method of proof is based on the Bubnov�Galerkin method ofprojecting the system onto a finite-dimensional subspace of the infinite-dimensional phase space, solving the resulting ODE, then taking the limitof the approximate solutions as the projection approaches the identity toget the final result. Such a proof involves two steps. First, one provescertain a priori estimates for the approximate solutions which show thatthey are bounded in the appropriate spaces, then one uses a compactnesslemma to get the existence of the limiting solution.

One advantage of this method of showing the existence of strong solu-tions is that the boundary conditions play no direct role in the analysis. Inparticular, the H1 and H 2 a priori estimates and the compactness dependonly on certain properties of the Laplacian with the given boundary condi-tions and certain L2 a priori estimates and elliptic regularity for the Laplacianwith the associated boundary conditions. This allows us to state the

274 Norman

hypotheses of our existence theorem without reference to the specificboundary conditions. Of course, in any specific case the verification of thehypotheses will depend on the boundary conditions, however, they aretypically straightforward extensions of the estimates needed for the existenceof weak solutions.

One novel feature of our presentation of the existence results is that westructure our argument around a general existence theorem, which is inturn a wrapper around a well-known compactness lemma. This method ofpresentation makes clear exactly what properties of the equations areneeded for existence, along with how the properties are used.

In the final section of the paper we show the existence of global attrac-tors for the reacting flow system in two-dimensional domains. Our proof isbased on the classical technique of showing that the system is both dis-sipative and compact. However, since showing dissipativity is typicallydone using energy estimates which depend heavily on the boundary condi-tions, we do not give a complete global attractor existence proof. Instead,our result essentially says that if the system is dissipative, then it has aglobal attractor. This is sufficient to get the existence of global attractorsfor the two sets of boundary conditions considered in the literature, sincein both cases the systems are known to be dissipative; see Norman(7) andManley et al.(5) Our global attractor existence results are basically the sameas those of Manley et al.(5) However they also give estimates of the dimen-sion of the global attractor, an issue which we do not study.

As an example of how one can use the results of this paper, we wouldlike to mention Norman, (6) where these solutions are used in comparingthe dynamics of the reacting flow model to the dynamics of a continuousflow stirred tank reactor (CSTR). A CSTR is an ODE approximation to areacting flow based on the assumption that the chemical concentrationsand temperature are spatially homogeneous. Norman (6) shows that, forlarge chemical and thermal diffusivities, the CSTR ODE is a good approxi-mation to the full reacting flow PDE and, in particular, that the globalattractor for the reacting flow converges to the global attractor for theCSTR in a suitable sense.

An outline of the paper is as follows. In Section 2 we give an exampleof the type of boundary conditions which one might consider for the reactingflow system. In Section 3 we give some notation and state some standingassumptions. In Section 4 we state the main assumptions for the theorems,then state the theorems themselves. In Section 5 we state certain preliminarylemmas which are used in the subsequent proofs. In Section 6 we state ourgeneral existence theorem and the compactness lemma on which it is based.In Section 7 we analyze the equations in order to show that the existencetheorem can be applied to them, and in Section 8 we derive the needed

275Chemically Reacting Fluid Flows

priori estimates. Then in Section 9 we give proofs of the main existencetheorems in the paper. Finally, in Section 10 we give our global attractorexistence proof, along with some background on global attractors.

2. BOUNDARY CONDITION EXAMPLE

In this section we want to give one example of the types of boundaryconditions to which the theorems in this paper could be applied. The examplewe give was originally studied by Norman.(7) Its main feature is that theamount of the various chemicals and heat entering the system are specifiedas parameters of the system, which is an advantage when comparing thereacting flow model to certain reduced models, i.e., continuously stirredtank reactors (CSTRs); see Norman.(6)

We begin by using Dirichlet boundary data for the velocity.

u(x, t)=,u(x) for all x # �0

where ,u is not identically zero. We then define the partition of the boundary

1I _ 1O _ 1W=�0

into inflow, outflow, and wall portions, respectively. If we define

q(x)={&u(x) } n0

if x # 1I

otherwise

where n(x) is the unit outward normal to �0, and note that q(x)�0. Nowwe can define the chemistry and temperature boundary conditions as

�T�n

+qT=,T

1Le

�Yi

�n+qY i=,Yi

where q(x)=0 implies ,T (x)=,Yi (x)=0. In the inflow portion of theboundary, i.e., where u(x) } n<0, these boundary conditions are of theRobin type, which has the effect of fixing the flux of the temperature andchemical species to be ,T and ,YI

, respectively; see Norman.(6) On the out-flow portion of the boundary, they are of Neumann type, which impliesthat the only flux across the boundary is due to fluid flow, with none dueto diffusion. Finally, there is no flux across the wall portion of the boundary.

276 Norman

3. NOTATION

In this section we give some notation that is used in the rest of thepaper. We also state some of the assumptions which we make about thestructure of the equations.

We begin by giving some notation. We let Bu , BT , and BY be thelinear boundary operators associated with the velocity, temperature, andchemistry equations, respectively. Then, since we allow the possibility ofinhomogeneous boundary conditions, we have that the solutions satisfy

Buu(x, t)=,u(x) for x # �0 and t�0

BTT (x, t)=,T (x) for x # �0 and t�0

BY Yi (x, t)=,Yi(x) for x # �0 and t�0

We handle the inhomogeneous boundary conditions in the usual wayby decomposing the solution into a time-dependent part which satisfieshomogeneous boundary conditions and a part constant in time whichsatisfies the inhomogeneous boundary conditions, i.e.,

u(x, t)=v(x, t)+w(x)

T (x, t)=%(x, t)+%0(x)

Yi (x, t)='i (x, t)+'i, 0(x)

where

Buw(x)=,u(x) for x # �0

BT %0(x)=,T(x) for x # �0

BY 'i, 0(x)=,Yi(x) for x # �0

hence

Buv(x)=0 for x # �0

BT %(x)=0 for x # �0

BY 'i (x)=0 for x # �0

We assume that w, %0 , and 'i, 0 are fixed throughout the paper.


We now define the spaces in which we will find the solutions v, %, and 'i .For the velocity we have

H=ClosureL2(0)3[v # C�(0)3 : { } v=0, Bu v=0]

V=ClosureH1(0)3[v # C�(0)3 : { } v=0, Buv=0]

i.e., H and V are the spaces of divergence free vector fields with the properboundary conditions in L2(0) and H 1(0), respectively. We also denote thedual of V by V &1. We then define the projection P to be the orthogonalprojection of L2(0) onto H. In the case where we have Dirichlet boundarydata, P is the usual Leray projection. In this case, using the divergencetheorem it can be shown that any gradient is orthogonal to H, hence if weapply P to (1.1a) the pressure term will disappear, leaving us with aparabolic evolutionary equation. It is critical to our analysis that thisproperty continue to hold, i.e., we assume that all gradients are orthogonalto H.

We define similar concepts for the chemistry and temperature equations.For the temperature equation we have

H1=ClosureL2(0)[% # C�(0) : BT %=0]

V1=ClosureH 1(0)[% # C�(0) : BT %=0]

and for the chemistry equations we have

H2=ClosureL2(0)[' # C�(0) : BY '=0]

We, in particular, assume that V1=D(A1�21 ) and V2=D(A1�2

2 ). We also usethe product spaces

H� =H_H1_ `N

i=1

H2

V� =V_V1_ `N

i=1

V2

We now want to define notation for some of the terms in the equations.We begin with the linear parts of the equations and define the linearoperators

A= &P 2 with boundary conditions Bu=0

A1= &2 with boundary conditions BT =0

A2= &2 with boundary conditions BY =0

278 Norman

as the unbounded linear operators associated with the velocity, tempera-ture, and chemistry equations, respectively, and where P is the projectionof L2 onto the space H. We note that now the unknowns v, %, and 'i arein the domains of their respective operators.

We also define the following bilinear and trilinear forms associatedwith the Navier�Stokes nonlinearity and chemistry and temperature trans-port terms:

B(u, v)=P(u } {) v

B1(u, �)=(u } {) �

b(u, v, w)=( (u } {) v, w)

b1(u, �1 , �2)=( (u } {) �1 , �2)

We note that if w # H, then (B(u, v), w)=b(u, v, w).Using this notation, after applying P to the velocity equation, the

reacting flow system becomes

�t v+Pr Av+B(v+w, v+w)=f (%) in 0 (3.4a)

�i%+A1%+B1(v+w, %+%0)=g('1 ,..., 'N , %) in 0 (3.4b)

�t 'i+1

LeA2' i+B1(v+w, 'i+'i, 0)=|i ('1 ,..., 'N , %) in 0 (3.4c)

Buv(x)=0 for x # �0 (3.4d)

BT %(x)=0 for x # �0 (3.4e)

BY 'i (x)=0 for x # �0 (3.4f )

where

f (%)=Pf0(%+%0)+Pr P 2w (3.5a)

g('1 ,..., 'N , %)=& :N

i=1

hiWi ('1+'1, 0 ,..., 'N+'N, 0 , %+%0)+2%0

(3.5b)

|i ('1 ,..., 'N , %)=Wi ('1+'1, 0 ,..., 'N+'N, 0 , %+%0)+1

Le2'i, 0 (3.5c)


For simplicity later in the paper, we introduce the notation

Fv(v, %, 'i )=&Pr Av&B(v+w, v+w)+ f (%)

F% (v, %, 'i )=&A1%&B1(v+w, %+%0)+ g('1 ,..., 'N , %)

F'i(v, %, 'i )=&

1Le

A2 'i&B1(v+w, ' i+'i, 0)+| i ('1 ,..., 'N , %)

where f, g, and |i are given by (3.5). Then the reacting flow system (3.4)becomes simply

�tv=Fv(v, %, 'i )

�t %=F% (v, %, ' i )

�t 'i =F'i(v, %, ' i )

As mentioned in the introduction we show the existence of solutionsusing the Bubnov�Galerkin approach of projecting the equations ontofinite-dimensional subspaces. To do this we assume that Pn

u , PnT , and Pn

Y

are orthogonal projections onto n-dimensional subspaces of H, H1 , and H2 ,respectively. We further assume that

limn � �

(I&Pnu) u=0 for all u # H

limn � �

(I&PnT ) �=0 for all � # H1

limn � �

(I&PnY ) �=0 for all � # H2

We can then define the nth-order Bubnov�Galerkin approximate solution(vn, %n, 'n

i ) as the solution of the ODE,

�t vn=Pnu Fv(vn, %n, 'n

i ) (3.6a)

�t %n=PnTF% (vn, %n, 'n

i ) (3.6b)

�t 'ni =Pn

YF'i (vn, %n, 'ni ) (3.6c)

vn(0)=Pnu v(0) (3.6d)

%n(0)=PnT %(0) (3.6e)

'ni (0)=Pn

Y ' i (0) (3.6f )

280 Norman

4. ASSUMPTIONS AND STATEMENTS OF THEOREMS

In this section we state the assumptions which we make on the system.Proving these assumptions will, in particular, involve the specific boundaryconditions of interest. We also give a brief sketch of the properties that arerequired to verify the assumptions for a specific set of boundary conditions.

In particular, if we define A, A1 , A2 as in Section 3, then we assumethe following.

Assumption 1. A, A1 , and A2 are positive, self-adjoint operators withcompact inverses satisfying

V=D(A1�2), V1=D(A1�21 ), V2=D(A1�2

2 )

Typically, the positivity and self-adjointness of the Ai are shown usingintegration by parts arguments, while the compact inverse follows fromelliptic regularity.

Assumption 2. Suppose that for all scalar functions , the projectionP satisfies

P{=0

This assumption is typically a direct consequence of the boundaryconditions for the velocity.

Assumption 3. One has the following equivalent norms betweenSobolev spaces and fractional power spaces for :=1, 2.

C 1e &v&2

H:(0)�&A:�2v&2�C 2e &v&2

H :(0) for all v # V : (4.1a)

C 1e &�&2

H:(0)�&A:�2i �&2�C 2

e &�&2H :(0) for all � # V :

i (4.1b)

For the case :=1, the equivalent norms follow from integration byparts arguments and a Poincare� inequality. For the case :=2, they followfrom elliptic regularity.

Assumption 4. All of the homogenizing terms w, %0 , and 'i, 0 are inH2(0).

This assumption depends on being able to extend the ,i smoothly intothe interior of the domain, which in turn depends on the smoothness of the,i and trace theorems.


Assumption 5. For each n the solutions of the Bubnov�Galerkinapproximate system (3.6) exist for all time.

This assumption and the following assumption would all be verifiedusing L2 energy estimates.

Assumption 6. Each Bubnov�Galerkin approximate solution (vn, %n, 'ni )

satisfies the estimate

&%n(t)&2�C1 \1+&%n(0)&2+&vn(0)&2+ :N

i=1

&'ni (0)&2+ for all t>0

(4.2)

where C1 is independent of n.Also, for our two-dimensional existence theorem and global attractor

existence theorem, we require the following estimates.

Assumption 7. Assume that for each {>0, there exists a constant C� 1 ,independent of n but possibly depending on the initial conditions and {,and a constant C2 , independent of n and { and the initial conditions, suchthat for all 0�t0�t�{ one has

&vn(t)&2�C� 1 (4.3)

|t

t0

&A1�2vn(s)&2 ds�C2max[1, t&t0]

_\&vn(t0)&2+&%n(t0)&2+ :N

i=1

&'ni (t0)&2+1+ (4.4)

We also need the following norm bounds for our uniqueness theorem.Typically, such bounds are proved using integration by parts, a Poincare�inequality, and the equivalent norms (4.1).

Assumption 8. Assume that there exists a constant C3 such that

C3 &A1�2v&2�Pr(Av, v) +b(u, v, v) (4.5a)

C3 &A1�21 %&2�(A1%, %)+b1(u, %, %) (4.5b)

C3 &A1�22 '&2�

1Le

(A2', ')+b1(u, ', ') (4.5c)

282 Norman

for all u, v, %, and ' satisfying

{ } u=0

Buu=,u

{ } v=0

v # D(A)

% # D(A1)

' # D&(A2)

This assumption would be shown using integration by parts and aPoincare� inequality.

Finally, for our global attractor existence theorem, we require thefollowing estimate.

Assumption 9. Assume that for each {>0, there exists a constant C4 ,independent of n but depending on the initial conditions, such that for all0�t0�t�{ one has

|t

t0

&A1�2%n(s)&2 ds

�C4 max[1, t&t0] \&%n(t0)&+&vn(t0)&+ :N

i=1

&'ni (t0)&+1+ (4.6)

|t

t0

&A1�2'ni (s)&2 ds

�C4 max[1, t&t0] \&%n(t0)&+&vn(t0)&+ :N

i=1

&'ni (t0)&+1+ (4.7)

This assumption would follow from an L2 energy estimate.We can now state our existence theorem.

Theorem }I Let 0/Rd, d=2, 3 be a C2 bounded domain. Assumethat f0 and W satisfy the assumptions given in the introduction and thatAssumptions 1 through 6 above are satisfied. Then for each initial condition,

v(0) # V/H1(0)

%(0) # V1/H1(0)

'i (0) # V2/H1(0)


there exists a time {>0 such that on [0, {] there exists a strong solution tothe homogenized chemically reacting flow system (3.4). Furthermore, if d=2and Assumption 7 is satisfied, then the result is true for all {>0.

We also have that the solution satisfies the following properties.

1. The components of solution satisfy

v # L2(0, T; D(A)) & L�(0, T; V ) (4.8a)

% # L2(0, T; D(A1)) & L�(0, T; V1) (4.8b)

'i # L2(0, T; D(A2)) & L�(0, T; V2) (4.8c)

2. The time derivatives of the solution satisfy

�t v # L2(0, T; H ) (4.9a)

�t% # L2(0, T; H1) (4.9b)

�t 'i # L2(0, T; H2) (4.9c)

3. The equations are satisfied in a weak integrated form in L2(0), i.e.,

(v(t)&v(t0), v� ) =|t

t0

(Fv(v, %, 'i ), v� ) ds (4.10a)

(%(t)&%(t0), %� ) =|t

t0

(F% (v, %, 'i ), %� ) ds (4.10b)

(' i (t)&'i (t0), '� ) =|t

t0

(F'i(v, %, 'i ), '� ) ds (4.10c)

for each v� # H, %� # H1 , '� # H2 and 0�t0<t�T.

We also have the following uniqueness theorems. We should point outthat the stated properties of weak solutions are typical for weak solutionsin two dimensions.

Theorem "I Suppose that Assumptions 1 through 6 and 8 above aresatisfied. Let (v1, %1, '1

i ) be a strong solution of the reacting flow system(3.4) on [0, {], and let (v2, %2, '2

i ) be a weak solution of the equations on[0, {] satisfying

v2 # L2(0, {; V ) & L�(0, {; H ) (4.11a)

%2 # L2(0, {; V1) & L�(0, {; H1) (4.11b)

'2i # L2(0, {; V2) & L�(0, {; H1) (4.11c)

284 Norman

and

�tv2 # L2(0, {; V &1) (4.12a)

�t %2 # L2(0, {; V &11 ) (4.12b)

�t '2i # L2(0, {; V &1

2 ) (4.12c)

Suppose that both solutions have the same initial condition. Then they coincideon [0, {].

We give proofs of Theorems 1 and 2 in Section 9.

Corollary /I Let 0/R2 and suppose that Assumptions 1 through 6

and 8 are satisfied. Then strong solutions to the reacting flow system areunique in the class of weak solutions.

Proof. In two dimensions, the Sobolev estimates imply that thederivative properties (4.12) in the statement of Theorem 2 are satisfied forall weak solutions, hence Theorem 2 gives the result. g

Corollary �I Let 0/R3 and suppose that Assumptions 1 through 6

and 8 are satisfied. Then strong solutions are unique.

Proof. The corollary follows from Theorem 2 and the fact that strongsolutions satisfy the properties (4.12) in the statement of Theorem 2. g

Assuming that suitable types of weak solutions exist in two dimensions,then we can show that the weak solutions immediately become strong solu-tions. This theorem becomes important in our global attractor existencetheorem, since there we want to allow for initial conditions in L2(0).

Corollary <I Let 0/R2 and suppose that the assumptions of Theorem 1

are satisfied. Suppose further that (v, %, 'i ) is a weak solution satisfying theconditions in the statement of Theorem 2. Then the weak solution is a strongsolution on any interval [t1 , t2] satisfying 0<t1<t2<�.

Proof. Fix t1>0. Then by condition (4.11) we have that there existsa { # (0, t1) such that

v({) # V, %({) # V1 , 'i ({) # V2

Therefore, we can apply Theorem 1 and Corollary 3 to get the result. g


We also have the following global attractor existence theorem.

Theorem >I Let 0/R2. Suppose that Assumptions 1 through 4 andAssumption 7 are satisfied. Suppose also that for each initial condition inL2(0) there exists a weak solution satisfying the properties in the statementof Theorem 2 and that the system is dissipative in L2(0), in particular, thatfor all t>0 the estimate

&v(t)&2+&%(t)&2+ :N

i=1

&'i (t)&2

�C5e&_t(&%(0)&2+&v(0)&2+ :N

i=1

&'i (0)&2)+C6 (4.13)

holds, where _, C5 , and C6 are independent of the initial conditions and t.Then the reacting flow system (3.4) has a global attractor which is compactin L2(0) and which attracts all sets bounded in L2(0).

We give the proof for these theorems in Section 9.We would also like to mention the issue of showing that the solutions

conserve mass and that the mass fractions remain between zero and one.In Norman(7) solutions possessing these properties were said to be physi-cally reasonable. When the solutions have enough smoothness, the proof ofphysical reasonableness follows directly from the maximum principle. Infact, with the degree of smoothness that these strong solutions posses, it ispossible to use maximum principle arguments to show physical reasonable-ness. The usual trick of multiplying the solution by

Y&(x, t)={Y(x, t)0

if Y(x, t)<0otherwise

and integrating in space is sufficient to prove the result; see Manley et al.(5)

for details.

5. PRELIMINARIES

In this section we state several preliminary lemmas which we use laterin the paper. We begin with the following lemma which will be used inproving a priori estimates.

286 Norman

Lemma �I Let V, H, V $ be three Hilbert spaces, each being includedand dense in the following and with V $ the dual of V. Suppose further that� # L2(0, {; V ) and �t� # L2(0, {; V $). Then one has

12 �t &�&2

H=(�t�, �) almost everywhere in [0, {] (5.1)

For a proof, see Sell and You(9) or Temam.(10)

We also need the following uniform Gronwall inequality.

Lemma �I Let y(t) be absolutely continuous and h(t) and g(t) belocally integrable functions, with h(t) positive. Then for 0<t one has

y(t)�\ 1t&{ |

t

{y(s) ds+|

t

{h(s) ds+ exp \|

t

{g(s) ds+ (5.2)

where {=max[0, t&1].

For a proof see Sell and You.(9)

We also need the usual Sobolev estimates for the bilinear and trilinearforms (see Sell and You(9) or Constantine and Foias(2)), namely, that thereis a constant Cs depending only on 0/Rd, d=2, 3 such that

&B(u, v)&�Cs &u&3�4H 1 &u&1�4

H 2 &v&3�4H 1 &v&1�4

H2 (5.3a)

&B1(u, �)&�Cs &u&3�4H 1 &u&1�4

H 2 &�&3�4H 1 &�&1�4

H2 (5.3b)

|b(u, v, w)|�Cs &u&H 1 &v&1�2H 1 &v&1�2

H 2 &w& (5.3c)

|b(u, v, w)|�Cs &u&H 2 &v&H 1 &w& (5.3d)

|b(u, v, w)|�Cs &u&1�2 &v&1�2H1 &v&H 2 &w& (5.3e)

|b(u, v, w)|�Cs &u&H 1 &v&1�2H 1 &v&1�2

H 2 &w& (5.3f )

|b1(u, ,, �)|�Cs &u&H 1 &,&1�2H 1 &,&1�2

H 2 &�& (5.3g)

for u, v, w vector fields and ,, � scalars which are in the appropriateSobolev spaces. We also have the estimates for 0/R2

|b(u, v, w)|�Cs &u&1�2 &u&1�2H 1 &v&1�2

H 1 &v&1�2H 2 &w& (5.4)

(5.5)


We also use the estimates

|b(u, v, w)|�Cs &u&H 1 &v&H2 &w& (5.6a)

|b1(u, ,, �)|�Cs &u&H 1 &,&H 2 &�& (5.6b)

which follow from (5.3f ) and (5.3g). Furthermore, we freely use the equiv-alent norms in (4.1) in these estimates instead of the corresponding Sobolevnorms, possibly by picking a larger Cs .

6. GENERAL EXISTENCE THEOREM

For ease of exposition of our existence proofs, we use the followinggeneral existence theorem, which is basically just a wrapper around a well-known compactness lemma. The main advantage of this approach is thatit makes clear exactly what properties of the equations we are using andhow they are used.

Our general existence theorem is set in the following situation. Foreach equation, we assume the existence of three Hilbert spaces,

X 1i

/�Xi/�X &1

i , i=1,..., N

where in each case the embedding is dense and compact. Although we donot assume that X &1

i is the dual of X 1i , we do assume that it is contained

in the dual. We also define

W� 1= `N

i=1

X 1i

W� = `N

i=1

Xi

W� &1= `N

i=1

X &1i

We further assume that we have a system of equations

�t ,i=Fi (,1 ,..., ,N ) for i=1,..., N

If we let Pni , i=1,..., N be an orthogonal projection onto an n-dimen-

sional subspace of X i1 consisting of smooth functions, then we can define

288 Norman

the n th-order Bubnov�Galerkin system for approximate solutions to thereal solution with initial condition (,i (0)) as

�t ,ni =Pn

i F i (,n1 ,..., ,n

N ) for i=1,..., N (6.1a)

,ni (0)=Pn

i ,i (0) for i=1,..., N (6.1b)

We note that (3.6) is a special case of this construction.Our existence theorem is basically a wrapper around the following

compactness lemma.

Lemma � (Compactness Lemma). Fix {>0. Assume that, for i=1,..., N we have three Hilbert spaces, X 1

i , Xi , and X &1i , as above. Let ,n

i bea sequence in L2(0, {; X 1

i ) such that ,ni is bounded in L2(0, {; X 1

i ) and thesequence �t ,n

i is bounded in L p(0, {; X &1i ), where 1<p��. Then for each

i=1,..., N, there exists a subsequence, which we also denote ,ni , and a func-

tion ,i # L2(0, {; X 1i ), with �t ,i # L p(0, {; X &1

i ), such that the followingproperties hold.

1. One has ,n w�w , weakly in L2(0, {; X 1i ).

2. One has �t,n w�w �t, weakly in L p(0, {; X &1i ).

3. One has ,n � , strongly in L2(O, {; Xi ).

4. For each t # (0, �), one has ,n(t) � ,(t) strongly in X &1i .

5. There is a set E in (0, �) having measure zero, such that fort # R+&E, one has ,n

i (t) � ,i (t) strongly in Xi .

This lemma has been used many times before; see, for example,Constantine and Foias(2) and Lions.(4) In particular, Sell (8) gives an outlineof the proof.

The key property that the right-hand sides of the equations need tosatisfy when applying the compactness lemma is the following.

Property } (Continuity Property). A function Fi (,1 ,..., ,N) is saidto satisfy the continuity property if, for each sequence , j=(, j

1 ,..., , jN ) in

L2(0, {; W� 1) and limit function ,0=(,01 ,..., ,0

N ) in L2(0, {; W� 1) satisfying theconclusions of the compactness Lemma 9 for each i=1,..., N, we have

limj � �

Fi (, ji ) � F i (,0

i ) weakly in L1(0, {; X &1i ) for 1�i�N


Theorem }� (General Existence Theorem). Fix {>0. Let X 1i , Xi ,

and X &1i be given as above and suppose that each of the Fi , i=1,..., N

satisfies the continuity Property 1 in the corresponding spaces. Further,assume that for each i=1,..., N, each of the Bubnov�Galerkin approximatesolutions ,n=(,n

1 ,..., ,nN ) exists on [0, {] and that (,n

i ; �t,ni ) is uniformly

bounded in L2(0, {; X 1i )_L p(0, {; X &1

i ) independent of n for some p>1.Then for each initial condition ,(0)=(,1(0),..., ,N(0)) there exists a

solution ,n=(,n1 ,..., ,n

N ) satisfying the following properties.

1. For each i, one has that ,i is the limit of a subsequence of the ,ni

in the sense of the compactness lemma. As a consequence one has that ifthe ,n

i satisfy a property which is preserved under the convergence of thecompactness lemma, then ,i satisfies the same property.

2. For each i=1,..., N and for every ,� # X &1i and t>t0 one has

(,i (t)&,i (t0), ,� ) X i&1=|

t

t0

(Fi (,1 ,..., ,N), ,� ) ds (6.2)

for every 0�t0<t�{.

Proof. Because of the boundedness assumption on the (,ni , �t ,n

i ), wecan apply the compactness lemma and take a subsequence N times to getlimiting functions ,i , i=1,..., N. If we continue to denote the subsequencewith n, then we have that

,ni � ,i

where the convergence is in the sense of the compactness lemma. Also,because of the continuity assumption on the Fi , we have that

Fi (,n1 ,..., ,n

N ) w�w F i (,1 ,..., ,N) weakly in L1(0, {; X &1i ) for 1�i�N

We now just need to check Property 2, i.e., that the limiting solutionssatisfy the equations. Because the Bubnov�Galerkin approximations satisfyan ODE, we have that, for 0<t<{,

,ni (t)&,n

i (t0)=|t

t0

Pni F i (,1 ,..., ,N) ds for i=1,..., N

or we can write, for each ,� # W� &1,

(,ni (t)&,n

i (t0), ,� ) =|t

t0

Pni Fi (,1 ,..., ,N) ds for i=1,..., N (6.3)

290 Norman

We want to take the limit of this equation as n � � to get (6.2). We notethat by Property 4 of the compactness lemma, the left-hand side of (6.3)converges to the left-hand side of (6.2) for every t and t0 .

For the right-hand side, we note, for each i,

|t

t0

(PnF i (,n1 ,..., ,n

N )&Fi (,1 ,..., ,N), ,� ) | ds

�|t

t0

|(PnF i (,n1 ,..., ,n

N )&F i (,n1 ,..., ,n

N ), ,� ) | ds

+|t

t0

|(F i (,n1 ,..., ,n

N )&F i (,1 ,..., ,N), ,� ) | ds

�|t

t0

|(F i (,n1 ,..., ,n

N ), (I&Pn) ,� ) | ds

+|t

t0

|(F i (,n1 ,..., ,n

N )&F i (,1 ,..., ,N), ,� ) | ds

�|t

t0

|(F i (,n1 ,..., ,n

N ), (I&Pn) ,� ) | ds

+|t

t0

|(F i (,n1 ,..., ,n

N )&F i (,1 ,..., ,N), ,� ) | ds

The first term goes to zero, since the Fi (,n1 ,..., ,n

N ) are bounded inL2(0, T; X &1

i ) and (I&Pn) ,� goes to zero in X &1 uniformly in t. Thesecond term also goes to zero because of the continuity property of Fi .Hence we have that (6.2) is true for every t0 and t, which implies thatProperty 2 is satisfied. g

7. ANALYSIS OF THE EQUATIONS

We now want to return to the reacting flow system and prove theproperties necessary to apply the general existence theorem. One of thehypotheses of the theorem is that the derivatives of the approximatesolutions are bounded in the space L p(0, {; W� &1). Since

�t ,ni =F i (,n

j )

we see that the boundedness of the derivative is really a question ofwhether Fi maps the ,n

i into bounded sets. The way one typically proves


this is to show that the ,n themselves are bounded in various spaces, thento show that F, maps such bounded sets into bounded sets. In our case,we can summarize the boundedness properties of the F, we will use in thefollowing way.

Property " (Boundedness Property). We say that the functionsFi (,1 ,..., ,N), i=1,..., N satisfy the boundedness property for a given p #(1, �] in the spaces X 1

i , Xi and X &1i if, for each i, F i maps sets which are

bounded in L2(t0 , t; W� 1) & L�(t0 , t; W� ) into sets which are uniformlybounded in L p(t0 , t; W� &1

i ). In particular, we assume that for some functionBF we have for t>t0

|t

t0

&F(,1(s),..., ,n(s)& pXi

&1 ds

�BF \(t&t0)+maxi |

t

t0

&,i (s)&2V2

ds+maxi

ess supt0�s�t

&, i (s)&+ (7.1)

for all i=1,..., N.

For strong solutions to our chemically reacting flow we use the followingspaces when we apply the existence theorem:

X 1v=D(A) X 1

v =VX &1v =H

X 1%=D(A1) X 2

%=V1X &1% =H1 (7.2)

X 1'=D(A2) X 2

'=V2X &1' =H2

We recall that this implies that for each of the equations the triple of spacesis contained in the triple of Sobolev spaces H2(0), H1(0), and L2(0).

In the following series of lemmas we prove the continuity and bounded-ness properties for the various terms in the equations, then combine them toget the properties for the right-hand sides.

Lemma }}I The maps (u, v) [ B(u, v) and (u, �) [ B1(u, �) satisfythe continuity property in the above spaces.

Proof. We will actually show strong convergence rather than justweak convergence. We begin with B(u, v). Pick ui � u0 and vi � v0 satisfyingthe conditions in the continuity property. Then one has

B(ui , vi )&B(u0 , v0)=B(ui&u0 , vi )&B(u0 , v0&vi )

292 Norman

Fixing t>t0 and using the Ho� lder inequality and (5.3a), one obtains theestimates

|t

t0

&B(ui&u0 , vi )& ds

�C7 |t

t0

&Au i&u0&1�4 &A1�2(ui&u0)&3�4 &Avi&1�4 &A1�2vi&3�4 ds

�C8 \|t

t0

&Aui&u0&2+1�8

\|t

t0

&1�2(u i&u0)&2+3�8

_\|t

t0

&Avi &2+1�8

\|t

t0

&A1�2vi &2+3�8

The second term goes to zero by Property 3 of the compactness lemma,while the other terms are bounded. The analysis of the term B(u0 , u0&ui )is identical. Hence the property is shown for B(u, v).

The same proof using (5.3b) instead of (5.3a) gives the result forB1(u, �). g

Lemma }"I The maps (u, v) [ B(u, v) and (u, �) [ B1(u, �) satisfythe boundedness property where p=2.

Proof. Fix u, v satisfying the hypotheses of the boundedness property.For all t>t0 , using the Sobolev estimate (5.3a) and the Ho� lder inequality,one obtains the estimate,

|t

t0

&B(u, v)&2 ds�C9 |t

t0

&Au&1�2 &A1�2u&3�2 &Av&1�2 &A1�2v&3�2 ds

�C9&u&3�2L�(0, �; V )&u&3�2

L�(0, �; V )

_\|t

t0

&Au& ds+1�2

\|t

t0

&Av& ds+1�2

�C10&u&3�2L�(0, �; V )&u&3�2

L�(0, �; V )

_\|t

t0

&Au&2 ds+1�2

\|t

t0

&Av&2 ds+1�2

where C10 depends on t&t0 . This final estimate is bounded by the assump-tions in the boundedness property.

The identical proof works for B1(u, �). g


Lemma }/I The maps u [ Au, � [ Ai �, for i=1, 2, satisfy thecontinuity property.

Proof. We first consider A. By the definition of the Sobolev norms,we have that A maps D(A) continuously into H. Hence Property 1 of thecompactness lemma implies the result.

The same proof works for the Ai �. g

Lemma }�I The maps u [ Au, � [ Ai �, i=1, 2 satisfy the bounded-ness property for p=2.

Proof. First we look at Au. Fix t>t0 . Then we need to consider

|t

t0

&Au&2 ds

The boundedness of this integral is one of the assumptions of the bounded-ness property, so the result is proved.

The same proof works for the Ai �. g

Lemma }<I Suppose that the Wi are Lipschitz functions. Then|k(,1 ,..., ,N) satisfies the continuity property for each k.

Proof. Pick , ji � ,i satisfying the conditions in the continuity

property for each i. We actually show that

|k(, j1 ,..., , j

N) � |k(,1 ,..., ,N)

strongly in L2(0, {; L2(0)). We note that since |k is Lipschitz in eachcoordinate, we have

||k(, j1 ,..., , j

N)&|k(,1 ,..., ,N)|�C11 :N

i=1

|, ji &, i |

for some constant C11 . This, plus Property 3 of the compactness lemma,gives the result after integrating in time and space. g

Lemma }>I The functions |k('1 ,..., 'N , %) satisfy the boundednessproperty for any p>1.

Proof. This is a direct consequence of the fact that the |k areeverywhere bounded. g

Corollary }�I The maps Fv , F% and F'iall satisfy the boundedness and

continuity properties.

294 Norman

Proof. The statement follows directly from the above lemmas and thefact that the sets of functions satisfying the two properties are closed underscalar multiplication and addition. g

8. A PRIORI ESTIMATES

In this section we prove a priori estimates for the solutions of theBubnov�Galerkin system (3.6), which we repeat here for convenience:

�t vn+Pr Avn+PnB(vn+w, vn+w)=Pn f (%n) (8.1a)

�t%n+A1%n+Qn B1(vn+w, %n+%0)=Qng('n1 ,..., 'n

N , %n) (8.1b)

�t 'ni +

1Le

A2'ni +RnB1(vn+w, 'n

i +'i, 0)=Rn|i ('n1 ,..., 'n

N , %n) (8.1c)

vn(0)=Pn v(0) (8.1d)

%n(0)=Qn%(0) (8.1e)

'ni (0)=Rn'i (0) (8.1f )

The estimates we derive will, along with the boundedness property, give thebounds needed to apply the general existence theorem. The key observationis contained in the following lemma, which says that whenever the velocityis bounded in the correct spaces, so axe the chemistry and temperature.

Lemma }�I Fix {>0 and suppose that Assumptions 1 through 4 aresatisfied. Suppose further that for the temperature and chemistry equationsone has initial conditions in H1(0), i.e.,

%(0) # V1

'i (0) # V2 for i=1,..., N

and that on the interval [0, {] one has

&vn(s)+2&2H 1(0)�C� 2 for all s # [0, {]

where C� 2 may depend on the initial conditions and { but not on n. Then onehas

&A1�21 %n(t)&2�C� 3 for 0�t�{ (8.2)

|t

t0

&A1%n&2 ds�C� 3 max[t&t0 , 1] for 0�t0�t�{ (8.3)

where C� 3 is independent of n but may depend on the initial conditions.


Proof. Throughout this proof, C12 denotes a constant independent ofn and the initial conditions but possibly depending on {, while C� 3 denotesa constant which is also independent of n but which may depend on theinitial conditions and {. We begin by taking the inner product of theequation for %, Eq. (8.1b), with A1%, to get

12 �t &A1�2

1 %n&2+&A1 %n&2+b1(vn+w, %n+%0 , A1%n)

=(g('n1 ,..., 'n

N , %n), A1%n) (8.4)

From the Sobolev estimates (5.3g) and (5.6b), the Young inequality, andthe fact that g is bounded, one obtains the following estimates:

|b1(vn+w, %n, A1%n)|�&vn+w&H 1 &A1�21 %n&1�2 &A1 %n&3�2

� 16 &A1%n&2+C12C� 4

2 &A1�21 %n&2

|b1(vn+w, %0 , A1%n)|�&vn+w&H 1 &%0&H2 &A1 %n&

� 16 &A1%n&2+C12C� 2

2

|(g('n1 ,..., 'n

N , %n), A1%n) |� 16 &A1%n&2+C12

Using these estimates in (8.4) gives

�t &A1�21 %n&2+&A1%n&2�C12+C12C� 2

2+C12C� 42 &A1�2

1 %N&2

(8.5)�t &A1�2

1 %n&2+(+1&C12C� 42) &A1�2

1 %n&2�C12+C12C� 22

where +1>0 is the smallest eigenvalue of A1 . This, by the Gronwallinequality, implies that on [0, {] there exists a C12 independent of n suchthat

&A1�21 %n(t)&2�C12 for t # [0, T]

i.e., (8.2) holds.We now turn to proving (8.3). Integrating (8.5) from t0 to t, where

0�t0<t�{, and using (8.2) gives

|t

t0

&A1%n&2 ds�C13(t&t0)

which gives (8.3).

296 Norman

Identical arguments give the results for the 'i equations. g

We are now ready to consider the velocity equation. Because of theassumption we make on &%n&2 and the definition of f (3.5a), we have that

& f (%n)&2�C� 4

where C� 4 is independent of the order of the approximation n. This impliesthat our velocity equation is just a usual Navier�Stokes equation withforcing term in L�(0, �; L2(0)). Therefore we expect the same qualitativeresults as for the usual Navier�Stokes equations. The only difference wehave from the standard literature is our inhomogeneous boundary condi-tions. However, we see that they present no problem. In fact, in the followingtwo lemmas our proofs consist of taking the inhomogeneous Navier�Stokesequations and deriving inequalities which are qualitatively identical to theinequalities one obtains in the homogeneous case. We can then appeal tothe standard literature to complete our results.

Lemma }�I Assume that Assumptions 1 through 4 are satisfied.Suppose further that 0/R3 and that the initial velocity is in H1(0), i.e.,

v(0) # V

Then there exists a time { and a constant C� 5 , depending on the initialconditions but independent of n such that

&A1�2vn(t)&2�C� 5 for all 0�t�{ (8.6)

|t

t0

&Avn(s)&2 ds�C� 5 max[t&t0 , 1] for all 0�t0�t�{ (8.7)

Proof. Throughout the proof we let C� 5 denote a constant independentof n, but possibly depending on the initial conditions. We begin by takingthe inner product of Avn with Eq. (8.1a) to get

12 &A1�2vn&2+Pr &Avn&2�( f (%n), Avn)+|b(vn+w, vn+w, Avn)| (8.8)

Next, we note that (4.2) and the definition of f implys that

& f (%n(t))&2�C1(1+&%n(0)&+&vn(0)&+ :N

i=1

&'ni (0)&) for all t>0

(8.9)


where C1 is independent of n and the initial conditions. Now from theSobolev estimates (5.3c) and (5.3d), the bound on f (%n) (8.9), and theYoung inequality, one obtains the estimates

|b(vn, vn, Avn)|�Cs &A1�2vn&3�2 &Avn&3�2

�Pr10

&Avn&2+C� 5 &A1�2vn&6

|b(w, vn, Avn)|�Cs &w&H 2 &A1�2vn& &Avn&

�Pr10

&Avn&2+C� 5 &A1�2vn&6+C� 5 &w&3H 2

|b(vn, w, Avn)|�Cs &A1�2vn& &w&1�2H 1 &w&1�2

H 2 &Avn&

�Pr10

&Avn&2+C� 5 &A1�2vn&6+C� 5 &w&3H 2

|b(w, w, vn)|�Cs &w&H2 &w&H 1 &Avn&

�Pr10

&Avn&2+C� 5 &w&4H 2

|( f (%n, Avn) |�Pr10

&Avn&2+C� 5

Substituting in these estimates into (8.8) gives

�t &A1�2vn&2+Pr &Avn&2�C� 5 &A1�2vn&6+C� 5(8.10)

�t &A1�2vn&2�C� 5 &A1�2vn&6+C� 5

Now this inequality is well known from the 3D Navier�Stokes equations(see Sell and You, (9) Temam, (10) or Constantin and Foias(2)) and impliesthat there is a time {>0 and a constant C� 5 satisfying the conclusions of thelemma. Furthermore, by integrating (8.10) from t0 to t one can show (8.7).

g

Lemma "�I Assume that Assumptions 1 through 4 and Assumption 7

are satisfied. Suppose further that 0/R2 and that the initial velocity is inH1(0), i.e.,

v(0) # V

298 Norman

Then for each {>0, there exists a constant C� 6 , depending on the initialconditions and { but independent of n such that

&A1�2vn(t)&2�C� 6 for all 0�t�{ (8.11)

|t

t0

&Avn(s)&2 ds�C� 6 max[t&t0 , 1] for all 0�t0�t�{ (8.12)

Proof. Throughout this proof we let C� 6 denote a constant independentof n, but possibly depending on the initial conditions and {, and let C14

denote a constant independent of n, the initial conditions and {. We againbegin by taking the inner product of the equation for vn with Avn andobtain

12 �t &A1�2vn&2+Pr &Avn&2�( f (%n), Avn)+|b(vn+w, vn+w, Avn)| (8.13)

Now we use the two-dimensional Sobolev estimate (5.4), the three-dimen-sional estimates (5.3e) and (5.3d), and the Young inequality to get

|b(vn, vn, Avn)|�Cs &vn&1�2 &A1�2vn& &Avn&3�2

�Pr10

&Avn&2+C� 14 &vn&2 &A1�2vn&4

|b(vn, w, Avn)|�Cs &vn&1�2 &A1�2vn&1�2 &w&H 2 &Avn&

�Pr10

&Avn&2+C� 14 &vn&2+&A1�2vn&4+C14 &v2&

|b(w, vn, Avn)|�Cs &w&H 2 &A1�2vn& &Avn&

�Pr10

&Avn&2+C� 14 &A1�2vn&2

|b(w, w, Avn)|�Cs &w&H 2 &Avn&

�Pr10

&Avn&2+C14

|( f (%n), Avn) |�Pr10

&Avn&2+C14


Substituting these estimates into (8.13) and using the assumption on &vn&(4.3) gives

�t &A1�2vn&2+Pr &Avn&2�C14 &vn&2 &A1�2vn&4+C14 &A1�2vn&2+C14 &v&+C14

(8.14)�t &A1�2vn&2�C� 6(&A1�2vn&2+1) &A1�2vn&2+C� 6

This last inequality is equivalent to the inequality one encounters whenstudying the 2D Navier�Stokes equations with homogeneous boundaryconditions. In particular, it, combined with the estimates in Assumption 7and the uniform Gronwall inequality from Lemma 8 inequality, implies thebound (8.11). The key point is that integrals of coefficient of &A1�2vn&2 onthe right-hand side are bounded in n, i.e.,

|t

t0

&vn(s)&2 &A1�2vn(s)&2+1 ds�C� 6 for 0�t0<�t�{

holds with C� 6 independent of n. In our case, this follows from the estimatesin Assumption 7. For more details, we refer the reader to the proof ofTheorem 6, where similar calculations are done.

Finally, the bound (8.12) follows from integrating (8.14) and using thebounds for &v& and &A1�2v& in (4.3) and (4.3), respectively. g

9. PROOFS OF THEOREMS

Proof of Theorem 1: Main Existence Theorem. We first note thatfrom the a priori estimates in Lemmas 18 and 19 or 20, we have that thereexists a { such that

vn is bounded in L2(0, {; D(A)) and L�(0, {; V )

%n is bounded in L2(0, {; D(A1)) and L�(0, {; V1)

'ni is bounded in L2(0, {; D(A2)) and L�(0, {; V2)

where the bounds are independent of n. Furthermore, in two dimensions,{>0 is arbitrary. The boundedness properties of the equations given inCorollary 17 then imply that

�tvn is bounded in L2(0, T; H )

�t%n is bounded in L2(0, T; H1)

�t'ni is bounded in L2(0, T; H2)

where the bounds are independent of n.

300 Norman

We now want to apply the general existence theorem, Theorem 10,where the X j

i are given in (7.2). We can do this in particular because theequivalent norms (4.1) imply the needed compact embeddings. This thengives the result.

Proof of Theorem 1: Uniqueness Theorem. We begin by defining

v� =v2&v1

%� =%2&%1

'� i ='2&'1

Then we have that

�t v� +Pr Av� +B(v2+w, v� )+B(v� , v1+w)= f (%� )

�i %� +A1%� +B1(v2+w, %� )+B1(v� , %1+%0)

=g('21 ,..., '2

N ; %2)& g('11 ,..., '1

N , %1)

�t '� i+1

LeA2'� i+B1(v2+w, '� i )+B1(v� , '1

i +'i, 0)

=|i ('21 ,..., '2

N , %2)&|i ('11 ,..., '1

N , %1)

Most importantly, we have that

�t v� # L2(0, T; V &1)

�t %� # L2(0, T; V &11 )

�t'� i # L2(0, T; V &12 )

These properties imply, by Lemma 7, that the following formal propertiesin fact hold:

12 �t &v� &2=(�tv� , v� )

12 �t &%� &2=(�t%� , %� )

12 �t &'� &2=(�t'� , '� )


From the Sobolov estimates (5.6a) and (5.6b) and the Younginequality, we get the estimates

|b(v� , v1+w, v� )|�C3

3&A1�2v� &2+C15 &v1+w&2

H2 &v� &2 (9.1a)

|b1(v� , %1+%0 , %� )|�C3

3&A1�2v� &2+C15 &%1+%0&2

H 2 &%� &2 (9.1b)

|b1(v� , '1i +'i, 0 '� i )|�

C3

3&A1�2v� &2+C15 &'1

i +'i, 0&2H 2 &'� i&2 (9.1c)

where C15 depends only on C3 and Cs . Furthermore, using (1.2), (3.5), andthe Lipschitz property of the Wi , we can estimate

|( f (%2)& f (%1), v� ) |�c0 &%� & &v� & (9.2a)

�c20 &%� &2+&v� &2 (9.2b)

|(g('2j , %2)& g('1

j , %1), %� ) |�C16 \&%� &2+ :N

j=1

&'� j &2+ (9.2c)

|(Wi ('2j , %2)&Wi ('1

j , %1), 'i) |�C16 \&%� &2+ :N

j=1

&'� j&2+ (9.2d)

where C16 is independent of t, but may depend on {.Now we can take the inner product of (3.4a), (3.4b), and (3.4c) with

v, %, and 'i , respectively, and use the equivalent norms from (4.5) to get

�t &v&+C3 &A1�2v&2�C3

3&A1�2v&2+C15 &v1+w&2

H2 &v� &2+&v� &2+c20 &%� &2

(9.3a)

�t &%&+C3 &A1�21 %&2�

C3

3&A1�2v&2+C15 &%1+%0&2

H2 &%� &2

+C16 \&%� 2+ :N

j=1

&'� j &2+ (9.3b)

�t &'i&+C3 &A1�21 %&2�

C3

3&A1�2v&2+C15 &'1

i +'i, 0&2H 2 &'i&2

+C16 \&%� 2+ :N

j=1

&'� j &2+ (9.3c)

302 Norman

Now from the equivalent norms (4.1) and (4.8), we get that &v1+w&2H 2 ,

&%1i +%0&2

H 2 , and &'1i +' i, 0&2

H 2 are integrable. Therefore, since the maxi-mum of a finite number of integrable functions is again integrable, we cansum the equations in (9.3) to get

�t \&v� &2+&%� &2+ :N

i=1

&'i&2+�h(t) \&v� &2+&%� &2+ :N

i=1

&'� i &2+where h(t) is an integrable function. Therefore, the Gronwall inequalityimplies

\&v� (t)&2+&%� (t)&2+ :N

i=1

&'� i (t)&2+�\&v� (0)&2+&%� (0)&2+ :

N

i=1

&'� i (0)&2+ exp \|t

0h(s) ds+

which implies that the solutions coincide, which is the desired result. g

10. GLOBAL ATTRACTORS

In this section we want to prove the two-dimensional global attractorexistence Theorem 6. The classical global attractor existence theorem wewant to use is based on two definitions. We say that a dynamical systemS(t) on a metric space X is dissipative if there exists a bounded set U/Xsuch that for all x # X, there exists a time {x such that S(t)x # U for allt>{x . We calf U an absorbing set. The second definition is that S(t) iscompact if for each bounded set V there exists a time t such that S(t) V iscompact. For more information on these definitions another dynamicalsystems theory; see Sell and You, (9) Hale, (3) and Temam.(11) Given thesedefinitions, we can state the following existence theorem.

Theorem "} (Billotti�LaSalle). Suppose that the dynamical systemS(t) is compact and point dissipative. Then S(t) has a global attractor.Furthermore, the attractor uniformly attracts all bounded sets.

For a proof, see, for example, Billotti and LaSalle(1) or Sell andYou.(9) This theorem was used, for instance, by Manley et al.(5) to show theexistence of global attractors to the reacting flow system (1.1) with bound-ary conditions representing a two-dimensional premixed flame in a tube.

Typically the application of this theorem in situations similar to oursinvolves two steps. First, one uses a priori L2 energy estimates to show the


existence of an absorbing set in L2(0). Second, one uses smoothing proper-ties of the equation to show compactness. The simplest way to do this isto show the existence of an absorbing set in H1(0), then use the fact thatH1(0) is compactly embedded in L2(0) to get compactness. It is thisstrategy that we will follow.

However, in this paper we do not show dissipativity for the reactingflow system. As mentioned above, dissipativity typically depends on apriori L2 energy estimates, which in turn depend intimately on the boundaryconditions. Instead, our global attractor existence Theorem 6 essentiallystates that dissipativity implies compactness, which in turn implies theexistence of global attractors. We should point out that dissipativity isknown for the two sets of boundary conditions which, we know of in theliterature; see Manley et al.(5) and Norman. (7)

The first step in proving the global attractor existence theorem is thefollowing theorem, which states that if the system is dissipative and v isuniformly bounded in H1(0), then there exists a bounded absorbing set inH1(0) for the chemistry and temperature equations.

Lemma ""I Let 0/Rd, d=2 or 3. Suppose that the reacting flowsystem (3.4) is dissipative, i.e., (4.13) is satisfied. Suppose further that foreach initial condition (v(0), %(0), 'i (0)) # H� there exists a { such that

&A1�2v(t)+w&2�C17 for t�{ (10.1)

where C17 is independent of the initial conditions. Then there exists anabsorbing set in H1(0) for the chemistry and temperature equations.

Proof. Fix a set of initial conditions (v(0), %(0), 'i (0)) # H� andassociated time {. Throughout the rest of the proof we work on the interval({+1, �).

We begin by taking the inner product of the equation for % (3.4b) withA1% and use Theorem 7 to get

12 �t &A1�2

1 %&2+&A1%&2+b1(v+w, %+%0 , A1%)=(g(%, ' j ), %) (10.2)

Now for t # [{+1, �), from the Sobolev estimates (5.3g) and (5.6b), thebound for g, and the Young inequality, one obtains the estimates

|b1(v+w, %, A1 %)|�Cs &A1�2v& &A1�21 %&1�2 &A1%&3�2

� 16 &A1 %&2+ 729

32 C 4s C 2

17 &A1�21 %&2

|b1(v+w, %0 , A1%)|� 16 &A1 %&2+ 3

2 C 2s C17 &%0&2

H2

(g, A1 %) � 16 &A1 %&2+C18

304 Norman

where C18 depends only on the bound for g. Substituting these estimatesinto (10.2) gives

�t &A1�21 %&2++1 &A1�2

1 %&2� 72916 C 4

s C 217 &A1�2

1 %&2+C19

where +1 is the smallest eigenvalue of A1 . Now we can apply the uniformGronwall inequality from Lemma 8 on the interval (t&1, t) for t>{+1 toget

&A1�21 %(t)&2�\C20 |

t

t&1&A1�2

1 %(s)&2 ds+C19+ e +1

where

C20= 72916 C 4

s C 217+1

We now want to apply the estimate (4.6) from Assumption 9. Since(4.6) is written in terms of the Bubnov�Galerkin approximation, we needto take the limit as n � � to get an estimate for the solution itself. We cando this in particular because the 2D uniqueness theorem, Corollary 3,implies that the specific solution we are working with is the limit of theBubnov�Galerkin approximations. However, this is straightforward. Inparticular, the left-hand side converges for all t and t0 because of Property 4and the right-hand side converges for all t and t0>0 by Property 3 of thecompactness lemma. This gives

|t

t0

&A1�2%(s)&2 ds�C4 max[1, t&t0]

_\&%(t0)&2+&vn(t0)&2+ :N

i=1

&'i (t0)&2+1+ (10.4)

for all t>t0>0.Now, from (10.4) and the dissipativity estimate (4.13), one obtains the

inequality

&A1�21 %(t)&2�\C20C4e&_t \&%n(0)&+&vn(0)&+ :

N

i=1

&'ni (0)&++C19+C4+ e +1

which implies the existence of an absorbing set for %.Exactly the same argument works for the 'i equation. We omit the

details. g


Proof of Global Attractor Existence Theorem 6. Throughout thisproof C21 denotes a constant independent of the initial conditions and t butpossibly depending on t0 , defined below. We begin by showing the exist-ence of an absorbing set for v in H1(0). In particular, using the samecalculations that led to (8.14) in the proof of Lemma 20, but applied to theoriginal v equation, we get

�t &A1�2v(t)&2++ Pr &A1�2v(t)&2�C14 g(t) &A1�2v(t)&2+C14 &v(t)&+C14

where

g(t)=&v(t)&2 &A1�2v(t)&2+1

and + is the smallest eigenvalue of A. Now, by the dissipativity, there existsa constant C22 and a time t0>0, both independent of the initial conditionssuch that

&v(t)&�C22 for all t�t0

Hence one has

�t &A1�2v(t)&2++ Pr &A1�2v(t)&2�C14 g(t) &A1�2v(t)&2+C21 for all t�t0

(10.5)

Now we want to derive estimates for the integral of &A1�2v(s)&2 on[t&1, t], where t0�t&1. As in the proof of the previous lemma, becauseof the compactness lemma convergence, we can apply (4.4) to get

|t

t&1&A1�2v(s)&2 ds�C2 \&v(t&1)&2+&%(t&1)&2+ :

N

i=1

&' i (t&1)&2+1+Now, again by dissipativity, we can assume, possibly by increasing t0 , that

|t

t&1&A1�2v(s)&2 ds�C21 (10.6)

This also implies that

|t

t&1g(s) ds�C21 (10.7)

306 Norman

Now we apply the uniform Gronwall inequality from Lemma 8 on[t&1, t] with t&1>t0 to (10.5) to get

&A1�2v(t)&2�C21 for t�t0+1

i.e., there exists an absorbing set for v in H1(0).Now, by the assumptions of the theorem, for each initial condition in

L2(0) there exists a unique weak solution, which immediately becomes astrong solution. This allows us to define a dynamical system S(t) based onthese solutions. Using a generalization to the arguments in the uniquenesstheorem, Theorem 2, one can show that S(t) is continuous in L2(0)_R.By the above calculations, this semiflow has an absorbing set in H1(0),which implies that S(t) is compact, since H1(0) is compactly embedded inL2(0). Since S(t) is dissipative by assumption, we can apply Theorem 21to get the result.

ACKNOWLEDGMENTS

This research was supported in part by the Army Research Office andthe IMA at the University of Minnesota.

REFERENCES

1. Billotti, J. E., and LaSalle, J. P. (1971). Dissipative periodic processes. Bull. Am. Math.Soc. 77, 1082�1088.

2. Constantin, P., and Foias, C. (1988). Navier�Stokes Equations, University of ChicagoPress, Chicago.

3. Hale, J. (1988). Asymptotic Behavior of Dissipative Systems, Vol. 25, Am. Math. Soc.,Providence, RI.

4. Lions, J. L. (1969). Quelques me� thodes de re� solution des proble� mes aux limites non line� aires,Gauthier-Villar, Paris.

5. Manley, O., Marion, M., and Teman, R. (1993). Equations of combustion in the presenceof complex chemistry. Ind. Univ. Math. J. 42, No. 3, 941�967.

6. Norman, D. E. (2000). Chemically reacting flows and continuous flow stirred tankreactions: The large diffusion limit. Dynam. Diff. Eq. 12, 309�355.

7. Norman, D. E. (1999). Chemically reacting fluid flows: Weak solutions and global attractors.Diff. Eq., in press.

8. Sell, G. R. (1996). Global attractors for the three-dimensional Navier�Stokes equations.J. Dynam. Diff. Eq. 8, No. 1, 1�33.

9. Sell, G. R., and You, Y. Dynamics of Evolutionary Equations, Springer-Verlag, New York(in press).

10. Teman, R. (1984). Navier�Stokes Equations, 3rd rev. ed., North Holland, Amsterdam.11. Teman, R. (1988). Infinite Dimensional Dynamical Systems in Mechanics and Physics,

Springer-Verlag, New York.


Chemically Reacting Fluid Flows: Strong Solutions and Global Attractors

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