Global existence for fully nonlinearreaction-diffusion systems describing

multicomponent reactive flow

Martine MARION, Ecole Centrale de Lyon

Dynamics and Differential Equations, IMA, June 2016

Dedicated to the memory of George Sell

1 - The governing equations

2 - The diffusive fluxes : mathematical study

3 - The fundamental energy estimate

4 - Existence of weak solutions

Joint work with R. Temam

1 - The governing equations

• Unknowns

v ,p : velocity and pressure

θ : (reduced) temperature

Y = (Y1, . . . ,YN) : mass fractions of the N species.

• Equations for hydrodynamics and combustion

∂v∂t

+ (v ·∇)v − Pr ∆v + ∇p = edσθ

div v = 0

∂θ

∂t+ (v ·∇)θ −∆θ = −

N∑i=1

hiωi(θ,Y1, . . . ,YN)

∂Yi

∂t+ (v ·∇)Yi + ∇ · Fi = ωi(θ,Y1, . . . ,YN), 1 ≤ i ≤ N

1 - The governing equations

• Chemical rates : Arrhenius law

R reversible reactions :∑N

i=1 νji Si ↔

∑Ni=1 λ

jiSi , 1 ≤ j ≤ R

Production rate of species i : ωi =∑R

j=1(λji − ν

ji )rj

Reaction rate of reaction j : rj = K dj∏N

i=1( YiMi

)νji − K r

j∏N

i=1( YiMi

)λji

• Mathematical properties

N∑i=1

ωi(θ,Y ) = 0 for θ ≥ 0, 0 ≤ Yk ≤ 1

ωi(θ,Y ) = αi(θ,Y )− Yiβi(θ,Y ), αi ≥ 0, βi ≥ 0, boundedN∑

i=1

hiωi(0,Y ) ≤ 0

1 - The governing equations

Diffusion in gaseous mixtures

∂Yi∂t + (v ·∇)Yi + ∇ · Fi = ωi(θ,Y1, . . . ,YN), 1 ≤ i ≤ N

Fi = YiVi Vi = diffusion velocity of species i = v i − v

•∑N

i=1 YiVi = 0

•∇Xi =∑N

j=1,j 6=iκDij

XiXj(Vj − Vi) i = 1, . . . ,N

Stefan-Maxwell equations

Xi = mole fraction

Dij = Dji > 0 binary diffusion coefficient for species i and j

If Dij = D for all i, j :

∇Xi = − κD Fi i = 1, . . . ,N Fick’s Law

1 - The governing equations

• Mathematical studies

Local in time solutions or simplified models :

Giovangigli-Massot (1998)Bothe (2011), Boudin-Grec-Salvarani (2012), Juengel-Stelzer (2013),Herberg-Meyries-Prüss-Wilke (2014)

Chen-Juengel (2015)

2 - The diffusive fluxes : mathematical study

• Diffusion equations

∂Yi∂t + (v ·∇)Yi + ∇ · Fi = ωi(θ,Y1, . . . ,YN), 1 ≤ i ≤ N

Fi = YiVi where

∑Ni=1 YiVi = 0∑Nj=1,j 6=i

κDij

XiXj(Vj − Vi) = ∇Xi i = 1, . . . ,N

• Determination of the diffusion velocities when Yi > 0, ∀i

Y − X relations : Xi = YiMi

1YM

where YM =∑N

j=1YjMj

(N + 1) equations :

∑Ni=1 YiVi = 0

B(Y )V = −Y 2M∇X

Bij(Y ) =

−d′ijYiYj for j 6= i,

N∑k=1;k 6=i

d′ikYiYk for j = i,d′ij = κ

Dij Mi Mj

2 - The diffusive fluxes : mathematical study ∑Ni=1 YiVi = 0

B(Y )V = −Y 2M∇X

• The matrix B(Y ) is symmetric semi-definite positive

• For an appropriate value of γ > 0 the matrix

Cij(Y ) = Bij(Y ) + γYiYj , 1 ≤ i, j ≤ N

is symmetric positive definite∑Ni,j=1 Cij(Y )Vj · Vi ≥ γ

(∑Nj=1 Yj

)(∑Ni=1 Yi |Vi |2

)• The N + 1 equations are equivalent to

C(Y )V = −Y 2M∇X

2 - The diffusive fluxes : mathematical study

• Determination of the fluxes when Yi ≥ 0 ∀i,∑N

i=1 Yi > 0Fi = YiVi if Yi > 0Fi = 0 if Yi = 0

Fi = −∑N

j=1 aij(Y1, . . . ,YN)∇Yj , aij = rational functions of Y1, . . . ,YN

• Diffusion equations

∂Yi∂t + (v ·∇)Yi + ∇ · Fi = ωi(θ,Y1, . . . ,YN), 1 ≤ i ≤ N

Boundary conditions :

x ∈ Ω = (0, `)× (0,h) or (0, `)× (0, L)× (0,h)

Yi = Y ui > 0 at xd = 0, ν · Fi = 0 otherwise

Initial conditions : Yi(x , 0) = Yi,0(x) ≥ 0

3 - The fundamental energy estimate

∂Yi∂t + (v ·∇)Yi + ∇ · Fi = ωi(θ,Y1, . . . ,YN), 1 ≤ i ≤ N

We suppose Yi > 0 ∀i and∑N

i=1 Yi = 1. Then µi = 1Mi

log Xi is defined.

µi − µui = ∂g

∂Yi, g(Y ) =

∑Nj=1

YjMj

[log Xj − log Xu

j

]entropy of the mixture

g is convex

• ddt

∫Ω

g(Y ) +∫

Γhg(Y )−

∑Ni=1

∫Ω

F i ·∇µi =∑N

i=1

∫Ωωi(θ,Y )(µi − µu

i )

• Thanks to ωi(θ,Y ) = αi(θ,Y )− Yiβi(θ,Y )

ωi(θ,Y )(µi − µui ) ≤ c

• −∑N

i=1

∫Ω

F i ·∇µi ≥ c1∫

Ω|∇Y |2

• ddt

∫Ω

g(Y ) + c1∫

Ω|∇Y |2 ≤ c

3 - The fundamental energy estimate

• −∑N

i=1

∫Ω

F i ·∇µi ≥ c1∫

Ω|∇Y |2

Fi = YiVi , C(Y )V = −Y 2M∇X , Yi

Mi Xi= YM

−∑N

i=1 Fi ·∇µi = −∑N

i=1 YiVi · ∇XiMi Xi

= 1YM

∑Ni,j=1 Cij(Y )Vj · Vi

−∑N

i=1 Fi ·∇µi ≥ γ(∑N

i=1 Yi |Vi |2)

≥ c|∇X |2

≥ c1|∇Y |2

4 - Existence of solutions

Introduction of normal unknowns

Kawashima-Shizuta (1988), Giovangigli-Massot (1996),Degond-Genieys-Juengel (1997)

Ei : ∂Yi∂t + (v ·∇)Yi + ∇ · Fi = ωi(θ,Y ) ; E1 + ....+ EN = 0

∑Ni=1 Ei

log XiMi

=∑N−1

i=1 Ei ( log XiMi−

log(1−∑N−1

j=1 Xj )

MN)

ξi = log XiMi−

log(1−∑N−1

j=1 Xj )

MN, i = 1, ...,N − 1 (∗)

• For ξ ∈ RN−1 the relations (∗) determine uniquely X = (X1, . . . ,XN)

with 0 < Xi < 1 ∀i and∑N

i=1 Xi = 1

• Y ∈ RN can be expressed as C∞ function of ξ ∈ RN−1 with 0 < Yi < 1

and∑N

i=1 Yi = 1 ; we set : Y = Y(ξ)

4 - Existence of solutions

• Equations for the unknown ξ = (ξ1, ξ2, ..., ξN−1) :

∂Yi∂t + (v ·∇)Yi + ∇ · Fi = ωi(θ,Y ), 1 ≤ i ≤ N − 1

• Time discretization with time step k > 0 :

1k (Y m

i − Y m−1i ) + (vm ·∇)Y m

i +∇ · Fmi −k∆ξm

i = ωi(θm,Y m), 1 ≤ i ≤ N − 1

The ξm are defined recursively and Y m = Y(ξm). Also

vm(x) = 1k

∫ mk(m−1)k v(x , t)dt , θm(x) = 1

k

∫ mk(m−1)k θ(x , t)dt

• Boundary conditions :

ξmi = ξu

i at xd = 0

4 - Existence of solutions

1k (Y m

i −Y m−1i ) + (vm ·∇)Y m

i +∇ · Fmi − k∆ξm

i = ωi(θm,Y m), 1 ≤ i ≤ N− 1

ξmi = ξu

i at xd = 0

• Existence of solution by the Galerkin method

A priori estimates = discrete analogs of the fundamental energyestimate∫

Ωg(Y m)−

∫Ω

g(Y m−1) + c1k∫

Ω|∇Y m|2 + k2 ∫

Ω|∇ξm|2 ≤ ck

• Passage to the limit k → 0 : add equation for Y mN = 1−

∑N−1i=1 Y m

i

1k (Y m

i − Y m−1i ) + (vm ·∇)Y m

i + ∇ · Fmi −k∆ξm

i = ωi(θm,Y m), 1 ≤ i ≤ N

ξNi = −

∑N−1i=1 ξi

4 - The full system

Add equations for the velocity, pressure and temperature :

1k (Y m

i −Y m−1i ) + (vm ·∇)Y m

i +∇ · Fmi − k∆ξm

i = ωi(θm,Y m), 1 ≤ i ≤ N− 1

1k (vm − vm−1) + (vm ·∇)vm − Pr ∆vm + ∇pm = edσθ

m, divvm = 0,

1k (θm − θm−1) + (vm ·∇)θm −∆θm = −

∑Ni=1 hiωi(θ

m+,Y m)

Existence of weak solutions for the equations coupled with thehydrodynamics and temperature equations