Towards Global Existence and Optimal Equilibriation Rates ... · Towards Global Existence and...

Towards Global Existence andOptimal Equilibriation Rates for

Reaction-Diffusion Models

Klemens Fellner

Institute of Mathematics and Scientific Computing,

University of Graz

with J. A. Canizo, L. Desvillettes, E. Latos, S. Rosenberger, Bao Q. Tang

Banff 02.07.2014 – p. 1/25

OverviewTowards bright sun and white snow for RD models

Single Reversible Reaction: Quadratic RD system

Entropy-Dissipation structure

Global existence (classical vs. weak)

Convergence to equilibrium (optimal?)

Networks of Reversible Reactions

Motivation

Entropy-Dissipation structure?

Quasi-Steady-State-Approximation

Volume-Surface RD models

Banff 02.07.2014 – p. 2/25

Reaction-Diffusion SystemsPrototypical quadratic system: A1 +A2 ↔ A3 +A4

One reversible reaction of four species Ai

A1 +A2 ↔ A3 +A4

mass action law kinetics: reaction rates ∼ a1a2 − a3a4

bounded, smooth domain Ω ⊂ RN , |Ω| = 1

homogeneous Neumann boundary conditions

Banff 02.07.2014 – p. 3/25

Reaction-Diffusion SystemsPrototypical quadratic system: A1 +A2 ↔ A3 +A4

Concentrations ai(t, x) of Ai, different diffusivities di > 0

∂ta1 − d1xa1 = −a1 a2 + a3 a4

∂ta2 − d2xa2 = −a1 a2 + a3 a4

∂ta3 − d3xa3 = +a1 a2 − a3 a4

∂ta4 − d4xa4 = +a1 a2 − a3 a4

quadratic non-linearities, Bootstrap?

no comparison principle, no invariant regions,Turing instability?

NO, there is an entropy functional!

Banff 02.07.2014 – p. 3/25

Entropy and Entropy DissipationPrototypical quadratic model: A1 +A2 ↔ A3 +A4

Kinetic (free energy) entropy

H(ai) =

∫

Ω

4∑

i=1

(ai ln(ai)− ai) dx

Entropy dissipation ddt

H = −D ≤ 0

D(ai) = 44

∑

i=1

∫

Ω

di|∇√

ai|2 dx +

∫

Ω

(a1 a2 − a3 a4) lna1 a2

a3 a4

dx ≥ 0

Programme

Global (classical, weak, renormalised) solutions

Large-time behaviour: explicit exponential convergence

Banff 02.07.2014 – p. 4/25

Entropy and Entropy DissipationPrototypical quadratic model: A1 +A2 ↔ A3 +A4

Equilibrium state ai∞i=1..4 is the unique vector of positiveconstants balancing the reaction rate

a1∞ a3∞ = a2∞ a4∞,

and satisfying the three (linear indep.) mass-conservationlaws (homogeneous Neumann boundary conditions)

a1∞ + a2∞ =1

|Ω|

∫

Ω

(a10 + a20) dx,

a1∞ + a4∞ =1

|Ω|

∫

Ω

(a10 + a40) dx,

a2∞ + a3∞ =1

|Ω|

∫

Ω

(a20 + a30) dx.

Banff 02.07.2014 – p. 5/25

The Entropy MethodQuantitative large-time behaviour

E(f) non-increasing convex entropy functional

D(f) entropy dissipation, f∞ entropy minimising equilibrium

d

dtE(f) =

d

dt(E(f)− E(f∞)) = −D(f) ≤ 0

provided conservation laws: D(f) = 0⇐⇒ f = f∞

D ≥ Φ(E(f)− E(f∞)), Φ(0) = 0, Φ ≥ 0

⇒ explicit convergence in entropy, exponential if Φ′(0) > 0

⇒ convergence in L1 : ‖f − f∞‖21 ≤ C(E(f)− E(f∞))

Cziszár-Kullback-Pinsker inequalities of convex entropies

Banff 02.07.2014 – p. 6/25

The Entropy MethodMy Personal Entropy Method Dictionary

Understanding Entropy-Dissipation Structure ⇐⇒Entropy Entropy-Dissipation (EDD) Estimate

D ≥ Φ(E(f)− E(f∞)), Φ(0) = 0, Φ ≥ 0

Really Understanding ED Structure ⇐⇒ OptimalRate/Constant in EED Estimate

Gradient flow structure?

Banff 02.07.2014 – p. 7/25

Entropy Entropy-Dissipation EstimateD ≥ C (E − E∞)

Theorem:a For any functions ai, i = 1, 2, 3, 4 measurable,non-negative, satisfying the conservation laws holds

D(ai) ≥ C(Mij)(E(ai|ai,∞)).

Proof: Additivity E(ai|ai,∞) = E(ai|ai) + E(ai|ai,∞)

E(ai|ai) =4

∑

i=1

∫

Ω

ai ln

(

ai

ai

)

dx ≤ L(Ω)4

∑

i=1

∫

Ω

|∇x

√ai|2 dx ,

+ Long long calculations + Conservation laws!

⇒ obtain Functional Inequality Not sharp!a [L. Desvillettes, K.F.]

Banff 02.07.2014 – p. 8/25

Entropy Entropy-Dissipation EstimateD ≥ C (E − E∞)

Entopry Entropy-Dissipation Estimate

D(ai) ≥ C(Mij)(E(ai|ai,∞))

+ Gronwall argument + Cziszár-Kullback inequality⇒ explicit exponential convergence to equilibrium in L1.

As long a solutions exists?

Banff 02.07.2014 – p. 9/25

Entropy A-priori EstimatesEntropy decay

The entropy decays

H(T ) = H(0)−∫ T

0

D(s) ds

with the entropy dissipation

D(ai) = 44

∑

i=1

∫

Ω

di|∇√

ai|2 dx +

∫

Ω

(a1 a2 − a3 a4) lna1 a2

a3 a4

dx

Banff 02.07.2014 – p. 10/25


H(T ) :

ai ∈ L∞([0,+∞);L log L(Ω)) ∀i = 1, .., 4∫ T

0

D(s) : √

ai ∈ L2([0,+∞);H1(Ω)) ∀i = 1, .., 4 : di > 0

in 1D: ‖ai‖3−εL3−ε([0,T ]×[0,1]) ≤ C(1 + T ) + parabolic bootstrap

⇒ ‖ai‖L∞([0,T ]×[0,1]) ≤ C(

1 + T212

)

⇒ global classical solutions

in 2D: a2i ≤ ai e

sai‖√ai(t)‖2H1(Ω) +

ai‖√

ai(t)‖2H1(Ω)

sln(a2

i ) for ln(ai) > 1

Trudinger ineq. ⇒ global L2 bound: ‖ai‖2L2([0,T ]×Ω) ≤ C(1 + T )

⇒ global weak (super-)solutions [M. Pierre 2003]a

a[Desvillettes F. Pierre Vovelle]

Banff 02.07.2014 – p. 10/25


H(T ) :

ai ∈ L∞([0,+∞);L log L(Ω)) ∀i = 1, .., 4

∫ T

0

D(s) :

√ai ∈ L2([0,+∞);H1(Ω)) ∀i = 1, .., 4 : di > 0

∫ T

0

∫

Ω(a1 a2 − a3 a4) ln

(

a1 a2

a3 a4

)

dxdt ≤ C

in 3+D: ‖ai‖1+2/N

L1+2/N ([0,T ]×Ω)≤ C(1 + T )

⇒ renormalised solutions (in the sense of [DiPerna, Lions]) a

a[Desvillettes F. Pierre Vovelle]

Banff 02.07.2014 – p. 10/25

Duality Argument for Entropy DensityEntropy density equation: A1 +A2 ↔ A3 +A4

Denote zi = ai ln(ai)− ai ⇒ H(ai) =∫

Ω

∑4i=1 zi dx

∂t

(∑4

i=1 zi

)

−∆x

(∑4

i=1 di zi

)

≤ 0, t ∈ [0, T ], x ∈ Ω,

n · ∇xzi = 0, t ∈ [0, T ], x ∈ ∂Ω,

Banff 02.07.2014 – p. 11/25


Denote zi = ai ln(ai)− ai ⇒ H(ai) =∫

Ω

∑4i=1 zi dx

∂t

(∑4

i=1 zi

)

−∆x

(∑4

i=1 di zi

)

≤ 0, t ∈ [0, T ], x ∈ Ω,

n · ∇xzi = 0, t ∈ [0, T ], x ∈ ∂Ω,

rewrite with z :=∑4

i=1 zi and M(t, x) :=P4

i=1 di zi

zas

∂tz −∆x[M z] ≤ 0, t ∈ [0, T ], x ∈ Ω,

n · ∇x[M z] = 0, t ∈ [0, T ], x ∈ ∂Ω,

coefficient M bounded by mindi ≤M(t, x) ≤ maxdi!

Banff 02.07.2014 – p. 11/25


Simplest case: 0 < d0 ≤ mindi ≤M(t, x) ≤ maxdi <∞

∂t

(

∑ki=1 zi

)

−∆x

(

∑ki=1 di zi

)

≤ 0, t ∈ [0, T ], x ∈ Ω,

n · ∇xzi = 0, t ∈ [0, T ], x ∈ ∂Ω,

uniform L2(log L)2 bound + quadratic non-linearities

⇒ uniform integrability of non-linearities

⇒ convergence in L1(QT ) of approximating sequence

⇒ global L2-weak solutions in all space dimensions!

Banff 02.07.2014 – p. 11/25

Duality Argument for Entropy DensityGeneral systems ∂tai −∇x ·(di∇xai) = fi(a)

Quadratic Lotka-Volterra systems in RN : For z ∈ (0,∞)q

∂tai = di∆xai + ai

∑qj=1 pij(aj − zj), i = 1 . . . q

∇xai ·n = 0 on ∂Ω, ai(0, ·) = ai,0(·) ∈ L2(Ω),

Then, there exists a global weak solution in L2(Ω) in RN .

Banff 02.07.2014 – p. 11/25

Improved Duality ArgumentAn improved duality argument

Let Ω ⊂ RN be a bounded domain with ∂Ω ∈ C2+α. Let T > 0,

∂tu−∆x(M(t, x)u) = 0 on ΩT ,

u(0, x) = u0(x) ∈ Lp(Ω) for x ∈ Ω,

∇xu · ν(x) = 0 on [0, T ]× ∂Ω,

with 0 < a ≤M(t, x) ≤ b < +∞ for (t, x) ∈ ΩT .

Then, any weak solution u satisfies (1/p + 1/p′ = 1)

‖u‖Lp(ΩT ) ≤ (1 + bDa,b,p′) T 1/p ‖u0‖Lp(Ω), p ∈ (2,+∞),

where Da,b,p′ :=C a+b

2 ,p′

1−C a+b2 ,p′

b−a2

as long as C a+b2

,p′b−a2

< 1. a

a[Canizo Desvillettes F., CPDE]Banff 02.07.2014 – p. 12/25

SummaryResults A1 +A2 ↔ A3 +A4

1D: global classical solutions based on entropy structure≈ alternative to theory of [Amann],... (more general)explicit exponential decay (rates) in all Sobolev norms.

2D: global classical solutions [Goudon, Vasseur] [CañizoDesvillettes F.]explicit exponential decay (rates) in L2.

allD: global weak L2-solutionsexplicit exponential decay (rates) in Lp, 1 ≤ p < 2.Blow-up example [M. Pierre D. Schmidt]

Banff 02.07.2014 – p. 13/25

Towards optimal constants/rates?Nonlinear mass action law system αA ↔ βB

∂ta− daa =

∂tb− dbb =

−α α

β −β

·

l aα

k bβ

Linearisation around equilibrium u = a− a∞ and v = b− b∞

⇒ Rescaled linearised system

∂tu− dau =

∂tv − dbv =

−α2 αβ

αβ −β2

·

u

v

Fourier expansion u =∑∞

k=0 uk(t)ϕk, v =∑∞

k=0 vk(t)ϕk

∆ϕk = λkϕk in Ω

n · ∇ϕk = 0 on ∂Ωk = 0, 1, . . .

Banff 02.07.2014 – p. 14/25


For any eigenmode k ∈ N, we have

∂t

(

uk

vk

)

=

daλk − α2 αβ

αβ dbλk − β2

·(

uk

vk

)

, k = 0, 1, . . .

a pair of eigenvalues µi, i = 1, 2 and eigenvectors ei, i = 1, 2:

µ1(0) = 0, µ2(0) = −(α2 + β2) < 0,

µ1(k) = da+db

2λk − α2+β2

2+

√

(λk(db−da)+α2−β2)2

4+ α2β2 < 0,

µ2(k) = da+db

2λk − α2+β2

2−

√

(λk(db−da)+α2−β2)2

4+ α2β2 < 0,

We have µ2(k) < µ1(k) and µ1(k + 1) < µ1(k) for all k ∈ N.Banff 02.07.2014 – p. 14/25


Two dominant negative eigenvalues

µ2(0) = −(α2 + β2) < 0,

µ1(1) =da + db

2λk−

α2 + β2

2+

√

(λk(db − da) + α2 − β2)2

4+ α2β2,

Special case: d1 = d2 = d ⇒ µ1(1) = dλk.

Optimal rate of convergence depends on

|µ1(1)| > |µ2(0)| ⇔ π2 >α2

d2+

β2

d1

Geometry + stoichiometric coefficients + diffusion rates

Banff 02.07.2014 – p. 14/25


Optimal constant in Entropy Entropy-Dissipation Estimatea

I = minβu+αv=0,u,v∈C∞

da

∫

Ω|∇u|2 + db

∫

Ω|∇v|2dx +

∫

Ω(αu− βv)2dx

∫

Ωu2 + v2

= minc0=0

∑∞k=0 |µ1(k)|(ck)2 + |µ2(k)|(dk)2

∑∞k=0(c

k)2 + (dk)2

≥ min|µ1(1)|, |µ2(0)|

Minimising functions are ∼ ϕ0 (“large diffusion”) and ∼ ϕ1

(“small diffusion”)!?

Conjecture: nonlinear constants/minimisers are the same!

a[G. Pissante, E. Latos, K. F.]

Banff 02.07.2014 – p. 14/25

OverviewTowards bright sun and white snow for RD models

Single Reversible Reaction: Quadratic RD system

Entropy-Dissipation Structure

Global existence (classical vs. weak)

Convergence to equilibrium (optimal?)

Networks of Reversible Reactions

Motivation

Entropy-Dissipation Structure? (beyond detailedbalance)

Quasi-Steady-State-Approximation

Volume-Surface RD models

Banff 02.07.2014 – p. 15/25

MotivationProtein-localisation before asymmetric stem-cell division

Asymmetric stem-cell division:

Cell-diversity by localisation of cell-fate determinants into oneside of the cell cortex and into one of two daughter cells.a

aGFP-Pon in SOP precursor cells in living Drosophila larvae [Meyer, Emery,

Berdnik, Wirtz-Peitz, Knoblich, Current Biology, 2005]

Banff 02.07.2014 – p. 16/25

MotivationProtein-localisation before asymmetric stem-cell division

Mathematical model:

“high” concentrations, insignificant stochastic effects

system of (reversible) reaction-diffusion equations

volume(cytoplasm)-surface(membran) dynamics

Banff 02.07.2014 – p. 16/25

Mixed Volume-Surface Reaction-Diffusion SystemsModel Assumptions and Quantities

Key protein Lgl in cell cortex and cytoplasm.Kinase aPKC phosphorylates Lgl on a part Γ2 of cell cortex.

L(t, x): cytoplasmic concentration of Lgll(t, x): cortical concentration of Lglp(t, x): cortical phosphorylated LglP (t, x): cytoplasmic phosphorylated Lgl

L(t, x)α←−−−β

P (t, x)

λ

y

γ ξ|Γ2

x

l(t, x)σ|Γ2−−−→ p(t, x)

Banff 02.07.2014 – p. 17/25

Mixed Volume-Surface Reaction-Diffusion SystemsA prototypical model I

Volume equations with diffusion coefficients dL, dP > 0

(V)

Lt − dL∆L = αP − βL, x ∈ Ω, t > 0,

Pt − dP ∆P = −αP + βL, x ∈ Ω, t > 0,

L(0, x) = L0(x), P (0, x) = P0(x), x ∈ Ω

Boundary conditions on ∂Ω = Γ =Γ1 ∪ Γ2 and Γ1 ∩ Γ2 = ∅

(BC)

dL∂L∂ν

= γl − λL, x ∈ Γ, t > 0,

dP∂P∂ν

= 0, x ∈ Γ1, t > 0,

dP∂P∂ν

= ξp, x ∈ Γ2, t > 0,

Reaction rates α, β, γ, λ, σ, ξ are positive constantsBanff 02.07.2014 – p. 17/25

Mixed Volume-Surface Reaction-Diffusion SystemsA prototypical model II

Boundary dynamics

(BD)

lt − dl∆Γl = λL− γl − σχΓ2l, x ∈ Γ, t > 0

pt − dp∆Γ2p = σl − ξp, x ∈ Γ2, t > 0,

dp∂p

∂νΓ2= 0, x ∈ ∂Γ2,

l(0, x) = l0(x), x ∈ Γ,

p(0, x) = p0(x), x ∈ Γ2,

∆ is the usual Laplacian in the domain Ω

∆Γ and ∆Γ2 are Laplace-Beltrami operator on Γ and Γ2

χΓ2 is the characteristic function of Γ2

Banff 02.07.2014 – p. 17/25

Mixed Volume-Surface Reaction-Diffusion SystemsLocal well-posedness

Conservation law:

d

dt

[

∫

Ω

(L(t, x) + P (t, x)) +

∫

Γ

l(t, x) +

∫

Γ2

p(t, x)]

= 0

For a T > 0 there exists a unique weak solution L,P, l, p on(0, T ), which remains nonnegative if the intital data are so.

Proof: [Bao Quoc Tang, S. Rosenberger, K. F.]

fix-point argument between (V),(BC) and (BD).

local weak solutions of (V),(BC) and (BD)

non-negativity of solutions of (V),(BC) and (BD)

uniqueness

Banff 02.07.2014 – p. 17/25

Mixed Volume-Surface Reaction-Diffusion SystemsGlobal existence

The unique solution (L,P, l, p) exists globally.

Proof: H(t) = 12

(

‖L(t)‖2Ω + ‖P (t)‖2Ω + ‖l(t)‖2Γ + ‖p(t)‖2Γ2

)

dH

dt≤ ηH

where η = max

dL + α+β2

, dP + α+β2

, CP (λ+γ)2

4dL+ σ

2, CP ξ2

4dP+ σ

2

Convergence to equilibrium for all initial data andparameter?

Problem: Decaying entropy functional?

Banff 02.07.2014 – p. 17/25

Simple 1D modelEquilibrium: Asymmetric Protein Localisation

1D Cell x ∈ [0, 1], aPKC is located at x = 0.

⇒ Explicit solution (β = 0): l(∞,1)l(∞,0)

= 1 + a(1 + bh(g))

with a = σγ, b = λ

dL, g2 = α

dP, h(x) = (ex−1)2

x(e2x−1),

Remark: ξ does not enter stationary state!Banff 02.07.2014 – p. 18/25

Systems of Reaction-Diffusion EquationsConvergence of Network of Reversible Linear Reactions

Deviation around the stationary state

X := L− L∞, Y := P − P∞, x := l − l∞, y := p− p∞

Network of four linear reversible reactions: general structure

Xα←−−−β

Y

λ

y

γ ξ

x

µ

xσ−−−→ν

y

Conservation law∫

[X + Y + x + y] = 0, ∀t ≥ 0.

Banff 02.07.2014 – p. 19/25

Systems of Reaction-Diffusion EquationsDetailed balance vs. complex kinetics

Then, the detailed balance condition αλσξβµνγ

= 1 is sufficient(and necessary?) for a convex entropy of the form:

d

dt

∫

[Φ1(X) + Φ2(Y ) + Φ3(x) + Φ4(y)]≤ 0.

Incompatibility of diffusion and reaction eigen-structure.

RD-systems allow potentially for oscillations ([BZ]).

From numerics: Decaying entropy after transient phase?

Banff 02.07.2014 – p. 19/25

Systems of Reaction-Diffusion EquationsNetwork of Reversible Linear Reactions

Closed system of reactions between n substances:

Ui Uj i, j = 1, 2, . . . , naij

aji

Concentrations U = [u1, u2, . . . , un]T , D = diag(d1, d2, . . . , dn)

Ut = DU + RU,

∂U∂ν

= 0, U(0) = [u1,0, u2,0, . . . , un,0]T ≥ 0.

aii = −∑nj=1,j 6=i aij, R =

a11 a12 . . . a1n

a21 a22 . . . a2n

. . . . . . . . . . . .

an1 an2 . . . ann

Banff 02.07.2014 – p. 20/25


Convergence to the unique equilibrium?

∑nj=1 ajiui,∞ = 0, i = 1, 2, . . . , n,

|Ω|∑ni=1 ui,∞ = M > 0

Exists a quadratic entropy E[U ](t) = 12

∑ni=1 αi

∫

Ω|ui(t, x)|2dx,

for a multiplier B = diag(α1, α2, . . . , αn) with αi > 0 to get

D[U ](t) = −dE

dt=

n∑

i=1

αidi‖∇ui‖2 − UT BRU ≥ 0 ?

Banff 02.07.2014 – p. 20/25


Algebra Theorem: Assume that R = (aij)ni,j=1 such that

(i) aij ≥ 0, i, j = 1, 2, . . . , n,

(ii)∑n

i=1 aji = 0 for all i = 1, 2, . . . , n,

(iii)∑n

j=1 a2ji ×

∑nk=1 a2

ik 6= 0 for all i = 1, 2, . . . , n.

Then, the matrix BR, where B = diag(1/ρ11, 1/ρ22, . . . , 1/ρnn),is negative semi-definite, ρij are some co-factors.

BR has n eigenvalues 0 = λ1 > λ2 ≥ λ3 ≥ . . . λn.

Interpretation: B transforms to relative entropy!

Banff 02.07.2014 – p. 20/25


Friedmann-Neumann-Rannacher modela

u1 u3 u4 u5

u6u7u2u0

rimp2 rdelay rdelay

rdelay

rdelayrdelay

ractpJAK

rexp

rimp

Yes, if assume that reactions occur only within one domain

aE. Friedmann, R. Neumann, R. Rannacher, Well-posedness of a linear spatio-

temporal model of the JAK2/STAT5 signaling pathway, Comm. Math. Anal. 15

(2013) 76-102.

Banff 02.07.2014 – p. 20/25

Quasi-steady-state approximationQSSA as ξ → +∞

Fast expulsion limit ξ → +∞:

L P

ℓ p

αβ

γλ

σ

ξ

ξ→+∞−−−−→

L P

l

α

β

σγλ

Convergence towards reduced QSSAa

Proof: ideas from duality method and entropy dissipation.

a[T.Q.Bao, K.F., S. Rosenberger]

Banff 02.07.2014 – p. 21/25

Nonlinear Boundary ModelToy model system

Volume-concentrations u(x, t), Surface-concentration v(x, t)

Nonlinear Robin-type boundary condition and matchingreversible reaction source term on Γ = ∂Ω

(NBV)

ut − δu∆u = 0, x ∈ Ω, t ≥ 0,

δu∂u∂ν

= −α(kuuα − kvv

β), x ∈ Γ, t ≥ 0,

vt − δv∆Γv = β(kuuα − kvv

β), x ∈ Γ, t ≥ 0,

u(0, x) = u0(x) ≥ 0, x ∈ Ω,

v(0, x) = v0(x) ≥ 0, x ∈ Γ

Stoichiometric coefficients α, β ∈ [1,+∞)

Reaction rates ku(t, x), kv(t, x)

Banff 02.07.2014 – p. 22/25

Nonlinear Boundary ModelModel properties

Mass conservation law

M = β

∫

Ω

u(t, x) dx + α

∫

Γ

v(t, x) dS, ∀t ≥ 0.

The unique equilibrium (u∞, v∞) balances the reaction,

uα∞ = vβ

∞,

and satisfies the mass conservation

β|Ω|u∞ + α|Γ|v∞ = M.

Uniqueness from monotonicity of uα∞ =

(

Mα|Γ| −

β|Ωα|Γ| |u∞)

)β

Banff 02.07.2014 – p. 22/25

Exponential Convergence to EquilibriumEntropy and entropy dissipation

Entropy functional:

E(u, v) =

∫

Ω

u(log u− 1)dx +

∫

Γ

v(log v − 1)dS

Entropy dissipation:

D(u, v) = − d

dtE(u, v)

= δu

∫

Ω

|∇u|2u

dx + δv

∫

Γ

|∇Γv|2v

dS

+

∫

Γ

(vβ − uα) logvβ

uαdS ≥ 0

Two cases: non-degenerate δv > 0 and degenerate δv = 0.Banff 02.07.2014 – p. 23/25

Exponential Convergence to EquilibriumExplicit exponential convergence to equilibrium

Theorem:

Assume Ω ⊂ Rn with smooth boundary Γ = ∂Ω.

Assume initial data (u0, v0) ∈ L∞(Ω)× L∞(Γ).

Then, the global bounded solution (u, v) satisfies the followingexponential convergence to equilibrium

‖u(t)− u∞‖2L1(Ω) + ‖v(t)− v∞‖2L1(Γ) ≤C1e

−C0t (E(u0, v0)− E(u∞, v∞)) ,

where the constants C0 > 0, C1 > 0 depending on α, β, Ω andthe initial mass M = β

∫

Ωu0dx + α

∫

Γv0dS.

Banff 02.07.2014 – p. 23/25

Exponential Convergence to EquilibriumEntropy entropy dissipation estimate

For all functions u : Ω→ R+ and v : Ω→ R+ satisfying massconservation

β

∫

Ω

u dx + α

∫

Γ

v dS = M,

there exists C0 > 0 such that

D(u, v) ≥ C0 (E(u, v)− E(u∞, v∞))

Proof for two cases:non-degenerate boundary diffusion δv > 0

degenerate boundary diffusion δv = 0: need L∞-bounds

Banff 02.07.2014 – p. 23/25

Conclusions and Open ProblemsTowards bright sun and white snow for RD models

Good existence theory in 2D for quadratic nonlinearities

Higher dimensions?

Algebraic Structure of Networks

How to combine entropy and duality method?

THANK YOU VERY MUCH!!

Banff 02.07.2014 – p. 24/25

ReferencesJ. A. Cañizo, L. Desvillettes, K. F., Improved duality estimates and

applications to reaction-diffusion equations, to appear in CPDE

L. Desvillettes, K. F., M. Pierre, J. Vovelle, About Global Existence

for Quadratic Systems of Reaction-Diffusion, J. Advanced

Nonlinear Studies, 7 no. 3 (2007) pp. 491–511.

L. Desvillettes, K. F., Entropy Methods for Reaction-Diffusion

Equations: Slowly Growing A-priori Bounds, Revista

Matemática Iberoamericana, 24 no. 2 (2008) pp. 407–431.

T. Goudon, A. Vasseur. Regularity analysis for systems of

reaction- diffusion equations, Ann. Sci. Ec. Norm. Super., (4)

43 no. 1 (2010) pp. 117–141.

M. Pierre, Weak solutions and supersolutions in L1 for

reaction-diffusion systems, J. Evol. Equ., 3, no. 1 (2003)

153–168.Banff 02.07.2014 – p. 25/25

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