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On Potential Methods for Porous Media Flows

Wolfgang L. Wendland

Institute for Applied Analysis and Numerical Simulation & SIMTECH, Univ. Stuttgart

joint work with Mirela Kohr (Babeş–Bolyai Univ., Cluj–Napoca, Romania)Massimo Lanza de Cristoforis (Univ. of Padova, Italy),

Sergey E. Mikhailov (Brunel Univ. London, United Kingdom)

Strobl 2016, 3.–10.7.

Contents1. The transmission problem2. Function spaces3. Solution properties for Brinkman and Stokes systems4. Potential operators5. The linear transmission problem6. The nonlinear Darcy–Forchheimer–Brinkman transmission problem7. The problem on manifolds8. More general spaces

Prof. Dr. Dr.h.c. Wolfgang L. Wendland Potential Methods 1/25

1. The transmission problem

∆u− αu− k|u|u− β(u · ∇)u−∇p = f and ∇ · u = 0 in Ω+;∆u−∇p = 0 and ∇ · u = 0 in Ω−;γ+u+ − γ−u− = h0 on Γ = ∂Ω connected boundary and∂+n;α(u+, p+; (f+ + E+(k|u+|u+ + β(u+ · ∇)u+)

)− µ∂−n (u−, p−; f−)

+ 12P(γ−u−+ γ+u+) = g0 on Γ;

u−(∞) = u∞ : lim|x|→∞

∫S2(x)

|u−(y(x)

)− u∞|dsy = 0, (Leray).

Ω+ ⊂ R3 bounded Lipschitz; Ω− := R3\Ω+;α, k, β ≥ 0;P(x) ∈ L∞(Γ)3×3 , P(x) = P∗(x) , ∀ ξ ∈ R3 : 〈Pξ, ξ〉R3 ≥ 0;S2(x) = y(x) : |y(x)− x| = 1 ; E(v) := 1

2(∇v + (∇v)>

);

〈∂±n;α(u±, p±; f±),ϕ〉L2(Γ) = 2〈E(u),Eγ−1± ϕ〉L2(Ω±)

+α〈u±, γ−1± ϕ〉L2(Ω±) − 〈p±,div (γ−1

± ϕ)〉L2(Ω±) + 〈f±, γ−1± ϕ〉L2(Ω±)


2. Function spaces

Hs(R3) := f | F−1(1 + |ξ|2)−s/2Ff : f ∈ L2(R3,Hs(Ω) := f ∈ D′(Ω) : ∃F ∈ Hs(R3) ∧ F |Ω = f,

Hs(Ω) := f ∈ Hs(R3) ∧ suppf ⊆ Ω,(Hs(Ω)

)′ = H−s(Ω) , H−s(Ω) =(Hs(Ω)

)′, L2(Ω)−duality;

%(x) := (1 + |x|2)1/2 , Lp(%−1; Ω−) := f | %−1f ∈ Lp(Ω−),

H1(Ω−) := f ∈ D′(Ω−) | %−1f ∈ L2(Ω−) ∧∇f ∈ L2(Ω−),‖ f ‖2H1(Ω−) := (‖ %−1f ‖2L2(Ω−) + ‖ ∇f ‖2L2(Ω−)),

H1(Ω−) = D(Ω−)

‖·‖H1(Ω−) ⊂ H1(Ω−),

H1(Ω−) := v ∈ H1(R3) ∧ supp v ⊆ Ω−,

H−1(Ω−) =(H1(Ω−)

)′, H−1(Ω−) =

(H1(Ω−)

)′;Embedding: H1(Ω−) → L6(Ω−) for Ω− ⊂ R3.


Traces:

γ± : H1(Ω±)→ H12 (Γ) , ∃ γ−1

± : H1/2(Γ)→ H1(Ω±),γ− : H1(Ω−) → H

12 (Γ) surjectively (Lang+Méndez 2006, Mikhailov 2011) .

3. Solution properties for Brinkman and Stokes systems

Lemma: (Amrouche+Razafison 2006) Let u ∈ D′(Ω−)3∧∇u ∈ L2(Ω)3×3.Then there exists u∞ ∈ R3 s.th. u− u∞ ∈ H1(Ω−)3,

u∞ = 14 lim|x|→∞

∫S2

u(y(x)

)dsy,

u ∈ L6(Ω−)3 , ‖u− u∞‖L6(Ω−)3 ≤ C‖∇u‖L2(Ω−)3 ,

lim|x|→∞

∫S2

|u(y(x)

)− u∞|dsy = 0.


Lα(u±, p±) := ∆u± − αu± −∇p±,

H1(Ω±,Lα) := (u±, p±, f±) ∈ H1(Ω±)3 × L2(Ω±)× H−1(Ω±)3.

Lemma: Let α ≥ 0 , Lα(u±, p±) = f±,div u± = 0 in Ω± and(u±, p±, f±) ∈ H1(Ω±,Lα).

Then Green’s formula holds for all w ∈ H1(Ω±)3:

±〈∂±n;α(u±, p±, f±) , γ±w〉Γ = 2〈E(u±),E(w)〉Ω± + α〈u±,w±〉Ω±− 〈p±,div w〉Ω± + 〈f±,w〉Ω±

and ∂±n;α(u±, p±, f±) ∈ H−1/2(Γ).

Lemma: Let α = 0 and (u, p) ∈ H1(Ω−,L0) be a solution of theStokes system. Then Green’s formula holds in Ω− and ∂n;0(u, p, f−) ∈H−1/2(Γ).If f− = 0 then u(x) = O(|x|−1) , ∇u(x) = O(|x|−2)and p(x) ∈ O(|x|−2) for |x| → ∞.


4. Potential operators

Brinkman fundamental solution in R3:(for n = 2 see e.g. Kohr & Pop 2004)

Gαjk(z) = 14π|z|

δjkA1(

√α|z|) + zjzk

|z|2 A2(√α|z|)

, P sj (z) = 1

2πzj

|z|3 ,

A1(%) = 2e−%(1 + %−1 + %−2)− 2%−2 = 1− 43%+ 3

4%2 − . . . ,

A2(%) = −2e−%(1 + 3%−1 + 3%−2) + 6%−2 = 1− 14%

2 − . . . ,

Sαijk(z) = P sj (z)δik + ∂kGαij(z) + ∂iGαkj(z)

Pd,αik (z) = 1

2π

δik(α|z|2 − 2) 1

|z|3 + 6 zizk

|z|5

.

Stokes fundamental solution: Set α = 0.


Newton potentials:

(Nα;Ω±Ψ)j(x) := −∫

y∈Ω±Gαjk(x− y)Ψk(y)dy: H−1(Ω±)3 → H1

div(Ω±)3,

(P sα;Ω±Ψ)(x) := −∫

y∈Ω±P sk (x− y)Ψk(y)dy : H−1(Ω±)3 → L2(Ω±)3.

Surface Potentials: Ψ(x) = (γ−1± g)(x) , x ∈ Ω±;

(Vα;Γ)j(x) := −∫

y∈ΓGαjk(x− y)gk(y)dsy : H−1/2(Γ)3 → H1

div(Ω±)3,

(P sα;Γg)(x) := −∫

y∈ΓP sk (x− y)gk(y)dsy : H−1/2(Γ)3 → L2(Ω±)3.

(Wα;Γ)k(x) := −∫

y∈ΓSαik`(x− y)n`(y)hi(y)dsy : H1/2(Γ)3 → H1

div(Ω±)3,

(P dα;Γh)(x) :=∫

y∈ΓP d,αik (x− y)nk(y)hi(y)dsy : H1/2(Γ)3 → L2(Ω±)3.


Boundary integral operators and jump relations at x ∈ Γ:

(γ±Vα;Γg)j(x) =: (Vα;Γg)j(x) =∫

y∈ΓGαjk(x− y)gk(y)dsy :

H−1/2(Γ)3 → H1/2(Γ)3,

(γ±Wα;Γh)j(x) =: ± 12δjkhk(x) + (Kα;Γh)j(x)

= ± 12δjkhk(x) + p.v.

∫x6=y∈Γ

Sαijk(x− y)nk(y)hi(y)dsy :

H1/2(Γ)3 → H1/2(Γ)3,

∂±nα;Γ(Vα;Γg, P sα;Γg)j(x) = ∓ 12δjkgk(x) + (K∗α;Γg)j(x) :(

H−1/2(Γ))3 → (

H1/2(Γ))3,

∂±nα;Γ(Wα;Γh, P dα;Γh)j(x) = (Dα;Γh)j(x) :(H1/2(Γ)

)3 → (H−1/2(Γ)

)3;

Ker Vα;Γ = Rn.


5. The linear transmission problem where k=β = 0

Theorem: The linear problem with k = β = 0 has for given dataf± ∈ H−1(Ω±)3 , h0 ∈ H1/2(Γ)3 , g0 ∈ H−1/2(Γ)3 , u∞ ∈ R3

a unique solution (u+, p+) ∈ H1div(Ω+)3 × L2(Ω+),

(u−, p−) ∈ H1div(Ω−)3 × L2(Ω−) and

‖u+‖H1(Ω+)3 + ‖p+‖L2(Ω+) + ‖u−− u∞‖H1(Ω−)3 + ‖p−‖L2(Ω−)

≤ C‖f+‖H1(Ω+)3 + ‖f−‖H−1(Ω−)3 + ‖h0‖H1/2(Γ)3 + ‖g0‖H−1/2(Γ)3

with C = C(Ω+,P, α, µ), satisfying

lim|x|→∞

∫S2|u−

(y(x)

)− u∞|dsy = 0

and(u−(x)− u∞) = O(|x|−1) , ∇u−(x) = O(|x|−2) , p−(x) = O(|x|−2)for |x| → ∞.


Sketch of the proof:1. Ansatz: u± = v± + w± , pv,± + π± ; α = 0 in Ω−;where v± = Nα,Ω± f± , pv,± = P sα;Ω± f± satisfying the homogeneousBrinkman system in Ω+ and Stokes system in Ω−.

2. w± = Vα;Γϕ+ Wα;ΓΦ, π± = P sα;Γϕ+ P dα;ΓΦ,γ+w+ − γ−w− = h00 = h0 − γ+v+ + γ−v−,∂+n;α(w+, π+)− µ∂−n;α(w−, π−) + 1

2P(γ+w+ + γ−w−) = g00= g0 − ∂+

n;α(v+, π+) + µ∂n;0(v−, π−)− 12P(γ+v+ + γ−v−

3. Transmission conditions on Γ yield for Φ ∈ H1/2(Γ) , ϕ ∈ H−1/2(Γ)the Fredholm equations Φ = Kα,0;ΓΦ + Vα,0;Γϕ− h00,

12 (1 + µ)I + (1− µ)K∗Γϕ = −(1− µ)D0,ΓΦ−Dα,0;ΓΦ−K∗α,0,Γϕ− 1

2P((Vα,Γ + V0,Γ)ϕ+ (Kα,Γ +K0,Γ)

)Φ + g00.

The operator . . . is invertible on H−1/2(Γ) andDα,0;Γ := Dα,Γ −D0,Γ , K

∗α,0,Γ := K∗α,Γ −K∗0,Γ− and P(. . . ) are

compact.

4, Green’s theorem and vanishing energy for h00 = g00 = 0 impliesuniqueness.


6. The nonlinear Darcy–Forchheimer–Brinkman transmissionproblem

Lemma: LetE±(w)(x) :=

w(x) for x ∈ Ω±,

0 for x ∈ Ω∓.Then the nonlinear

mappings Ik,β,Ω+(v) :=E+(k|v|v + β(v · ∇)v

)are positively homo-

geneous of order 2 and for v ∈ H1div (Ω+)3 into

L3/2(Ω+)3 and H−1(Ω+)3 are bounded:‖Ik,β,Ω+(v)‖L3/2(Ω+)3 ≤ c′1‖v‖2H1(Ω+)3 ,

‖Ik,β,Ω+(v)‖H−1(Ω+)3 ≤ c1‖v‖2H1(Ω+)3

and for v,w ∈ H1(Ω+)3 it holds‖Ik,β,Ω+(v)− Ik,β,Ω+(w)‖

H−1(Ω+)3

≤ c1(‖ v ‖H1(Ω+)3 + ‖ w ‖H1(Ω+)3)‖v−w‖H1(Ω+)3 .


Theorem: For the nonlinear problem∆u− αu− k|u|u− β(u · ∇)u−∇p = f and ∇ · u = 0 in Ω+;∆u−∇p = 0 and ∇ · u = 0 in Ω−;γ+u+ − γ−u− = h0 on Γ = ∂Ω and∂+n;α(u+, p+; (f+ + E+(k|u+|u+ + β(u+ · ∇)u+)

)− µ∂−n (u−, p−; f−)

+ 12P(γ−u−+ γ+u+) = g0 on Γ;

lim|x|→∞

∫S2(x)

|u−(y(x)

)− u∞|dsy = 0

there are two positive constants ζ and η depending on Ω+,P, α, µ, k, βsuch that for all given data with

‖(f+, f−,h0,u∞)‖X0 :=‖f+‖H−1(Ω+)3+‖f−‖H−1(Ω+)3+‖h0‖H1/2(Γ)3+‖g0‖H−1/2(Γ)3+|u∞|

≤ζ,

the nonlinear problem has a unique solution satisfying‖(u+, p+,u− − u∞, p−)‖Y :=

‖u+‖H1(Ω+)3 + ‖p+‖L2(Ω+)+ ‖u−−u∞‖H1(Ω−)3 + ‖p−‖L2(Ω−)≤ η


Corollary: The solution satisfies the apriori estimate

‖(u+, p+,u− − u∞, p−)‖Y ≤ C‖(f+, f−,h0,g0,u∞)‖X0

with C = C(Ω+,P, α, µ). Moreover

lim|x|→∞

∫S2|u−

(y(x)

)− u∞|dsy = 0.


Sketch of the proof:

Let v ∈ H1(Ω+) be given and solve for u:∆u− αu−∇p = f+ + k|v|u + β(v · ∇)u , ∇ · u = 0 in Ω+,∆u−∇p = 0 , ∇ · u = 0 in Ω−.γ+u+ − γ−u− = h0,∂+n;α(u+, p+;

(f+ + E+(k|v|u + β(v · ∇)u)

)− µ∇−n;0(u−, p−, f−)+

+P(γ−u− + γ+u+) = g0 on Γ,lim|x|→∞

∫S2(x)

|u−(y(x)

)− u∞|dsy = 0,

This defines a mapping v 7→ u =: U(v) : Bζ → Bζwhere Bζ ⊂ H1

div(Ω+)3 is a closed ball of radius ζ. The a priori estimatesare uniform with respect to v and they allow us to find ζ > 0 such thatU becomes in Bζ a contraction if the given data are small enough. Thenthe Banach–Cacciapoli theorem yields the unique existence of u ∈ Bζwith U(u) = u.


7. The problem on a compact Riemannian connected boundarylessorientable manifold Mn−1 ⊂ Rn.

In every chart of a finite atlas let for ξ ∈ Uτ ⊂ Rn−1 , x = x(ξ),the basis of the tangent bundle be: ∂j = ∂

∂ξjx , j = 1, . . . , n− 1;

fundamental tensor: gjk = ∂jx · ∂kx , g`mgmk = δ`k ; g = det gjk.For u(ξ) = uj∂j tangential field in the tangent bundle,div u = 1√

g∂k(√guk) , (grad p) = (gjk∂jp)∂k and u` = g`kuk.

Riemannian or Levi–Civita connection:Γ`jk := g`m(∂j∂kx) · ∂mx Christoffel symbol of 2nd kind,covariant derivative: ∇uv := uj(∂jvk + Γkj`v`)∂kdeformation tensor: (Def u)`k = 1

2 (∂ku` − Γm`kum + ∂ùk − Γmkùm)adjoint: (Def∗A)k = −(divA)k = −g`tA`k;t = −g`t(∂tA`k − ΓmtÀmk)


Assumption: Mn−1 does not admit any nontrivial Killing fieldX, i.e. satisfying Def X = 0.Darcy–Forchheimer–Brinkman system onMn−1:(Lu)k := 2(Def∗Defu)k = −g`s

∂s(∂ku` − Γm`kum + ∂`uk − Γmk`um)

−Γms`(∂kum−Γjmkuj+∂muk−Γjkmuj)

Lu + Pu + k|u|u + β∇uu + grad p = f ,div u = 0 in Ω ⊂Mn−1.

Let Ω+ ⊂M be a Lipschitz domain with exterior normal vector n onΓ = ∂Ω+ and Ω− := Ω \ Ω+ also Lipschitz.

Conormal derivative (hydrodynamic tension) t±Pand Stokes theorem:

t±P : H1(Ω±)× L2(Ω±)× H−1(Ω±)→ H−1/2(Γ),

±∫Γ

t±P(u, p; f±) ·ϕdσΓ = 2∫

Ω±E±Def u : Def(γ−1

± ϕ)ds+

+∫

Ω±E±Pu · γ−1

± ϕds+∫

Ω±E±p div (γ−1

± ϕ)ds−∫

Ω+

f± · γ−1± ϕds.


Let λ onMn−1 be a symmetric semidefinit matrix–valued function.

Theorem: The transmission problem onMn−1:

Lu+ + Pu + grad p+ = f+|Ω+ − (k|u+|u+ +β∇u+u+),div u+ = 0 in Ω+,

Lu− + grad p− = f−|Ω−−∇u−u− , div u− = 0 in Ω−,

µγ+u+ − γ−u− = h on Γ,t+P(u+.p+, f+ − E+(k|u+|u+ + β∇u+ u+)

)−t−0

(u−, p−; f− − E−(∇u−u)

)+ λγ+u+ = g

with sufficiently small given data(f+, f−,h,g) ∈ H−1(Ω+)× H−1(Ω−)×H1/2(Γ)×H−1/2(Γ)has a unique solution (u±, p±) ∈ H1(Ω±)× L2(Ω±).

The proof is based on Newton and boundary potentials in Ω±, and thatthe Killing fields are zero. Fundamental tensors can be constructed withA. Pomp’s Levi functions as approximation of the fundamental tensors.


8. More general spaces, 0 < s < 1:

Hs+ 1

2div (Ω±) :=

u ∈ Hs+ 1

2 (Ω±) ∧ div u = 0 in Ω±,

Hs− 1

2∗ (Ω+) :=

p ∈ Hs+ 1

2 (Ω+) ∧∫

Ω+

pds = 0,

Hsn(Γ) :=

v ∈ Hs(Γ) ∧

∫Γ

v·n dσ = 0,

Solutions: (u+, p+) ∈ Hs+ 12

div (Ω+)×Hs− 12

∗ (Ω+),(u−, p−) ∈ Hs+ 1

2div (Ω−)×Hs− 1

2 (Ω−);

Given data:(f+, f−,h,g) ∈ Hs−3/2(Ω+)× Hs−3/2(Ω−)×Hs

n(Γ)×Hs−1(Γ).


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