Maximum estimates for generalized Forchheimer flows in
heterogeneous porous media
Emine Celik
Joint with Luan Hoang
April 9, 2016
Texas State University, San Marcos, Texas
The 2016 Texas Differential Equations Conference
Outline
Fluid flows in porous media
Characters of PDE
Maximum estimates for the pressure
Maximum estimates for the pressure’s time derivative
Open problems
Generalized Forchheimer flows in heterogeneous porous media
Fluid flows in porous media
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Generalized Forchheimer flows in heterogeneous porous media
Porous medium
• A material containing pores (voids)
• e.g. porous or fissured rocks, soils, ceramics, sands, cemented sandstone, karstic
limestone, foam rubber, bread, lungs or kidneys
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Generalized Forchheimer flows in heterogeneous porous media
Homogeneous/Heterogeneous
• Porous medium is homogeneous wrt a macroscopic quantity if that parameter
has the same value throughout the domain.
• Otherwise it is heterogeneous: soil, geological media, multi-layer media
Goal
• Investigate maximum estimates for generalized Forchheimer fluid flows in
heterogeneous porous media
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Generalized Forchheimer flows in heterogeneous porous media
velocity: v(x, t) pressure: p(x, t) viscosity: µ constants: a, b, c, d
density: ρ(x, t) porosity: φ ∈ (0, 1)
permeability: k
• Darcy’s law (1856): v = −a∇p, where a =
k
,
µ
• The Forchheimer two-term law (1901): av + b|v|v = −∇p,
• The Forchheimer three-term law (1901): av + b|v|v + c|v|2 v = −∇p,
• The Forchheimer power law (1930): av + d|v|m−1 v = −∇p, m ∈ [1, 2],
• Generalized Forchheimer equations: gF (|v|)v = −∇p
where gF (s) = a0 + a1 sα1 + · · · + aN sαN , ai ≥ 0 for i = 1, . . . , N − 1,
a0 , aN > 0 and αj ≥ 0.
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Generalized Forchheimer flows in heterogeneous porous media
• U (bounded domain in Rn ) models a porous medium.
• Γ = ∂U : C 1 -boundary of U .
• Generalized Forchheimer equation for heterogeneous porous media:
g(x, |v|)v = −∇p,
where g(x, s) = a0 (x) + a1 (x)sα1 + · · · + aN (x)sαN , s ≥ 0,
a1 (x), a2 (x), . . . , aN −1 (x) ≥ 0, and a0 (x), aN (x) > 0.
⇓
g(x, |v|)|v| = |∇p|.
• s 7→ sg(x, s) is strictly increasing, mapping [0, ∞) onto [0, ∞).
⇒ ∀ξ ∈ [0, ∞), ∃! non-negative solution s = s(x, ξ) of the equation sg(x, s) = ξ.
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Generalized Forchheimer flows in heterogeneous porous media
• ⇒ |v| = s(x, |∇p|) and
v=−
∇p
∇p
=−
.
g(x, |v|)
g(x, s(x, |∇p|))
• ⇒ Nonlinear generalization of Darcy’s equation:
v = −K(x, |∇p|)∇p,
where K : Ū × R+ → R+ :
K(x, ξ) =
1
g(x, s(x, ξ))
for x ∈ Ū , ξ ≥ 0.
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Generalized Forchheimer flows in heterogeneous porous media
PDE
• Continuity equation:
φρt + ∇ · (ρv) = 0.
• Equation of state for (isothermal) slightly compressible fluids:
dρ
ρ
= ,
dp
κ
where
1
>0
κ
small compressibility.
• Rewrite continuity:
φ
dρ ∂p
dρ
+ ρ∇ · v + ∇p · v = 0,
dp ∂t
dp
• Combine with equation of state:
φ
ρ ∂p
ρ
= −ρ∇ · v − ∇p · v.
κ ∂t
κ
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Generalized Forchheimer flows in heterogeneous porous media
• Use generalized Darcy’s equation:
φ(x)
∂p
= κ∇ · (K(x, |∇p|)∇p) + K(x, |∇p|)|∇p|2 .
∂t
• κ large & t → κt ⇒ φ(x)
∂p
= ∇ · (K(x, |∇p|)∇p).
∂t
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Generalized Forchheimer flows in heterogeneous porous media
• The initial boundary value problem (IBVP):

∂p


φ(x) ∂t = ∇ · (K(x, |∇p|)∇p) on U × (0, ∞),
p = ψ on Γ × (0, ∞),



p(x, 0) = p0 (x) on U,
where p0 (x) and ψ(x, t) are given initial and boundary data.
• We consider any φ(x) > 0.
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Generalized Forchheimer flows in heterogeneous porous media
Properties of K(ξ):
• Define
a=
−aK(x, ξ) ≤ ξ
αN
∈ (0, 1).
αN + 1
∂K(x, ξ)
≤0
∂ξ
∀ξ ≥ 0.
• ⇒ K(x, ξ) is decreasing in ξ.
K(x, ξ) ≤ K(x, 0) =
1
1
=
.
g(x, 0)
a0 (x)
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Generalized Forchheimer flows in heterogeneous porous media
Lemma
For ξ ≥ 0, one has
ξa
2W1 (x)
W2 (x)
≤ K(x, ξ) ≤
a
+ aN (x)
ξa
and, consequently,
W1 (x)ξ 2−a −
aN (x)
≤ K(x, ξ)ξ 2 ≤ W2 (x)ξ 2−a .
2
M (x) = max{aj (x) : j = 0, . . . , N },
m(x) = min{a0 (x), aN (x)},
a
W1 (x) =
aN (x)
,
2N M (x)
and
W2 (x) =
N M (x)
.
m(x)aN (x)1−a
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Generalized Forchheimer flows in heterogeneous porous media
Characters of PDE
φ(x)
∂p
= ∇ · (K(x, |∇p|)∇p)
∂t
• φ(x) → 0 then LHS is degenerate
• K(x, |∇p|) is degenerate if |∇p| is large
• K(x, |∇p|) can be small or large at different x. (singular-degenerate PDE)
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Generalized Forchheimer flows in heterogeneous porous media
Poincaré-Sobolev’s inequality: If 1 ≤ q < n then
kf kLq∗ (U ) ≤ c k∇f kLq (U ) for all f ∈ W̊ 1,q (U ),
c depends on q, n and the domain U , and q ∗ = nq/(n − q).
• Let γ1 (x), γ2 (x) > 0 be two functions on U .
• Let X̊γr,q
(U ) be a certain class, containing functions which vanish on the
1 ,γ2
boundary Γ. Then there is a positive constant c0 such that
kukLrγ
1
(U )
≤ c0 k∇ukLqγ2 (U ) .
(Sawyer and Wheeden 1992; Duc, Phuc and Nguyen 2007)
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Generalized Forchheimer flows in heterogeneous porous media
e.g.
X̊γr,q
(U ) = W̊ 1,q0 (U ) ∩ {u : ∇u ∈ Lqγ2 (U )}.
1 ,γ2
Then the following weighted Poincaré-Sobolev inequlaity holds:
Z
r1
Z
q1
|u|r γ1 dx
≤ c0
|∇u|q γ2 dx
U
U
with
∗
q∗
0 Z
Z
q−q
qq0∗−r
0
q0
qq0
r
∗
c0 = c
γ2 (x)− q−q0 dx
γ1 (x) q0 −r dx 0 < ∞.
U
U
n
o
r,q
X̊γr,q
(U
×
(0,
T
))
:=
u(x,
t)
:
u(·,
t)
∈
X̊
(U
)
for
almost
all
t
∈
(0,
T
)
.
γ1 ,γ2
1 ,γ2
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Generalized Forchheimer flows in heterogeneous porous media
Lemma
Let r, q be two numbers satisfying
r > 2,
r > q ≥ 1.
Set
p = 2 + q(1 − 2/r) = q + 2(1 − q/r).
If T > 0 and u(x, t) ∈ X̊γr,q
(U × (0, T )), then
1 ,γ2
q
kukLpγ1 (U ×(0,T )) ≤ c0p ess sup ku(t)kL2γ
0<t<T
(U )
1
+ k∇ukLqγ2 (U ×(0,T )) .
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Generalized Forchheimer flows in heterogeneous porous media
Maximum estimates for the pressure
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Generalized Forchheimer flows in heterogeneous porous media
• Ψ(x, t) be an extension (in x) of ψ(x, t) from boundary Γ to U .
• Let p̄ = p − Ψ, then we have
φ(x)
∂ p̄
= ∇ · (K(x, |∇p|)∇p) − φ(x)Ψt
∂t
p̄ = 0 on Γ × (0, ∞).
on U × (0, ∞),
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Generalized Forchheimer flows in heterogeneous porous media
• Strict Degree Condition (SDC):
deg(g) <
4
.
n−2
Assumption: φ(x) ∈ L1 (U ), and there are r > 2 and c2 > 0 (under SDC)
such that
kukLrφ (U ) ≤ c2 k∇ukL2−a (U )
W1
for functions u(x) that vanish on the boundary Γ.
Note that r0 =
r
r−1
< 2.
2
> 2.
r
The following estimates use a fixed parameter r1 , which is a number in interval
(1, r0 /2).
r0 = 2 + (2 − a) 1 −
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Generalized Forchheimer flows in heterogeneous porous media
• Let p̄(k) = max{p̄ − k, 0}, k ≥ 0.
• Multiply PDE by p̄(k) ζ.
• Integrate over U , integration by parts, use bounds for K(·), Cauchy’s and
triangle inequalities, integrating in t and with other calculations we have
Z
|p̄(k) (x, t)|2 ζ(t)φ(x)dx
sup
0<t<T
U
Z
T
Z
+
0
Z
T
Z
+ 16
0
U
W1 (x)|∇p̄(k) |2−a ζdxdt ≤ 4
Z
0
T
Z
|p̄(k) |2 |ζt |φdxdt
U
χk · (aN (x) + W1 (x)|∇Ψ|2−a + T |Ψt |2 φ + a0 (x)−1 |∇Ψ|2 )ζdxdt.
U
• Let ti = θT 1 − 21i .
• t0 = 0 < t1 < t2 < ... < θT and ti → θT as i → ∞.
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Generalized Forchheimer flows in heterogeneous porous media
• Let ζ(t) : R → [0, 1] such that
(
0 for t ≤ ti
and
ζi (t) =
1 for t ≥ ti+1
0 ≤ ζi0 (t) ≤
2
2i+2
=
ti+1 − ti
θT
∀t ∈ R.
• Define ki = M0 (1 − 2−i )
• Set Ai,j = {(x, t) : p(x, t) > ki , t ∈ (tj , T )} for i, j ≥ 0.
• Apply k = ki+1 and ζ = ζi ≤ 1 with
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Generalized Forchheimer flows in heterogeneous porous media
Z
ωT0 ,T = T
r10
aN (x) φ(x)
1−r10
dx + T
r10
Z
U
Z
T0 +T
Z
+
T0
T0
Z
0
|Ψt (x, t)|2r1 φ(x)dxdt
U
W1 (x)|∇Ψ(x, t)|2−a +a0 (x)−1 |∇Ψ(x, t)|2
r10
0
φ(x)1−r1 dxdt.
U
Z
|p̄
sup
0<t<T
T0 +T
U
(ki+1 )
2
Z
T
Z
(x, t)| ζi (t)φ(x)dx +
0
W1 (x)|∇p̄(ki+1 ) (x, t)|2−a ζi (t)dxdt
U
1
≤ C̄
2
i
− 2
2i (ki ) 2
r0
kp̄ kL2 (Ai,i ) + C̄4 r1 M0 r1 ωT1 kp̄(ki ) kLr12 (Ai,i ) .
φ
φ
θT
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Generalized Forchheimer flows in heterogeneous porous media
• Apply Poincare-Sobolev inequality to r > 2, q = 2−a, the weights γ1 (x) = φ(x),
γ2 (x) = W1 (x), and the function u(x, t) = p̄(ki+1 ) (x, t)ζi (t) and use Hölder’s
inequality
1
1
kp̄(ki+1 ) kL2φ (Ai+1,i+1 ) ≤ µ̄(Ai+1,i+1 ) 2 − r0 kp̄(ki+1 ) kLr0 (Ai+1,i+1 )
φ
2−a
r0
≤ C̄c2
(4i+1 M0−2 )
1
1
2 − r0
1− r2
0
kp̄(ki ) kL2 (A
i,i )
φ
n 2i 12
i − 2 12 2r10
1
kp̄(ki ) kL2φ (Ai,i ) + 4 r1 M0 r1 ωT 1 kp̄(ki ) kLr12 (Ai,i )
φ
θT
1
1
1
2i 2−a
i − 2 2−a
o
2
2
0
r (2−a)
r1
(ki ) r1 (2−a)
1
r1
+
+
4
kp̄
k
.
kp̄(ki ) kL2−a
M
ω
2 (A
2 (A
0
T
)
L
)
i,i
i,i
φ
φ
θT
·
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Generalized Forchheimer flows in heterogeneous porous media
Lemma (Hoang, Kieu and Phan 2014)
Let {Yi }∞
i=0 be a sequence of non-negative numbers satisfying
Yi+1 ≤
m
X
Ak B i Yi1+µk ,
i = 0, 1, 2, · · · ,
k=1
where B > 1, Ak > 0 and µk > 0 for k = 1, 2, . . . , m.
Let µ = min{µk : 1 ≤ k ≤ m}.
1
−µ
If Y0 ≤ min (m−1 A−1
)1/µk : 1 ≤ k ≤ m then lim Yi = 0.
k B
i→∞
⇒
Z
T
θT
Z
|p̄(M0 ) |2 φ(x)dxdt = 0 ⇒ p̄(M0 ) (x, t) = 0
U
⇒ p̄(x, t) ≤ M0
a.e. in
U × (θT, T )
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Generalized Forchheimer flows in heterogeneous porous media
Proposition
For any T0 ≥ 0, T > 0 and θ ∈ (0, 1), one has
2−a 1 κ1
1
kp̄(t)kL∞ (U ×(T0 +θT,T0 +T )) ≤ C̄ max{1, c2 } r0 −2 (θT )− 2 + (θT )− 2−a
· (1 + ωT0 ,T )κ2 kp̄kνL12 (U ×(T0 ,T0 +T )) + kp̄kνL22 (U ×(T0 ,T0 +T )) ,
φ
φ
where C̄ > 0 is independent of c2 , T0 , T , and θ,
κ1 =
r0 (r1 − 1)
r0
, κ2 =
,
r0 − 2
2r0 + (r0 − 2)r1 (2 − a)
ν1 =
r0 − 2r1
2(r0 − 2 + a)
, ν2 =
,
r0 + (r0 − 2)r1
(2 − a)(r0 − 2)
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Generalized Forchheimer flows in heterogeneous porous media
We assume
Two-weight Poincaré-Sobolev’s inequality: There is c1 > 0 such that
if u(x) vanishes on Γ then
kukL2φ (U ) ≤ c1 k∇ukL2−a (U ) .
W1
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Generalized Forchheimer flows in heterogeneous porous media
Theorem (C., Hoang 2016)
(i) If t > 0 then
Z
Z
2
p̄2 (x, t)φ(x)dx ≤
p̄2 (x, 0)φ(x)dx + CM(t) 2−a .
U
U
where
Z
G(t) = G[Ψ](t) := max{
Z
aN (x)dx, 1} +
U
Z
+
U
W1 (x)|∇Ψ(x, t)|2−a dx +
a0 (x)−1 |∇Ψ(x, t)|2 dx
U
Z
2−a
2(1−a)
,
|Ψt (x, t)|2 φ(x)dx
U
M(t) = M[Ψ](t) be a continuous function on [0, ∞) that satisfies
M(t) is increasing and M(t) ≥ G(t) ∀t ≥ 0,
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Generalized Forchheimer flows in heterogeneous porous media
A = A[Ψ] := lim sup G(t) and
B = B[Ψ] := lim sup[G0 (t)]− .
t→∞
t→∞
Theorem (cont.)
(ii) If A < ∞ then
Z
lim sup
t→∞
2
p̄2 (x, t)φ(x)dx ≤ CA 2−a .
U
(iii) If B < ∞ then there is T > 0 such that for all t > T
Z
1
2
p̄2 (x, t)φ(x)dx ≤ C(B 1−a + G(t) 2−a )
U
Note that
M(t) ≥ 1 ∀t ≥ 0,
and A ≥ 1.
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Generalized Forchheimer flows in heterogeneous porous media
Theorem
(i) If t ∈ (0, 1), then
ν2
1
,
kp̄kL∞ (U ×(t/2,t)) ≤ Ct−κ3 N1 (0, t)κ2 kp̄(0)kL2φ (U ) + M(t) 2−a
where
κ3 =
κ1
ν1
r0
r0 − 2r1
−
=
−
> 0.
2−a
2
(2 − a)(r0 − 2) 2(r0 + (r0 − 2)r1 )
If t ≥ 1, then
ν2
1
kp̄kL∞ (U ×(t− 21 ,t)) ≤ CN1 (t − 1, t)κ2 kp̄(0)kL2φ (U ) + M(t) 2−a
.
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Generalized Forchheimer flows in heterogeneous porous media
Theorem (cont.)
(ii) If A < ∞ then
lim sup kp̄kL∞ (U ×(t− 12 ,t)) ≤ C lim sup N1 (t − 1, t)
t→∞
κ2
ν2
A 2−a .
t→∞
(iii) If B < ∞ then there is T > 0 such that for all t > T
1
1
kp̄kL∞ (U ×(t− 12 ,t)) ≤ CN1 (t − 1, t)κ2 B 2(1−a) + G(t) 2−a
ν2
.
o
i
n Z
0
0
0
N1 (s, t) = max 1,
aN (x)r1 φ(x)1−r1 dx + |Ψt (x, τ )|2r1 φ(x) dxdτ
U
Z t Z h
r10
0
+
W1 (x)|∇Ψ(x, τ )|2−a + a0 (x)−1 |∇Ψ(x, τ )|2 φ(x)1−r1 .
s
U
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Generalized Forchheimer flows in heterogeneous porous media
Maximum estimates for the pressure’s time
derivative
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Generalized Forchheimer flows in heterogeneous porous media
IBVP
• Let q(x, t) = pt (x, t) and
⇒
φ(x)
q̄(x, t) = p̄t (x, t) = pt (x, t) − Ψt .
∂ q̄
= ∇ · (K(x, |∇p|)∇p)t − φ(x)Ψtt
∂t
q̄ = 0 on Γ × (0, ∞).
on U × (0, ∞),
• Fix a number r2 such that
r2 >
2
2(r − 1)
=
.
2 − r0
r−2
• Note that r20 belongs to (1, 2/r0 ).
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Generalized Forchheimer flows in heterogeneous porous media
Proposition
There is a constant C̄ > 0 independent of c2 such that for any T0 ≥ 0, T > 0
and θ ∈ (0, 1) we have
1
1
1
kp̄t kL∞ (U ×(T0 +θT,T0 +T )) ≤ C̄ [(θT )− 2 ST0 ,T,θ ] δ1 + (ZT0 ,T ST0 ,T,θ ) 1+δ2
δ2
r
2
· max{1, c2 } r−2 · kp̄t kL2φ (U ×(T0 ,T0 +T )) + kp̄t kL1+δ
2 (U ×(T ,T +T )) ,
0
0
φ
where
δ1 = 1 −
r0
,
2
ST0 ,T,θ = B1 +
1
r0
−
,
r20
2
Z
ar 0
4(2−a)
sup
W1 (x)|∇p(x, t)|2−a dx
,
δ2 =
t∈[T0 +θT,T0 +T ]
U
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Generalized Forchheimer flows in heterogeneous porous media
Proposition (cont.)
ZT0 ,T = ka0 (x)−1/2 ∇Ψt kL2r2 (U ×(T0 ,T0 +T )) + T 1/2 kΨtt kL2r2 (U ×(T0 ,T0 +T )) .
φ
φ
For t > s ≥ 0, define
−1/2
N2 (s, t) = 1 + ka0
∇Ψt kL2r2 (U ×(s,t)) + kΨtt kL2r2 (U ×(s,t)) .
φ
φ
Then N2 (s, t) ≥ 1 and
1 + ZT0 ,T ≤ (max{1, T })1/2 N2 (T0 , T0 + T ).
Z
G1 (t) = G1 [Ψ](t) :=
a0 (x)−1 |∇Ψt (x, t)|2 dx.
U
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Generalized Forchheimer flows in heterogeneous porous media
Theorem
(i) If t ∈ (0, 23 ) then
1
1
kp̄t kL∞ (U ×(t/2,t)) ≤ Ct− 2δ1 N2 (0, t) 1+δ2
κ4
Z t
2
2−a
· A0 + M(t)
+
G1 (τ )dτ
,
0
where
1
ar
+
,
2 2(2 − a)(r − 2)
Z
Z
A0 =
H(x, |∇p(x, 0)|)dx +
p̄2 (x, 0)φdx.
κ4 =
U
U
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Generalized Forchheimer flows in heterogeneous porous media
Theorem
If t ≥ 32 then
1
kp̄t kL∞ (U ×(t− 14 ,t)) ≤ CN2 (t − 12 , t) 1+δ2
Z
2
· kp̄(0)k2L2 + M(t) 2−a +
φ
t
G1 (τ )dτ
κ 4
.
t− 54
(ii) If A < ∞ then
1
lim sup kp̄t kL∞ (U ×(t− 41 ,t)) ≤ C lim sup N2 (t − 12 , t) 1+δ2
t→∞
t→∞
Z t
2
κ 4
G1 (τ )dτ
.
A 2−a + lim sup
t→∞
t−1
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Generalized Forchheimer flows in heterogeneous porous media
Theorem
(iii) If B < ∞ then there is T > 0 such that for all t > T ,
Z
1
1
2
kp̄t kL∞ (U ×(t− 14 ,t)) ≤ CN2 (t− 12 , t) 1+δ2 B 1−a +G(t) 2−a +
t
G1 (τ )dτ
κ4
.
t− 54
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Generalized Forchheimer flows in heterogeneous porous media
Publications related heterogeneous Porous Media
• Generalized Forchheimer flows in heterogeneous porous media (with Luan
Hoang), 29pp, Nonlinearity, Vol. 29, No. 3 (March 2016), 1124-1155.
• Maximum estimates for generalized Forchheimer flows in heterogeneous porous
media (with Luan Hoang), 28pp, submitted. [arXiv Preprint]
Open problems
• Higher integrability for gradient of pressure.
• Continuous dependence on ai (x).
• Gas flows in heterogeneous porous media.
Emine Celik | The 2016 Texas Differential Equations Conference
39/40
Thank you!
Questions?