Forchheimer Equations in Porous Media - Part III
Luan Hoang, Akif Ibragimov
Department of Mathematics and Statistics, Texas Tech University
http://www.math.umn.edu/∼lhoang/
luan.hoang@ttu.edu
Applied Mathematics Seminar
Texas Tech University
September 15&22, 2010
Luan H. - Texas Tech
Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
Outline
1
Introduction
2
Bounds for the solutions
3
Dependence on the boundary data
4
Dependence on the Forchheimer polynomials
Luan H. - Texas Tech
Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
Introduction
Darcy’s Law:
αu = −∇p,
the “two term” law
αu + β|u| u = −∇p,
the “power” law
c n |u|n−1 u + au = −∇p,
the “three term” law
Au + B |u| u + C|u|2 u = −∇p.
Here α, β, c, A, B, and C are empirical positive constants.
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General Forchheimer equations
Generalizing the above equations as follows
g (|u|)u = −∇p.
Let G (s) = sg (s). Then G (|u|) = |∇p| ⇒ |u| = G −1 (|∇p|). Hence
u=−
∇p
g (G −1 (|∇p|))
where
K (ξ) = Kg (ξ) =
= −K (|∇p|)∇p,
1
1
=
,
g (s)
g (G −1 (ξ))
sg (s) = ξ.
We derived non-linear Darcy equations from Forchheimer equations.
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Forchheimer Equations - III
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Equations of Fluids
Let ρ be the density. Continuity equation
dρ
= −∇ · (ρu),
dt
For slightly compressible fluid it takes
dρ
1
= ρ,
dp
κ
where κ 1. Substituting this into the continuity equation yields
dρ dp
dρ
= −ρ∇ · u −
u · ∇p,
dp dt
dp
dp
= −κ∇ · u − u · ∇p.
dt
Since κ 1, we neglect the second term in continuity equation
dp
= −κ∇ · u .
dt
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Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
Combining the equation of pressure and the Forchheimer equation, one
gets after scaling:
dp
= ∇ · K (|∇p|) ∇p .
dt
Consider the equation on a bounded domain U in R3 . The boundary of U
consists of two connected components: exterior boundary Γ2 and interior
(accessible) boundary Γ1 .
Dirichlet condition on Γ1 : p(x, t) = ψ(x, t) which is known for x ∈ Γi .
On Γ2 :
u·N =0⇔
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∂p
= 0.
∂N
Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
Class FP(N, α
~)
We introduce a class of “Forchheimer polynomials”
Definition
A function g (s) is said to be of class FP(N, α
~ ) if
g (s) = a0 s α0 + a1 s α1 + a2 s α2 + . . . + aN s αN =
N
X
aj s α j ,
j=0
where N > 0, 0 = α0 < α1 < α2 < . . . < αN , and
a0 , aN > 0, a1 , . . . , aN−1 ≥ 0. Notation αN = deg(g ).
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Forchheimer Equations - III
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Let N ≥ 1 and α
~ be fixed. Let
R(N) = {~a = (a0 , a1 , . . . , aN ) ⊂ RN+1 : a0 , aN > 0, a1 , . . . , aN−1 ≥ 0}.
Let g = g (s,~a) be in FP(N, α
~ ) with ~a ∈ R(N). We denote
a=
αN
a
αN
∈ (0, 1), b =
=
∈ (0, 1),
αN + 1
2−a
αN + 2
n
1 1 o
≥ 1.
χ(~a) = max a0 , a1 , . . . , aN , ,
a0 aN
Many estimates below will have constants depending on χ(~a).
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Lemma
Let g (s,~a) be in class FP(N, α
~ ). One has for any ξ ≥ 0 that
C0−1 χ(~a)−1−a
C0 χ(~a)1+a
≤
K
(ξ,~
a
)
≤
,
(1 + ξ)a
(1 + ξ)a
and for any m ≥ 1, δ > 0 that
C0−1 χ(~a)−1−a
δa
(ξ m−a − δ m−a ) ≤ K (ξ,~a)ξ m ≤ C0 χ(~a)1+a ξ m−a ,
(1 + δ)a
where C0 = C0 (N, αN ) depends on N, αN only.
In particular, when m = 2, δ = 1, one has
2−a C0−1 χ(~a)−1−a (ξ 2−a − 1) ≤ K (ξ,~a)ξ 2 ≤ C0 χ(~a)1+a ξ 2−a .
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The Monotonicity
Proposition
(i) For any y , y 0 ∈ Rn , one has
K (|y |,~a)y − K (|y 0 |,~a)y 0 · y − y 0 ≥ aK (|y | ∨ |y 0 |,~a)|y − y 0 |2 .
(ii) For any functions p1 and p2 one has
Z
K (|∇p1 |,~a)∇p1 − K (|∇p2 |,~a)∇p2 · ∇p1 − ∇p2 dx
U
Z
2
≥a
K (|∇p1 | ∨ |∇p2 |,~a)|∇p1 − ∇p2 | dx
U
Z
≥ C5
2−a
|∇p1 − ∇p2 |
2
2−a
dx
1 + k∇p1 kL2−a (U) ∨ k∇p2 kL2−a (U)
−a
,
U
where C5 = C5 (N, deg(g ), χ(~a)).
Luan H. - Texas Tech
Forchheimer Equations - III
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Poincaré–Sobolev inequality
Let f (x) and ξ(x) be two functions on U with f (x) vanishing on Γ1 and
ξ(x) ≥ 0 . Then
Z
|f (x)|
(2−a)∗
dx
2
(2−a)∗
≤ C6
Z
U
K (ξ(x),~a)|∇f (x)|2 dx
U
Z
a
2−a
× 1+
H(ξ(x),~a)dx
,
U
where (2 − a)∗ = n(2 − a)/(n − (2 − a)) and C6 = C6 (N, deg(g ), χ(~a), U).
Subsequently, when deg(g ) ≤ 4/(n − 2) one has
Z
Z
Z
a
2−a
|f (x)|2 dx ≤ C7
K (ξ(x),~a)|∇f (x)|2 dx 1 +
H(ξ(x),~a)dx
,
U
U
U
where C7 = C7 (N, deg(g ), χ(~a), U).
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Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
Degree Condition (DC)
deg(g ) ≤
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4
n(2 − a)
⇐⇒ 2 ≤ (2 − a)∗ =
.
n−2
n − (2 − a)
Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
Extension of the boundary data
Let Ψ(x, t), x ∈ U, t ∈ [0, ∞) be an extension of ψ(x, t) from Γ1 to U.
For instance, one can use the following harmonic extension Ψ of ψ:
∆Ψ = 0 on U, Ψ = ψ and Ψ = 0.
Γ1
Γ2
We denote such Ψ by H(ψ). Then the Sobolev norms of H(ψ) on U can
be bounded by the Sobolev or Besov norms of ψ on Γ1 . In particular, we
have
∂k
∂k H(ψ)
≤
C
(k,
r
)
k
r ,2
k ψ r ,2 ,
∂t
∂t
W (U)
W (Γ1 )
for k = 0, 1, 2, r = 0, 1.
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Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
Shift of solutions
Shifted solution: Let p = p − Ψ, then p satisfies
∂p
= ∇ · (K (|∇p|)∇p) − Ψt
∂t
p=0
on U × (0, ∞),
Γ1 × (0, ∞).
on
Define:
Z
H(ξ,~a) =
ξ2
√
K ( s,~a)ds.
0
We denote H(x, t) = H[p](x, t) = H(|∇p(x, t)|,~a).
Constants in estimates below depend on χ(~a).
Luan H. - Texas Tech
Forchheimer Equations - III
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A priori Estimates - I(a)
Lemma
One has
1d
2 dt
Z
Z
2
p (x, t)dx ≤ −C
U
H(x, t)dx + CG1 (t),
U
where
Z
G1 (t) =
Z
2−a Z
1
r (1−a)
r
r0
|∇Ψ(x, t)| dx+
|Ψt (x, t)| dx 0
+
|Ψt (x, t)|r0 dx 0
U
2
U
U
with r0 denoting the conjugate exponent of (2 − a)∗ , thus explicitly having
the value
n(2 − a)
n(2 + αN )
r0 =
=
.
(2 − a)(n + 1) − n
n + 2 + αN
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Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
A priori Estimates - I(b)
Corollary
One has for t ≥ 0 that
Z
Z
p 2 (x, t)dx ≤
p 2 (x, 0)dx + C Λ1 (t),
U
U
where
Z
Λ1 (t) =
t
G1 (τ )dτ.
0
In the case deg(g ) ≤ 4/(n − 2) one has
Z
Z
2
p (x, t)dx ≤
p 2 (x, 0)dx + C Λ2 (t),
U
U
2
where Λ2 (t) = (1 + Env (G1 )(t)) 2−a , with Env (G1 )(t) being a continuous,
increasing envelop of the function G1 (t) on [0, ∞).
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A priori Estimates - I: Proofs
No Degree Condition: minimal information
With Degree Condition: Sobolev-Poincare inequality and non-linear
differential inequality (weaker than the usual Gronwall’s)
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Non-linear differential inequality
Lemma
Suppose
y 0 ≤ −Ay α + f (t),
for all t > 0, with A, α > 0 and y (t), f (t) ≥ 0.
Let F (t) be a continuous, increasing envelop of f (t) on [0, ∞). Then one
has
y (t) ≤ y (0) + A−1/α F (t)1/α , ∀t ≥ 0.
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A priori Estimates - II(a)
Lemma
For any ε > 0, one has
Z
Z
Z
d
2
p t (x, t)dx ≤ ε
H(x, t)dx +
H(x, t)dx + Cε G2 (t),
dt U
U
U
where Cε is positive and
Z
Z
G2 (t) =
|∇Ψt (x, t)|2 dx +
|Ψt (x, t)|2 dx.
U
U
Consequently, one has
Z
Z
Z
d
2
2
H(x, t) + p (x, t)dx + p t (x, t)dx ≤ −C
H(x, t)dx +CG3 (t),
dt U
U
U
where G3 (t) = G1 (t) + G2 (t).
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Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
A priori Estimates - II(b)
Corollary
(i) Given δ > 0, there is Cδ > 0 such that for all t ≥ 0 one has
Z t
Z
Z
δt
H(x, t)dx ≤ e
H(x, 0)dx + Cδ
e δ(t−τ ) G2 (τ )dτ.
U
U
0
(ii) For t ≥ 0, one has
Z
2
Z tZ
H(x, t) + p (x, t)dx +
U
0
p 2t (x, τ )dxdτ
U
Z
≤
H(x, 0) + p 2 (x, 0)dx + C
U
Luan H. - Texas Tech
Forchheimer Equations - III
Z
t
G3 (τ )dτ.
0
Applied Math. Seminar, Sept.15&22.2010
A priori Estimates - II(c)
Corollary
For t ≥ 0, one has
Z
Z
Z
−C1 t
H(x, t)dx ≤e
H(x, 0)dx + C
p 2 (x, 0)dx
U
U
U
Z t
+C
e −C1 (t−τ ) Λ1 (τ ) + G3 (τ ) dτ.
0
In case deg(g ) ≤ 4/(n − 2), one has for t ≥ 0 that
Z
Z
Z
−C1 t
p 2 (x, 0)dx
H(x, t)dx ≤e
H(x, 0)dx + C
U
U
Z Ut
e −C1 (t−τ ) Λ2 (τ ) + G3 (τ ) dτ.
+C
0
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Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
A priori Estimates - II(d)
Lemma
Suppose deg(g ) ≤
4
n−2 .
Then
Z
Z
2
H(x, t)dx +
|∇Ψ(x, t)| dx ,
p (x, t)dx ≤ Ch(t)
Z
2
U
U
Z
2
p (x, t)dx
U
where
2−a
2
U
Z
Z
≤C 1+
H(x, t)dx + C
U
|∇Ψ(x, t)|2 dx,
U
Z
a
2−a
h(t) = 1 +
H(x, t)dx
.
U
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Forchheimer Equations - III
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A priori Estimates - II(e)
Proposition
Suppose deg(g ) ≤
Z
4
n−2 .
One has the following two estimates
2
H(x, t) + p (x, t)dx ≤ e
−C1
Rt
0
h−1 (τ )dτ
Z
H(x, 0) + p 2 (x, 0)dx
U
U
Z
+C
t
e −C1
Rt
τ
h−1 (θ)dθ
G3 (τ )dτ,
0
and
Z
2
Z
H(x, t)+p (x, t)dx ≤
U
2
H(x, 0)+p 2 (x, 0)dx +C 1+Env (G3 )(t) 2−a .
U
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Forchheimer Equations - III
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A priori Estimates - III(a)
Let q = pt , q = p t .
Lemma
One has for t > 0 that
Z
Z
Z
Z
d
2
2
2
q dx ≤ −C
K (|∇p|)|∇q| dx + C
|∇Ψt | dx +
|qΨtt |dx.
dt U
U
U
U
Note: May happen
Z
lim sup
t→0
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p 2t (x, t)dx = ∞.
U
Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
A priori Estimates - III(b)
Proposition
One has for t ≥ t0 > 0 that
Z
≤
Z t Z
U
+C
0
If deg(g ) ≤
Z
4
n−2 ,
t
e
−C1
Rt
τ
0
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Z
2
U
then for t ≥ t0 > 0 one has
≤
U
+C
hZ
U
p 2t (x, t)dx
Z
i
t0
H(x, 0) + p (x, 0)dx + C
G3 (τ )dτ
U
0
Z
2
r
2
r0
|∇Ψt (x, τ )| dx + h(τ )
|Ψtt (x, τ )| dx 0 dτ.
t0−1
p 2t (x, t)dx
−C
t0−1 e 1
Rt
1
t0 h(τ ) dτ
hZ
2
Z
t0
H(x, 0)+p (x, 0)dx+C
G3 (τ )dτ
U
0
Z
Z
1
dθ
2
h(θ)
|∇Ψt (x, τ )| dx + h(τ )
|Ψtt (x, τ )|2 dx dτ.
U
U
Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
A priori Estimates - IV(a)
Proposition
Suppose deg(g ) ≤ 4/(n − 2).
(i) For any t ≥ t0 > 0 one has
Z
R
hZ
−C t h−1 (τ )dτ
H(x, 0)
H(x, t) + p 2t (x, t) + p 2 (x, t)dx ≤ (1 + t0−1 )e 1 t0
U
U
Z t0
Z t
i
R t −1
2
+ p (x, 0)dx + C
G3 (τ )dτ + C
e −C1 τ h (θ)dθ G4 (τ )dτ,
0
where G4 (t) = G3 (t) +
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0
R
U
Ψ2tt (x, t)dx.
Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
A priori Estimates - IV(b)
Proposition (continued)
(ii) Assume, in addition, that
Z ∞
def
M0 ==
e −C1 (t−τ ) Λ2 (t) + G3 (t) dτ < ∞.
0
Then there is d0 > 0 depending on the initial data of the solution and the
value M0 above so that
Z
hZ
−1 −d0 (t−t0 )
2
2
H(x, 0)
H(x, t) + p t (x, t) + p (x, t)dx ≤ (1 + t0 )e
U
U
Z t0
Z t
i
+ p 2 (x, 0)dx + C
G3 (τ )dτ + C
e −d0 (t−τ ) G4 (τ )dτ,
0
0
for all t ≥ t0 > 0.
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A priori Estimates - IV(c): No condition on h(t)
Proposition
Suppose deg(g ) ≤
Z
U
4
n−2 .
One has for t ≥ t0 > 0 that
H(x, t) + p 2t (x, t) + p 2 (x, t)dx
Z
2
−1
≤ (1 + t0 )
H(x, 0) + p 2 (x, 0)dx + C 1 + Env (G4 )(t) 2−a .
U
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A priori Estimates - V(a): Uniform bounds
Theorem
Suppose deg(g ) ≤ 4/(n − 2). Assume that
sup kψ(·, t)kW 1,2 (Γ1 ) , sup kψt (·, t)kW 1,2 (Γ1 ) < ∞.
[0,∞)
[0,∞)
Then
Z
sup
Z
2
H(x, 0) + p 2 (x, 0)dx + C 1+
H(x, t) + p (x, t)dx ≤ 4
[0,∞) U
U
2−a
2
sup kψ(·, t)k2W 1,2 (Γ1 ) + sup kψt (·, t)kL1−a
2 (Γ ) + sup kψt (·, t)kW 1,2 (Γ1 )
1
[0,∞)
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2
2−a
.
[0,∞)
[0,∞)
Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
A priori Estimates - V(b): Uniform bounds
Theorem (continued)
If, in addition,
sup kψtt (·, t)kL2 (Γ1 ) < ∞,
[0,∞)
then for any t0 > 0 one has
Z
sup
H(x, t) +
pt2 (x, t)
2
+ p (x, t)dx ≤ C (1 +
[t0 ,∞) U
2
+ p (x, 0)dx +
Ct0−1
Z
Γ1
t0−1 )
Z
H(x, 0)
U
|ψ(x, 0)|2 dσ + C 1 + sup kψ(·, t)k2W 1,2 (Γ1 )
[0,∞)
2−a
2
2
+ sup kψt (·, t)kL1−a
2 (Γ ) + sup kψt (·, t)kW 1,2 (Γ1 ) + sup kψtt (·, t)kL2 (Γ1 )
1
[0,∞)
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[0,∞)
2
2−a
.
[0,∞)
Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
Dependence on the boundary data
Let p1 (x, t) and p2 (x, t) be two solutions of the IBVP with the boundary
profiles ψ1 (x, t) and ψ2 (x, t), respectively. Let Ψk (x, t) be an extension of
ψk (x, t), for k = 1, 2.
We denote
z(x, t) = p1 (x, t) − p2 (x, t),
p k = pk − Ψk ,
Ψ(x, t) = Ψ1 (x, t) − Ψ2 (x, t),
k = 1, 2,
z = p 1 − p 2 = z − Ψ.
Let Hk (x, t) = H[pk ](x, t) = H(|∇pk (x, t)|,~a) for k = 1, 2.
deg(g )
deg(g )
a
Recall that a = deg(g
)+1 and b = 2−a = deg(g )+2 .
def R
We will establish various estimates for Z (t) == U z 2 (x, t)dx, for t ≥ 0.
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First, we derive a general differential inequality for Z (t).
Lemma
One has for all t > 0,
Z
Z
2
1d
2−a
2
z dx ≤ − C
|∇z|2−a dx
(1 + kH1 kL1 + kH2 kL1 )−b
2 dt U
U
+ C (1 + kH1 kL1 + kH2 kL1 )−b k∇Ψk2L2−a
+ C (kH1 kL1 + kH2 kL1 )1/2 k∇ΨkL2
+ C (1 + kH1 kL1 + kH2 kL1 )b kΨt k2Lr0 ,
where r0 is the conjugate of (2 − a)∗ .
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Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
Let Gj [Ψk ], k = 1, 2, j = 1, 2, 3, 4, denote the quantity Gj for
corresponding solution pk with boundary data extension Ψk . Similarly, let
Λj [Ψk ], k = 1, 2, j = 1, 2, denote the corresponding quantity Λj defined for
Ψk .
Let m(t) = m1 (t) + m2 (t), where for k = 1, 2,
Z
Z
−C1 t
mk (t) = e
Hk (x, 0)dx +
p 2k (x, 0)dx
U
U
Z t
+
e −C1 (t−τ ) Λ1 [Ψk ](τ ) + G3 [Ψk ](τ ) dτ.
0
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Estimate for finite time interval
Theorem
One has for all t ≥ 0 that
Z
Z
2
z (x, t)dx ≤
U
2
z (x, 0)dx + C
U
Z t
0
1/2
+ m(τ )
k∇Ψ(·, τ )k2L2−a
k∇Ψ(·, τ )kL2 + (1 + m(τ ))b kΨt (·, τ )k2Lr0 dτ.
Consequently, for any given T > 0,
Z
2
Z
z (x, t)dx ≤ 4
sup
[0,T ] U
U
z 2 (x, 0)dx+6 sup kΨ(·, t)k2L2 +CT sup k∇Ψ(·, t)
[0,T ]
[0,T ]
+ CT (1 + A∗ + D∗ (T ))δ sup k∇Ψ(·, t)kL2 + sup kΨt (·, t)k2Lr0 ,
[0,T ]
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[0,T ]
Applied Math. Seminar, Sept.15&22.2010
Above, δ = max{1/2, b},
Z
Z
Z
Z
2
A∗ =
H1 (x, 0)dx +
p 1 (x, 0)dx +
H2 (x, 0)dx +
p 22 (x, 0)dx,
U
D∗ (T ) =
U
2
X
Z
sup
U
t
U
e −C1 (t−τ ) Λ1 [Ψk ](τ ) + G3 [Ψk ](τ ) dτ.
k=1 [0,T ] 0
Luan H. - Texas Tech
Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
Corollary
Suppose for both k = 1, 2 and t ≥ 0 one has
2−a
r1
k∇Ψk (·, t)k2L2 + k(Ψk )t (·, t)kL1−a
r0 ≤ C (1 + t)
and
k∇(Ψk )t (·, t)k2L2 + k(Ψk )t (·, t)k2L2 ≤ C (1 + t)r2 ,
where r1 , r2 > 0. Let r3 = 1 + max{r1 + 1, r2 }. Then
Z
2
Z
z (x, t)dx ≤
z 2 (x, 0)dx
U
Z t
+ C (1 + A∗ )
(1 + τ )r3 /2 k∇Ψ(·, τ )kL2 + (1 + τ )r3 b kΨt (·, τ )k2Lr dτ,
U
0
where A∗ is defined previously.
Luan H. - Texas Tech
Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
Estimate for all time under the Degree Condition
R
Using appropriate estimates of U Hk (x, t)dx, we denote
Z
Z
−C1 t
mk (t) = e
p 2k (x, 0)dx
Hk (x, 0)dx +
U
Z t U
+
e −C1 (t−τ ) Λ2 [Ψk ](τ ) + G3 [Ψk ](τ ) dτ,
k = 1, 2.
0
Also, let
0
m(t) = m1 (t) + m2 (t) and S(t , t) =
Luan H. - Texas Tech
Forchheimer Equations - III
Z
t
(1 + m(τ ))−b dτ.
t0
Applied Math. Seminar, Sept.15&22.2010
Theorem
Suppose deg(g ) ≤ 4/(n − 2). Then for all t ≥ 0 one has
Z
Z
2
−C1 S(0,t)
z (x, t)dx ≤ e
z 2 (x, 0)dx
U
U
Z t
+C
e −C1 S(τ,t) k∇Ψ(·, τ )k2L2−a
0
+ m(τ )1/2 k∇Ψ(·, τ )kL2 + (1 + m(τ ))b kΨt (·, τ )k2Lr0 dτ.
Luan H. - Texas Tech
Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
Corollary
Suppose deg(g ) ≤ 4/(n − 2). Assume that
2 X
k=1
sup Λ2 [Ψk ](t) + sup G3 [Ψk ](t) < ∞.
[0,∞)
[0,∞)
Then one has
Z
Z
sup
z 2 (x, t)dx ≤ 4
z 2 (x, 0)dx + C sup kΨ(·, t)k2L2
[0,∞) U
U
+ C sup
[0,∞)
[0,∞)
Ab∗ k∇Ψ(·, t)k2L2−a +
b+ 1
A∗ 2 k∇Ψ(·, t)kL2
2
kΨ
(·,
t)k
+ A2b
r
t
∗
L0 ,
where
2 Z
X
A∗ = 1+
|∇pk (x, 0)|2−a +p 2k (x, 0)dx+ sup Λ2 [Ψk ](t)+ sup G3 [Ψk ](t)
k=1
U
Luan H. - Texas Tech
[0,∞)
Forchheimer Equations - III
[0,∞)
Applied Math. Seminar, Sept.15&22.2010
Corollary
Suppose deg(g ) ≤ 4/(n − 2). Assume that
lim S(0, t) = ∞,
t→∞
1
lim (1 + m(t))b+ 2 k∇Ψ(·, t)kL2 = 0,
R
Then limt→∞ U z 2 (x, t)dx = 0.
t→∞
Luan H. - Texas Tech
lim (1 + m(t))b kΨt (·, t)kLr0 = 0.
t→∞
Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
Dependence on the Forchheimer polynomials
Let the boundary data ψ(x, t) on Γ1 be fixed. For each coefficient vector
~a, we denote by p(x, t;~a) the solution of
∂p pt = ∇ · (K (|∇p|)∇p), p|Γ1 = ψ,
= 0,
∂N Γ2
with K = K (ξ,~a).
Let g1 = g (s,~a(1) ) and g2 = g (s,~a(2) ) be two functions of class FP(N, α
~ ).
Let pk = pk (x, t;~a(k) )) for k = 1, 2. Let p = p1 − p2 , then
∂p
= ∇ · (K (|∇p1 |,~a(1) )∇p1 ) − ∇ · (K (|∇p2 |,~a(2) )∇p2 ).
∂t
Multiplying this equation by p and integrating by parts over the domain
yield
1d
2 dt
Z
2
Z
p dx = −
U
Luan H. - Texas Tech
(K (|∇p1 |,~a(1) )∇p1 −K (|∇p2 |,~a(2) )∇p1 )·(∇p1 −∇p2 )dx
U
Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
Perturbed Monotonicity
Let ~a and ~a0 be two arbitrary vectors. We denote by ~a ∨ ~a0 and ~a ∧ ~a0 the
maximum and minimum vectors of the two, respectively, with components
(~a ∨ ~a0 )j = max{aj , aj0 }
and
(~a ∧ ~a0 )j = min{aj , aj0 }.
Then component-wise one has ~a ∧ ~a0 ≤ ~a,~a0 ≤ ~a ∨ ~a0 .
Define χ(~a,~a0 ) = max{χ(~a), χ(~a0 )}. Note that
χ(~a ∨ ~a0 ), χ(~a ∧ ~a0 ) ≤ χ(~a,~a0 ),
χ(t~a + (1 − t)~a0 ) ≤ χ(~a,~a0 )
Luan H. - Texas Tech
Forchheimer Equations - III
∀t ∈ [0, 1].
Applied Math. Seminar, Sept.15&22.2010
Perturbed Monotonicity
Lemma
Let g (s,~a) and g (s,~a0 ) belong to class FP(N, α
~ ). Then for any y , y 0 in
n
R , one has
(K (|y |,~a)y − K (|y 0 |,~a0 )y 0 ) · (y − y 0 ) ≥ (1 − a)K (|y | ∨ |y 0 |,~a ∨~a0 )|y − y 0 |2
− Nχ(~a,~a0 )|~a − ~a0 |K (|y | ∨ |y 0 |,~a ∧ ~a0 )(|y | ∨ |y 0 |)|y − y 0 |,
where a ∈ (0, 1).
Luan H. - Texas Tech
Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
Let
Z
Z
M k (t) = 1 + e −C1 t
p 2k (x, 0)dx
|∇pk (x, 0)|2−a dx +
U
U
Z t
+
e −C1 (t−τ ) Λ1 (τ ) + G3 (τ ) dτ, k = 1, 2,
0
and M = M 1 + M 2 . One has
Z
Z
(1)
(2)
2
K (|∇pk |,~a ∧ ~a )|∇pk | dx ≤ C + C
|∇pk |2−a dx ≤ C M k ,
U
U
where C depends on χ(~a(1) ∧ ~a(2) ) and χ(~a(k) ).
Denote Hk (x, t) = H(|∇pk (x, t)|,~a(k) ) for k = 1, 2.
Luan H. - Texas Tech
Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
Finite time estimate
First, we obtain the continuous dependence for finite time.
Proposition
For t ≥ 0 one has
Z
Z
2
|p1 (x, t) − p2 (x, t)| dx ≤
|p1 (x, 0) − p2 (x, 0)|2 dx
U
U
Z t
+ C |~a(1) − ~a(2) |
M(τ )dτ,
0
where C > 0 depends on N, αN , and χ(~a(1) ,~a(2) ). Consequently, the
solution p(x, t;~a) depends continuously (in finite time intervals) on the
initial data and the coefficient vector ~a ∈ R(N).
Luan H. - Texas Tech
Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
Large time estimate under the Degree Condition
Under the Degree Condition,
where
−C1 t
R
U
Z
|∇pk (x, t)|2−a dx is bounded by CMk (t)
Z
p 2k (x, 0)dx
Mk (t) = 1 + e
|∇pk (x, 0)| dx +
U
U
Z t
+
e −C1 (t−τ ) Λ2 (τ ) + G3 (τ ) dτ, k = 1, 2,
2−a
0
and M(t) = M1 (t) + M2 (t).
Luan H. - Texas Tech
Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
Proposition
Suppose αN ≤ 4/(n − 2). Then for t ≥ 0,
Z
Z
R
2
−C1 0t M(τ )−b (τ )dτ
|p1 (x, t) − p2 (x, t)| dx ≤ e
|p1 (x, 0) − p2 (x, 0)|2 dx
U
U
Z t
Rt
−b
+ C2 |~a(1) − ~a(2) |
e −C1 τ M(θ) (θ)dθ M(τ )dτ.
0
def
Assume, in addition, that M0 == sup[0,∞) M(t) < ∞ then
Z
2
|p1 (x, t) − p2 (x, t)| dx ≤ e
−C1 t/M0b
U
+
Z
|p1 (x, 0) − p2 (x, 0)|2 dx
U
C2 C1 M01+b |~a(1) − ~a(2) |
for all t ≥ 0, and consequently
Z
lim sup
|p1 (x, t) − p2 (x, t)|2 dx ≤ C2 C1 M01+b |~a(1) − ~a(2) |.
t→∞
U
Luan H. - Texas Tech
Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
Uniform estimate in the coefficients
Let D be a compact set in R(N). Define
χ̂(D) = max{χ(~a), ~a ∈ D}.
Note that χ̂(D) ∈ (0, ∞) and for any ~a ∈ D, one has
χ̂(D)−1 ≤ χ(~a)−1 ≤ χ(~a) ≤ χ̂(D).
Let ~a(k) belong to D for k = 1, 2. Set
Z
2 Z
X
2−a
2
|∇pk (x, 0)| dx +
A∗ = 2 +
p k (x, 0)dx
k=1
U
Z
U
t
λ(t) = 1 +
e −C1 (t−τ ) Λ1 (τ ) + G3 (τ ) dτ,
0
Z
λ(t) = 1 +
t
e −C1 (t−τ ) Λ2 (τ ) + G3 (τ ) dτ,
0
where C1 depends on N, αN and χ̂(D).
Then M(t) ≤ A∗ λ(t), M(t) ≤ A∗ λ(t).
Luan H. - Texas Tech
Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
Theorem
Assume, additionally, that αN ≤ 4/(n − 2).
(i) One has for t ≥ 0 that
Z
Z
Rt
−b dτ
2
−C1 A−b
λ(τ
)
∗
0
|p1 (x, t) − p2 (x, t)| dx ≤ e
|p1 (x, 0) − p2 (x, 0)|2 dx
U
U
Z t
−b R t
−b
(1)
(2)
~
e −C1 A∗ τ λ(θ) dθ λ(τ )dτ.
+ C2 A∗ |~a − a |
0
def
(ii) Assume, in addition, that M0 == sup[0,∞) λ(t) < ∞. Then
Z
2
|p1 (x, t) − p2 (x, t)| dx ≤e
−C1 t/(A∗ M0 )b
U
Z
|p1 (x, 0) − p2 (x, 0)|2 dx
U
+ C2 C1−1 (A∗ M0 )1+b |~a(1) − ~a(2) |.
Consequently
Z
lim sup
|p1 (x, t) − p2 (x, t)|2 dx ≤ C2 C1−1 (A∗ M0 )1+b |~a(1) − ~a(2) |.
t→∞
U
Luan H. - Texas Tech
Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
Next Semester - Spring 2011
Forchheimer Equations in Porous Media - Part IV
Boundary Flux condition
Refined asymptotic estimates: limsup estimates for nonlinear
differential inequalities, uniform Gronwall inequality, ...
Continuous dependence for pressure gradients and velocity
Luan H. - Texas Tech
Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010
Next Semester - Spring 2011
Forchheimer Equations in Porous Media - Part IV
Boundary Flux condition
Refined asymptotic estimates: limsup estimates for nonlinear
differential inequalities, uniform Gronwall inequality, ...
Continuous dependence for pressure gradients and velocity
THANK YOU!
Luan H. - Texas Tech
Forchheimer Equations - III
Applied Math. Seminar, Sept.15&22.2010