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Physica D
journal homepage:www.elsevier.com/locate/physd
An ODE for boundary layer separation on a sphere and a
hyperbolic space
Chi Hin Chan
a,∗, Magdalena Czubak
b, Tsuyoshi Yoneda
caDepartment of Applied Mathematics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, 30010, Taiwan, ROC bDepartment of Mathematical Sciences, Binghamton University (SUNY), Binghamton, NY 13902-6000, USA
cDepartment of Mathematics, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551, Japan
a r t i c l e i n f o
Article history:
Received 9 February 2014 Accepted 9 May 2014 Available online 24 May 2014 Communicated by R. Temam
Keywords:
Navier–Stokes equation Riemannian manifolds Boundary layer separation Coriolis effect
a b s t r a c t
Ma and Wang derived an equation linking the separation location and times for the boundary layer separation of incompressible fluid flows. The equation gave a necessary condition for the separation (bifurcation) point. The purpose of this paper is to generalize the equation to other geometries, and to phrase it as a simple ODE. Moreover we consider the Navier–Stokes equation with the Coriolis effect, which is related to the presence of trade winds on Earth.
© 2014 Elsevier B.V. All rights reserved.
1. Introduction
In the beginning of the 20th century, Prandtl proposed the boundary layer theory. Since then there has been a lot of exten-sive developments in the theory (see Rosenhead [1] for example). In general, the laminar flow in the boundary layer should be gov-erned by a boundary layer equation, which is deduced from the Navier–Stokes equations. The existence of the singularity in the steady boundary layer flow along fixed wall has led to important advances in the understanding of the steady boundary layer sep-aration. In this point of view, Van Dommelen and Shen [2] made a key observation of shock singularities with numerical computa-tions. In the beginning of the 21st century, Ghil, Ma and Wang [3–6] have developed a rigorous theory on the boundary layer separa-tion of incompressible fluid flows. Their articles are oriented to-wards the structural bifurcation and boundary layer separation of the solution to the Navier–Stokes equations. In particular, in [6] au-thors established a simple equation, which they call a ‘‘separation equation’’, linking the separation location and times. Furthermore, they showed that the structural bifurcation occurs at a degenerate singular point with integer index of the velocity field at the criti-cal bifurcation time. Their theory is based on the classification of
∗Corresponding author.
E-mail addresses:cchan@math.utexas.edu,cchan@math.nctu.edu.tw (C.H. Chan),czubak@math.binghamton.edu(M. Czubak),
yoneda@math.titech.ac.jp(T. Yoneda).
the detailed orbit structure of the velocity field near the bifurca-tion time and locabifurca-tion (see also [7]). On the other hand, Ghil, Liu, Wang and Wang [8] gave a new rigorous argument of ‘‘adverse pressure gradient’’ mathematically under certain conditions. The conditions were consistent with the careful numerical experiment also found in [8]. The appearance of the adverse pressure gradient is well known to be the main mechanism for the boundary-layer separation in physics.
The purpose of this paper is to obtain the separation equation of Ma and Wang’s in other geometries and to phrase it as an ODE. In order to state our main result, we need to explain Ma and Wang’s separation equation precisely. Let K be a compact domain in R2
with Cr+1boundary,
∂
K , for r≥
2. Consider the Navier–Stokesequation on K given by ut
+ ∇
uu−
1u+ ∇
p=
0,
div u=
0,
u|∂K=
0,
u(
x,
0) = φ(
x),
φ
|∂K=
0.
(1.1)Since we only consider the flow near the boundary, we can replace
K by R2
−
K .We call a point p
∈
∂
K ‘‘∂
-regular point of u’’ if the normal derivative of the tangential component of u at p is nonzero, i.e.,∂(
u·
τ)(
p)/∂
n̸=
0, otherwise, p∈
∂
K is called a∂
-singular point (bifurcation point) of u.Theorem 1.1 ([6]). Let K be a compact domain in R2 with Cr+1 boundary,
∂
K , for r≥
2. Let p0∈
∂
K , and t0≥
0. If(
p0,
t0)
is a http://dx.doi.org/10.1016/j.physd.2014.05.004∂
-singular point (bifurcation point) of the solution u of (1.1), then∂φτ
(
p0)
∂
n=
t0 0∇ ×
1u−
k1u·
τ
dt,
(1.2)where
∇ ×
1u=
∂τ
(
∆·
n) − ∂
n(
1u·
τ)
, and k(
p0)
is the curvatureof
∂
K at p0.Eq.(1.2)is called the ‘‘separation equation’’.
We now give an example which tells us imposing inflow pro-file is useful. A wind turbine system consisting of a diffuser shroud with a broad-ring at the exit periphery and a wind turbine inside it was developed by Ohya and Karasudani [9]. Their experiments show that a diffuser-shaped (not nozzle-shaped) structure can ac-celerate the wind at the entrance of the body. This is called ‘‘wind-lens phenomena’’. A strong vortex formation with a low-pressure region is created behind the broad brim. The wind flows into a low-pressure region, and the wind velocity is increased more near the entrance of the diffuser. In general, creation of a vortex needs sepa-ration phenomena near a boundary (namely, bifurcation phenom-ena), and before separating from the boundary, the flow moves towards the reverse direction near the boundary against the lami-nar flow (inflow) direction. In order to consider such phenomena in pure mathematics, imposing inflow profile at the entrance should be reasonable.
We moreover consider the situation on a sphere and a hyper-bolic space (we can easily deduce the ODE in the Euclidean case). In the case of the sphere, one motivation comes from studying the flow on the Earth (seeCorollary 1.4).
Now, we write the equation on a Riemannian manifold, M, where M is taken either to be a sphere S2
(
a2)
or a hyperbolic spaceM
=
H2(−
a2)
. We write the equation in the language ofdifferen-tial 1-forms as follows.
Let O be the base point in M. Let
(
r, θ)
be the normal polar co-ordinates on M. Then we have the following orthonormal moving frame e1=
∂
r,
(1.3) e2=
1 sa(
r)
∂θ
,
(1.4)where sa
(
r) =
sin(aar) if M=
S2(
a2)
or sa(
r) =
sinha(ar) if M=
H2
(−
a2)
. We also introduce ca(
r)
, where ca(
r) =
cos(
ar)
if M=
S2
(
a2)
or ca(
r) =
cosh(
ar)
if M=
H2(−
a2)
. Note ddrsa
(
r) =
ca(
r).
(1.5) For simplicity, in the sequel, we omit the writing of subscripts a insaand caand simply write s
,
c. The associated dual frame to{
e1,
e2}
can be written as
e1
=
dr,
(1.6)e2
=
s(
r)
dθ.
(1.7)Hence the volume form on M is given by VolM
=
e1∧
e2=
sdr∧
dθ
.Let
∇
be the Levi-Civita connection on M. We have∇
∂r∂
r=
0,
(1.8)∇
∂r∂θ
= ∇
∂θ∂
r=
ce2=
c s∂θ
,
(1.9)∇
∂θ∂θ
= −
cs∂
r.
(1.10) These imply∇
e1e1= ∇
e1e2=
0,
∇
e2e2= −
c se1,
∇
e2e1=
c se2.
(1.11)Let d be the distance function on M. Define an obstacle K on M by
K
=
(
BO(δ)) = {
p∈
M:
d(
p,
O) ≤ δ}
. Consider a smooth vectorfield u defined on a neighborhood near
∂
K . Then u can be written as u=
ure1+
uθe2,
for some locally defined smooth functions ur
,
uθ. By ‘‘lowering theindex’’ we can obtain a 1-form u∗
=
ure1
+
uθe2. For simplicitywe just write u for both the vector field and the 1-form. Recall the Hodge star operator,
∗
, is a linear operator that sends k-forms ton
−
k-forms and is defined byα ∧ ∗β =
g(α, β)
VolM.
(1.12)Then
∗ ∗
α = (−
1)
nk+kα,
(1.13)where n is the dimension of the manifold, and k the degree of
α
. Here, by a direct computation∗
e1=
e2,
∗
e2= −
e1,
∗
VolM=
1.
Recall
d∗
α = (−
1)
nk+n+1∗
d∗
α.
(1.14)So for two dimensional manifolds we have d∗
= − ∗
d∗
. Then theNavier–Stokes equation on M
−
K is given by ut+ ∇
uu−
1u−
2Ric u+
dp=
0,
d∗u
=
0,
u|∂K
=
0,
u
(
x,
0) =
u0(
x),
u0|∂K=
0.
(1.15)
For fixed p0
∈
∂
K , and u a solution of(1.15), let us give the keyparameters k
=
ka,δ:=
ca(δ)
sa(δ)
,
α
1(
t) := ∂
ruθ(
t,
p0),
α
2(
t) := ∂
r2uθ(
t,
p0),
α
3(
t) := ∂
r3uθ(
t,
p0),
η(
t) :=
1 s2(δ)
∂
r∂
2 θuθ(
t,
p0).
Note that k includes both curvature of the manifold and curvature of the boundary. Our main theorem is the following:
Theorem 1.2. Let
α
1(
0) >
0 (initial data), andα
2(
t)
,α
3(
t)
andη(
t)
be given functions. Then
α
1(
t)
satisfies the following ODE:∂
tα
1(
t) = −
k2α
1(
t) + α
3(
t) +
2kα
2(
t) +
2η(
t).
Remark 1.3. We give five remarks.•
A∂
-singular point (bifurcation point) occurs at t0iff a functionα
1(
t)
satisfiesα
1(
t0) =
0.•
The above result is a generalization of [10] which is considered in the Euclidean space R2.•
We can regardα
1(
t)
,α
2(
t)
andα
3(
t)
as a part of the inflowprofile. However
η(
t)
is not. Let us be more precise. Choose˜
p
∈
∂
K close to p0∈
∂
K , and let˜
K
:= {
p∈
M−
K:
d(
p, ˜
p) <
d(
p0, ˜
p)}.
Then
α
1(
t)
,α
2(
t)
andα
3(
t)
can be determined by u(·,
t)
on∂ ˜
K∩
K near p0∈
∂
K (boundary value).η(
t)
can be determinedby u
(·,
t)
inK˜
∩
K near p0∈
∂
K (interior flow).•
We can find a geometric meaning ofη(
t)
(see also [10]).– Convexing streamlines: We can see (geometrically) convex-ing streamlines near the boundary iff
η(
t) <
0.
– Almost parallel streamlines: We can see (geometrically) almost parallel streamlines near the boundary iff
η(
t) =
0.
– Concaving streamlines: We can see (geometrically) concav-ing streamlines near the boundary iff
•
It is reasonable to assume uθ does not grow polynomially for the r direction (this is due to the observation of the ‘‘boundary layer’’, since the flow should be a uniform one away from the boundary). Thus, it should be reasonable to focus on the following two cases:– (Poiseuille type profile)
−
k2α
1
(
t) +
2kα
2(
t) <
0 (α
1(
t) >
0,α
2(
t) <
0) andα
3(
t)
is small comparing withα
1(
t)
andα
2(
t)
.– (Before separation profile) 2k
α
2(
t) + α
3(
t) <
0 (α
2(
t) >
0,α
3(
t) <
0) andα
1(
t)
is small comparing withα
2(
t)
andα
3(
t)
.In this point of view, the well-known physical phenomena of ‘‘adverse pressure gradient’’ occurs in ‘‘before separation profile’’, since
α
2(
t) >
0 and dp=
1u on the boundary.Our method can be applied to geophysics, in particular, to the ‘‘trade winds’’ on Earth. The trade winds are the easterly surface winds that can be found in the tropics, within the lower por-tion of the Earth’s atmosphere near the equator. The Coriolis ef-fect is responsible for deflecting the surface air, which flows from subtropical high-pressure belts towards the Equator, towards the west in both hemispheres. In the corollary below, we consider the Navier–Stokes equation with the Coriolis effect on a rotating sphere. The equation is
ut
+ ∇
uu− △
u−
2Ric u+
β
cos(
ar) ∗
u+
dp=
0,
(1.16)d∗u
=
0,
(1.17)u
(
x,
0) =
u0,
(1.18)where
β ∈
R is a Coriolis parameter. The termβ
cos(
ar) ∗
u repre-sents the effect upon the velocity u due to the rotation of the sphere with constant speedβ
. It is worthwhile to mention that the exis-tence and uniqueness of parallel laminar flows satisfying the sta-tionary version of the above system has been considered in [11]. More details on the Coriolis effect on a sphere can be found in [12], and in the vorticity formulation, for example, in [13].Corollary 1.4. Let u satisfy(1.16)–(1.18)and the following condi-tions
uθ
|
∂K=
0,
ur|
∂K=
λ
0∈
R,
∂
rur|
∂K=
0.
Then
α
1(
t)
satisfies∂
tα
1(
t) = −
k(
k+
λ
0)α
1(
t) + (
2k−
λ
0)α
2(
t) + α
3(
t) +
2η(
t)
+
λ
0β(
a sin(
aδ) −
k cos(
aδ)).
This is proved in Section3.
Remark 1.5. If
λ
0(
a sin(
aδ)−
k cos(
aδ))
is strictly positive, andβ
issufficiently large compared with
α
1(
0) >
0,λ
0, k,α
2(
t)
,α
3(
t)
andη(
t)
, thenα
1(
t)
can never be zero. This expresses that the south (ornorth) flow deflects towards the east (or west). Moreover we can find an asymptotic behavior of
α
1(
t)
:lim t→∞ α1( t) −(2k−λ0) ˜α2+ ˜α3+2η + λ˜ 0β(a sin(aδ) −k cos(aδ)) k(k+λ0) =0, if
α
2(
t) → ˜α
2,α
3(
t) → ˜α
3andη(
t) → ˜η
. 2. Proof of the main theoremWe first prepare the necessary computations and then put them together in Section2.6.
2.1. Divergence free condition in coordinates
If u is divergence free, then d∗u
=
0. Compute 0=
d∗u= − ∗
d∗
u= − ∗
d∗
u= − ∗
d(
ure2−
uθe1)
= − ∗
(∂
r(
sur) + ∂θ
uθ)
dr∧
dθ
= −
1 s(∂
r(
sur) + ∂θ
uθ).
This implies∂
r(
sur) + ∂θ
uθ=
0.
(2.1)In addition on
∂
K , thanks to the no-slip boundary condition, from(2.1)we can deduce
0
=
∂θ
uθ|
∂K= {−
cur−
s∂
rur}|
∂K= −
s∂
rur|
∂K.
(2.2)2.2. Computing normal and tangential components of1u
The goal is to compute g
(
1u,
e1)
and g(
1u,
e2)
. First,−
1u=
dd∗u
+
d∗du=
d∗du. Next du= {
∂
r(
suθ) − ∂θ
ur}
dr∧
dθ =
1 s{
∂
r(
suθ) − ∂θ
ur}
VolM.
It follows 1u= −
d∗du= ∗
d∗
1 s{
∂
r(
suθ) − ∂θ
ur}
VolM= ∗
d1 s{
∂
r(
suθ) − ∂θ
ur}
= ∗
∂
r
1 s∂
r(
suθ)
−
∂
r
1 s∂θ
ur
e1+
1 s∗ {
∂θ
∂
r(
suθ) − ∂
2 θur}
dθ
=
∂
r
1 s(∂
r(
suθ) − ∂θ
ur)
e2−
1 s2{
∂
r(
s∂θ
uθ) − ∂
2 θur}
e1.
Then g(
1u,
e1) =
1 s2{
∂
2 θur−
∂
r(
s∂θ
uθ)}
=
1 s2{
∂
2 θur−
c∂θ
uθ−
s∂
r∂θ
uθ}
,
(2.3)which can be rewritten using(2.1)as follows
g
(
1u,
e1) =
1 s2{
∂
2 θur−
c∂θ
uθ+
s∂
r2(
sur)}
=
1 s2∂
2 θur−
c s2∂θ
uθ∓
a 2u r+
2 c s∂
rur+
∂
2 rur,
(2.4)where
∓
depends on the choice of M. Here, and in the sequel, the upper sign refers to the sphere and the lower sign to the hyperbolic plane. Next g(
1u,
e2) = ∂
r
1 s(∂
r(
suθ) − ∂θ
ur)
=
∂
r
c suθ+
∂
ruθ−
1 s∂θ
ur
= −
1 s2uθ+
c s∂
ruθ+
∂
2 ruθ+
c s2∂θ
ur−
1 s∂
r∂θ
ur (2.5) since∂
r(
c/
s) = −(
1/
s2)
and∂
r(
1/
s) = −(
c/
s2)
. 2.3. Computing1s∂θ
g(
1u,
e1)
on∂
KFirst observe that on
∂
K , from the no-slip boundary condition,(2.2)and(2.4)we have g
(
1u,
e1)|∂
K=
∂
r2ur|
∂K.
Hence 1 s∂θ
g(
1u,
e 1)|∂
K=
1 s∂θ
∂
2 rur|
∂K.
(2.6)We need this formula to estimate the pressure term on the boundary.
2.4. Computing
∂
rg(∇
uu,
e2)
on∂
KFirst, by the properties of the connection and(1.11)
∇
uu= ∇
ure1+uθ e2(
ure1+
uθe2)
=
ur∇
e1(
ure1+
uθe2) +
uθ∇
e2(
ure1+
uθe2)
=
ur(∂
rure1+
∂
ruθe2) +
uθ
1 s∂θ
ure1+
ur c se2+
1 s∂θ
uθe2−
uθ c se1
.
(2.7) Then g(∇
uu,
e2) =
ur∂
ruθ+
uθur c s+
1 suθ∂θ
uθ.
(2.8)Differentiating and evaluating on the boundary and using the no-slip boundary condition, we reduce(2.8)to
∂
rg(∇
uu,
e2)|∂
K=
∂
rur∂
ruθ+
1
s
∂
ruθ∂θ
uθ.
But then the divergence free condition(2.1)again with the no-slip boundary condition imply
∂
rg(∇
uu,
e2)|∂
K=
∂
rur∂
ruθ−
1 s∂
ruθ∂
r(
sur)
∂K=
0.
(2.9) 2.5. Computing∂
rg(∇
p,
e2)
on∂
K First dp=
∂
rpdr+
∂θ
pdθ = ∂
rpe1+
1 s∂θ
pe 2.
Hence∇
p=
∂
rpe1+
1 s∂θ
pe2,
and∂
rg(∇
p,
e2) = ∂
r
1 s∂θ
p
= −
c s2∂θ
p+
1 s∂
r∂θ
p= −
c sg(∇
p,
e2) +
1 s∂
r∂θ
p.
(2.10)Equivalently, we can write(2.10)as
∂
rg(∇
p,
e2) = −
c
sg
(∇
p,
e2) +
1
s
∂θ
g(∇
p,
e1).
(2.11) 2.6. Proof of the formulaWe now follow the proof in [6], but without assuming that p0is
a bifurcation point.
Let p0
∈
∂
K , and t0>
0. Begin by writing∂
r|
p0g(
u(
t0, ·),
e2) − ∂
r|
p0g(
u0,
e2)
=
t0 0 d dt{
∂
r|
p0g(
u(
t, ·),
e2)}
dt.
(2.12) From(1.15)it follows d dt{
∂
rg(
u(
t, ·),
e 2)} = ∂
rg(
1u(
t, ·),
e2) + ∂
rg(
2Ric(
u(
t, ·)),
e2)
−
∂
rg(∇
u(t,·)u(
t, ·),
e2) − ∂
rg(
dp(
t, ·),
e2).
We can simplify by using Ric
(
u) =
a2u if M=
S2(
a2)
and Ric(
u) =
−
a2u if M=
H2(−
a2)
, and write Ric(
u) = ±
a2u. Also on theboundary we can use(2.9)to write
d dt
{
∂
rg(
u(
t, ·),
e2)} = ∂
rg(
1u(
t, ·),
e 2) ±
2a2∂
rg(
u(
t, ·),
e2)
−
∂
rg(∇
p(
t, ·),
e2)
=
∂
rg(
1u(
t, ·),
e2) ±
2a2∂
ruθ−
∂
rg(∇
p(
t, ·),
e2).
Next from(2.11)we have d dt
{
∂
rg(
u(
t, ·),
e2)} = ∂
rg(
1u(
t, ·),
e 2) ±
2a2∂
ruθ+
c sg(∇
p(
t, ·),
e2) −
1 s∂θ
g(∇
p(
t, ·),
e1).
Since on∂
K dp=
1u,
(2.13)going back to(2.12)we obtain (compare this with(1.2))
∂
r|
p0g(
u0,
e2) − ∂
r|
p0g(
u(
t0, ·),
e2)
= −
t0 0∂
rg(
1u(
t,
p0),
e2) ±
2a2∂
ruθ+
c sg(∇
p(
t,
p0),
e2) −
1 s∂θ
g(∇
p(
t,
p0),
e1)
dt= −
t0 0∂
rg(
1u(
t,
p0),
e2) −
1 s∂θ
g(
1u(
t,
p0),
e 1)
+
c sg(
1u(
t,
p0),
e 2) ±
2a2∂
ruθdt.
(2.14)To obtain the necessary and sufficient condition, we write(2.14)
more explicitly as follows. From(2.5)
∂
rg(
1u(
t,
p0),
e2)
=
∂
r
−
1 s2uθ+
c s∂
ruθ+
∂
2 ruθ+
c s2∂θ
ur−
1 s∂
r∂θ
ur
( t,p0)=
−
2 s2∂
ruθ+
c s∂
2 ruθ+
∂
3 ruθ−
1 s∂
2 r∂θ
ur
( t,p0).
(2.15) And again from(2.5)c sg
(
1u(
t,
p0),
e 2) =
c2 s2∂
ruθ+
c s∂
2 ruθ
( t,p0).
(2.16) Then(2.14)–(2.16)and(2.6)give∂
r|
p0g(
u0,
e2) − ∂
r|
p0g(
u(
t0, ·),
e2)
=
t0 0
2 s2∂
ruθ−
c s∂
2 ruθ−
∂
3 ruθ+
1 s∂
2 r∂θ
ur+
1 s∂θ
∂
2 rur−
c2 s2∂
ruθ−
c s∂
2 ruθ∓
2a 2∂
ruθ
dt=
t0 0
2−
c2 s2∓
2a 2
∂
ruθ−
2 c s∂
2 ruθ−
∂
3 ruθ+
2 s∂
2 r∂θ
urdt=
t0 0 c2 s2∂
ruθ−
2 c s∂
2 ruθ−
∂
3 ruθ+
2 s∂θ
∂
2 rurdt.
Finally, by using(2.1)again, we can rewrite the last term as 2 s
∂θ
∂
2 rur|
∂K= −
2 s2∂
r∂
2 θuθ|
∂K.
It follows∂
r|
p0g(
u0,
e2) − ∂
r|
p0g(
u(
t0, ·),
e2)
=
t0 0 c2(δ)
s2(δ)
∂
ruθ(
t,
p0) −
2 c(δ)
s(δ)
∂
2 ruθ(
t,
p0)
−
∂
r3uθ(
t,
p0) −
2 s2(δ)
∂
r∂
2 θuθ(
t,
p0)
dt,
(2.17)or equivalently
α
1(
0) − α
1(
t0) =
t00
k2
α
1(
t) − α
3(
t) −
2kα
2(
t) −
2η(
t)
dt,
which gives the desired ODE.
3. With Coriolis force and the in-flow condition case
Recall here we have
uθ
|
∂K=
0,
ur|
∂K=
λ
0∈
R,
∂
rur|
∂K=
0 (3.1)and we work with(1.16)–(1.18). Then note that
g
(β
cos(
ar) ∗
u,
e2) = β
cos(
ar)
g(∗
u,
e2) = β
cos(
ar)
urand
∂
r|
p0g(β
cos(
ar) ∗
u,
e2) = −
aβλ
0sin(
aδ).
(3.2) Next, recall(2.8) g(∇
uu,
e2) =
ur∂
ruθ+
uθur c s+
1 suθ∂θ
uθ.
Then differentiate, evaluate on the boundary, and this time use
(3.1)to obtain
∂
rg(∇
uu,
e2)|∂
K=
λ
0∂
r2uθ+
λ
0∂
ruθ c s+
1 s∂
ruθ∂θ
uθ.
From the divergence free condition and(3.1)we have
∂
rg(∇
uu,
e2)|∂
K=
λ
0∂
r2uθ+
λ
0∂
ruθ c s−
1 s∂
ruθ∂
r(
sur)
∂ K=
λ
0∂
r2uθ|
∂K.
(3.3) Another place where we obtain an extra term is in(2.13), where due to(2.7),(3.1)and the Coriolis term in(1.16), now we have dp=
1u−
λ
0∂
ruθe2+
2a2λ
0e1−
β
cos(
aδ)λ
0e2on
∂
K . With the Coriolis term(3.2), then(2.14)becomes∂
r|
p0g(
u0,
e2) − ∂
r|
p0g(
u(
t0, ·),
e2)
= −
t0 0
∂
rg(
1u(
t,
p0),
e2) −
1 s∂θ
g(
1u(
t,
p0),
e 1)
+
c sg(
1u(
t,
p0) − λ
0(∂
ruθ+
β
cos(
aδ))
e 2,
e2)
+
2a2∂
ruθ−
λ
0∂
r2uθ+
aβλ
0sin(
aδ)
dt.
(3.4)We then repeat the same computations that followed(2.14). The results are the same except that we have the four extra terms that appeared in(3.4). This turns(2.17)into
∂
r|
p0g(
u0,
e2) − ∂
r|
p0g(
u(
t0, ·),
e2)
=
t0 0 k2α
1(
t) − α
3(
t) −
2kα
2(
t) −
2η(
t)
+
λ
0(
kα
1(
t) + α
2(
t)) − λ
0β(
a sin(
aδ) −
k cos(
aδ))
dt.
AcknowledgmentsThe authors would like to thank the referee for their helpful comments. The first author is partially supported by a grant from the National Science Council of Taiwan (NSC 101-2115-M-009-016-MY2). The second author is partially supported by a grant from the Simons Foundation #246255. The third author is partially sup-ported by JSPS KAKENHI Grant Number 25870004. The authors also would like to thank their three institutions for the hospitality dur-ing the visits when this work was carried out. Also, this paper was developed during a stay of the third author as an assistant profes-sor of the Department of Mathematics at Hokkaido University.
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