Singularity formation and global existence of classical solutions for one dimensional rotating shallow water system

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by  Bin Cheng

SINGULARITY FORMATION AND GLOBAL EXISTENCE OF CLASSICAL
SOLUTIONS FOR ONE DIMENSIONAL ROTATING SHALLOW WATER
SYSTEM
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1 BIN CHENG, PENG QU, ANDCHUNJINGXIE
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2
Abstract. Westudyclassicalsolutionsofonedimensionalrotatingshallowwatersystemwhich
n
a playsanimportantroleingeophysicalfluiddynamics. Themainresultscontaintwocontrasting
J aspects. First, when the solution crosses certain threshold, we prove finite-time singularity

0
formation for the classical solutions by studying the weighted gradients of Riemann invariants
1
andutilizingconservationofphysicalenergy. Infact,thesingularityformationwilltakeplacefor

] alargeclassofC1 initialdatawhosegradientsandphysicalenergycanbearbitrarilysmalland
P
higher order derivativesshould be large. Second,when theinitial data haveconstant potential
A
vorticity, global existence of small classical solutions is established via studying an equivalent
.
h
form of a quasilinear Klein-Gordon equation satisfying certain null conditions. In this global
t
a existence result, the smallness condition is in terms of the higher order Sobolev norms of the
m
initial data.
[

1
v
6 1. Introduction and Main Results
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5
The one-dimensional rotating shallow water system plays an important role in the study
2
0 of geostrophic adjustment and zonal jets (e.g. [28, 10]). Upon suitable rescaling, the one-
.
1 dimensional rotating shallow water system in Eulerian form reads
0
7 ∂ h+∂ (hu) = 0,
t x
1
v: ∂t(hu)+∂x(hu2)+∂xhγ/γ = hv, (1)

Xi ∂t(hv)+∂x(huv) = hu,
−
r 
a where h denotes the height of the fluid surface, u denotes the velocity component in the x-
direction, and v is the other horizontal velocity component that is in the direction orthogonal to
the x-direction. TheCoriolis force, caused by a rotating frame, is represented by the (hv, hu)T
−
terms on the right-hand side of the last two equations of (1). For the rotating shallow water
model, one has γ = 2 in the pressure law ([6]), but in this paper we will prove results that are
valid for the general case γ 1.
≥
The system (1) is a typical one dimensional system of balance laws, which has attracted
plenty of studies since Riemann [7]. Oneof the most important features of nonlinear hyperbolic
2010 Mathematics Subject Classification. 35L40, 35L67, 35L72, 35Q86, 76U05.
Key words and phrases. Rotating shallow water system, formation of singularity, Riemann invariants, global
existence, Klein-Gordon equation.
1
2
system of conservation laws is that the wave speed depends on the solution itself so that the
classical solutions in general are expected to form singularity in finitetime (cf. [17, 14]). In fact,
the system (1) can also be regarded as one dimensional compressible Euler system with source
terms. Theformation ofsingularity andcritical thresholdphenomenafor thecompressibleEuler
system without or with certain special source terms were studied in [19, 27, 3] and references
therein. Theadditional sourceterms very often demandsubstantial novel techniques inaddition
to the classical singularity formation theory developed in [17, 14] and subsequent literature.
On the other hand, with (h,u,v) depending only on (t,x)-variables, the system (1) can be
regarded as a special case of the two dimensional rotating shallow water system ([23]). The
latter is a widely used approximation of the three dimensional incompressible Euler equations
and the Boussinesq equations in the regime of large scale geophysical fluid motion. In the two
dimensional setting, it is shown in [5] that there exist global classical solutions for a large class
of small initial data subject to the constant potential vorticity constraint, which is analogue
of the irrotational constraint for the compressible Euler equations. Since classical solutions of
two dimensional compressible Euler system in general form singularity (c.f. [24]), the result of
[5] evidences that the Coriolis forcing term plays a decisive role in the global well-posednesss
theory for classical solutions of compressible flows. A key ingredient in the proof of [5] is that
the rotating shallow water system and in fact also its one dimensional reduction, when subject
to the (invariant) constant potential vorticity constraint, can be reformulated into a quasilinear
Klein-Gordon system. Note that the solutions of Klein-Gordon equations are of faster dispersive
decay than those of thecorrespondingnonlinear wave system, thelatter of which is derived from
non-rotatingfluidmodels. Therateofthisdispersivedecay,however,istiedtospatialdimension.
In fact, in theone dimensional setting, thedecay rate of Klein-Gordon system is not fast enough
for the global existence theory of [5] to be directly applied to the system (1).
In short, for studying the lifespan and global existence of classical solution for the one dimen-
sional rotating shallow water system, we have to introduce novel techniques and investigate the
nonlinear structure of the system (1) more carefully.
The main objectives of this paper are two-fold. First, we study the formation of singularities
for a general class of C1 initial data by capturing the nonlinear interactions in the system
(1). These initial data can have arbitrary small gradients and physical energy but higher order
derivatives should be large. Second, we take a careful look at the structure of the system (1),
exploit the dispersion provided by the Coriolis forcing terms, and show the global existence of
classical solutions for a class of small initial data that are of small size in terms of higher order
Sobolev norms.
Before stating the main theorems, we first rewrite the system (1) in the Lagrangian coordi-
nates, which takes a simpler form in one dimensional setting.
Assume the initial height field h(0,x) = h (x) C1(R) is strictly away from vacuum, i.e.,
0
∈
0 < c h (x) = h(0,x) C . It then induces a “coordinate stretching” at t = 0 specified by
h 0 h
≤ ≤
3
a C2 bijection φ :R R defined as
→
x
ξ = φ(x) := h(0,s)ds, (2)
Z0
whose inverse function can be written as x = φ−1(ξ). Assume u C1([0,T) R). Let σ(t,ξ) be
∈ ×
the unique particle path determined by
∂ σ(t,ξ) =u(t,σ(t,ξ)),
t
(3)
(σ(0,ξ) = φ−1(ξ),
so that for each fixed t [0,T), we have a C1 bijection x =σ(t,ξ) :R R. Define
∈ →
h(t,ξ) := h(t,σ(t,ξ)), u(t,ξ) := u(t,σ(t,ξ)), and v(t,ξ) := v(t,σ(t,ξ)). (4)
In the Appendix A, we show that (h,u,v) solves the system (1) if and only if (h,u,v) defined
e e e
in (4) is a solution of the following rotating shallow water system in the Lagrangian form,
e e e
∂ h+h2∂ u= 0,
t ξ
∂ u+∂ hγ/γ v = 0, (5)
 te eξ e −


∂ v+u= 0.
te e e

Fortherestofthepaper,wedealwiththesystem(5). Forconvenience, wedropthetildesigns

e e
in h, u, v when there is no ambiguity for the presentation. Thus, in the Lagrangian coordinates,
the rotating shallow water system (5) can be written as
e e e
∂ h+h2∂ u= 0,
t ξ
∂tu+∂ξ hγ/γ v = 0, (6)
−


∂ v+u=(cid:0) 0. (cid:1)
t

The objective in this paper is thento study the Cauchy problem for the system (6) with initial

data
(h,u,v)(0,ξ) = (h (ξ),u (ξ),v (ξ)) for ξ R. (7)
0 0 0
∈
For any C1 solution of (6), it follows from (6) that one has
1
∂ +∂ v = 0. (8)
t ξ
h
(cid:16) (cid:17)
This is a key geophysical property of rotating fluid which is conservation of potential vorticity.
Therefore, we have the invariance of the potential vorticity
1 1
+∂ v(t,ξ) = +∂ v(0,ξ) := ω (ξ) (9)
ξ ξ 0
h(t,ξ) h(0,ξ)
in the Lagrangian form.
Before stating the main theorems, some notations are in order. Inspired by the techniques of
[27] by Tadmor and Wei, we introduce the “weighted gradients of Riemann invariants”,
Zj = √h ∂ξu+( 1)jhγ−23∂ξh for j = 1,2. (10)
−
(cid:2) (cid:3)
4
Also, define
♯ ♯
Z := sup max Z (0,ξ) and ω := supω (ξ). (11)
0 i 0 0
ξ 1≤i≤2 ξ
The first main result is on the formation of singularities for classical solutions and consists of
two theorems.
Theorem 1. Fix T′ 0. Consider a classical solution (h,u,v) C1([0,T′] R) to the rotating
≥ ∈ ×
shallow water system (6) with initial data satisfying inf h >0 and (h 1,u ,v ) C1(R). If
ξ 0 0− 0 0 ∈ 0
inf min Z (T′,ξ) 2ω♯, (12)
ξ 1≤i≤2 i ≤ − 0
q
then the solution must develop a singularity in finite time t = T♯ > T′ in the following sense
inf h(t,ξ) > 0, sup max h(t,ξ), u(t,ξ), v(t,ξ) < , (13)
0≤t<T♯, 0≤t<T♯, { | | | |} ∞
ξ∈R ξ∈R
and
sup max Z (t,ξ) < , lim inf min Z (t,ξ) = . (14)
j j
0≤t<T♯,1≤j≤2 ∞ tրT♯ξ∈R1≤j≤2 −∞
ξ∈R
The proof of Theorem 1 is given in Section 3.1.
Remark 1. It follows from the conservation of potential vorticity and the bounds for h in
(13) that ∂ v is bounded even when the singularity is formed. Furthermore, it follows from the
ξ
definition of Z (j = 1, 2) in (10) and the estimates in (13)-(14) that we have
j
lim inf ∂ u(t,ξ) = .
ξ
tրT♯ξ∈R −∞
Obviously,iftheinitialdatasatisfy(12),thenthesolutionoftheproblem(6)formsingularities
in finite time. In fact, we can also characterize a class of initial data which does not satisfy (12)
at the initial time, but rather evolve to satisfy (12), and eventually form a singularity according
to the above theorem.
We define physical energy of the rotating shallow water system as
∞ 1
E(t) := (u2+v2)(t,ξ)+ (h(t,ξ))dξ, (15)
2 Q
Z−∞
where
1 h
(h) := (sγ−2 s−2)ds 0. (16)
Q γ − ≥
Z1
Finally, define
♯ ♯ ♯
E = E(0) and G := 2ω +max Z , 2ω , (17)
0 0 0 0 0
q n q o
♯ ♯
where ω and Z are defined in (11).
0 0
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Theorem 2. Consider the Cauchy problem for the system (6) subject to initial data (7) satis-
fying (h 1,u ,v ) C1(R) and inf h > 0. Suppose
0 − 0 0 ∈ 0 ξ 0
−2
inf min Z (0,ξ) < √2 ω♯ −1 G E +1 γ (18)
ξ 1≤i≤2 i − r 0− Fγ 0 0
h i
(cid:0) (cid:1)
where −1 is the inverse function of the function () defined by
Fγ Fγ ·
16 α3 3−2 3−2−1
Fγ(α) : =3γ3 (α+1)3 (α+1) γ +(α+1) γ . (19)
n o
Then the solution must develop a singularity in finite time t = T♯ in the sense of (13) and (14).
The proof of Theorem 2 is a straightforward combination of Theorems 1 and 9. We have the
following remarks.
Remark 2. Straightforward computations show that defined in (19) is a monotonically
γ
F
increasing function mapping (0, ) to (0, ). Hence the inverse function −1 is always well-
∞ ∞ Fγ
defined on (0, ).
∞
Remark 3. If (h 1,v ) are compactly supported, it follows from (9) that one always has
0 0
−
♯
ω 1, so all the square roots in (12), (17), and (18) are always real.
0 ≥
Remark 4. We consider only theinitial data suchthat (h 1,u ,v )arecompactly supported.
0 0 0
−
As the propagation of information for general data is at a finite speed, the results in Theorems
1 and 2 can be easily extended to general initial data without compact support. Also thanks
to the finite speed of propagation, when the initial data are indeed compactly supported and a
singularity does develop in finite time as in (12) and (14), we actually have the singularity occur
at a finite location as well.
Remark 5. The singularity formation criterion (18) allows arbitrarily small initial gradients at
1/3
the order of O(E ). Indeed, by definition of in (19), we have
0 Fγ
(α) 32
γ
lim F = .
αց0 α3 3γ3
Therefore, with G > 0 bounded above by a constant, we can find positive constants C and E
0 0
so that
C−1E 1/3 −1 G E CE 1/3 for all E < E . (20)
0 ≤ Fγ 0 0 ≤ 0 0 0
♯
Then, by choosing arbitrarily small E(cid:0)0 < E0(cid:1)and choosing ω0 to be arbitrarily close to 1 (with
the most convenient choice being h 1,v 0), we make the right hand side of (18) at order
0 0
≡ ≡
1/3
O(E ). This in turn allows us to choose initial data having small gradients, i.e.,
0
Zj t=0 = inxf √h ∂ξu+(−1)jhγ−23∂ξh t=0 ∼ O(E01/3)
(cid:12) n o(cid:12)
which satisfy the cond(cid:12)ition (18) for th(cid:2)e singularity formati(cid:3)on(cid:12).
(cid:12) (cid:12)
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Remark 6. Thesingularityformationcriterion(18)alsoreflectsthefactthatweutilizephysical
energy and its conservation to prove pointwise singularity formation.
In fact, if the initial data has not only small gradients, but also small higher derivatives in
Sobolevspaces,thereisaglobalexistenceofclassicalsolutionsforrotatingshallowwatersystem.
This is our next main result on the global existence of classical solutions for rotating shallow
water system.
Theorem 3. Consider the Cauchy problem (6) and (7) subject to compactly supported initial
data (h 1,u ,v ) with inf h > 0. Suppose that ω 1. Then, there exists a small positive
0 0 0 ξ 0 0
− ≡
number δ so that if the Sobolev norm ku0kHk(R) +kv0kHk+1(R) < δ for some sufficiently large
integer k, then there is a global classical solution for the problem (6) and (7) for all time t 0.
≥
Theorem 3 is proved in Section 4. There are a few remarks in order.
Remark 7. In Theorem 3, we consider only the data close to the constant state (1,0,0). In
fact, the results also hold for any data close to (H¯ ,0,0) with a constant H¯ .
0 0
Remark 8. Although the singularity formation result in Theorem 2 allows arbitrarily small
1/3
initial gradients at the order of O(E ) for any sufficiently small E , Theorem 2 and Theorem
0 0
3 are compatible, or more precisely, they characterize different sets of initial data. To see this,
we recall Gagliardo-Nirenberg interpolation inequality to have
2 1−2 2 1−1
k∂ξu0kL∞(R) ≤ Cku0kH2(R) ≤ Cku0kHkk(R)ku0kL2(kR) ≤ Cδk E02 k .
♯
For any initial data satisfying the assumptions in Theorem 3 with k 7 so that ω 1 = ω
≥ 0 ≡ 0
and small E < δ, one has
0
2 5/14
k∂ξu0kL∞(R) ≤CδkE0 .
Applying similar argument to 1 1 = ω 1 = ∂ v shows that
− h0 0− h0 ξ 0
2 5/14
kh0 −1kL∞(R)+k∂ξh0kL∞(R) ≤ CδkE0 ,
1/3
which is much smaller than O(E ). By the lower boundin (20), it is impossiblefor such initial
0
data to also satisfy the assumption (18) of Theorem 2 as long as we choose δ in Theorem 3 to
be sufficiently small.
Remark 9. By the Theorem 3 above, with sufficiently small initial data, there is a global
solution for the rotating shallow water system, which is fundamentally different from the non-
rotating, compressible Euler system [14]. This shows that the rotation plays an important role
in the well-posedness theory of classical solutions to the partial differential equations modeling
compressible flows.
7
Remark 10. Infact, theresultsonbothsingularityformationandglobalexistenceinthispaper
also work in the similar fashion for the one dimensional Euler-Poisson system with a nonzero
background charge for hydrodynamical model in semiconductor devices and plasmas. We would
also like to mention the recent work [11] where the global existence of classical solutions for the
Euler-Poisson system with small initial data was proved via a different method.
Therestofthepaperisorganizedasfollows. InSection2,weintroducetheRiemanninvariants
andweightedgradientsofRiemanninvariants,andgivesomebasicestimatesforthesequantities.
In Section 3, we prove the finite time formation of singularity via investigating the weighted
gradients of Riemann invariants and utilizing conservation of physical energy. In Section 4, we
reformulatetheLagrangian rotatingshallow watersystemsubjecttoconstant potentialvorticity
into a one dimensional Klein-Gordon equation which is then shown to satisfy the null conditions
in [8]. The results in [8] help establish the global existence of small classical solutions. We
also provide two appendices. Appendix A contains the proof for the equivalence between the
Eulerian form of rotating shallow water system (1) and its Lagrangian form (5). In Appendix
B, we present two elementary lemmas which are used to prove the singularity formation for the
rotating shallow water system.
2. Riemann Invariants and Their Basic Estimates
The system (6) is a typical 3 3 system of balance laws. One way to diagonalize the system
×
of balance laws is to write the system in terms of the Riemann invariants. However, a 3 3
×
system usually does not have 3 full Riemann invariant coordinates [7]. Fortunately, the system
(6) has 3 full Riemann invariant coordinates R (i = 1,2,3), so that the system (6) is recast into
i
a “diagonalized” form,
γ+1
∂tR1 h 2 ∂ξR1 R3 = 0,
− −
 γ+1
∂tR2+h 2 ∂ξR2−R3 = 0, (21)

 R1+R2
∂ R + = 0,
t 3
2

where, by borrowing notations from the so-called p-system, we let p(1) = hγ, i.e., p(s) := s−γ
 h γ γ
and define Riemann invariants as
1
h
R := u+ p′(s)ds = u (h),
1
− −K
 Z1
 h1 p (22)
R2 := u− −p′(s)ds = u+K(h),
Z1
p
R := v,
 3



with, apparently, 
h
γ−3
(h) := s 2 ds. (23)
K
Z1
8
Note that h can be expressed in terms of the Riemann invariants as
2
R R γ−1 z+1 γ−1 , γ > 1,
h= ϑ 2− 1 with ϑ(z) = −1(z) = 2 (24)
2 K e(cid:16)z, (cid:17) γ = 1.
(cid:16) (cid:17) 
Based on the Riemann invariants formulation alone, we have the following estimates related
to the L∞ bounds of the solutions which then lead to an important upper bound for h and
consequently the finite speed of propagation.
Lemma 4. Fix T > 0. Let (h,u,v) C1([0,T] R) with h> 0 solve system (6) and equivalently
∈ ×
(21). Suppose
♯ ♯
Z , ω , infh and sup h , u , v are all finite and positive (25)
0 0 ξ 0 ξ { 0 | 0| | 0|}
and
M := sup R (0,ξ), R (0,ξ), R (0,ξ) < .
0 1 2 3
ξ∈R{| | | | | |} ∞
Then, at any t [0,T], we have
∈
sup R (t,ξ), R (t,ξ), R (t,ξ) M et (26)
1 2 3 0
ξ∈R{| | | | | |} ≤
and
2
γ−1M et+1 γ−1 , γ > 1,
suph(t,ξ) θ♯(t):= ϑ(M et) = 2 0 (27)
0
ξ∈R ≤ (cid:16)e(M0et), (cid:17) γ = 1.

Proof. Obviously, the estimate (27) is a consequence of the representation (24) and the
estimate (26). So we need only to prove the estimate (26). Then, it suffices to show that, for
any ε> 0, N >0, we have
max max e−tR (t,ξ) < M := M +ε, (28)
i 0ε 0
(t,ξ)∈AN,T,ε1≤i≤3| |
where A (see Fig. 1) is the trapezoid
N,T,ε
γ+1
A := (t,ξ) [0,T] R ξ N +(T t) ϑ(eT(M +ε)) 2 . (29)
N,T,ε 0ε
∈ × | | ≤ −
n (cid:12) o
Suppose that the estimate (28) is not t(cid:12)rue. By the comp(cid:2)actness of A (cid:3), there must exist
(cid:12) N,T,ε
an earliest time t′ so that
max max e−tR (t,ξ) = M . (30)
i 0ε
(t,ξ)∈AN,T,ε 1≤i≤3| |
t∈[0,t′]
Note the definition of M implies that t′ > 0. The speeds of the characteristics of (21) are
0
γ+1 γ+1
h 2 , 0, and h 2 , respectively. It follows from (24) and (30) that one has
−
hγ+21 < ϑ(eT(M0ε+ε)) γ+21 for (t,ξ) AN,T,ε and t [0,t′].
∈ ∈
(cid:2) (cid:3)
9
Thus, the above estimate together with the definition of A in (29) guarantees that all
N,T,ε
characteristics of (21) emitting from (t′,ξ′) and going backward in time always stay within
A . Now, introduce
N,T,ε
m♯(t) := max max R (t,ξ) and m♭(t) := min max R (t,ξ).
i i
(t,ξ)∈AN,T,ε1≤i≤3 (t,ξ)∈AN,T,ε1≤i≤3
Then,foranyt (0,t′],uponintegratingeachequationof (21)alongtheassociatedcharacteristic
∈
from 0 to t, we have
t
m♯(t) m♯(0)+ max m♯(s), m♭(s) ds
≤ { − }
Z0
and
t
m♭(t) m♭(0)+ min m♯(s), m♭(s) ds.
≥ {− }
Z0
Therefore, we have
t
max m♯(t), m♭(t) max m♯(0), m♭(0) + max m♯(s), m♭(s) ds.
{ − } ≤ { − } { − }
Z0
Therefore, max m♯(t), m♭(t) satisfies a Gronwall’s inequality which leads to
{ − }
max m♯(t), m♭(t) et max m♯(0), m♭(0) etM for all t (0,t′].
0
{ − } ≤ { − } ≤ ∈
This contradicts (30). Hence the lemma is proved. ✷
The upper bound of h in (27) allows us to define the following trapezoidal regions, similar to
(29), in the spirit of domain of dependence and domain of influence.
γ+1
Ωbw := (t,ξ) [0,T] R ξ N +(T t) ϑ(eT(M +1)) 2 , (31)
N,T ∈ × | |≤ − 0
Ωfw := n(t,ξ) [0,T] R(cid:12)(cid:12) ξ N +t ϑ(eT((cid:2)M +1)) γ+21 .(cid:3) o (32)
N,T ∈ × (cid:12)| |≤ 0
n (cid:12) o
(cid:12) (cid:2) (cid:3)
t (cid:12) t
T T
Ωfw
Ωbw N,T
N,T
A
N,T,ε
−N N ξ −N N ξ
Fig. 1 Ωbw and A Fig. 2 Ωfw
N,T N,T,ε N,T
10
γ+1
Under the assumptions of Lemma 4, we have that characteristics with speed h 2 or 0
±
emitting from within Ωbw (resp. Ωfw ) and going backward (resp. forward) in time always
N,T N,T
stay within Ωbw (resp. Ωfw ) till t = 0 (resp. t = T).
N,T N,T
Note that it is crucial that we shall also prove the lower bound of h to be strictly above 0,
which will be dealt with later.
2.1. Dynamics of gradients of Riemann invariants. Here, we follow the original idea of
Lax ([17])to studythedynamics of gradients of Riemann invariants andwill furtherreformulate
the system inspired by the method in [27] by Tadmor and Wei. Note that despite the similarity
ofourequations withthoseof[27], theirODEs[27,(3.12)] forweightedgradients oftheRiemann
invariants do not have a term that corrsponds to 1/h term in our ODEs (38). This is in fact
one of the main technical difficulties we have to tackle here.
First, differentiating the first two equations of (21) with respect to ξ gives
γ+1
D1(∂ξR1) ∂ξ(h 2 )(∂ξR1)= ∂ξR3,
− (33)
 γ+1
D2(∂ξR2)+∂ξ(h 2 )(∂xR2)= ∂ξR3,
where 
γ+1 γ+1
D1 := ∂t h 2 ∂ξ and D2 := ∂t +h 2 ∂ξ.
−
It follows from (23) and (24) that one has
∂ξh = ′1(h) ∂ξR2−2 ∂ξR1 = h−γ−23 ∂ξR2−2 ∂ξR1,
K
so
γ+1 γ+1 γ−1 γ+1 ∂ξR2 ∂ξR1
∂ξ(h 2 )= h 2 ∂ξh = h − .
2 2 2
Combine this with potential vorticity conservation (9) and transform (33) into
γ+1 1
D (∂ R ) = h(∂ R ∂ R )(∂ R )+ω ,
1 ξ 1 ξ 2 ξ 1 ξ 1 0
4 − − h
(34)
 γ+1 1
D2(∂ξR2) = h(∂ξR1 ∂ξR2)(∂ξR2)+ω0 .
4 − − h
We also use ∂ R ,∂ R and the first equation in (6) to rewrite the dynamics of h as
ξ 1 ξ 2
h2(∂ R +∂ R )
ξ 2 ξ 1
∂ h+ = 0. (35)
t
2
It follows from the definition of R and R that one has
1 2
h2(∂ξR2−∂ξR1) = h2∂ξ (h) = hγ+21∂ξh. (36)
2 K
Substituting (36) into (35) gives
D h = h2∂ R and equivalently D h = h2∂ R . (37)
1 ξ 2 2 ξ 1
− −