The Non–Radial Velocity Theorem revisited
R. Kaiser∗
Fakultät für Mathematik und Physik
Universität Bayreuth
D-95440 Bayreuth, Germany
(received 8 December 2006; in final form 18 April 2007)
May 25, 2007
Abstract
The paper presents evidence that non-radial flows, i.e. flows v with v · r ≡ 0, in a fluid
of radially symmetric conductivity contained in a spherical volume BR of radius R are
1
Rnot capable of magnetic field generation. More precisely, it is proved that the L -norm
BR |P (· , t)| dv of the poloidal scalar P (r, t) of the magnetic field is bounded for all times
by its initial value in the L2 -norm. Moreover, the L1 -norm of P over an arbitrary sphere
Sr with radius r decays (in time) to zero uniformly with respect to r. In the case that the
poloidal field has died out, we prove, furthermore, decay to zero of the toroidal scalar T
in the norm (max|r|=r T (r, t) − min|r|=r T (r, t)) uniformly with respect to r. This implies,
in particular, uniformly pointwise decay to zero of the toroidal scalar in BR .
Key Words: Dynamo theory, antidynamo theorem.
1
Introduction
An important class of antidynamo theorems forbids dynamo action in a fluid sphere if the flow
field is subject to certain kinematic restrictions. The most prominent example in this class is
the toroidal velocity theorem, asserting that there is no dynamo action for a purely toroidal
flow field (Elsasser, 1946; Bullard and Gellman, 1954). This theorem excludes e.g. dynamo
action in stably stratified planetary cores, where (e.g. precession-driven) non-radial motions
may well be present. Another example is the radial velocity theorem which does not allow
dynamo action if the flow field has only a spherically symmetric radial component (Namikawa
and Matsushita, 1970; Ivers and James, 1986). An antidynamo theorem also holds for the
combination of both types of flow fields (Ivers and James, 1988b).
The non-radial-velocity theorem is a natural extension of the toroidal velocity theorem
in that the divergence constraint on the flow field is lifted whereas the radial component is
still vanishing. The name of the theorem has been coined by Ivers and James, who proved
among other things boundedness of the poloidal scalar in the L1 -norm for all times and,
once the poloidal field has died out, pointwise decay of the toroidal scalar to zero (Ivers and
∗
Email: [email protected]
1
James, 1988a). From a rigorous point of view, however, these results are not yet completely
satisfying. The well-established rigorous methods, energy estimates and maximum principles,
which yield decay results in the L2 - and L∞ -norm, respectively, and which apply well e.g. to
the case of toroidal flows, are not (directly) applicable to non-radial flows.
Instead, Ivers and James divide all space along zero-level-sets of the poloidal scalar P into
countably many subvolumes of uniform sign (of P ). Integrating the equation of motion for P
separately over any subvolume, applying Gauss’ theorem, and summing up, finally, over all
subvolumes
contained in the fluid volume, Ivers and James are able to deduce the inequality
R
d/dt BR |P (· , t)| dv < 0 implying boundedness of the L1 -norm for all times. Unfortunately,
level-sets of even arbitrarily smooth functions can be very nasty, and it is neither for sure
that the subvolumes are such that Gauss’ theorem applies nor that subdivision interchanges
with integration. So, it is not guaranteed that the result from above applies to any (smooth)
solution of the induction equation and a proof avoiding these shortcomings would be desirable.
Such a proof is provided in this paper. We adapt for this purpose a method to our needs,
which has already successfully been applied to (toroidal) dynamo fields with plane symmetry
or axisymmetry
(Lortz and Meyer-Spasche, 1982). We prove, moreover, decay to zero of the
R
quantity Sr |P (· , t)| r −2 ds uniformly with respect to r with a decay time not larger than 4×
the free decay time. This result seems to be new. It is valid for nonsteady compressible flows
(with r · v ≡ 0, of course) and nonsteady spherically symmetric conductivities.
Concerning the toroidal field our equation of motion for the toroidal scalar T (and vanishing poloidal scalar) differs from that in (Ivers and James, 1988a) by a radially varying,
nonlocal term, which secures a vanishing mean value of T over spheres. This term prevents a
straightforward application of maximum principles. Instead, one has to consider the quantity
(max|r|=r T (r, t) − min|r|=r T (r, t)). Solving an auxiliary problem and applying comparison
functions similar to those used by Ivers and James lead, finally, to uniformly pointwise decay
of T to zero.
Concerning the magnetic field itself the decay result for P = B · r implies decay of the
radial magnetic field component. However, our results do not allow conclusions about the nonradial field components. So, the possibility of dynamo fields with decaying radial component
as well as decaying poloidal and toroidal scalars, but non-decaying non-radial components is
not excluded. These fields, if they exist, develop necessarily locally steep gradients. Such a
behaviour is hardly compatible with the parabolic character of the induction equation; its
rigorous exclusion, however, seems to be beyond the methods used in this paper.
The paper is organized as follows: In section 2 we derive from the induction equation
scalar subproblems for the poloidal scalar P and, if P is zero, for the toroidal scalar T .
These derivations depend crucially on the non-radial character of the flow and a conductivity
which varies at most radially in space. Section 3 deals with the poloidal problem: a mixed
parabolic-elliptic initial-value problem in all space. We formulate a maximum principle for
this situation, discuss an auxiliary problem and prove two decay results. The toroidal problem
(section 4) takes the form of a parabolic initial-boundary-value problem in the fluid sphere.
In this case a decay result for the toroidal scalar is proved in a nonstandard maximum-norm.
2
Equations of motion for poloidal and toroidal scalars
Given a volume V ⊂ IR3 filled with a fluid of conductivity λ > 0, which is in prescribed motion
according to a flow field v, the kinematic dynamo problem asks for solutions B, which do not
2
decay in time, (so-called dynamo solutions) of the following initial-value problem:

∂t B = ∇ × (v × B) − ∇ × (λ∇ × B) , ∇ · B = 0






∇ × B = 0, ∇ · B = 0


B continuous




|B(r, ·)| = O(|r|−3 )




B(· , 0) = B0 , ∇ · B0 = 0
in V × (0, ∞),
in IR3 \ V × (0, ∞),
in IR3 × (0, ∞),
(1)
for |r| → ∞,
on V × {t = 0}.
The induction equation (1)1 describes the generation of the magnetic field B by the motion
of a conducting fluid. B continues outside the fluid volume as a vacuum field, which decays
(spatially) at least as fast as a dipole field (cf. Backus, 1958). Note that the initial magnetic
field B0 is prescribed only in the fluid volume V .
We are concerned in this paper with the special case where the fluid volume is a ball BR
of radius R, the conductivity λ = λ(|r|, t) is spherically symmetric with lower bound
0 < λ0 ≤ λ(|r|, t),
(2)
and, most important, the flow field v = v(r, t) has no radial component: v · r ≡ 0.
We use the well-known poloidal-toroidal decomposition of solenoidal vector fields in the
form
B = ∇ × (∇ × Sr) + ∇ × T r = −∇ × ΛS − ΛT
(3)
with Λ := r × ∇ (Mie, 1908; Backus, 1958; Schmitt, 1995). The poloidal scalar S and the
toroidal one T are uniquely determined by B:
r · B = −L S,
r · ∇ × B = −L T,
(4)
R
if S and T have vanishing mean-value over spheres Sr of radius r, hT is := (4πr 2 )−1 Sr T ds =
0, hSis = 0, and if B is sufficiently regular (e.g. B ∈ C 1 ). In fact, the Laplace-Beltramioperator L := Λ · Λ is then invertible and referring to P := r · B = −L S instead of S as
poloidal scalar is justified.
In our situation problem (1) contains the following scalar sub-problem for the poloidal
scalar P :

∂t P = λ∆P − ∇ · (vP )
in BR × (0, ∞),






0 = ∆P
in IR3 \ BR × (0, ∞),


(5)
P and r · ∇P continuous
in IR3 × (0, ∞),




|P (r, ·)| = O(|r|−2 )
for |r| → ∞,




P (· , 0) = P0 , hP0 is = 0 on BR × {t = 0}.
(5)1 follows from the radial component of (1)1 using (4)1 and the identities ∇ × (v × B) · r =
−∇ · (vP ) (for non-radial flows) and ∇ × (λ∇ × B) · r = −λ∆P (for spherically symmetric
conductivities). (5)2 is the radial component of ∆B = 0, and (5)3 follows from (1)3 and
the divergence-constraint. The notation in (5)4 means spatial decay, which is uniform with
respect to t in any finite interval [T1 , T2 ] with 0 < T1 < T2 . Note, finally, that any solution
P of (5) satisfies hP is = 0. This follows in BR × (0, ∞) from the initial condition and in
IR3 \ BR × (0, ∞) from (5)2,4 .
3
A sub-problem for the toroidal scalar T is obtained if S resp. P is set to zero. Applying Λ
′
on (1)1 , using (4)2 , and the identity Λ · ∇ × (λ∇ × B) = L[λ∆T + λr rr · ∇(rT )] (for spherically
symmetric conductivities) yields
L [∂t T ] = L [λ∆T +
λ′ r
· ∇(rT ) − v · ∇T ]
r r
(6)
with r = |r|, λ′ := rr · ∇λ. Note that the operator L can only be removed from eq. (6) if the
bracket on the right-hand side has mean-value zero. So, one obtains from (6) 1
∂t T = λ∆T +
λ′ r
· ∇(rT ) − v · ∇T + hv · ∇T is .
r r
(7)
It is convenient to introduce the variable T := rT , as well as radial and non-radial derivatives,
∂r :=
r
·∇,
r
∇nr := ∇ −
r
∂r .
r
(8)
Since T vanishes outside the fluid volume the toroidal sub-problem takes, finally, the form:


 ∂t T = ∂r (λ ∂r T ) + λ∇nr · ∇nr T − v · ∇T + hv · ∇T is in BR × (0, ∞),

T =0
on SR × (0, ∞),
(9)


 T (· , 0) = T , hT i = 0
on B × {t = 0}.
0
0 s
R
Remarks: 1. If the conductivity is allowed to vary on spheres, the term −(Λ λ) · (∇nr ∂r T +
Λ∆S) has to be added on the right-hand side of eq. (5)1 , i.e. there is a coupling between
poloidal and toroidal fields. Numerical calculations for a model problem of this type indicate
that an antidynamo result can not be expected in this case (Busse and Wicht, 1992).
2. If the poloidal field has not yet died out, the nonlocal term rL−1 [Λ · ∇ × (v × (∇ × Λ S))]
has to be added on the right-hand side of eq. (9)1 . Decay of the toroidal field is expected in
this case, too. Note that L−1 is a bounded linear operator on, e.g., the space of continuous
zero-mean functions (cf. Schmitt, 1997). The decay result for the poloidal scalar, derived
below, however, seems to be too weak to obtain decay results for the toroidal scalar in this
case.
3
Decay of the poloidal scalar
Let us start with formulating a maximum principle for mixed parabolic-elliptic problems of
the type (5), viz.:
1

∂t q ≤ λ∆q + b · ∇q + c q , c ≤ 0






0 ≤ ∆q


q and r · ∇q continuous




q(r, t) → 0




q(· , 0) = q0
in BR × (0, T ),
in IR3 \ BR × (0, T ],
in IR3 × (0, T ],
for |r| → ∞, t ∈ (0, T ] fixed ,
on BR × {t = 0}.
The nonlocal term in (7) is missing in (Ivers and James, 1988a)!
4
(10)
Problem (10) differs from (5), besides (10)1,3 being now inequalities, by a general first order
term and a zeroth-order term c(·, ·) with sign condition in (10)1 . Moreover, the decay condition
(5)4 has been relaxed and q is no longer required to have a vanishing mean-value over spheres.
We call a solution q of (10) with q0 ∈ C(BR ) classical if
q ∈ C(BR × [0, T ]) ∩ C 1 (IR3 × (0, T ]) ∩ C12 (BR × (0, T )) ∩ C 2 (IR3 \ BR × (0, T ]).
In this notation the upper index at “C” refers to the order of spatial derivatives and the lower
one (omitted if zero) to temporal derivatives; so, C12 means q, ∂t q, ∇q, ∇∇q are all continuous
functions.
Lemma 1 Let q be a classical solution of problem (10) with bounded coefficients λ, b, and c,
and λ satisfying, moreover, condition (2). Then
sup
q = max q0 .
(11)
BR
IR3 ×(0,T ]
Moreover, if supIR3 ×(0,T ] q is attained at some point (r0 , t0 ) with t0 > 0, then
q ≡ 0 in IR3 × (0, t0 ] .
Proof: The proof is based on elliptic and parabolic maximum principles and their corresponding boundary versions (cf. Protter and Weinberger, 1967, Theorems 6,7 p. 64f. and Theorems 5-7, p. 167f.). Let M denote supIR3 ×(0,T ] q, which is nonnegative because of (10)4 . We
deal with the case M > 0 first and show that M cannot be attained at any point ∈ IR3 ×(0, T ].
Suppose M is attained at some point (r0 , t0 ) with |r0 | ≥ R, t0 > 0. Because of (10)4 we can
choose R1 > |r0 | such that q < M on SR1 × {t0 } and applying the (strong version of the)
elliptic maximum principle to BR1 \ BR × {t0 } yields |r0 | = R. On the other hand, assuming
|r0 | ≤ R, the (strong version of the) parabolic maximum principle applied to BR ×[0, T ] yields
either t0 = 0 (which is excluded) or |r0 | = R or q ≡ M in BR × [0, t0 ]. The boundary version
of the parabolic maximum principle yields in the second case r · ∇q(r0 , t0 ) > 0, and in the
third case we have r · ∇q(r0 , t0 ) = 0 at any point (r0 , t0 ) with |r0 | = R. In any case there
is a contradiction to the boundary version of the elliptic maximum principle, which yields
r · ∇q(r0 , t0 ) < 0, and condition (10)3 . Thus, M has to be attained for t = 0. Recalling that
q(r, t) < maxSR ×{t} q for any (r, t) ∈ IR3 \ BR × (0, T ], assertion (11) is proved in the case
M > 0.
If M is attained at some point (r0 , t0 ) ∈ IR3 × (0, T ], we have necessarily M = 0. In the
case |r0 | < R the parabolic maximum principle yields q ≡ 0 in BR × [0, t0 ], and with (10)3
and the boundary version of the elliptic maximum principle follows q ≡ 0 in IR3 × (0, t0 ]. If
|r0 | > R we apply the elliptic maximum principle first to obtain q ≡ 0 on IR3 \ BR × {t0 }.
From the boundary version of the parabolic maximum principle together with (10)3 follows
now that M = 0 has to be attained also in some interior point (r1 , t1 ) of BR × (0, T ], which
can be chosen arbitrarily close to SR × {t0 }. Thus, we proceed as in the situation |r0 | < R
and conclude q ≡ 0 in IR3 × (0, t1 ] with t1 ≥ t0 . Similar arguments apply to the case |r| = R.
This proves the second assertion of the lemma.
2
Remark: A corresponding minimum principle is obtained by reversing the inequalities in
(10)1,2 (but keeping c ≤ 0) and replacing sup and max by inf and min, respectively.
5
We shall need later on positive solutions of the

∂t p = λ∆p − ∇ · (vp)






0 = ∆p


p and r · ∇p continuous




|p(r, ·) − a| = O(|r|−2 )




p(· , 0) = p0 > 0 , hp0 is = a
following auxiliary problem
in BR × (0, ∞),
in IR3 \ BR × (0, ∞),
in IR3 × (0, ∞),
(12)
for |r| → ∞,
on BR × {t = 0}.
λ and v are here as in problem (5) and a denotes a positive constant. The existence of
weak solutions for problems of type (12) has been proved in (Stredulinsky, Meyer-Spasche
and Lortz, 1986). The existence of classical solutions, which we presuppose in the following,
is proved in a forthcoming paper (Kaiser, 2007). Concerning λ and v we assume some Hölder
regularity (λ ∈ C α (BR × [0, T ]), v ∈ C 1+α (BR × [0, T ]) for any T > 0 is enough here2 ). We
list some properties of classical solutions p of (12), which are important in the following:
(i) p is unique,
(ii) p is positive,
(iii) |∂r p| = O(|r|−3 ) for |r| → ∞,
R
R
(iv) BR p(· , t) dv = BR p0 dv for any t ∈ [0, ∞), provided ∂r λ ≡ 0.
To prove (i), (ii) consider q := p e−dt , d > 0, which satisfies the system

∂t q = λ∆q − v · ∇q − (∇ · v + d) q






0 = ∆q


q and r · ∇q continuous




|q(r, ·) − a e−d · | = O(|r|−2 )




p(· , 0) = p0 > 0 , hp0 is = a
in BR × (0, ∞),
in IR3 \ BR × (0, ∞),
in IR3 × (0, ∞),
(13)
for |r| → ∞,
on BR × {t = 0}.
Choosing d > −∇ · v, q meets all requirements of system (10) (except (10)4 ) and Lemma
1 (with subsequent remark) implies as usual uniqueness of q and hence p. To prove q > 0
choose T > 0 and consider m := inf IR3 ×(0,T ] q. The case m ≤ 0 can be excluded completely
analogously to the proof of Lemma 1. This proves (ii). (iii) follows from the following
representation of harmonic fields exterior to BR and satisfying (12)4 in spherical coordinates
(r, θ, φ) (cf. Backus, 1958):
p(r, t) =
∞ X
n
X
n=1
R n+1
Ynm (θ, φ) + a .
cnm (t)
r
m=−n
(14)
This series as well as that of ∂r p are uniformly converging in any domain IR3 \ BR1 × [T1 , T2 ]
with R1 > R, 0 < T1 < T2 . To prove (iv) one computes for t ∈ (0, ∞), R1 > R:
Z
Z
Z
∂r p(· , t) ds .
∂r p(· , t) ds −
∆p(· , t) dv =
0=
BR1 \BR
2
S R1
Concerning notation cf. section 4.
6
SR
Thus, with (iii) in the limit R1 → ∞:
Z
∂r p(· , t) ds = 0 .
SR
With (12)1 follows, therefore:
Z Z
d
λ(t) ∆p(· , t) − ∇ · v(· , t) p(· , t) dv
p(· , t) dv =
dt BR
ZBR
λ(t) ∂r p(· , t) ds = 0 .
=
SR
The following theorem is a rigorous version of the poloidal bound in (Ivers and James, 1988a).
We assume here (and only here) uniform conductivity in space, i.e. ∂r λ ≡ 0.
Theorem 1 Let P be a classical solution of problem (5) and p a positive classical solution of
the auxiliary problem (12) with v denoting a non-radial flow field and λ a spatially homogeneous conductivity satisfying condition (2), then
Z
BR
|P (· , t)| dv ≤
Z
p0 dv
BR
!1/2
Z
BR
P02
dv
p0
!1/2
for any t ∈ (0, ∞).
Proof: With (5)1 and (12)1 one computes for t ∈ (0, ∞):
Z Z
d
P
P2
P2
2 ∂t P − 2 ∂t p dv
dv =
dt BR p
p
p
BR
)
Z (
P2
P
λ ∆P − ∇ · (v P ) − 2 λ ∆p − ∇ · (v p) dv
2
=
p
p
BR
Z Z
P 2
P
P2
2 ∂r P − 2 ∂r p ds .
p ∇ dv + λ
= −2 λ
p
p
p
SR
BR
On the other hand, with (5)2 , (12)2 :
Z
Z
P 2
P
P2
2 ∆P − 2 ∆p dv = −2
0=
p ∇ dv
p
p
p
B \B
BR1 \BR
Z R1 R
Z P
P
P2
P2
2 ∂r P − 2 ∂r p ds −
+
2 ∂r P − 2 ∂r p ds .
p
p
p
p
SR
SR
1
Thus, in the limit R1 → ∞:
d
dt
Z
P2
dv = −2 λ
p
BR
which implies
Z
BR
Z
P 2
p ∇ dv ≤ 0 ,
p
IR3
P 2 (· , t)
dv ≤
p(· , t)
7
Z
BR
P02
dv .
p0
(15)
Now (15) follows from
Z
BR
|P (· , t)| dv
!2
≤
Z
p(· , t) dv
Z
BR
BR
P 2 (· , t)
dv
p(· , t)
and property (iv).
2
Remarks: 1. Choosing p0 = const yields the bound
kP (· , t)kL1 (BR ) ≤ |BR |1/2 kP0 kL2 (BR ) ,
t ∈ (0, ∞) .
(16)
2. Stredulinsky, Meyer-Spasche and Lortz (1986) derive also an upper bound on solutions of
problem (12). However, the positivity of the initial value seems to be crucial in their proof
and is clearly incompatible with the zero-mean condition in problem (5).
The next theorem proves decay of the poloidal scalar to zero.
Theorem 2 Let P be a classical solution of problem (5) and p a positive classical solution
of the auxiliary problem (12) with v denoting a non-radial flow field and λ a conductivity
satisfying condition (2), then
 πr

 j0
for r ≤ R
DP2 E
2R
π
R
(r, t) ≤ C e−d t
(17)

p s
 j0
for r > R
2 r
with
λ0 π 2
and C := max
d :=
0≤r≤R
4 R2
(
hP02 /p0 is
j0 (πr/2R)
)
for any (r, t) ∈ [0, ∞) × (0, ∞). j0 denotes the zeroth spherical Bessel function.
Proof: With ∆ = r −2 ∂r (r 2 ∂r ) + ∇nr · ∇nr a computation analogous to that in the proof of
Theorem 1 yields for r ≤ R:
Z d DP2 E
1
P
P2
=
2
∂
P
−
∂
p
ds
t
t
dt p s 4πr 2 Sr
p
p2
)
Z (
2
2 1
P 1
P
P
λ
2
∂r (r 2 ∂r P ) − 2 2 ∂r (r 2 ∂r p) − 2 p ∇nr ds
=
4πr 2 Sr
p r2
p r
p
D
E
D
D
E
E
2
2
λ
P
P
λ
P 2
= 2 ∂r r 2 ∂r
− 2 λ p ∇ ,
≤ 2 ∂r r 2 ∂r
r
p s
p
r
p s
s
and similarly for r > R:
0≤
1 2 DP2 E .
∂r r ∂r
r2
p s
Defining the radially symmetric function Q by Q(r, t) := hP 2 /pis (|r|, t), Q satisfies the system

∂t Q ≤ λ ∆Q
in BR × (0, ∞),





 0 ≤ ∆Q
in IR3 \ BR × (0, ∞),
(18)
Q and r · ∇Q continuous
in IR3 × (0, ∞),





 |Q(r, ·)| = O(|r|−2 )
for |r| → ∞.
8
On the other hand, defining the bounding function B by
 π|r| 
 j0
for |r| ≤ R
,
B(r, t) := e−d t
R
π2R

 j0
for |r| > R
2 |r|
B satisfies the system

∂t B = λ0 ∆B





 0 = ∆B
in BR × (0, ∞),
in IR3 \ BR × (0, ∞),
B and r · ∇B continuous





 |Q(r, ·)| = O(|r|−1 )
in IR3 × (0, ∞),
(19)
for |r| → ∞.
Thus, with λ0 ∆B = −λ0 π 2 /(4R2 )B ≥ λ∆B the function D := Q − CB satisfies system (10)
and we conclude with Lemma 1:
(
)
DP2 E
πr 0
− C j0
≤0
sup D ≤ max D(· , 0) = max
0≤r≤R
p0 s
2R
BR
IR3 ×(0,T ]
for any T > 0. This proves (17).
2
Remark: Let p be the unique solution of the auxiliary problem (12) with initial value p0 =
a = const. The radially symmetric function p̃(r, t) := hpis (|r|, t) is then the unique solution
of (12) with v ≡ 0 and initial value p̃0 = a = const, thus p̃ ≡ a. This leads to a refinement of
the bound (16):
 πr

 j0
DP2 E
2R
≤ (4π)2 a C e−d t
kP (· , t)k2L1 (Sr ) ≤ (4π)2 hpis
π R

p s
 j0
2 r
 πr
)
(

 j0
for r ≤ R
kP0 k2L2 (Sr )
2R
−d t
R
π
≤ 4π max
e
,

0≤r≤R
j0 (πr/2R)
 j0
for r > R
2 r
in particular:
kP (· , t)kL1 (Sr ) ≤
√
d
2 π max kP0 kL2 (Sr ) e− 2 t .
0≤r≤R
(20)
Note for the last bound that j0 is a decreasing function on [0 , π/2] with j0 (0) = 1 and
j0 (π/2) = 2/π.
4
Decay of the toroidal scalar
Substituting the abbreviations (8), system (9) turns out as a standard parabolic initialboundary-value problem – with the exception of the nonlocal term hv · ∇T is . The question of
9
existence and uniqueness of solutions for problems of that type have been treated in (Schmitt,
1997). According to this reference the following regularity assumptions:3
λ ∈ C 1+α (BR × [0, T ]) ,
v ∈ C α (BR × [0, T ]) ,
T0 ∈ C 2+α (BR )
(21)
for any T > 0, together with the compatibility conditions
T0 = ∂r (λ∂r T0 ) = 0 on SR
(22)
T ∈ C 2+α (BR × [0, T ])
(23)
guarantee an unique solution
of problem (9).4 Recalling the defining relation T = r T , the further condition
|T0 (r)| = o(r)
for r → 0
(24)
on the initial field T0 is reasonable.
The following theorem is a correct version of the toroidal bound in (Ivers and James,
1988a).
Theorem 3 Let T be a solution of problem (9), where the non-radial flow field v, the conductivity λ, and the initial field T0 satisfy the conditions (21), (22), and (24). λ satisfies,
moreover, condition (2) and the bound Rλ′ /λ < λ1 , λ1 > 0. The spherical maximum norm
kT ks (r, t) := max|r|=r T (r, t) − min|r|=r T (r, t) then satisfies the bound
kT ks (r, t) ≤ C b1 (r) e−d t
(25)
λ1
λ1 (1−r/R) −λ r/R), and d := λ /(R2 b (R))
with C := max0≤r≤R {kT0 ks /b1 (r)}, b1 (r) := λ−2
1
0
1
1 (e −e
for any (r, t) ∈ [0, R] × [0, ∞).
Proof: Fix an arbitrary T > 0 and consider the two-dimensional auxiliary problems


∂ B ± = ∂r (λ ∂r B ± ) + f in (0, R) × (0, T ),

 t
B± = 0
on {r = 0, R} × [0, T ],


 B ± (· , 0) = B ±
on [0, R] × {t = 0}
(26)
0
with f := hv · ∇T is and B0+ := max|r|= · T0 (r), B0− := min|r|= · T0 (r). Note that with (21),
(23) we have f ∈ C α ([0, R] × [0, T ]) and at least B0± ∈ C([0, R]). Thus, there are unique
solutions B ± ∈ C 2+α ((0, R) × (0, T )) ∩ C([0, R] × [0, T ]) of (26) (cf. Friedman, 1964, Theorem
9, p. 69). Setting B̃ ± (r, t) := B ± (|r|, t) the function D ± := T − B̃ ± satisfy
(
∂t D ± = ∂r (λ ∂r D ± ) + λ∇nr · ∇nr D ± − v · ∇D ± in BR × (0, T ),
D± = 0
on SR × [0, T ),
3
Concerning (uniform) Hölder continuity and Hölder norms in parabolic problems we refer to (Friedman,
1964). Our notation differs only slightly from that used in this reference.
4
Note that Schmitt treated a more general problem involving a nonlinear term and obtained, consequently,
a solution which is only local in time. The techniques used in the proof yield, however, in the case of a linear
problem (such as problem (9)) a global solution.
10
and the parabolic maximum principle yields
max D+ =
BR ×[0,T ]
min
D− = 0 .
BR ×[0,T ]
Thus, we have for kT ks the bound
kT ks (r, t) = max T (r, t) − min T (r, t) ≤ B + (r, t) − B − (r, t) .
|r|=r
|r|=r
Setting B := B + − B − and using (26), B satisfies now the initial-boundary-value problem


∂ B = ∂r (λ ∂r B) in (0, R) × (0, T ),

 t
B=0
on {r = 0, R} × [0, T ],


 B(· , 0) = kT k
on [0, R] × {t = 0} .
0 s
On the other hand, B1 (r, t) := b1 (r) e−d t satisfies the system5
(
∂t B1 ≥ ∂r (λ ∂r B1 ) in (0, R) × (0, T ),
B1 ≥ 0
on {r = 0, R} × [0, T ] .
Thus, B1 can be used as a bounding function for B, which proves (25). Note that condition
(24) ensures a finite constant C.
2
Remarks: 1. If λ′ does not change the sign on [0, R] × [0, ∞), simpler bounding functions with
larger decay rates are available, viz.:
r 2 λ0
e−2 R2 t
B2 (r, t) := 1 −
R r 2 λ0
B3 (r, t) := 1 − 1 −
e−2 R2 t
R
if λ′ ≥ 0 ,
if λ′ ≤ 0 .
2. In terms of the poloidal scalar T , the bound (25) takes the form
n kT k o b (r)
1
0 s
e−d t .
0≤r≤R b1 (r)/r
r
kT ks (r, t) ≤ max
(27)
Noting the equivalence of norms,
k · kL∞ (Sr ) ≤ k · ks ≤ 2 k · kL∞ (Sr ) ,
as well as the estimate b1 (R)/R ≤ b1 (r)/r ≤ b′1 (0), (27) implies a bound in the standard
maximum norm:
eλ1 − 1
kT kL∞ (BR ) ≤ 2 λ1 λ1
kT0 kL∞ (BR ) e−d t .
e − 1 − λ1
5
B1 is a “reversed” version of the bounding function used by Ivers and James (1988a).
11
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