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arXiv:1708.00694v2 [math.AP] 25 Apr 2018
AXISYMMETRIC FLOWS IN THE EXTERIOR OF A CYLINDER
K. ABE AND G. SEREGIN
Abstract. We study an initial-boundary value problem of the three-dimensional Navier-
Stokes equations in the exterior of a cylinder Π = {x=(xh,x3)| |xh|>1}, subject to the
slip boundary condition. We construct unique global solutions for axisymmetric initial data
u0∈L3∩L2(Π) satisfying the decay condition of the swirl component ruθ
0∈L∞(Π).
1. Introduction
We consider the three-dimensional Navier-Stokes equations:
(1.1) ∂tu−∆u+u· ∇u+∇p=0
div u=0in Π×(0,∞).
It is well known that for small initial data u0∈L3
σ(R3), there exists a unique global solution
u∈BC([0,∞); L3) of (1.1) [22]. However, unique existence of a global solution is unknown
in general for large initial data in L3with finite energy. Here, BC([0,∞); X) denotes the
space of all bounded and continuous functions from [0,∞) to a Banach space Xand Lp
σ(Π)
denotes the Lp-closure of compactly supported smooth solenoidal vector fields in a domain
Π⊂R3.
For initial data with finite energy u0∈L2(R3), it is well known that global Leray-Hopf
weak solutions exist [29], [19]. However, their regularity and uniqueness are unknown. For
large initial data in L3(R3), weak solutions are constructed in [7], [27]. See [40] for weak
L3-solutions.
The purpose of this paper is to construct unique global solutions of (1.1) for large ax-
isymmetric initial data in L3∩L2. We say that a vector field uis axisymmetric if
u(x)=tRu(Rx)x∈R3, η ∈[0,2π],
for R=(er(η),eθ(η),ez) and er(η)=t(cos η, sin η, 0), eθ(η)=t(−sin η, cos η, 0), ez=
t(0,0,1). We say that a scaler function pis axisymmetric if p(x)=p(Rx) for x∈R3
and η∈[0,2π]. We set the cylindrical coordinate (r, θ, z) by x1=rcos θ,x2=rsin θ,x3=z
and decompose the axisymmetric vector field into three terms:
Date: April 26, 2018.
2010 Mathematics Subject Classification. 35K20, 35B07, 35K90.
Key words and phrases. Navier-Stokes equations, Axisymmetric solutions, exterior of a cylinder.
1
2
u(x)=ur(r,z)er(θ)+uθ(r,z)eθ(θ)+uz(r,z)ez.
The azimuthal component uθis called swirl velocity (see, e.g., [34]).
Unique global solutions of (1.1) for axisymmetric initial data without swirl were first
constructed in [24], [42] by the Galerkin approximation. Later on, unique global solutions
are constructed in [28] by a strong solution approach for axisymmetric data without swirl
in H2(R3). See also [1] for H1/2(R3). For axisymmetric solutions of (1.1), the vorticity
ω=curl uis expressed by
ω=ωrer+ωθeθ+ωzez
=(−∂zuθ)er+(∂zur−∂ruz)eθ+∂ruθ+uθ
rez,
and for v=urer+uzez, the azimuthal component ωθsatisfies the vorticity equation
(1.2) ∂tωθ
r+v· ∇ωθ
r−∆ + 2
r∂rωθ
r=∂zuθ
r2.
For axisymmetric solutions without swirl, the right-hand side vanishes and the global a priori
estimate
ωθ
r
L2(R3)≤
ωθ
0
r
L2(R3)t>0,
holds. The above vorticity estimate implies existence of unique global solutions for axisym-
metric data without swirl u0∈L3∩L2(R3). (We may assume the condition ωθ
0/r∈L2(R3)
since local-in-time solutions belong to H2(R3).) In other words, unique global solutions
exist for large axisymmetric initial data in L3∩L2(R3), provided that without swirl. For
axisymmetric data with swirl, unique existence of global solutions in R3is unknown.
In this paper, we study axisymmetric solutions with swirl in the exterior of a cylinder
Π = {x=(x1,x2,x3)∈R3| |xh|>1,xh=(x1,x2)},
subject to the slip boundary condition
(D(u)n)tan =0,u·n=0 on ∂Π.(1.3)
Here, n=−erdenotes the unit outward normal vector field on ∂Π,D(u)=(∇u+∇Tu)/2 is
the deformation tensor and ftan =f−n(f·n) is a tangential component of a vector field f
on ∂Π. Since axisymmetric vector fields u=urer+uθeθ+uzezsatisfy
3
ur=0, ∂ruθ−uθ=0, ∂ruz=0 on {r=1},
subject to the slip boundary condition (1.3), the azimuthal component of vorticity ωθvan-
ishes on the boundary (see Remarks 6.1 (ii) for the Dirichlet boundary condition).
By the partial regularity result [6], it is expected that axisymmetric solutions are smooth
in the interior of Π. Moreover, as noted in [11], they will not develop singularities on the
boundary due to viscosity. See [37], [18] for partial regularity results up to the boundary
subject to the Dirichlet boundary condition. The regularity theory for the slip boundary
condition (1.3) may be simpler than that for the Dirichlet boundary condition. In fact, for a
half space regularity results are deduced from a whole space case by a reflection argument;
see [3]. In this paper, we prove that axisymmetric solutions are sufficiently smooth in the
exterior of the cylinder Π×(0,∞), subject to the slip boundary condition (1.3). We impose
the slip boundary condition in order to construct approximate solutions for R3; see Remarks
1.2 (iii).
Our goal is to construct unique global mild solutions of (1.1) for axisymmetric initial data
with swirl in L3∩L2(Π). Since the boundary of the cylinder Π⊂R3is uniformly regular,
we construct mild solutions by using the ˜
Lp-theory. We set
˜
Lp(Π)=Lp∩L2(Π)
(resp. ˜
Lp
σ(Π)=Lp
σ∩L2
σ(Π)) for p∈[2,∞). It is proved in [14] ( [15]) that that the Helmholtz
projection Pacts as a bounded operator on ˜
Lp(Π). Moreover, it is recently shown in [17] that
the Stokes operator subject to the slip boundary condition A=P∆generates a C0-analytic
semigroup on ˜
Lp
σ(Π) (see also [14], [16] for the Dirichlet boundary condition). We construct
mild solutions for u0∈˜
L3
σ(Π) of the form
u(t)=etAu0−Zt
0
e(t−s)AP(u· ∇u)(s)ds.(1.4)
Since the swirl component satisfies the Robin boundary condition, axisymmetric solutions
of (1.4) satisfy the energy equality
ZΠ|u|2dx+2Zt
0ZΠ|∇v|2+|∇uθ|2+
uθ
r
2dxds+2Zt
0Z∂Π|uθ|2dHds=ZΠ|u0|2dx,(1.5)
where dHdenotes the surface element on ∂Π.
We construct unique global solutions for large axisymmetric data with swirl u0∈˜
L3
σ(Π)
satisfying the decay condition of the swirl component ruθ
0∈L∞(Π). The main result of this
paper is the following:
Theorem 1.1. Let u0∈˜
L3
σ(Π)be an axisymmetric vector field. Assume that ruθ
0∈L∞(Π).
Then, there exists a unique axisymmetric mild solution u ∈BC([0,∞); ˜
L3(Π)) satisfying
(1.5) for t ≥0.
4
Remarks 1.2.(i) It is unknown in general whether axisymmetric solutions in R3for u0∈
˜
L3
σ(R3) satisfying ruθ
0∈L∞(R3) are globally bounded for all t>0. See [35], [36], [8],
[21] for regularity criteria of axisymmetric solutions. For axisymmetric solutions, an upper
bound of the form |u(x,t)| ≤ Cr−1,r<1, is called type I condition. It is proved in [9], [10] by
De Giorgi method and [23], [39] by the Liouville-type theorem that axisymmetric solutions
do not develop type I singularities. See [38] about type I singularities. Recently, it is shown
in [26] ( [31]) that axisymmetric smooth solutions in R3×(−T,0) for u(·,−T)∈L2(R3)
and ruθ(·,−T)∈L∞(R3) satisfy an upper bound of the form |u(x,t)| ≤ C|log r|1/2r−2near
(r,t)=0 with some constant C.
(ii) It is known that solutions of (1.1) in R3are smooth if the direction of vorticity is Lip-
schitz continuous for spatial variables in regions of high vorticity magnitude [13] (called a
geometric regularity criterion). For axisymmetric flows without swirl, vorticity varies only
in the azimuthal direction and is identified with a scalar function. On the other hand, for
axisymmetric flows with swirl vorticity varies also in the radial and vertical directions. We
constructed unique global solutions whose vorticity may become large and vary in three di-
rections. For a half space R3
+, a geometric regularity criterion is proved in [4], subject to the
slip boundary condition. See also [5] for the Dirichlet boundary condition.
(iii) Theorem 1.1 implies existence of approximate solutions for R3. Since the exterior of
the cylinder Πε={r> ε}approaches R3as ε→0, axisymmetric solutions in R3can be
viewed as limits of solutions in Πε. Indeed, axisymmetric solutions without swirl in Πε
are uniformly bounded in L∞
tH1
xfor ε > 0 and approach those in R3[24, p.78, l.7]. See
Remarks 6.1 (iii). For the case with swirl, unique existence of global solutions is proved
in [43] ( [44]) in a bounded cylindrical domain for sufficiently smooth initial data. It is
unknown whether global solutions with swirl are uniformly bounded for all ε > 0. We
constructed unique global mild solutions for u0∈˜
L3
σ(Πε) satisfying the uniform estimate
for the swirl component (1.6).
Let us sketch the proof of Theorem 1.1. We first construct local-in-time mild solutions
of (1.4) for u0∈˜
L3
σand prove that mild solutions are axisymmetric and satisfy the energy
equality (1.5) for axisymmetric initial data. The major step of the proof is to derive a global
L4-bound for axisymmetric solutions u=v+uθeθ. Once we obtain the global bound, it is
not difficult to see that u∈BC([0,∞); ˜
L3) by local solvability and the energy equality (1.5).
We first prove the global L∞-estimate for the swirl component
||ruθ||L∞(Π)≤ ||ruθ
0||L∞(Π)t>0.(1.6)
Since r≥1 in the exterior of the cylinder Π, the L∞-estimate (1.6) and the energy equality
(1.5) implies the global L4-bound for uθof the form
||uθ||4≤ ||ruθ
0||
1
2
∞||u0||
1
2
2t>0.(1.7)
In order to prove (1.6), we study the drift-diffusion equation subject to the Robin boundary
condition:
5
(1.8)
∂tΓ + b· ∇Γ−∆Γ + 2
r∂rΓ = 0 in Π×(0,T),
∂nΓ + 2Γ = 0 on ∂Π×(0,T),
Γ = Γ0on Π× {t=0}.
Here, ∂n=−∂rdenotes the normal derivative. The function Γ = ruθis a solution of (1.8) for
b=v. We prove the L∞-estimate
||Γ||L∞(Π)≤ ||Γ0||L∞(Π)t>0,(1.9)
for solutions to (1.8). Since the sign of the coefficient is plus in the Robin boundary condi-
tion, a maximum principle holds if the coefficient band Γare bounded in Π×[0,T]. Then
the L∞-estimate (1.9) easily follows from a maximum principle (see Lemma 3.1). If Γis
decaying sufficiently fast as |x| → ∞, we are able to obtain (1.9) by estimating Lp-norms
of Γfor p=2mand sending m→ ∞. Since we assume that ruθ
0is merely bounded, the
function ruθmay not decay as |x| → ∞ . We shall prove (1.9) for non-decaying solutions Γ.
We apply the L∞-estimate (1.9) for ruθand obtain (1.6). Note that the boundedness of
ruθdoes not follow from properties of local-in-time solutions to (1.1) for u0∈˜
L3
σ. For this
purpose, we first extend the L∞-estimate (1.9) for mild solutions to (1.8) for Γ0∈L∞and
the coefficient bsuch that t1/2−3/2pb∈C([0,T]; Lp) vanishes at time zero for p∈(3,∞]. We
then deduce from the integral form (1.4) that ruθis a mild solution to (1.8) (see Lemma 4.7).
We next estimate a global L4-norm of v=urer+uzez. We apply an interpolation inequality
||v||4≤C||v||
1
4
2(||v||2+||ωθ||2)3
4,(1.10)
and estimate an energy norm of the vorticity ωθ. Since ωθvanishes on the boundary, we
control the external force ∂z(uθ/r)2by using viscosity and estimate
(1.11) ZΠ
ωθ
r
2dx+Zt
0ZΠ∇ωθ
r
2dxds≤ZΠ
ωθ
0
r
2dx+||ruθ
0||2
∞||u0||2
2
=:E t >0.
Since the above vorticity estimate implies the global bound
(1.12) ZΠ|ωθ|2dx+Zt
0ZΠ |∇ωθ|2+
ωθ
r
2!dxds≤ZΠ|ωθ
0|2dx
+C(E3
4||u0||
1
2
2+||ruθ
0||2
∞)||u0||2
2,t>0,
the local-in-time solution u=v+uθeθis globally bounded on L4.
This paper is organized as follows. In Section 2, we state a local existence theorem of
mild solutions for u0∈˜
L3
σand prove axial symmetry of mild solutions. In Section 3, we
6
study the drift-diffusion equation (1.8) for a bounded coefficient and prove the L∞-estimate
(1.9) by a maximum principle. In Section 4, we extend (1.9) for mild solutions to (1.8)
under the weak regularity condition of a coefficient, and apply (1.9) for the swirl component
of axisymmetric solutions. In Section 5, we prove the a priori estimates (1.11) and (1.12).
In Section 6, we prove Theorem 1.1. In Appendix A, we give a proof for a local solvability
result stated in Section 2. In Appendix B, we prove some interpolation inequalities used in
Section 5.
2. Local existence of axisymmetric solutions on ˜
L3
In this section, we construct local-in-time axisymmetric solutions of (1.1) for u0∈˜
L3
σ
satisfying the energy equality (1.5). Local solvability for u0∈˜
L3
σis known for R3[22,
Theorem 3]. We give a proof for the exterior of the cylinder by using ˜
Lp-theory in Appendix
A.
2.1. Local solvability. Let Cα([δ, T]; X) denote the space of all α-th H¨older continuous
functions f∈C([δ, T]; X) for a Banach space X. Let Cα((0,T]; X) denote the space of
functions in Cα([δ, T]; X) for all δ∈(0,T). For the convenience, we denote by ˜
Lp=Lp∩L2
also for p=∞.
Lemma 2.1. For u0∈˜
L3
σ, there exist T >0and a unique mild solution of (1.4) satisfying
t3
2(1
3−1
p)u∈C([0,T]; ˜
Lp),3≤p≤ ∞,(2.1)
t3
2(1
3−1
r)+1
2∇u∈C([0,T]; ˜
Lr),3≤r<∞,(2.2)
t3/2(1/3−1/p)u and t3/2(1/3−1/r)+1/2∇u vanish at time zero except for p =3. Moreover,
(2.3) u∈Cα((0,T]; ˜
L3),
∇u∈Cα
2((0,T]; ˜
L3),0< α < 1.
We show that mild solutions satisfy (1.1) by applying an abstract regularity result [32,
4.3.1 Theorem 4.3.4].
Proposition 2.2. Let B be a generator of an analytic semigroup in a Banach space X with
a domain D(B). Assume that f ∈L1(0,T;X)∩Cβ((0,T]; X)for β∈(0,1). Then,
w=Zt
0
e(t−s)Bf(s)ds
belongs to Cβ((0,T]; D(B)) ∩C1+β((0,T]; X).
7
Proposition 2.3. The mild solution u in Lemma 2.1 satisfies
u∈Cγ((0,T]; D(A)) ∩C1+γ((0,T]; L2),0< γ < 1
2,(2.4)
for D(A)={u∈L2
σ∩H2|(D(u)n)tan =0,u·n=0∂Π}. In particular, u satisfies the
equations (1.1) and (1.3).
Proof. We set f=−Pu· ∇u. It follows from (2.1)-(2.3) that
||f||2≤ ||u· ∇u||2≤ ||u||3||∇u||6≤C
t3
4
,
||f(t)−f(τ)||2≤ ||(u(t)−u(τ)) · ∇u(t)||2+||u(τ)· ∇(u(t)−u(τ))||2
≤ ||u(t)−u(τ)||3||∇u(t)||6+||u(τ)||6||∇u(t)− ∇u(τ)||3
≤C|t−τ|α
t3
4
+|t−τ|α
2
τ1
4for 0 < τ < t≤T.
Thus f∈L1(0,T;L2)∩Cα/2((0,T]; L2) for α∈(0,1). Applying Proposition 2.2 yields
(2.4).
2.2. Axial symmetry. We show that mild solutions are axisymmetric and satisfies the en-
ergy equality (1.5) for axisymmetric initial data.
Lemma 2.4. Assume that u0is axisymmetric. Then, the mild solution u in Lemma 2.1 is
axisymmetric and satisfies
(2.5)
∂tur+v· ∇ur−|uθ|2
r−∆−1
r2ur+∂rp=0
∂tuθ+v· ∇uθ+ur
ruθ−∆−1
r2uθ=0
∂tuz+v· ∇uz−∆uz+∂zp=0
∂rur+ur
r+∂zuz=0
in Π×(0,T),
ur=0, ∂ruθ−uθ=0, ∂ruz=0on ∂Π×(0,T),(2.6)
and the energy equality (1.5).
Proposition 2.5. Assume that a vector field u =urer+uθeθ+uzezsatisfies (1.3). Then,
(ur,uθ,uz)satisfies (2.6). The converse also holds.
8
Proof. By fundamental calculations using the cylindrical coordinate, we observe that
D(urer)er=∂rurer+1
2r∂θureθ+1
2∂zurez,
D(uθeθ)er=1
2∂ruθ−uθ
reθ,
D(uzez)er=1
2∂ruzez,
D(u)er=∂rurer+1
21
r∂θur+∂ruθ−uθ
reθ+1
2(∂zur+∂ruz)ez.
By (1.3), (ur,uθ,uz) satisfies (2.6). Conversely, suppose that (2.6) holds. Then,
D(u)er=∂rurer,u·er=0 on {r=1}.
Thus (1.3) holds for u=urer+uθeθ+uzez.
Proposition 2.6. Set the rotation operator U =Uη:L2(Π)−→ L2(Π)by
f(x)7−→ tR f (Rx)
and R =(er(η),eθ(η),ez)for η∈[0,2π]. Then, we have
UetA f=etAU f,(2.7)
UPg=PUg,(2.8)
U(h· ∇h)=(Uh)· ∇(U h),(2.9)
for f ∈L2
σ, g ∈L2and h ∈H1satisfying h · ∇h∈L2.
Proof. We give a proof for (2.7). We are able to prove (2.8) and (2.9) by a similar way.
We set w=etA fand wη=Uηw. Since the Stokes equations are rotationally invariant, wη
satisfies
∂twη−∆wη+∇qη=0
div wη=0in Π×(0,∞),
with some associated pressure qη. It follows that
wη(x)=tRw(Rx)
=wr(r, θ +η, z)er(θ)+wθ(r, θ +η, z)eθ(θ)+wz(r, θ +η, z)ez.
Since (wr,wθ,wz) satisfies (2.6) by Proposition 2.5, wηsatisfies the slip boundary condition
9
(1.3). Since wηis a unique solution of the Stokes equations for fη=Uηf, we have wη=
etA fη.
Proof of Lemma 2.4. We multiply Uby (1.4). It follows from (2.7)-(2.9) that
Uu =UetAu0−Zt
0
Ue(t−s)AP(u· ∇u)(s)ds
=etAUu0−Zt
0
e(t−s)AP(Uu · ∇U u)(s)ds.
Since u0is axisymmetric, u0=Uu0. Hence Uu is a mild solution of (1.1) for u0. By the
uniqueness of the mild solution, we have u=Uηufor η∈[0,2π]. Thus uis axisymmetric.
Since usatisfies (1.1) and (1.3) by Proposition 2.3, (ur,uθ,uz) satisfies (2.5) and (2.6). The
energy equality (1.5) follows from integration by parts.
3. A maximum principle
We consider the drift-diffusion equation (1.8) with a bounded coefficient and prove the
L∞-estimate (1.9) by a maximum principle. Let C(Π×[0,T]) denote the space of all bounded
and continuous functions in Π×[0,T]. Let C2,1(Π×[δ, T]) denote the space of all functions
f∈C(Π×[δ, T]) such that ∂s
t∂k
xf∈C(Π×[δ, T]) for 2s+|k| ≤ 2. We denote by C2,1(Π×(0,T])
the space of all functions in C2,1(Π×[δ, T]) for all δ∈(0,T). The goal of this section is:
Lemma 3.1. Let Γ∈C2,1(Π×(0,T]) ∩C(Π×[0,T]) be a solution of (1.8). Assume that
b∈C(Π×[0,T]). Then, the L∞-estimate (1.9) holds for t ≥0.
We prove Lemma 3.1 by a maximum principle. When Πis bounded, a maximum princi-
ple with the Robin boundary condition is known [30, Lemma 2.3]. We give a proof for the
unbounded domain Π.
Proposition 3.2 (Maximum principle).Assume that Γ∈C2,1(Π×(0,T]) ∩C(Π×[0,T])
satisfies
∂tΓ + b· ∇Γ−∆Γ + 2
r∂rΓ≤0in Π×(0,T],(3.1)
∂nΓ + 2Γ≤0on ∂Π×(0,T],(3.2)
Γ≤0on Π× {t=0}.(3.3)
Then,
10
Γ≤0in Π×[0,T].(3.4)
Corollary 3.3. Assume that the reverse inequalities of (3.1)-(3.3) hold. Then, Γ≥0in
Π×[0,T].
Proof of Lemma 3.1. We set
M=sup
x∈Π
Γ0(x),
m=inf
x∈ΠΓ0(x).
We first show (1.9) when m≤0. We set
Γm=m−Γ.
The function Γmsatisfies (3.1) and (3.3). Since m≤0, it follows that
(∂n+2)Γm=2m−(∂n+2)Γ
=2m≤0.
Hence the condition (3.2) is satisfied. Applying Proposition 3.2 implies that
m≤Γ(x,t) in Π×[0,T].(3.5)
We next estimate Γfrom above. We first consider the case M≤0. Since Γ0≤M≤0, we
apply Proposition 3.2 to Γand observe that Γ≤0. It follows from (3.5) that
||Γ||∞=−inf
x∈Π
Γ(x,t)
≤ −m=||Γ0||∞.
Thus (1.9) holds. We next consider the case M>0. We set
ΓM=M−Γ.
Since (∂n+2)ΓM=2M>0, the reverse inequalities of (3.1)-(3.3) hold for ΓM. Applying
Corollary 3.3 implies that
Γ(x,t)≤Min Π×[0,T].(3.6)
11
By (3.5) and (3.6), we obtain
||Γ||∞=max n−inf
x∈ΠΓ(x,t),sup
x∈Π
Γ(x,t)o
≤max{−m,M}=||Γ0||∞.
We proved (1.9) when m≤0.
It remains to show (1.9) when m>0. Since Γ0≥m>0, we observe that Γ≥0 by
Corollary 3.3. Applying Corollary 3.3 for ΓM=M−Γimplies that 0 ≤Γ≤M. Thus (1.9)
holds when m>0. The proof is complete.
We prove Proposition 3.2 from the following:
Proposition 3.4. We set
L=∂t+b· ∇ − ∆ + 2
r∂r,
N=n· ∇.
Assume that Γ∈C2,1(Π×(0,T]) ∩C(Π×[0,T]) satisfies
(L+1)Γ≤0in Π×(0,T],(3.7)
(N+2)Γ≤0on ∂Π×(0,T],(3.8)
Γ≤0on Π× {t=0}.(3.9)
Then,
Γ≤0in Π×[0,T].
Proof of Proposition 3.2. Applying Proposition 3.4 for ˜
Γ = Γe−timplies (3.4).
We first consider the case when the function Γattains a maximum in Π. When Γattains
the maximum as |x| → ∞, we modify Γso that it attains a maximum in Π.
Proof of Proposition 3.4. We argue by contradiction. Suppose on the contrary that there
exists a point (x0,t0)∈Π×[0,T] such that
Γ(x0,t0)>0.(3.10)
12
We set
M=sup Γ(x,t)|x∈Π,t∈[0,T]>0.
Case 1. The function Γattains the maximum in Π×[0,T].
We take a point (x1,t1)∈Π×[0,T] such that
M= Γ(x1,t1)>0.
By (3.9), we may assume that t1>0. Then, there are two cases whether x1∈Πor x1∈∂Π.
(a) x1∈Π. We observe that
∂tΓ(x1,t1)≥0,
∇Γ(x1,t1)=0,
∆Γ(x1,t1)≤0.
Hence we have
((L+1)Γ)(x1,t1)≥Γ(x1,t1)>0.
This contradicts (3.7). Thus the function Γdoes not attain the maximum in the interior of Π.
(b) x1∈∂Π. Since the function Γincreases along the normal direction near the boundary,
we have
∂Γ
∂n(x1,t1)≥0.
It follows that
((N+2)Γ)(x1,t1)≥2Γ(x1,t1)>0.
This contradicts (3.8). Thus the function Γdoes not attain the maximum on the boundary.
Case 2. The function Γattains the maximum at space infinity.
We modify Γand reduce the problem to Case 1. We set
Γε(x,t)= Γ(x,t)−ε(At +|x|2),
13
by positive constants A, ε > 0. We shall show that, by choosing A−1and εsufficiently small,
depending on b,x0,t0and Γ(x0,t0), the function Γεsatisfies the conditions (3.7)-(3.10).
Once we verify these conditions, it is not difficult to derive a contradiction. In fact, the
function Γεis negative in Π∩ {|x|>R} × [0,T] for R=√M/ε. The condition (3.10) for Γε
implies the existence of some point (x1,t1)∈Π∩ {|x| ≤ R} × [0,T] such that
Mε=sup Γε(x,t)|x∈Π,t∈[0,T]
= Γε(x1,t1)>0.
However, by the same way as we have shown in Case 1, the conditions (3.7)-(3.10) for Γε
imply that such the point (x1,t1) does not exist. Thus we are able to conclude that Case 2
does not occur neither.
It remains to show (3.7)-(3.10) for Γε. It follows that
(∂n+2)(At +|x|2)=(−∂r+2)(At +r2+|z|2)
=2(At +|z|2)+2r(r−1)
≥0,
(N+2)Γε=(N+2)Γ−ε(∂n+2)(At +|x|2)≤0.
Thus the conditions (3.8) and (3.9) are satisfied for A, ε > 0. We show that (3.7) holds for
Γεand sufficiently large A. Since
L(At +|x|2)=∂t+b· ∇ − ∆ + 2
r∂r(At +|x|2)
=A+2b·x−2,
it follows that
(L+1)Γε=(L+1)Γ−ε(L+1)(At +|x|2)
=(L+1)Γ−ε(A(1 +t)+|x|2+2b·x−2).
Since the function Γsatisfies (3.7), the first term of the right-hand side is negative. We set
A0=sup 2+2||b||L∞(Π×[0,T])|x| − |x|2|x∈Π>0.
It follows that
14
A(1 +t)+|x|2+2b·x−2≥A−(2 +2||b||∞|x| − |x|2)
≥A−A0.
Thus the condition (3.7) holds for Γεand A≥A0. Since
Γε(x0,t0)= Γ(x0,t0)−ε(At0+|x0|2),
the condition (3.10) holds for Γε,ε < ε0and ε0= Γ(x0,t0)(At0+|x0|2)−1>0. We proved
that (3.7)-(3.10) holds for Γε. The proof is now complete.
4. An a priori L∞-estimate for swirl
We prove the a priori L∞-estimate for the swirl component (1.6) (Lemma 4.7). Since the
boundedness of ruθdoes not follow from properties of local-in-time solutions to (1.1), we
extend the L∞-estimate (1.9) for mild solutions to (1.8). In the subsequent section, we show
that ruθis a mild solution to (1.8) and obtain the desired estimate (1.6).
4.1. Mild solutions. We define a mild solution of (1.8). We set the elliptic operators by
L0γ= ∆γ−1
r2γ,
L1Γ = ∆Γ −2
r∂rΓ,
subject to the Robin boundary conditions, ∂nγ+γ=0 and ∂nΓ + 2Γ = 0 on ∂Π. We also
set the operator L′
0= ∆ −r−2, subject to the Dirichlet boundary condition. By the classical
Lp-estimates for elliptic operators [2], it is known that the operators B=L0,L1,L′
0generate
C0-analytic semigroups on Lpfor p∈(1,∞) [32, Theorem 3.1.3]. Moreover, the semigroups
are analytic also for p=∞(see [32, Corollary 3.1.24]). By analyticity of the semigroups,
they satisfy the regularizing estimate
||∂k
xetB f||∞≤C
t3
2p+|k|
2||f||p
(4.1)
for 0 <t≤T0, 3 <p≤ ∞ and |k| ≤ 1. By using the semigroup et L1, we consider the integral
equation
Γ = etL1Γ0−Zt
0
e(t−s)L1(b· ∇Γ)(s)ds.(4.2)
We assume that the coefficient bsatisfies the regularity condition
15
(4.3) t1
2−3
2pb∈C0([0,T]; Lp) for 3 <p≤ ∞.
Here, C0([0,T]; Lp) denotes the space of all functions in C([0,T]; Lp), vanishing at time
zero. Note that solutions of (1.4) satisfies the condition (4.3) by Lemma 2.1. We prove
the L∞-estimate (1.9) for mild solutions Γ∈Cw([0,T]; L∞) of (4.2), where Cw([0,T]; L∞)
denotes the space of all weakly-star continuous functions from [0,T] to L∞.
We first recall that mild solutions of (4.2) are H¨older continuous up to second orders in
Π×[0,T] for sufficiently smooth Γ0and bby the H¨older regularity results for second order
equations [25, Chapter IV], [32, Chapter 5].
Let C(Π) denote the space of all bounded and continuous functions in Π. Let Cm(Π)
denote the space of all functions f∈C(Π) such that ∂k
xf∈C(Π) for |k| ≤ mwith non-
negative integer m. We denote by C∞(Π) the space of all functions in Cm(Π) for all m≥1.
We denote by Cµ(Π) the space of all µ-th H¨older continuous functions f∈C(Π) for µ∈
(0,1). For m=[m]+µ,Cm(Π) denotes the space of all functions f∈C[m](Π) such that
∂k
xf∈Cµ(Π) for |k|=[m], where [m] is the greatest integer smaller than m>0. We denote
by Cµ,µ/2(Π×[0,T]) the parabolic H¨older space for µ∈(0,2), which is the space of all
functions f∈C(Π×[0,T]) such that f(·,t)∈Cµ(Π) for t∈[0,T] and f(x,·)∈Cµ/2[0,T]
for x∈Π. We denote by C2+µ,1+µ/2(Π×[0,T]) the space of all functions f∈C2,1(Π×[0,T])
such that ∂s
t∂k
xf∈Cµ,µ/2(Π×[0,T]) for 2s+|k| ≤ 2.
Proposition 4.1. Let T >0. Let b satisfy (4.3).
(i) For Γ0∈L∞, there exists a unique mild solution Γ∈Cw([0,T]; L∞)of (4.2) such that
t1/2∇Γ∈Cw([0,T]; L∞). If Γ0and b are axisymmetric, the mild solution Γis axisymmetric.
(ii) Assume that
b∈Cµ,µ/2(Π×[0,T]), µ ∈(0,1),(4.4)
Γ0∈C2+µ(Π)and ∂nΓ + 2Γ = 0on ∂Π.(4.5)
Then, the mild solution belongs to C2+µ,1+µ/2(Π×[0,T]). In particular, the L∞-estimate
(1.9) holds for t ≥0.
Proof. The assertion (i) follows from a standard iteration argument. We are able to prove
axial symmetry by a similar way as we did in the proof of Lemma 2.4. The assertion (ii)
follows from a H ¨older regularity result for second order equations [32, Theorem 5.1.21,
Corollary 5.1.22]. The L∞-estimate (1.9) follows from Lemma 3.1.
16
4.2. Approximation of initial data. We prove the L∞-estimate (1.9) without the conditions
(4.4) and (4.5) by approximation. For this purpose, we prepare H ¨older norms for space-time
functions [25]. We set the µ-th H¨older semi-norm in Q= Ω ×(δ, T] for µ∈(0,1) by
[f](µ, µ
2)
Q=sup
t∈(δ,T]
[f](µ)
Ω(t)+sup
x∈Ω
[f](µ
2)
(δ,T](x),
[f](µ)
Ω(t)=sup (|f(x,t)−f(y,t)|
|x−y|µ
x,y∈Ω,x,y),
[f](µ
2)
(δ,T](x)=sup (|f(x,t)−f(x,s)|
|t−s|µ
2
t,s∈(δ, T],t,s).
When µ=1, we set
[f](1,1
2)
Q=||∇f||L∞(Q)+sup
x∈Ω
[f](1
2)
(δ,T](x).
For m=[m]+µ, we set
[f](m,m
2)
Q=X
2s+|k|=[m]
[∂s
t∂k
xf](µ, µ
2)
Q,
|f|(m,m
2)
Q=X
2s+|k|≤[m]||∂s
t∂k
xf||L∞(Q)+[f](m,m
2)
Q.
We first remove the condition (4.5) by approximation of Γ0∈L∞.
Proposition 4.2. For Γ0∈L∞(Π), there exists a sequence {Γ0,ε} ⊂ C∞(Π)supported in Π
such that
(4.6) ||Γ0,ε||∞≤ ||Γ0||∞
Γ0,ε →Γ0a.e. in Π.
Proof. For x=rer(θ)+zez, we set
˜
Γ0,ε(x)=
Γ0((r−ε)er(θ)+zez)r≥1+ε,
0 0 ≤r<1+ε.
By mollification of ˜
Γ0,ε, we obtain the desired sequence.
Proposition 4.3. In Proposition 4.1 (ii), the estimate (1.9) holds without the condition (4.5).
17
Proof. For Γ0∈L∞, we take a sequence {Γ0,ε}satisfying (4.6). Since Γ0,ε is smooth in Π
and supported in Π, it satisfies the condition (4.5). Since the estimate (1.9) holds for the
mild solution Γεof (4.2) for Γ0,ε by Proposition 4.1 (ii), it follows from (4.6) that
||Γε||∞≤ ||Γ0||∞t>0.(4.7)
We shall show that Γεconverges to a mild solution of (4.2) for Γ0. We use the H ¨older
continuity of the coefficient bin (4.4). We apply the local H¨older estimate for parabolic
equations [25, Chapter IV, Theorem 10.1] and estimate
|Γε|(2+µ,1+µ
2)
Q≤C||Γε||L∞(Π×(0,T))
(4.8)
for Q=(B∩Π)×(δ, T] and δ > 0 with some constant C, independent of ε. Here, B⊂R3
denotes an open ball satisfying B∩Π,∅. By (4.7) and (4.8), Γεsubsequently converges to
a limit Γlocally uniformly in Π×(0,T] up to second derivatives.
It is not difficult to see that the limit Γis a mild solution of (4.2) for Γ0. In fact, by
choosing a subsequence, we have
etL1Γ0,ε →etL1Γ0locally uniformly in Π×(0,T].
Since ∇Γεconverges to ∇Γlocally uniformly in Π×(0,T], similarly for each 0 <s<t, we
have
eρL1b· ∇Γε→eρL1b· ∇Γlocally uniformly in Π×(0,T].
Hence sending ε→0 to (4.2) implies the limit Γis a mild solution for Γ0. The estimate
(4.7) is inherited to the limit Γ.
4.3. Approximation of a coefficient. We next remove the condition (4.4).
Proposition 4.4. For b satisfying (4.3), there exists a sequence {bε} ⊂ C∞(Π×[0,T]) satis-
fying (4.3) and
lim
ε→0sup
0≤t≤T
t1
2−3
2p||b−bε||p(t)=0for 3<p<∞.(4.9)
Proof. We may assume that bis smooth in Πby mollification by spatial variables. Since g=
t1/2−3/(2p)bvanishes at time zero by (4.3), by shifting gby a time variable, and mollification,
we obtain a sequence {gε} ⊂ C∞(Π×[0,T]) such that gε(·,t) is supported in (0,T] and
18
lim
ε→0sup
0≤t≤T||gε−g||p(t)=0.(4.10)
Since gε(·,t) is supported in (0,T], the function bε=t−1/2+3/2pgεis smooth in Π×[0,T]
and satisfies (4.3). The convergence (4.9) follows from (4.10).
Lemma 4.5. The estimate (1.9) holds for mild solutions of (4.2) for Γ0∈L∞and t >0.
Proof. We shall show the estimate (1.9) between 0 <t≤T1for some T1>0. Once we
have (1.9) near time zero, it is extendable for all t>0 by taking t=T1as an initial time.
We take a sequence {bε}satisfying (4.9). Since the estimate (1.9) holds for a mild solution
Γεfor Γ0∈L∞and the coefficient bεby Proposition 4.3, we have
||Γε||∞≤ ||Γ0||∞t>0.(4.11)
We shall show that Γεconverges to a mild solution Γin the sense that
lim
ε→0sup
0<t≤T1n||Γ−Γε||∞+t1
2||∇(Γ−Γε)||∞o=0.(4.12)
The desired estimate follows from (4.11) and (4.12) by sending ε→0.
We set ρε= Γ −Γεand aε=b−bε. It follows from (4.2) that
ρε=−Zt
0
e(t−s)L1(b· ∇Γ−bε· ∇Γε)ds
=−Zt
0
e(t−s)L1(aε· ∇Γ + bε· ∇ρε)ds.
For p∈(3,∞), we set the constants
Kε=sup
0<t≤T1||ρε||∞+t1
2||∇ρε||∞,
K=sup
0<t≤T1||Γ||∞+t1
2||∇Γ||∞,
Nε=sup
0≤t≤T1
t1
2−3
2p||aε||p,
Lε=sup
0≤t≤T1
t1
2−3
2p||bε||p.
It follows from (4.1) that
19
||ρε||∞≤Zt
0
C
(t−s)3
2p||aε||p||∇Γ||∞+||bε||p||∇ρε||∞ds
≤C(NεK+LεKε)Zt
0
ds
(t−s)3
2ps1−3
2p
=C0(NεK+LεKε).
Similarly, we estimate ∇ρεand obtain
Kε≤C0(NεK+LεKε).
We take an arbitrary δ > 0. By (4.3), there exists T1>0 such that
sup
0<t≤T1
t1
2−3
2p||b||p≤δ.
By (4.9), that there exits ε0>0 such that
sup
0<t≤T1
t1
2−3
2p||b−bε||p≤δfor ε≤ε0.
We estimate
Lε=sup
0<t≤T1
t1
2−3
2p||bε||p≤sup
0<t≤T1
t1
2−3
2p||bε−b||p+sup
0<t≤T1
t1
2−3
2p||b||p
≤2δfor ε≤ε0.
By taking δ=(4C0)−1, we estimate
Kε≤2C0NεK.
Since Nε→0 as ε→0 by (4.9), we proved (4.12).
4.4. An application to axisymmetric solutions. We now prove the a priori L∞-estimate
for the swirl component (1.6). It suffices to show that the swirl component ruθis a mild
solution of (4.2).
Proposition 4.6. The semigroups satisfy
20
eθ·etA f=etL0fθ,(4.13)
eθ·curl etAg=et L′
0eθ·curl g,(4.14)
retL0γ=etL1(rγ),(4.15)
for axisymmetric f , g ∈L2
σand γ∈L∞satisfying eθ·curl g ∈L2and rγ∈L∞.
Proof. We set w=etA f. Since wθ=eθ·wsatisfies
∂twθ−∆−1
r2wθ=0 in Π×(0,∞),
∂nwθ+wθ=0 on ∂Π×(0,∞),
wθ=fθon Π× {t=0},
the function wθagrees with etL0fθby the uniqueness of the heat equation. Similarly, we are
able to prove (4.14) and (4.15).
Lemma 4.7 (A priori L∞-estimate).Let u =v+uθeθbe an axisymmetric mild solution of
(1.4) in Lemma 2.4. Assume that ruθ
0∈L∞. Then, ruθis a mild solution of (4.2) for b =v.
In particular, the L∞-estimate (1.6) holds.
Proof. We set
(4.16) h=u· ∇u=v· ∇ur−|uθ|2
rer+v· ∇uθ+ur
ruθeθ+(v· ∇uz)ez,
∇Φ = (I−P)h,
for axisymmetric mild solutions uin [0,T]. Since his axisymmetric, the function Φis
independent of θand we have
eθ·Pu· ∇u=eθ·(h− ∇Φ)
=eθ·(h−er∂rΦ−ez∂zΦ)
=v· ∇uθ+ur
ruθ.
We multiply eθby (1.4). It follows from (4.13) that
uθ=etL0uθ
0−Zt
0
e(t−s)L0v· ∇uθ+ur
ruθds.
21
Since uθ
0∈L∞by ruθ
0∈L∞, the above integral form implies that uθ∈Cw([0,T]; L∞) and
t1/2∇uθ∈Cw([0,T]; L∞).
On the other hand, there exists a unique axisymmetric mild solution Γfor Γ0=ruθ
0and
b=vby Proposition 4.1 (i). We multiply r−1by (4.2). It follows from (4.15) that γ= Γ/r
satisfies
γ=etL0uθ
0−Zt
0
e(t−s)L0v· ∇γ+ur
rγds.
Since γ∈Cw([0,T]; L∞) and t1/2∇γ∈Cw([0,T]; L∞), it is not difficult to show that uθ
agrees with γby estimating the difference uθ−γ. Thus ruθis a mild solution of (4.2). The
proof is now complete.
5. Energy estimates for the azimuthal component of vorticity
We prove the global estimates (1.11) and (1.12).
(5.1)
∂tωθ+v· ∇ωθ−ur
rωθ−∆−1
r2ωθ=∂z|uθ|2
rin Π×(0,T),
ωθ=0 on ∂Π×(0,T),
ωθ=ωθ
0on Π× {t=0}.
Proposition 5.1. Axisymmetric mild solutions of (1.4) in Lemma 2.4 satisfy
ωθ∈Cγ((0,T]; D(L′
0)) ∩C1+γ((0,T]; L2),0< γ < 1
2.(5.2)
In particular, ωθsatisfies the vorticity equation (5.1), where D(L′
0)=H2∩H1
0denotes the
domain of the operator L′
0= ∆ −r−2on L2.
Proof. We recall that the mild solution uis expressed by
u(η+δ)=eηAu(δ)+Zη
0
e(η−s)Af(δ+s)ds,
for 0 ≤δ≤η≤T−δand f=−Pu· ∇u. It follows from (4.16) that
22
g:=eθ·curl f
=−eθ·curl (h− ∇Φ)
=−∂zhr+∂rhz
=−v· ∇ωθ+ur
rωθ+∂z|uθ|2
r.
We multiply eθ·curl by u. It follows from (4.14) that
ωθ(η+δ)=eηL′
0ωθ(δ)+Zη
0
e(η−s)L′
0g(δ+s)ds.
Since u∈Cγ((0,T]; H2) for γ∈(0,1/2) by (2.4), we have g∈Cγ((0,T]; L2). By Proposi-
tion 2.2, ωθsatisfies (5.2).
Proposition 5.2. The function v =urer+uzezsatisfies
||∇v||2=||ωθ||2,(5.3)
||ur||4≤C||ur||
1
4
2||ωθ||
3
4
2,(5.4)
||uz||4≤C||uz||
1
4
2(||uz||2+||ωθ||2)3
4,(5.5)
||ωθ||4≤C||ωθ||
1
4
2||∇ωθ||
3
4
2,(5.6)
||∇(ωθeθ)||2=||∇ωθ||2
2+
ωθ
r
2
21
2,(5.7)
with some constant C. In particular, the estimate (1.10) holds.
Proof. Since vsatisfies
−∆v=curl curl v− ∇div v
=curl (ωθeθ),
it follows from (2.6) that
ZΠ|∇v|2dx=−ZΠ
∆v·vdx+Z∂Π
∂v
∂n·vdH
=ZΠ
curl (ωθeθ)·vdx−Z∂Π
(∂rurur+∂ruzuz)dH
=ZΠ|ωθ|2dx.
23
Thus (5.3) holds. Similarly, we obtain (5.7) by integration by parts. Since vrand ωθvanish
on ∂Π, applying the interpolation inequality (B.1) implies (5.4) and (5.6). We apply (B.2)
for vzand obtain (5.5).
Lemma 5.3. The estimates (1.11) and (1.12) hold for t >0for axisymmetric mild solutions
for u0∈˜
L3
σsatisfying ruθ
0∈L∞and ∇u0∈L2with some constant C.
Proof. We prove (1.11). Since ωθ/rsatisfies (1.2) and vanishes on ∂Π, by multiplying 2ωθ/r
by (1.2) and integration by parts, we have
d
dtZΠ
ωθ
r
2dx+2ZΠ∇ωθ
r
2dx=−2ZΠuθ
r2∂zωθ
rdx.
Since r≥1, it follows that
uθ
r
4≤ ||uθ||4=
(ruθ)1
2uθ
r1
2
4≤ ||ruθ||
1
2
∞
uθ
r
1
2
2.(5.8)
By the Young’s inequality, we estimate
2ZΠuθ
r2∂zωθ
rdx≤
uθ
r
4
4+
∇ωθ
r
2
2
≤ ||ruθ||2
∞
uθ
r
2
2+
∇ωθ
r
2
2.
Hence
d
dtZΠ
ωθ
r
2dx+ZΠ∇ωθ
r
2dx≤ ||ruθ||2
∞
uθ
r
2
2.
We integrate the both sides between (0,t). By applying (1.5) and (1.6), we obtain (1.11).
We prove (1.12). We multiply 2ωθby (5.1) to see that
d
dtZΠ|ωθ|2dx+2ZΠ|∇ωθ|2+
ωθ
r
2dx=2ZΠ
ur
rωθωθdx+2ZΠ
∂z|uθ|2
rωθdx
=:I+II.
It follows from (5.4), (5.6) and (5.7) that
24
|I|=2ZΠ
ur
rωθωθdx≤2
ωθ
r
2||ur||4||ωθ||4
≤C
ωθ
r
2||ur||
1
4
2||ωθ||2||∇ωθ||
3
4
2
≤C
ωθ
r
3
4
2||ur||
1
4
2||ωθ||2||∇(ωθeθ)||2
≤C′
ωθ
r
3
2
2||ur||
1
2
2||ωθ||2
2+1
2||∇(ωθeθ)||2
2.
Since r≥1, it follows from (5.8) that
|II|=2ZΠ
∂z|uθ|2
rωθdx≤2||uθ||2
4||∇ωθ||2
≤2||ruθ||∞
uθ
r
2||∇(ωθeθ)||2
≤2||ruθ||2
∞
uθ
r
2
2+1
2||∇(ωθeθ)||2
2.
By combining the estimates for Iand II, we obtain
d
dtZΠ|ωθ|2dx+ZΠ|∇ωθ|2+
ωθ
r
2dx≤C
ωθ
r
3
2
2||ur||
1
2
2+||ruθ||2
∞||ωθ||2
2+
uθ
r
2
2.
We integrate the both sides between (0,t). By (1.11), (1.5) and (1.6), we obtain (1.12).
6. Global bounds on L4
Proof of Theorem 1.1. For an axisymmetric u0∈˜
L3
σsatisfying ruθ
0∈L∞, the axisymmetric
mild solution u∈C([0,T]; ˜
L3) satisfies (1.6) by Lemma 4.7. It follows from (1.6) and (2.2)
that ruθ
0(·,t0)∈L∞and
∇u(·,t0)∈L2for t0∈(0,T].
We may assume that ∇u0∈L2by taking t=t0as an initial time. It follows from (1.7) and
(1.10)-(1.12) that u∈L∞(0,∞;L4). By (1.5) and Lemma 2.1, the mild solution belongs to
BC([0,∞); ˜
L3). The proof is now complete.
25
Remarks 6.1.(i) (The Euler equations) We constructed global solutions in the exterior do-
main by using viscosity. For the Euler equations, existence of global solutions is unknown.
We refer to [33], [20] for a one-dimensional blow-up model of axisymmetric Euler flows on
the boundary. See [12], [11] for blow-up results of models.
(ii) (The Dirichlet boundary condition) The statement of Lemma 2.1 is valid also for the
Dirichlet boundary condition. However, in this case unique existence of global solution
is unknown even for axisymmetric data without swirl. Since the azimuthal component of
vorticity ωθdoes not vanish on the boundary subject to the Dirichlet boundary condition,
the global vorticity estimates (1.11) and (1.12) are not available unlike the slip boundary
condition.
(iii) (Uniform estimates) The assertion of Theorem 1.1 is valid also for the exterior of the
cylinder Πε={r> ε}and we are able to construct global solutions u=uεsatisfying (1.6)
and the energy equality
(6.1) ZΠε|u|2dx+2Zt
0ZΠε|∇v|2+|∇uθ|2+
uθ
r
2dxds
+2
εZt
0Z∂Πε|uθ|2dHds=ZΠε|u0|2dx,t≥0.
For the case without swirl, the a priori estimates
ZΠε
ωθ
r
2dx+2Zt
0ZΠε∇ωθ
r
2dxds≤ZΠε
ωθ
0
r
2dx,(6.2)
(6.3) ZΠε|ωθ|2dx+Zt
0ZΠε|∇ωθ|2+
ωθ
r
2dxds
≤ZΠε|ωθ
0|2dx+C
ωθ
0
r
3
2
L2(Πε)||u0||
5
2
L2(Πε),t>0,
hold with some constant C, independent of ε. Hence we have a uniform bound
sup
ε≤ε0||u||L∞(0,∞;H1(Πε)) <∞,
provided that L2-norms of uθ
0,ωθ
0/rand ωθ
0in Πεare uniformly bounded for ε≤ε0.
Appendix A. Mild solutions on ˜
L3
We give a proof for local solvability of (1.1) on ˜
L3(Lemma 2.1).
Proposition A.1. The Stokes semigroup satisfies
26
||∂k
xetA f||˜
Lp≤C
t3
2(1
q−1
p)+|k|
2||f||˜
Lq,(A.1)
||∂k
x(eρA−1)eηAf||˜
Lq≤Cρα
ηα+|k|
2||f||˜
Lq,(A.2)
for f ∈˜
Lq
σ,2≤q≤p<∞,0<t, ρ, η ≤T0,α∈(0,1),|k| ≤ 1and T0>0.
Proof. The estimates (A.1) and (A.2) follow from estimates of the Stokes semigroup on
˜
Lq[17, Theorem 1.2] and the interpolation inequality (B.2).
Proposition A.2. For u0∈˜
L3
σ, there exists T >0and a unique mild solution of (1.4)
satisfying (2.1) and (2.2).
Proof. We set
uj+1=etAu0−Zt
0
e(t−s)AP(uj· ∇uj)ds,
u1=etAu0,
Kj=sup
0≤t≤T
tγ{||uj||˜
Lp+t1
2||∇uj||˜
Lp},
for γ=1/2−3/(2p) and p∈(3,∞). We take q∈[2,p]. By applying the Young’s inequality,
we estimate
||uj· ∇uj||q≤ ||uj||η||∇uj||p
for 1/η =1/q−1/p. Since q∈[2,p] and p>3, we observe that σ=3(1/p−1/η)=
3(2/p−1/q)≤3/p<1. By applying the interpolation inequality (B.2), we estimate
||uj||Lη≤C||uj||1−σ
Lp||∇uj||σ
W1,p.
We take T≤1 and estimate
||uj· ∇uj||Lq≤C||uj||1−σ
Lp||∇uj||1+σ
W1,p≤C Kj
sγ!1−σ 2Kj
sγ+1
2!1+σ
≤C′K2
j
s3
2(1−1
q).
Since the above estimate holds for s≤T≤1, we have
||uj· ∇uj||˜
Lq=max{||uj· ∇uj||Lq,||uj· ∇uj||L2} ≤
CK2
j
s3
2(1−1
q).(A.4)
Applying (A.1) implies
27
||∂k
xe(t−s)APuj· ∇uj||˜
Lp≤
CK2
j
(t−s)3
2(1
q−1
p)+|k|
2s3
2(1−1
q).(A.5)
We estimate Kj+1. We set p0=max{3p/(p+3),2}and fix q∈(p0,3) so that the right
hand-side of (A.5) is integrable near s=tfor |k| ≤ 1. It follows from (A.1) and (A.5) that
||uj+1||˜
Lp≤ ||etAu0||˜
Lp+C
t1
2−3
2p
K2
j.
Similarly, we estimate ∇uj+1and obtain Kj+1≤K1+C0K2
j. Since the Stokes semigroup is
strongly continuous on ˜
L3, we have K1→0 as T→0. We take T>0 sufficiently small so
that K1≤(4C0)−1and
Kj≤2K1for j=1,2,···.
By a similar way, we estimate the difference uj+1−ujand obtain
sup
0≤t≤T
tγ(||uj+1−uj||˜
Lp+t1
2||∇(uj+1−uj)||˜
Lp)→0 as j→ ∞.
Thus the sequence {uj}converges to a mild solution usatisfying (2.1) and (2.2) for p,r∈
(3,∞). In particular, we have
K=sup
0≤t≤T
tγ{||u||˜
Lp+t1
2||∇u||˜
Lp} ≤ 2K1.(A.6)
The uniqueness follows from the integral form since tγuand tγ+1/2∇uvanish at time zero.
It remains to show (2.1) and (2.2) at end points. The property (2.1) for p=∞follows
from the interpolation inequality (B.2). It follows from (A.1), (A.4) and (A.6) that
||u−etAu0||˜
L3≤Zt
0||e(t−s)APu· ∇u||˜
L3ds
≤CK2
1Zt
0
ds
(t−s)3
2(1
q−1
3)s3
2(1−1
q)=C′K2
1.
Since K1→0 as T→0, the mild solution uis strongly continuous on ˜
L3at time zero.
Thus (2.1) holds for p=3. By a similar way, we estimate t1
2||∇u− ∇etAu0||˜
L3≤CK2
1. Since
t1/2∇etAu0vanishes on ˜
L3at time zero, (2.2) holds for r=3.
Proof of Lemma 2.1. It remains to show the H ¨older continuity (2.3). We set f=−Pu· ∇u.
It follows from (A.4) that
28
||f||˜
Lq≤CK2
s3
2(1−1
q)for 2 ≤q≤3.(A.7)
We take an arbitrary δ∈(0,T) and α∈(0,1). For δ≤τ < t≤T, we estimate
||u(t)−u(τ)||˜
L3≤ ||etAu0−eτAu0||˜
L3+Zt
τ||e(t−s)Af||˜
L3ds+Zτ
0||e(t−s)Af−e(τ−s)Af||˜
L3ds
=:I+II +III.
It follows from (A.2), (A.1) and (A.7) that
I≤Cδ−α(t−τ)α||u0||˜
L3,
II ≤CZt
τ||f||˜
L3ds≤C′δ−1K2(t−τ).
We estimate III. Since
e(t−s)Af−e(τ−s)Af=(e(t−τ)A−1)e(τ−s)
2Ae(τ−s)
2Af,
it follows from (A.2), (A.1) and (A.6) that
||e(t−s)Af−e(τ−s)Af||˜
L3≤Ct−τ
τ−sα||e(τ−s)
2Af||˜
L3
≤C′(t−τ)α
(τ−s)α+3
2(1
q−1
3)||f||˜
Lq
≤C′′K2(t−τ)α
(τ−s)α+3
2(1
q−1
3)s3
2(1−1
q).
We take q∈[2,3) so that 3/2(1/q−1/3) <1−αand obtain
III ≤Cδ−αK2(t−τ)α.
Thus u∈Cα([δ, T]; ˜
L3) for α∈(0,1). By a similar way, ∇u∈Cα/2([δ, T]; L2) follows. We
proved (2.3). The proof is now complete.
29
Appendix B. Interpolation inequalities
We give a proof for interpolation inequalities used in Proposition 5.2.
Lemma B.1. The estimates
||ϕ||p≤C||ϕ||1−σ
q||∇ϕ||σ
q, ϕ ∈W1,q
0,(B.1)
||φ||p≤C||φ||1−σ
q||φ||σ
1,q, φ ∈W1,q,(B.2)
hold for 1≤q≤p≤ ∞ satisfying σ=3(1/q−1/p)<1, where W1,q
0denotes the space of
functions in W1,q, vanishing on ∂Π.
Proof. The estimate (B.1) for Π = R3holds by estimates of the heat semigroup. Since the
trace of ϕ∈W1,q
0vanishes on ∂Π, we apply (B.1) to the zero extension of ϕto R3and obtain
the desired estimate for Π⊂R3. For functions φ∈W1,qwith non-trivial traces, we use an
extension operator E:W1,q(Π)−→ W1,q(R3) acting as a bounded operator also from Lq(Π)
to Lq(R3) [41, Chapter VI, 3.1 Theorem 5]. By applying (B.1) for R3and Eφ, we obtain
(B.2).
Acknowledgements
The first author would like to thank Oxford University for their hospitality from October
2015 to January 2016. The first author was supported by JSPS through the Grant-in-aid
for Research Activity Start-up 15H06312, Young Scientist (B) 17K14217 and Scientific
Research (B) 17H02853. The second author was supported by the Ministry of Education
and Science of the Russian Federation (grant 14.Z50.31.0037).
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(K. Abe) Departmen t of Math ematics, Graduate School of Sc ience, Osaka City University, Sugimoto 3-3-
138, Sumiyoshi-ku Osaka 558-8585, Japan
E-mail address:kabe@sci.osaka-cu.ac.jp
(G. Seregin) Mathematical Institute, Oxford University, Oxford, 24-29 StGiles OX1 3LB UK and
Voronezh State University, Voronezh, Russia
E-mail address:seregin@maths.ox.ac.uk
