RIMS Workshop on

Mathematical Analysis of Viscous Incompressible Fluid
Date: December 6 (Mon) – 8 (Wed), 2021
Venue: Online via Zoom

Organizers: Yasunori Maekawa (Kyoto University)
Yoshihiro Shibata (Waseda University)
This workshop will be held online via Zoom.
To participate in the workshop please register from the following form.
https://forms.gle/JgNuiBvyJeqGqYUu7
Deadline for registration: December 3 (Fri), 2020
Zoom URL information will be sent by email to all registrants on December 4 (Sat).

ProgramMonday, December 6
14:00 - 14:50 Koji Ohkitani (Kyoto University)
Self-similar profiles of solutions to hypo-viscous fluid equations
15:05 - 15:55 Hirokazu Saito (The University of Electro-Communications)
On the two-phase Navier-Stokes equations with a sharp interface
16:10 - 16:40 Ryosuke Nakasato (Tohoku University)
Well-posedness for the magnetohydrodynamics with Hall-effect near non-zeromagnetic equilibrium states
16:50 - 17:20 Zhongyang Gu (The University of Tokyo)
The Helmholtz decomposition of a space of vector fields with bounded mean
oscillation in a bounded domain

Tuesday, December 7
10:00 - 10:50 Tai-Peng Tsai (The University of British Columbia)
Weak and mild solutions to the Navier-Stokes equations in Wiener amalgam spaces
11:05 - 11:55 Ken Abe (Osaka City University)
Rigidity of Beltrami fields with a non-constant proportionality factor

14:00 - 14:50 Jan Brezina（Kyushu University）
On barotropic Navier-Stokes system with general boundary conditions
15:05 - 15:55 Erika Ushikoshi (Yokohama National University / Osaka University)
Hadamard variational formula for the fundamental solution of the non stationary
Stokes equations
16:10 - 16:40 Tatsu-hiko Miura (Kyoto University)
Enhanced dissipation for the two-jet Kolmogorov type flow on the unit sphere

Wednesday, December 8
10:00 - 10:50 Yoshifumi Kimura (Nagoya University)
Vortex reconnection and a finite-time singularity of the Navier-Stokes equations
11:05 - 11:55 Ryo Takada (Kyushu University)
Fast rotation limit for the incompressible Navier-Stokes equations in a 3D layer

RIMS workshop「Mathematical Analysis of Viscous Incompressible Fluid 」
December, 6-8, 2021 (Online via Zoom)
Organizers：Yasunori Maekawa（Kyoto University）
Yoshihiro Shibata (Waseda University)

Abstracts

Self-similar proﬁles of solutions to hypo-viscous ﬂuid equations
Koji Ohkitani
Research Institute for Mathematical Sciences, Kyoto University
ohkitani@kurims.kyoto-u.ac.jp

1 Motivation
We consider the source-type solution to the hypoviscous Burgers equation and provide aheuristic argument for ﬁnding a “Cole-Hopf-like transform” for its possible linearisation.

2 Burgers equation with standard dissipativity (review)
We consider the Burgers equation
ut + uux = νuxx ,(1)which satisﬁes static scale-invariance under x → λx, t → λ2 t, u → λ−1 u, for any λ(> 0).
This means that if u(x, t) is a solution, so is uλ (x, t) := λu(λx, λ2 t). E.g. it is readilyp−1checked that ∥uλ ∥Lp = λ p ∥u∥Lp , which shows that the L1 -norm is scale-invariant.
Let us clarify the two kinds of critical scale-invariance. Type 1 critical scale-invarianceis achieved with the velocity potential ϕ, deﬁned by u = ϕx , which obeys
1
ϕt + ϕ2x = νϕxx ,(2)2
where [ϕ] = [ν] = L2 /T . If ϕ(x, t) is a solution, so is ϕ(λx, λ2 t). Under dynamic scaling1
for the velocity potential ϕ(x, t) = Φ(ξ, τ ), ξ = √x2at , τ = 2alog t, (a > 0) we have1(3)Φτ + Φ2ξ = νΦξξ + aξΦξ .
2Type 1 critical scale-invariance is deterministic in nature, where the number of additionalterms is minimised, that is, only the drift term.
The other type 2 critical scale-invariance concerns the nth spatial derivative of theunknown for the type 1 scale-invariance (n is the spatial dimension.) For the Burgers
1equation, under dynamic scaling for velocity u(x, t) = √2at
U (ξ, τ ), we ﬁndUτ + U Uξ = νUξξ + a (ξU )ξ ,(4)

whose linearisation is the Fokker-Planck equation. The type 2 scale-invariance is statistical in nature where the number of additional terms is maximised in the sense that adivergence form is completed with the addition of aU term. A steady solution to (4)()
aξ 2
C exp − 2ν() .
U (ξ) =
(5)
∫ξ
aη 2
C
exp
−
dη
1 − 2ν
2ν
0
is known as the source-type solution where C = C(M ) is a constant, see e.g. [1, 2].
Observe that (5) is a near-identity transformation of the Gaussian function.

3 Burgers equation with hypo-viscous dissipativity
We now turn our attention to the hypo-viscous Burgers equation of the following form
ut + uux = −ν ′ Λu,(6)where Λ ≡ (−∂xx )1/2 = ∂x H[·] denotes the Zygmund operator and H[·] the Hilberttransform. The equation is known to be well-posed [3]. We are interested in deriving asource-type solution analogous to (5).
xType 1 critical scale-invariance: under dynamic scaling u(x, t) = U (ξ) , ξ = at,τ =1′
a log t, where [ν ] = [u] = L/T , (6) is transformed to
Ut + U Uξ = −ν ′ ΛU + aξUξ .

(7)

Type 2 critical scale-invariance: the governing equation for velocity gradient w = ux
wt + uwx = −w2 − ν ′ Λw,
is transformed under dynamic scaling toWτ + U Wξ = −W 2 − ν ′ ΛW + a(ξW )ξ ,(8)

where W = Uξ . Note that the diﬀerence in dissipativity oﬀsets the pair of criticalvariables (U, W ) by one derivative from that of the standard Burgers equation’s (Φ, U ).
Self-similar proﬁle: late-time behaviour is determined by the self-similar proﬁle W (ξ) =limτ →∞ W (ξ, τ ), if it exists. Its equation is obtained by setting Wτ = 0 in (8)(U W )ξ = −ν ′ ΛW + a(ξW )ξ ,which is integrated to give11ξW − H[W ] = ′ U W,
(9)
µ
ν
µ
where µ = ν ′ /a. This is the equation we ought to solve. It is clear that W = ξ2 +µ2ξ
and H[W ] = ξ2 +µ
2 solves it, when the nonlinearity (that is, RHS) is discarded. We willhandle (9) perturbatively assuming that the nonlinearity is small. A numerical approachfor handling (8) will also discussed. Time permitting, the SQG equation will also be
addressed together with comparisons to other works.

References[1] T.-P. Liu and M. Pierre, “Source-solutions and asymptotic behavior in conservation
laws,” J. Diﬀ. Eq. 51 419–441 (1984).
[2] M. Escobedo and E. Zuazua, “Large time behavior for convection-diﬀusion equations in RN ,” J. Func. Anal. 100, 119–161 (1991).
[3] A. Kiselev, F. Nazarov, R. Shterenberg, “Blow up and regularity for fractal Burgers
equation,” Dyn. Partial Diﬀer. Equ. 5, 211–40 (2008).

On the two-phase Navier-Stokes equations with a shape interface
Hirokazu Saito
The University of Electro-Communications
hsaito@uec.ac.jp
Let Ω± (t) and Γ(t), t > 0, be given byΩ± (t) = {x = (x′ , x3 ) ∈ R3 : x′ = (x1 , x2 ) ∈ R2 , ±(x3 − η(x′ , t)) > 0},
Γ(t) = {x = (x′ , x3 ) ∈ R3 : x′ = (x1 , x2 ) ∈ R2 , x3 = η(x′ , t)}.
In this talk, we consider the motion of two immiscible, viscous, incompressible ﬂuids,
ﬂuid+ and ﬂuid− , which occupy Ω+ (t) and Ω− (t), respectively. The positive constants
ρ± and µ± denote the densities and the viscosity coeﬃcients of the respective ﬂuids.
Deﬁne ρ = ρ+ 1Ω+ (t) + ρ− 1Ω− (t) and µ = µ+ 1Ω+ (t) + µ− 1Ω− (t) , where 1A is the indicatorfunction of A ⊂ R3 .
Let v = v(x, t) = (v1 (x, t), v2 (x, t), v3 (x, t))T be the ﬂuid velocity and q = q(x, t) thepressure at position x ∈ Ω+ (t) ∪ Ω− (t) with time t > 0. The motion is governed by thetwo-phase Navier-Stokes equations with the sharp interface Γ(t) as follows:∂t η − v3 = −v′ · ∇′ ηon Γ(t),ρ(∂t v + (v · ∇)v) = µ∆v − ∇q in Ω+ (t) ∪ Ω− (t),(1)
div v = 0in Ω+ (t) ∪ Ω− (t);the boundary conditions(−[[(µD(v) − qI)nΓ(t) ]] − [[ρ]]γa ηnΓ(t) = σκΓ(t) nΓ(t) on Γ(t),[[v]] = 0on Γ(t);(2)

the initial conditions
η|t=0 = η0

on R2 ,

v|t=0 = v0

in Ω0+ ∪ Ω0− .

(3)

The positive constants γa and σ denote the acceleration of gravity and the surface
tension coeﬃcient,
respectively. Let ∂t = ∂/∂t and ∂j = ∂/∂xj for j = 1, 2, 3. ThenPv′ ·∇′ η = 2j=1 vj ∂j η and D(v) = (∂i vj +∂j vi )1≤i,j≤3 . Let I be the 3×3 identity matrix.
In addition, nΓ(t) is the unit normal vector on Γ(t), pointing from Ω− (t) into Ω+ (t), andκΓ(t) the mean curvature of Γ(t). For functions f = f (x, t), x ∈ Ω+ (t) ∪ Ω− (t) witht > 0, we set[[f ]] = [[f ]](x0 , t) = lim f (x0 + εnΓ(t) , t) − f (x0 − εnΓ(t) , t)(x0 ∈ Γ(t)).
ε→0+0The η0 = η0 (x′ ) and v0 = v0 (x) are given initial data, andΩ0± = {x = (x′ , x3 ) | x′ = (x1 , x2 ), ±(x3 − η0 (x′ )) > 0}.

√∓ 1+|ξ ′ |2 x3
Deﬁne the extension operators E± by E± η = Fξ−1ηb(ξ ′ , t)](x′ ), ±x3 > 0,
′ [e
R
R
′
′
−ix ·ξ f (x′ ) dx′ and F −1 [g(ξ ′ )](x′ ) = (2π)−2ix′ ·ξ ′ g(ξ ′ ) dξ ′ . Let
where fb(ξ ′ ) =
′
2 e
2 e
ξ

R

R

Θ± be diﬀeomorphisms such that
Θ± : R3± × (0, ∞) 3 (x, t) = (x′ , x3 , t)[7→ Θ± (x, t) := (x′ , x3 + (E± η)(x′ , x3 , t), t) ∈Ω± (τ ) × {τ },τ ∈(0,∞)

where R3± = {±x3 > 0}. One sets u± = u± (x, t) = v(Θ± (x, t)) and p± = p± (x, t) =q(Θ± (x, t)), and then (1) becomes∂t η − u− · e3 = D(η, u− )on R30 × (0, ∞),ρ± ∂t u± − µ± ∆u± + ∇p± = F± (η, u± )in R3± × (0, ∞),(4)e ± (η, u± ) in R3 × (0, ∞),div u± = G± (η, u± ) = div G±

where R30 = {x3 = 0}; the boundary conditions in (2) become′
−{(µ+ D(u+ ) − p+ I)e3 − (µ− D(u− ) − p− I)e3 } − (ρ+ − ρ− )γa ηe3 − σ∆ ηe3= H+ (η, u+ ) − H− (η, u− ) − σHκ (η)e3 on R30 × (0, ∞), (5)u+ − u− = 0 on R30 × (0, ∞),P
where e3 = (0, 0, 1)T and ∆′ η = 2j=1 ∂j2 η; the initial conditions in (3) becomeon R2 ,

η|t=0 = η0

u± |t=0 = u0±

in R3± .

(6)e ± (η, u± ), H± (η, u± ), and Hκ (η) are nonlinear
Here D(η, u− ), F± (η, u± ), G± (η, u± ), G
terms. Our main result then reads as follows.
Theorem 1. Suppose ρ− > ρ+ . Then there exist a large number p0 > 2 and small
positive numbers q0 , r0 , ε0 such that for any p, q1 , and q2 satisfying
p0 ≤ p < ∞,and for any(η0 , u0+ , u0− ) ∈\2 < q 1 ≤ 2 + q0 ,3 < q2 < ∞,3−1/p−1/r
2−2/p
2−2/p
Br,p
(R2 ) × Br,p(R3+ )3 × Br,p(R3− )3r∈{q1 /2,q2 }satisfying some compatibility condition and the smallness conditionXk(η0 , u0+ , u0− )kB 3−1/p−1/r (R2 )×B 2−2/p (R3 )3 ×B 2−2/p (R3 )3 ≤ ε0 ,r∈{q1 /2,q2 }

r,p

r,p

−

r,p

+

(4)–(6) admits a unique solution (η, u+ , u− ), with pressures p± , satisfying
X 
khti1/2 (η, u+ , u− )kL (R ,W 3−1/q (R2 )×H 2 (R3 )3 ×H 2 (R3 )3 )q∈{q1 ,q2 }

p

+

q

q

+

q

−


+ khti1/2 (∂t η, ∂t u+ , ∂t u− )kL (R ,W 2−1/q (R2 )×L (R3 )3 ×L (R3 )3 ) ≤ r0 .
q
q
p
+
q
+
−
√
Here hti = 1 + t2 and the compatibility condition is introduced in the talk.

Well-posedness for the magnetohydrodynamics with Hall-eﬀect near
non-zero magnetic equilibrium states
Ryosuke Nakasato
Graduate school of Science, Tohoku University
ryosuke.nakasato.e4@tohoku.ac.jp
We consider the initial-value problem for the incompressible magnetohydrodynamic
system with the Hall-eﬀect (we also call it the Hall-magnetohydrodynamic system) inthe 3-dimensional Euclidean space R3 :e × B,e∂t u − ∆u + (u · ∇)u + ∇p = (∇ × B)
t > 0, x ∈ R3 ,







∂t B
e − ∆Be + ∇ × (∇ × B)e ×B
e = ∇ × (u × B),et > 0, x ∈ R3 ,
(1)

3
e

∇ · u = ∇ · B = 0,
t > 0, x ∈ R ,
e t=0 = (u0 , Be0 ),(u, B)|x ∈ R3 ,e = B(t,e x) :
where u = u(t, x) : R+ × R3 → R3 , p = p(t, x) : R+ × R3 → R and B
3
3
R+ × R → R denote the velocity of the ﬂuid, the pressure and the magnetic ﬁeld,
respectively. The third term of the second equations is often called the Hall-term. The
system (1) is used to model the magnetic reconnection phenomenon, that is not able to
be explained by the well-known magnetohydrodynamic system (namely, the system (1)without the Hall-term).
The aim of this talk is to obtain a solution as a perturbation from a constant equilibrium state (0, B̄), where B̄ ∈ R3 . In the case B̄ = 0, Danchin–Tan [1] have obtained a
global-in-time solution of (1) in critical Besov spaces.
Under the assumption B̄ ̸= 0, let us reformulate the system (1) by introducing the new
e − B̄ to get the following initial-value problem:
unknown vector-valued function B := B



|B|2


∂t u − ∆u + ∇ p +− (∇ × B) × B̄ = F,
t > 0, x ∈ R3 ,



2





∂t B − ∆B + ∇ × (∇ × B) × B̄ − ∇ × (u × B̄) = ∇ × G, t > 0, x ∈ R3 ,
(2)



3

∇ · u = ∇ · B = 0,
t > 0, x ∈ R ,(u, B)|t=0 = (u0 , B0 ),x ∈ R3 ,where the nonlinear terms F , G are given by
F := −(u · ∇)u + (B · ∇)B,

G := u × B − (∇ × B) × B,

(3)

respectively. In this talk, we shall state the results on the global well-posedness for thesystem (2) in a critical Lp -framework corresponding to Danchin–Tan [1].

References[1] Danchin, R., Tan, J., On the well-posedness of the Hall-magnetohydrodynamics
system in critical spaces, Comm. Partial Diﬀerential Equations, 46 (2021) 31–65.

The Helmholtz decomposition of a space of vector ﬁelds withbounded mean oscillation in a bounded domain
Zhongyang Gu
Yoshikazu Giga
The University of Tokyozgu@ms.u-tokyo.ac.jp
The Helmholtz decomposition of a vector ﬁeld is a fundamental tool to analyse the
Stokes and the Navier-Stokes equations. It is formally a decomposition of a vector ﬁeld
v in a domain Ω of Rn intov = v0 + ∇q
where v0 is a divergence free vector ﬁeld with supplemental condition like the boundary
condition and ∇q denotes the gradient of a scalar ﬁeld q. If v is in Lp in Ω for 1 < p < ∞,such a decomposition is well-studied; see e.g.[1]. However, if the vector ﬁeld is of boundedmean oscillation (BMO for short), such a problem is only studied when Ω is a half space
Rn+ [2], where the boundary is ﬂat. The goal of our result is to establish the Helmholtz
decomposition of BMO vector ﬁelds in a bounded domain, which is a typical example ofa domain with curved boundary.
Although the space of BMO functions in Rn is well-studied, the situation is less clear
if we consider such a space in a domain since there are several possible deﬁnitions due tothe presence of the boundary ∂Ω. Here is our setting. Let Ω be a bounded C 3 domain.
Let µ ∈ (0, ∞], for f ∈ L1loc (Ω) we deﬁne()
Z
1[f ]BM Oµ (Ω) := sup|f (y) − fBr (x) | dy Br (x) ⊂ Ω, r < µ|Br (x)| Br (x)where
1fBr (x) :=|Br (x)|Zf (y) dy.
Br (x)

Then we deﬁne the space BM Oµ (Ω) as
BM Oµ (Ω) := {f ∈ L1loc (Ω) | [f ]BM Oµ (Ω) < ∞}.
For ν ∈ (0, ∞], we set([f ]bν := sup

r

−n

)

Z
|f (y)| dy x ∈ ∂Ω, 0 < r < ν
Br (x)∩Ω

and let dΩ (x) denote the distance of x from the boundary ∂Ω, i.e.,dΩ (x) = inf {|x − y|, y ∈ ∂Ω}.

We then deﬁne the spacevBM Oµ,ν (Ω) = {v ∈ (BM Oµ (Ω))n | [∇dΩ · v]bν < ∞}where for v ∈ vBM O µ,ν (Ω),[v]vBM Oµ,ν (Ω) := [v]BM Oµ (Ω) + [∇dΩ · v]bν .
This space is introduced in our companion paper [3] in which the seminorm [·]vBM Oµ,ν (Ω)
is shown to be equivalent as far as each index is ﬁnite. In particular, in the case fora bounded domain, [·]vBM Oµ,ν (Ω) is indeed a norm and the space vBM Oµ,ν (Ω) is a
Banach space. Moreover, this space is independent of µ, ν including ∞. Hence, when Ωis bounded, without loss of generality we denote vBM O µ,ν (Ω) by vBM O(Ω). The main
theorem of our research reads as follow.
Theorem 1. Let Ω be a bounded C 3 domain in Rn . Then the topological direct sum
decompositionvBM O(Ω) = vBM Oσ (Ω) ⊕ GvBM O(Ω)
holds withvBM Oσ (Ω) := {v ∈ vBM O(Ω) | div v = 0 in Ω, v · n = 0 on ∂Ω},
GvBM O(Ω) := {∇q ∈ vBM O(Ω) | q ∈ L1loc (Ω)}
where n denotes the exterior unit normal vector ﬁeld. In other words, for v ∈ vBM O(Ω),
there exist unique v0 ∈ vBM Oσ (Ω) and ∇q ∈ GvBM O(Ω) satisfying v = v0 + ∇q.
Moreover, the mappings v 7→ v0 and v 7→ ∇q are bounded in vBM O(Ω).
For the space vBM O(Ω) that we consider, it is essential that we only require thenormal component of v, i.e., ∇dΩ · v, to be bν bounded in order to have the Helmholtz
decomposition. Requiring every component vi of v to be bν bounded is too strict to havethe Helmholtz decomposition.

References[1] G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state problems. Second edition. Springer Monographs in Mathematics.
Springer, New York, (2011). xiv+1018 pp.
[2] Y. Giga and Z. Gu, On the Helmholtz decompositions of vector ﬁelds of boundedmean oscillation and in real Hardy spaces over the half space. Adv. Math. Sci. Appl.
29 (2020), 87-128.
[3] Y. Giga and Z. Gu, Normal trace for vector ﬁelds of bounded mean oscillation.
arXiv:2011.12029 (2020).

Weak and mild solutions to the Navier-Stokes equations in Wiener
amalgam spaces
Tai-Peng Tsai
University of British Columbia
ttsai@math.ubc.ca
For the three dimensional incompressible Navier-Stokes equations in the Lp setting,the classical theories give existence of weak solutions for data in L2 and mild solutions
for data in Lp , p ≥ 3. These were extended to Lpuloc spaces, the space of functions withuniform local Lp norms, by Lemarié-Rieusset for weak solutions, and by Maekawa andTerasawa for mild solutions. Our goal is to build existence theorems in intermediate
spaces that bridge Lp and Lpuloc .
The Wiener amalgam space E(p, q) consists of functions whose local Lp -norms at
lattice points are globally in ℓq . Thus a function in E(p, q) has local integrability in Lp
and global decay in ℓq .
For weak solutions, we establish global existence in E(2, q) for q between 2 and ∞. For
q close to 2, the solutions are shown to satisfy some properties known in the Leray classbut not the Lemarié-Rieusset class, namely eventual regularity and long time estimateson the growth of the local energy.
For mild solutions, we establish local existence in E(p, q) for p ∈ (3, ∞] and q ∈ [2, ∞].
When p = 3, a further smallness assumption on the initial data ensures local existence
if q > 3, and global existence if q ≤ 3. We also prove local spacetime integral bounds ofthe solutions using Giga’s estimates.
This talk is based on joint work with Zachary Bradshaw and Chen-Chih Lai.

References[1] Zachary Bradshaw and Tai-Peng Tsai, Local energy solutions to the Navier-Stokes
equations in Wiener amalgam spaces, SIAM J. Math. Anal., to appear. Preprint:
arXiv:2008.09204[2] Zachary Bradshaw, Chen-Chih Lai and Tai-Peng Tsai, in preparation.

Rigidity of Beltrami ﬁelds with a non-constant proportionality
factorKen AbeOsaka City University
kabe@osaka-cu.ac.jp
Beltrami ﬁelds curl u = f u, div u = 0 appear as steady states of ideal incompressible
ﬂows or plasma equilibria. I will discuss existence and non-existence issues on them fornon-constant factor f . In the ﬁrst half of the talk, I will explain existence of axisymmetric
Beltrami ﬁelds forming vortex rings and their construction via a variational principle.
In the second half, I will discuss a rigidity problem on symmetry of u for symmetric f
and a relation with Grad’s conjecture. This talk is based on preprints arXiv:2008.09345,
arXiv:2108.03870.

On barotropic Navier-Stokes system with general boundary
conditions
Jan Březina
Kyushu University
brezina@artsci.kyushu-u.ac.jp
We consider the barotropic Navier–Stokes system describing the motion of a compressible Newtonian ﬂuid in a bounded domain with in and out ﬂux boundary conditions.
That is, the time evolution of the mass density ϱ = ϱ(t, x) and the velocity u = u(t, x)
is governed by∂t ϱ + divx (ϱu) = 0,∂t (ϱu) + divx (ϱu ⊗ u) + ∇x p(ϱ) = divx S(Dx u) + ϱ∇x G,
2t
S(Dx u) = µ ∇x u + ∇x u − divx uI + λ divx uI, µ > 0, λ ≥ 0,d1
∇x u + ∇tx u ,with Dx u ≡2(1)
on a bounded domain Ω ⊂ Rd , d = 1, 2, 3 and we consider the realistic situation with agiven boundary velocity,u|∂Ω = ub ,(2)and, decomposing the boundary asno∂Ω = Γin ∪Γout , Γin = x ∈ ∂Ω the outer normal n(x) exists, and ub (x) · n(x) < 0 ,we prescribe the density on the in–ﬂow component,
ϱ|Γin = ϱb .

(3)

Note that Γout includes the part of the boundary on which the ﬁeld ub is tangential,
meaning ub · n = 0.
We show that if the boundary velocity coincides with that of a rigid motion, allsolutions converge to an equilibrium state for large times. In other words, we study
stability and convergence to the static states in the multi–dimensional case, with thevelocity uE associated to a rigid motion, meaning
Dx uE = 0.

(4)

The corresponding density ϱE satisﬁes
divx (ϱE uE ) = 0,
divx (ϱE uE ⊗ uE ) + ∇x p(ϱE ) = ϱE ∇x G.

(5)

Accordingly, we consider the problem (1)–(3) with the boundary conditions
ub = uE on ∂Ω, ϱb = ϱE on Γin .

(6)

Under the hypothesis (6), and if the stationary density ϱE is strictly positive, theproblem (1)–(3) admits a Lyapunov function, namely the relative energy

Z



 1
2
′
ϱ|u − uE | + P (ϱ) − P (ϱE )(ϱ − ϱE ) − P (ϱE ) .
E ϱ, u ϱE , uE , E ϱ, u ϱE , uE ≡2Ω
The situation becomes more delicate if ϱE vanishes on a non–trivial part of Ω. In thatcase, the stationary problem may admit more (inﬁnitely many) solutions even if the total
mass is prescribed.
Our main result asserts that any weak solution of the problem (1)–(3), satisfying
a suitable form of energy inequality, approaches the equilibrium solution [ϱE , uE ] ast → ∞ as long as the stationary problem (5) admits a unique solution. To the best of
our knowledge, this is the ﬁrst result of this kind in the multi–dimensional case under
the non–zero in/out ﬂow boundary conditions. Note that such a result does not follow
from “standard” arguments, even if ϱE > 0, as the Lyapunov functionZE ϱ, u ϱE , uE (t, ·)t 7→Ω

is not continuous on the trajectories generated by weak solutions.

Hadamard variational formula for the fundamental solutionof the nonstationary Stokes equations
Erika Ushikoshi
Yokohama National University & Osaka University
ushikoshi-erika-ng@ynu.ac.jp

1 Introduction
Let Ω ⊂ R3 be a bounded domain with a smooth boundary ∂Ω. We consider the
nonstationary Stokes equations with the Dirichlet boundary conditions in Ω; dv (x, t) − ∆x v(x, t) + ∇x p(x, t) = f (x, t), x ∈ Ω, t > 0,



 dt
divx v(x, t) = 0,x ∈ Ω, t > 0,
(1)


v(x,
t)
=
0,
x
∈
∂Ω,
t
>
0,



v(x, 0) = a (x),x ∈ Ω,
0where v(x, t) = (v 1 (x, t), v 2 (x, t), v 3 (x, t)) and p(x, t) denote unknown velocity and pressure, respectively. Furthermore, f (x) = (f 1 (x), f 2 (x), f 3 (x)) is a given external force
and a(x) = (a1 (x), a2 (x), a3 (x)) is an initial data. It is known that the solution of (1)can be expressed by∫∫ t∫v m (x, t) =a0 (y) · Um (y − x, t) dy +f (x, τ ) · Um (y − x, t − τ ) dydτ, m = 1, 2, 3Ω0Ω

for a given f and a0 , where Um (x, t) is the fundamental solution for that.
The purpose of this article is to analyze the domain dependence of Um (x, t). Such aproblem was firstly considered in Hadamard [1] for the Green function of the Laplace
equation, and in recent years, Kozono-Ushikoshi [2] and Ushikoshi [5] investigated thatfor the stationary Stokes equations. On the other hand, Ozawa [3] presented the variational formula for the fundamental solution of the heat equation, which was applied todetermine the topological type of the domain by the variation of the eigenvalues for the
Laplace operator in Ozawa [4]. We establish the Hadamard variationl formula for thefundamental solution of the nonstationary Stokes equations with the Dirichlet boundary conditions. This is the joint work with Mr. Masaru Kamiya who is a graduate of
Yokohama National University.

2 Main Result
We assume that for every ε ≥ 0, there is a diﬀeomorphism Φε : Ω → Ωε satisfying thefollowing conditions.
(A.1) Φε = (ϕ1ε , ϕ2ε , ϕ3ε ) ∈ C ∞ (Ω)3 .

(A.2) Φ0 (x) = x for all x ∈ Ω.
(A.3) There exists S = (S 1 , S 2 , S 3 ) ∈ C ∞ (Ω)3 such that K(x; ε) := Φε (x) − x − S(x)εsatisfies supx∈Ω |K(x; ε)| + supx∈Ω |∇K(x; ε)| = O(ε2 ) as ε → 0.
(A.4) It holds that(det

∂ϕiε (x)∂xj)
for all x ∈ Ω and all ε ≥ 0.

=1
i,j=1,2,3

For the fundamental solution Uε,m (x, t) of (1) in Ωε ×(0, T ), the following theorem holds;
Theorem 1. Let {Uε,m }m=1,2,3 be the fundamental solution of (1) in Ωε × (0, T ). Thenfor any y, z ∈ Ω with y ̸= z, there exists
k (y, z, t) − U k (y, z, t)
Uε,m
m
ε→0
ε
for all t > 0. Moreover, it is expressed by
)
∫ t∫
3 (
i
∑
∂Uki
∂Um
k
δUm (y, z, t) =S(x) · νx(x, z, τ )(x, y, t − τ ) dσx dτ
∂νx
∂νx
0
∂Ω
k
δUm
(y, z, t) := lim

i=1

for k, m = 1, 2, 3, where S is as in (A.3), νx = (νx1 , νx2 , νx3 ) is the unit outer normal to∂Ω at x ∈ ∂Ω and σx denotes the surface element of ∂Ω.
Remark 1. We assume that the domain is smoothly perturbed with keeping its volume.
In order to remove this assumption, we need to simplify the method to construct its
formula and make use of the piola transform.
The key lemma is as follows.
Lemma 2.1. Let {Vε,m }m=1,2,3 be the function defined in ΩT := Ω × (0, T ) bykVε,m(x, y, t) :=

3
∑
∂xk
j=1

∂ x̃j

j
Uε,m
(Φε (x), Φε (y), t),

k = 1, 2, 3.

Then, for any 0 < θ < 1 and for any y ∈ Ω，it holds that2+θ, θ2∥(Vε,m − Um )(· − y, ·)∥ΩT→0

as ε → 02+θ, θ2for m = 1, 2, 3, where {Um }m=1,2,3 is the fundamental solution of (1) in ΩT and ∥·∥ΩTθ

denotes the norm of C 2+θ, 2 (ΩT ).

References[1] Hadamard, J., Memoires des Savants Etrangers, 33, (1908)[2] Kozono, H., Ushikoshi, E., Arch. Ration. Mech. Anal., 208, 1005–1055(2013)[3] Ozawa, S., Proc. Japan Acad. Ser. A Math. Sci., 54, 322-325(1978)[4] Ozawa, S., Proc. Japan Acad. Ser. A Math. Sci., 55, 328-333(1979)[5] Ushikoshi, E., Manuscripta Math., 146, 85-106(2015)

Enhanced dissipation for the two-jet Kolmogorov type ﬂow
on the unit sphere
Tatsu-Hiko Miura
Department of Mathematics, Kyoto University
t.miura@math.kyoto-u.ac.jp
A part of this talk is based on a joint work with Professor Yasunori Maekawa.
We consider the vorticity equation for a viscous ﬂuid on the 2D unit sphere S 2 in R3of the form∂t ω + u · ∇ω − ν(∆ω + 2ω) = f,u = nS 2 × ∇∆−1 ω

on

S 2 × (0, ∞).

(1)

Here ω is the scalar vorticity, u is the tangential velocity ﬁeld, and f is a given external
force. Also, ν > 0 is the viscosity coeﬃcient, ∇ is the gradient on S 2 , ∆ is the Laplace–
Beltrami operator on S 2 which is invertible on the space L20 (S 2 ) of L2 functions on S 2with vanishing mean, nS 2 is the unit outward normal vector ﬁeld of S 2 , and a · b anda × b are the inner and vector products of a, b ∈ R3 . Note that the zeroth order term2νω appears in (1) since we take the viscous them in the Navier–Stokes equations forthe velocity ﬁeld u as twice of the divergence of the deformation tensor for u (see [6]).
Let Ynm with n ≥ 0 and |m| ≤ n be the spherical harmonics and λn = n(n + 1) the
eigenvalue of −∆ corresponding to Ynm with |m| ≤ n. For n ≥ 1 and a ∈ R, the vorticity
equation (1) with external force f = aν(λn − 2)Yn0 has a stationary solution ωna = aYn0 .
Such a stationary ﬂow can be seen as a spherical version of the Kolmogorov ﬂow on the
2D ﬂat torus and we call it the n-jet Kolmogorov type ﬂow. In this talk, we focus on
the case n = 2. The linearized equation of (1) around the two-jet Kolmogorov type ﬂow
ω2a = ac02 (3 cos2 θ − 1) (here c02 is a constant) is of the form (after relabeling a ∈ R)∂t ω = Lν,a ω = νAω − iaΛω,
A = ∆ + 2,

ω|t=0 = ω0in

L20 (S 2 ),

Λ = cos θ(−i∂φ )(I + 6∆−1 ),(2)where θ and φ are the colatitude and longitude.
ν,aWe are interested in the behavior of a solution ω(t) = etL ω0 to (2) as ν → 0. Whenν,a(ω0 , Y1m )L2 (S 2 ) = 0 for m = 0, ±1, a standard energy method shows that etL ω0 decays
at the rate O(e−νt ). In the case of the plane Kolmogorov ﬂow [1, 3, 2, 8, 9], however, it
is shown that a solution to the linearized equation decays at a rate faster than O(e−νt )when ν is suﬃciently small. Such a phenomenon is called the enhanced dissipation, andν,aour aim is to study the enhanced dissipation for the solution etL ω0 to (2).
ν,atL ω ﬁrst without a precise decay
In fact, we obtained the enhanced dissipation
0√ for e−νtrate in [5] and then with the decay rate O(e) in [4], which is the same as in the planecase [1, 2, 8, 9]. There results themselves, however, are proved just by applications of theabstract results given by Ibrahim–Maekawa–Masmoudi [2] and of the Gearhart–Prüss
type theorem shown by Wei [7]. So in this talk we would like to focus on a somewhat
new idea for the spectral analysis of the perturbation operator Λ arising in the study ofthe enhanced dissipation. To apply the abstract results of [2], we need to show that Λdoes not have nonzero eigenvalues. In order to prove it, one typically analyzes an ODE
associated with Λ and applies the uniqueness of a (smooth) solution to the ODE to showthat a solution to the eigenvalue problem identically vanishes. Such an ODE approachis used in the plane case [3, 2, 9], but in our case we encounter a diﬃculty due to the sizeof the coeﬃcient of ∆−1 in Λ, and it seems to be too diﬃcult to deal with this diﬃcultyby the ODE approach. Instead, to overcome this diﬃculty, we make use of the mixing
property of Λ expressed by the recurrence relation for the spherical harmonicsmmm
cos θ Ynm = amn Yn−1 + an+1 Yn+1

(3)

with nonnegative coeﬃcients am
n . In the actual proof, we use (3), a Hardy type inequality
on S 2 , and λn → ∞ as n → ∞ to show that the high frequency part (with respect tothe index n in the expansion by Ynm ) of a solution to the eigenvalue problem vanishes,
and then apply (3) again to ﬁnd that the low frequency part also vanishes.

References[1] M. Beck and C. E. Wayne. Metastability and rapid convergence to quasi-stationary
bar states for the two-dimensional Navier-Stokes equations. Proc. Roy. Soc. Edinburgh Sect. A, 143(5):905–927, 2013.
[2] S. Ibrahim, Y. Maekawa, and N. Masmoudi. On pseudospectral bound for nonselfadjoint operators and its application to stability of Kolmogorov ﬂows. Ann.
PDE, 5(2):Paper No. 14, 84, 2019.
[3] Z. Lin and M. Xu. Metastability of Kolmogorov ﬂows and inviscid damping of shear
ﬂows. Arch. Ration. Mech. Anal., 231(3):1811–1852, 2019.
[4] Y. Maekawa and T.-H. Miura. Rate of the enhanced dissipation for the two-jet
Kolmogorov type ﬂow on the unit sphere. arXiv:2109.13435.
[5] T.-H. Miura. Linear stability and enhanced dissipation for the two-jet Kolmogorov
type ﬂow on the unit sphere. arXiv:2105.07964.
[6] M. E. Taylor. Partial diﬀerential equations III. Nonlinear equations, volume 117 ofApplied Mathematical Sciences. Springer, New York, second edition, 2011.
[7] D. Wei. Diﬀusion and mixing in ﬂuid ﬂow via the resolvent estimate. Sci. China
Math., 64(3):507–518, 2021.
[8] D. Wei and Z. Zhang. Enhanced dissipation for the Kolmogorov ﬂow via the hypocoercivity method. Sci. China Math., 62(6):1219–1232, 2019.
[9] D. Wei, Z. Zhang, and W. Zhao. Linear inviscid damping and enhanced dissipation
for the Kolmogorov ﬂow. Adv. Math., 362:106963, 103, 2020.

Vortex reconnection and a ﬁnite-time singularity of theNavier-Stokes equations
Yoshifumi Kimura
Graduate School of Mathematics, Nagoya University
Furo-cho, Chikusa-ku, Nagoya 464-8602, JAPANkimura@math.nagoya-u.ac.jp
As a fundamental process in both classical and quantum turbulence, vortex reconnection has intensively been studied over recent decades. Recently we have developed
an analytical model of vortex reconnection challenging to study the ﬁnite singularity
problem for the Naiver-Stokes equations [1],[2]. In this model, two circular vortex ringsof circulation ±Γ and radius R = 1/κ are symmetrically placed on two planes inclined tothe plane x = 0 at angles ±α. Under an assumption that the vortex Reynolds number,
RΓ = Γ/ν, is very large, we have derived a nonlinear dynamical system for the local
behavior near the points of closest approach of the vortices (tipping points). Carefulnumerical investigation of the dynamical system reveals that the magnitude of vorticity
could take any large value for small viscosity but remains ﬁnite since the minimum coreradius never becomes zero.
In this talk, the analytic model will be illustrated after a brief review of the problem
and some preliminary numerical model [3] are presented. The assumptions for the analysis are far beyond the ones that the current DNS could attain, but we try to compare
the results of the analytic model and the DNS and other numerical simulations. Finallysome new development of the problem will be introduced [4].

References[1] H. K. Moﬀatt & Y. Kimura, Towards a ﬁnite-time singularity of the Naiver-Stokes
equations Part 1. Derivation and analysis of dynamical system, J. Fluid Mech.
(2019) 861, 930-967.
[2] H. K. Moﬀatt & Y. Kimura, Towards a ﬁnite-time singularity of the Naiver-Stokes
equations Part 2. Vortex reconnection and singularity evasion, Journal of Fluid
Mech. (2019) 870, R1.
[3] Y. Kimura & H. K. Moﬀatt, A tent model of vortex reconnection under Biot-Savart
evolution. J. Fluid Mech. (2018) 834, R1.
[4] P. J. Morrison & Y. Kimura, A Hamiltonian description of ﬁnite-time singularity
in Euler’s ﬂuid equation. arxiv.org/abs/2011.10864

Fast rotation limit for the incompressible Navier-Stokes equations in a
3D layer
Ryo Takada
Faculty of Mathematics, Kyushu University
takada@math.kyushu-u.ac.jp
Let D := R2 × T be a three-dimensional layer. Here, T = R/Z ≃ [0, 1] is the onedimensional torus, and the point of D is denoted by (x, z) with the horizontal variable x =(x1 , x2 ) ∈ R2 and the vertical variable z ∈ T.
In this talk, we consider the initial value problem for the rotating Navier-Stokes equations, describing the motion of incompressible viscous ﬂuids around the rotating vector ﬁeldΩ/2(−x2 , x1 , 0) in D:t > 0, (x, z) ∈ D,∂t u − ∆u + Ω(e3 × u) + (u · ∇)u + ∇p = 0
(1)
∇·u=0
t ⩾ 0, (x, z) ∈ D,
u(0, x, z) = u0 (x, z)(x, z) ∈ D.
Here, u = u(t, x, z) = (u1 (t, x, z), u2 (t, x, z), u3 (t, x, z)) and p = p(t, x, z) denote the unknown
velocity ﬁeld and the unknown pressure, respectively, while u0 = u0 (x, z) = (u0,1 (x, z), u0,2 (x, z), u0,3 (x, z))denotes the initial velocity ﬁeld. The constant Ω ∈ R represents the rotating speed aroundthe vertical unit vector e3 = (0, 0, 1).
The main purpose of this talk is to prove the unique existence of global in time solutions
to (1) for the initial data in scaling critical spaces, and study the asymptotics of solutions
when the rotating speed |Ω| tends to inﬁnity.
Before stating our results, we review the known results on the global existence of solutions
to (1). In the whole space R3 , Chemin, Desjardins, Gallagher and Grenier [1, 2] proved that1for given initial velocity u0 = v0 + w0 ∈ L2 (R2 )3 + Ḣ 2 (R3 )3 , there exists a positive parameterΩ0 = Ω0 (u0 ) such that for every Ω ∈ R with |Ω| ⩾ Ω0 the rotating Navier-Stokes equations (1)possesses a unique global solution. Furthermore, they [1, 2] showed that the global solution
to (1) converges to that of the 2D Navier-Stokes equations with the initial data v0 in thelocal in time norm L2loc (0, ∞; Lq (R3 )) for 2 < q < 6 as |Ω| → ∞. In the 3D inﬁnite layer D,
Gallay and Roussier-Michon [3] proved the global existence and the long-time asymptotics
of inﬁnite-energy solutions ∫to (1) for large |Ω|. They [3] decomposed the initial data as
u0 = ū0 + ũ0 with ū0 (x) = T u0 (x, z) dz and ũ0 = u0 − ū0 , and showed that for given initial
1 (D)3 satisfying ũ ∈ H 1 (D)3 , ū
1
2
1
2
2
data u0 ∈ Hloc
00,3 ∈ H (R ), ∂1 ū0,2 − ∂2 ū0,1 ∈ (L ∩ L )(R ),there exists a Ω0 = Ω0 (u0 ) > 0 such that (1) has a unique global solution u provided that|Ω| ⩾ Ω0 . Moreover, it is shown in [3] that the global solution converges to the two-dimensional
Lamb-Oseen vortex in L1 (R2 ) as t → ∞.
Following the idea in [3], we decompose the velocity ﬁelds as u(t, x, z) = ū(t, x)+ ũ(t, x, z),where∫u(t, x, z) dzū(t, x) = (Qu)(t, x) :=T

is the average of u with respect∫ to the vertical variable z, and we set ũ := (1 − Q)u. Note thatũ has zero vertical average T ũ(t, x, z) dz = 0. Similarly to the whole space R3 case in [1, 2],
the limit equation is the 2D incompressible Navier-Stokes equations for the three-components

velocity ﬁelds:

∞
∞
∞
∞

∂t u − ∆h u + (uh · ∇h )u + (∇h q, 0) = 0∇h · u ∞h =0 ∞u (0, x) = ū0 (x)t > 0, x ∈ R2 ,t ⩾ 0, x ∈ R2 ,x ∈ R2 ,(2)∞ ∞where ∇h = (∂1 , ∂2 ), ∆h = ∂12 + ∂22 , u∞h = (u1 , u2 ) and q = q(t, x) is a 2D pressure.
The main result of this talk reads as follows:
1

Theorem 1. For u0 = ū0 + ũ0 ∈ L2 (R2 )3 + (1 − Q)Ḣ 2 (D)3 with ∇h · ū0,h = ∇ · ũ0 = 0,
there exists a Ω0 = Ω0 (ū0 , ũ0 ) > 0 such that, for all Ω ∈ R with |Ω| ⩾ Ω0 , the rotating
Navier-Stokes equation (1) admits a unique global solution u = ū + ũ satisfyingū ∈ C([0, ∞); L2 (R2 ))3 ∩ L2 (0, ∞; Ḣ 1 (R2 ))3 ,1

3ũ ∈ C([0, ∞); (1 − Q)Ḣ 2 (D))3 ∩ L2 (0, ∞; (1 − Q)Ḣ 2 (D))3 .
Furthermore, for 2 < p, q < ∞ with 2/p + 2/q = 1, there holds
lim ∥u − u∞ (t)∥Lp (0,∞;Lq (D)) = 0.

|Ω|→∞

(3)

Here, u∞ is the unique global solution of the limit equation (2) with the initial data ū0 in theclass u∞ ∈ C([0, ∞); L2 (R2 ))3 ∩ L2 (0, ∞; Ḣ 1 (R2 ))3 .
Remark 1. Let p, q satisfy 2 < p, q < ∞ and 2/p+2/q = 1. Then, by the Sobolev embeddings3
1− 2(1− 2q )Ḣ q (R2 ) ,→ Lq (R2 ), H 2(D) ,→ Lq (D) and the Poincaré inequality in D, we have thefollowing relations:
L∞ (0, ∞; L2 (R2 )) ∩ L2 (0, ∞; Ḣ 1 (R2 )) ,→ Lp (0, ∞; Lq (R2 )),L∞ (0, ∞; (1 − Q)Ḣ 2 (D)) ∩ L2 (0, ∞; Ḣ 2 (D)) ,→ Lp (0, ∞; (1 − Q)Lq (D)).
1

3

In the proof of Theorem 1, we adapt the ideas in [1, 2, 3], and decompose the solution to(1) as u = ū + ũ = u∞ + (ū − u∞ ) + λ + (ũ − λ), where λ is the linear solution associated with
the initial data ũ0 whose Fourier transform is compactly supported in some ball. The key
ingredients of the proof are the Strichartz estimates for the linear solution λ, and the global
energy estimates for the perturbations ū − u∞ and ũ − λ.
This talk is based on the joint work with Hiroki Ohyama (Kyushu University).

References[1] J.-Y. Chemin, B. Desjardins, I. Gallagher, and E. Grenier, Anisotropy and dispersion in rotating ﬂuids,
Stud. Math. Appl., vol. 31, North-Holland, Amsterdam, 2002, pp. 171–192.
[2] J.-Y. Chemin, B. Desjardins, I. Gallagher, and E. Grenier, Mathematical geophysics, The Clarendon
Press, Oxford University Press, Oxford, 2006.
[3] T. Gallay and V. Roussier-Michon, Global existence and long-time asymptotics for rotating ﬂuids in a
3D layer, J. Math. Anal. Appl. 360 (2009), 14–34.