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Blowup analysis for a quasi-exact 1D model of 3D
Euler and Navier–Stokes
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Nonlinearity
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Nonlinearity
Nonlinearity
37
(2024) 035001 (28pp)
https://doi.org/10.1088/1361-6544/ad1c2f
Blowup analysis for a quasi-exact 1D model
of 3D Euler and Navier–Stokes
Thomas Y Hou and Yixuan Wang
∗
Applied and Computational Mathematics, Caltech, Pasadena, CA 91125, United
States of America
E-mail:
roywang@caltech.edu
Received 29 June 2023; revised 18 November 2023
Accepted for publication 8 January 2024
Published 22 January 2024
Recommended by Dr Yao Yao
Abstract
We study the singularity formation of a quasi-exact 1D model proposed by Hou
and Li (2008
Commun.PureAppl.Math.
61
661–97). This model is based on an
approximation of the axisymmetric Navier–Stokes equations in the
r
direction.
The solution of the 1D model can be used to construct an exact solution of the
original 3D Euler and Navier–Stokes equations if the initial angular velocity,
angular vorticity, and angular stream function are linear in
r
. This model shares
many intrinsic properties similar to those of the 3D Euler and Navier–Stokes
equations. It captures the competition between advection and vortex stretching
as in the 1D De Gregorio (De Gregorio 1990
J. Stat. Phys.
59
1251–63; De
Gregorio 1996
Math. Methods Appl. Sci.
19
1233–55) model. We show that the
inviscid model with weakened advection and smooth initial data or the original
1D model with Hölder continuous data develops a self-similar blowup. We
also show that the viscous model with weakened advection and smooth initial
data develops a finite time blowup. To obtain sharp estimates for the nonlocal
terms, we perform an exact computation for the low-frequency Fourier modes
and extract damping in leading order estimates for the high-frequency modes
using singularly weighted norms in the energy estimates. The analysis for the
viscous case is more subtle since the viscous terms produce some instability
∗
Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the
Creative Commons Attribution
3.0 licence
. Any further distribution of this work must maintain attribution to the author(s) and the
title of the work, journal citation and DOI.
© 2024 IOP Publishing Ltd & London Mathematical Society
1
Nonlinearity
37
(2024) 035001
T Y Hou and Y Wang
if we just use singular weights. We establish the blowup analysis for the vis-
cous model by carefully designing an energy norm that combines a singularly
weighted energy norm and a sum of high-order Sobolev norms.
Supplementary material for this article is available
online
Keywords: fluid dynamics
,
blowup
,
computer-assisted proof
,
Fourier analysis
Mathematics Subject Classification numbers: 35Q35
,
76D03
1. Introduction
Whether the 3D incompressible Euler and Navier–Stokes equations can develop a finite time
singularity from smooth initial data is one of the most outstanding open questions in nonlin-
ear partial differential equations. An essential difficulty is that the vortex stretching term has
a quadratic nonlinearity in terms of vorticity. A simplified 1D model was proposed by the
Constantin–Lax–Majda model (CLM model for short) [
11
] to capture the effect of nonlocal
vortex stretching. The CLM model can be solved explicitly and can develop a finite time sin-
gularity from smooth initial data. Later on, De Gregorio (DG) incorporated the advection term
into the CLM model to study the competition between advection and vortex stretching [
14
,
15
], see [
9
] for singularity formulation in the distorted Euler equations with transport neglected
and also [
24
,
25
] for a related study on the stabilizing effect of advection for the 3D Euler and
Navier–Stokes equations. There have been recent studies on the effect of advection and vortex
stretching in other related models; see [
40
] for the generalized inviscid Proudman–Johnson
equation, [
20
] with a Riesz transform added to the vorticity formulation of 2D Euler equation,
and [
38
] with advection term dropped in the vorticity formulation of 3D Euler equation. In
[
39
], Okamoto
et al
further introduced a parameter for the advection term to measure the rel-
ative strength of the advection in the DG model. These simplified 1D models have inspired
many subsequent studies. Interested readers may consult the excellent surveys [
10
,
21
,
28
,
33
] and the references therein. Very recently, Huang
et al
[
26
] established self-similar blowup
for the whole family of gCLM models with
a
⩽
1 using a fixed-point argument. On the other
hand, these 1D scalar models are phenomenological in nature and cannot be used to recover
the solution of the original 3D Euler equations.
For the line of research on the singularity formulation for the 3D Euler equations, Luo–
Hou [
31
] presented in 2014 convincing numerical evidence that the 3D axisymmetric Euler
equations with smooth initial data and boundary develop a potential finite time singularity.
Inspired by Elgindi’s recent breakthrough for finite time singularity of the axisymmetric Euler
with no swirl and
C
1
,α
velocity [
16
], Chen and Hou proved the finite time blowup of the 2D
Boussinesq and 3D Euler equations with
C
1
,α
initial velocity and boundary [
6
]. For other
recent works on singularity formulation of 3D Euler with limited regularity, see also [
5
,
13
]
for initial data that is smooth except at the origin, [
12
] for more smooth data but with a
C
1
/
2
−
ε
force, and [
17
,
19
] for settings with nonsmooth boundary. Very recently, Chen and Hou proved
stable and nearly self-similar blowup of the 2D Boussinesq and 3D Euler with smooth initial
data and boundary using computer assistance [
7
].
In 2008, Hou and Li [
25
] proposed a new 1D model for the 3D axisymmetric Euler and
Navier–Stokes equations. This model approximates the 3D axisymmetric Euler and Navier–
Stokes equations along the symmetry axis based on an approximation in the
r
direction. The
solution of the 1D model can be used to construct an exact solution of the original 3D Navier–
Stokes equations if the initial angular velocity, angular vorticity, and angular stream func-
tion are linear in
r
. This model shares many intrinsic properties similar to those of the 3D
2
Nonlinearity
37
(2024) 035001
T Y Hou and Y Wang
Navier–Stokes equations. Thus, it captures some essential nonlinear features of the 3D Euler
and Navier–Stokes equations. In the same paper [
25
], the authors proved the global regular-
ity of the Hou–Li model by deriving a new Lyapunov functional, which captures the exact
cancellation between advection and vortex stretching.
The purpose of this paper is to study the singularity formation of a weak advection version
of the Hou–Li model for smooth data. We introduce a parameter
a
to characterize the relative
strength between advection and vortex stretching, just like the gCLM model. Both inviscid and
viscous cases are considered. We also prove the finite time singularity formation of the original
inviscid Hou–Li model (
a
=
1 and
ν
=
0) with
C
α
initial data. Inspired by the recent work of
Chen [
3
] for the DG model, we consider the case of
a
<
1 and treat
1
−
a
as a small parameter.
For the
C
α
initial data, we consider the original Hou–Li model with
a
=
1 and
1
−
α
small. By
using the dynamic rescaling formulation and analysing the stability of the linearised operator
around an approximate steady state of the original Hou–Li model (
a
=
1), we prove finite time
self-similar blowup.
We follow a general strategy that we have established in our previous works [
6
,
8
].
Establishing linear stability of the approximate steady state is the most crucial step in our
blowup analysis. To obtain sharp estimates for the nonlocal terms, we carry out an exact com-
putation for the low-frequency Fourier modes and extract damping in leading order estimates
for the high-frequency modes using singularly weighted norms in the energy estimates. The
blowup analysis for the viscous model is more subtle since the viscous terms do not provide
damping and produce some bad terms if we use a singularly weighted norm. We establish the
blowup analysis for the viscous model by carefully designing an energy norm that combines
a singularly weighted energy norm and a sum of high-order Sobolev norms.
1.1. Problem setting
In [
25
], Hou–Li introduced the following reformulation of the axisymmetric Navier–Stokes
equation:
u
1
,
t
+
u
r
u
1
,
r
+
u
z
u
1
,
z
=
2
u
1
ψ
1
,
z
+
ν
∆
u
1
,
(1.1)
ω
1
,
t
+
u
r
ω
1
,
r
+
u
z
ω
1
,
z
=
u
2
1
z
+
ν
∆
ω
1
,
(1.2)
−
∂
2
r
+(
3
/
r
)
∂
r
+
∂
2
z
ψ
1
=
ω
1
,
(1.3)
where
u
1
=
u
θ
/
r
, ω
1
=
ω
θ
, ψ
1
=
ψ
θ
/
r
,
and
u
θ
,
ω
θ
, and
ψ
θ
are the angular velocity, angu-
lar vorticity, and angular stream function, respectively. By the well-known Caffarelli–Kohn–
Nirenberg partial regularity result [
2
], the axisymmetric Navier–Stokes equations can develop
a finite time singularity only along the symmetry axis
r
=
0. To study the potential singularity
or global regularity of the axisymmetric Navier–Stokes equations, Hou–Li [
25
] proposed the
following 1D model along the symmetry axis
r
=
0:
u
1
,
t
+
2
ψ
1
u
1
,
z
=
2
ψ
1
,
z
u
1
+
ν
u
1
,
zz
,
ω
1
,
t
+
2
ψ
1
ω
1
,
z
=
u
2
1
z
+
νω
1
,
zz
,
−
ψ
1
,
zz
=
ω
1
.
(1.4)
Such a reduction is exact in the sense that if (
ω
1
,
u
1
,
ψ
1
) is an exact solution of the 1D model, we
can obtain an exact solution of the 3D Navier–Stokes equations by using a constant extension
in
r
. This corresponds to the case when the physical quantities
u
θ
=
ru
1
,
ω
θ
=
r
ω
1
are linear in
3
Nonlinearity
37
(2024) 035001
T Y Hou and Y Wang
r
. We assume that the solutions are periodic in
z
on
[
0
,
2
π
]
. We already know from the original
Hou–Li paper that this system is well-posed for
C
m
initial data with
m
⩾
1. In [
25
], the authors
also used the well-posedness of the Hou–Li model to construct globally smooth solutions to
the 3D equations with large dynamic growth.
In two recent papers by the first author [
22
,
23
], the author presented new numerical evid-
ence that the 3D axisymmetric Euler and Navier–Stokes equations develop potential singular
solutions at the origin. This new blowup scenario is very different from the Hou–Luo blowup
scenario, which occurs on the boundary. In this computation, the author observed that the
axial velocity
u
z
=
2
ψ
1
+
r
ψ
1
,
r
near the maximal point of
u
1
is significantly weaker than
2
ψ
1
.
This is due to the fact that
ψ
1
reaches the maximum at a position
r
=
r
ψ
that is smaller than
the position
r
=
r
u
in which
u
1
achieves its maximum, i.e.
r
ψ
<
r
u
. Therefore
ψ
1
,
r
is negat-
ive near the maximal position of
u
1
. Thus the axial velocity
u
z
is actually weaker than
2
ψ
1
,
which corresponds to
u
z
|
r
=
0
. Thus, the original Hou–Li model along
r
=
0 does not capture
this subtle phenomenon, which is three-dimensional in nature. To gain some understanding of
this potentially singular behaviour, we introduce the following 1D weak advection model.
u
t
+
2
a
ψ
u
z
=
2
u
ψ
z
+
ν
u
zz
,
ω
t
+
2
a
ψω
z
=
u
2
z
+
νω
zz
,
−
ψ
zz
=
ω,
(1.5)
where
a
is a parameter that measures the relative strength of advection in the Hou–Li model.
Remark 1.1.
For simplicity, we drop the subscript 1 in the above weak advection model. The
proposed model (
1.5
) in the inviscid case
ν
=
0 resembles the generalized Constantin–Lax–
Majda model (gCLM) [
39
]
ω
t
+
au
ω
x
=
u
x
ω,
u
x
=
H
ω,
where
H
ω
(
x
)=
1
π
p
.
v
.
ˆ
R
ω
(
y
)
x
−
y
d
y
is the Hilbert transform. They share similar structures of competition between advection and
vortex stretching. The case when
a
=
1 corresponds to the DG model. We obtain an explicit
steady-state to the inviscid Hou–Li model (
1.4
)
(
ω,
u
,ψ
)=(
sin
x
,
sin
x
,
sin
x
)
, similar to the
steady state
(
ω,
u
)=(
−
sin
x
,
sin
x
)
of the DG model on
S
1
. Many of the results we present in
this paper have analogies for the gCLM model; see in particular [
3
,
4
].
1.2. Main results
We summarize the main results of the paper below and devote the subsequent sections to
proving these results. Our first result is on the finite-time blowup of the weak inviscid advection
model; for its proof see section
2
and
3
.
Theorem 1.2.
For the weak advection model
(
1.5
)
in the inviscid case
ν
=
0
, there exists
a constant
δ >
0
such that for a
∈
(
1
−
δ,
1
)
, the weak advection model
(
1.5
)
develops a
finite time singularity for some C
∞
initial data. Moreover, there exists a self-similar profile
4
Nonlinearity
37
(2024) 035001
T Y Hou and Y Wang
(
ω
∞
,ω
∞
,ω
∞
)
corresponding to a blowup that is neither expanding nor focusing. More pre-
cisely, the blowup solution to (
1.5
) has the form
ω
(
x
,
t
)=
1
1
+
c
u
,
∞
t
ω
∞
,
u
(
x
,
t
)=
1
1
+
c
u
,
∞
t
u
∞
,ψ
(
x
,
t
)=
1
1
+
c
u
,
∞
t
ψ
∞
,
for some negative constant c
u
,
∞
with a blowup time given by T
=
−
1
c
u
,
∞
.
Remark 1.3.
Such self-similar blowup that is neither expanding nor focusing is observed
numerically for
a
∈
[
0
.
6
,
0
.
9
]
. See also a similar phenomenon observed for the gCLM model
in [
32
] for
a
∈
[
0
.
68
,
0
.
95
]
. The blowup result for the gCLM model has been proved in [
4
]
for
a
sufficiently close to 1. We remark that for
a
very close to 1, since we can show that
c
u
,
∞
=
2
(
a
−
1
)+
o
(
a
−
1
)
, the blowup time becomes very large due to the very small coeffi-
cient
1
−
a
in the vortex stretching term which slightly dominates the advection term. It would
be extremely difficult to compute such singularity numerically since it takes an extremely long
time for the singularity to develop. For
a
below a critical value
a
0
, i.e.
a
<
a
0
, we observe that
the weak advection Hou–Li model develops a focusing singularity.
The second result is on the blowup of the original Hou–Li model with
C
α
initial data; for
its proof see section
4
. In [
18
], the authors made an important observation that advection can
be weakened by
C
α
data. Intuitively if
u
=
O
(
x
α
)
in the origin, since
ψ
is
C
2
, we have that
ψ
u
x
≈
αψ
x
u
near the origin, the vortex stretching term is stronger than the advection term if
α<
1. See [
3
,
8
] on results of blowup of the DG model with Hölder continuous data.
Theorem 1.4.
Consider the Hou–Li model
(
1.4
)
in the inviscid case
ν
=
0
. There exists a con-
stant
δ
0
>
0
such that for
α
∈
(
1
−
δ
0
,
1
)
,
(
1.4
)
develops a finite time singularity for some C
α
initial data. Moreover, there exists a C
α
self-similar profile corresponding to a blowup that is
neither expanding nor focusing, similar to the setting in theorem
1.2
.
Remark 1.5.
This theorem establishes blowups of type
C
α
for any
α
close to 1, which of
course implies blowups in less regular classes since
C
α
⊂
C
α
1
for
α
1
< α
. The regularity of
the profile determines the speed of the blowup since our constructed
C
α
profile has blowup time
T
=
O
(
−
1
/
(
2
(
α
−
1
)))
. We remark that however, we do not have blowup for data intrinsically
in a low regularity class
C
ε
for
ε
close to 0; that is, data that is
C
ε
but not in any higher
C
α
classes. We conjecture that such blowup might be focusing, which is beyond the scope of this
paper.
Remark 1.6.
The above two theorems imply that the result of the wellposedness in [
25
] of the
Hou–Li model for
C
1
initial data is sharp. As long as the advection is weakened or slightly
less smooth data is allowed, we would have a self-similar blowup.
The third result is on the finite-time blowup of the weak advection model with viscosity.
The dynamic rescaling formulation implies that the viscous terms are asymptotically small.
Thus, we can build on theorem
1.2
to establish theorem
1.7
. We remark that there is no exact
self-similar profile due to the viscous term. We will provide more details of the blowup analysis
for the viscous case in section
5
.
Theorem 1.7.
Consider the weak advection model
(
1.5
)
with viscosity. There exists a constant
δ
1
>
0
such that for a
∈
(
1
−
δ
1
,
1
)
, the weak advection model
(
1.5
)
develops a finite time sin-
gularity for some C
∞
initial data.
We use the framework of the dynamic rescaling formulation to establish the blowups. This
formulation was first introduced by McLaughlin
et al
in their study of self-similar blowup
5
Nonlinearity
37
(2024) 035001
T Y Hou and Y Wang
of the nonlinear Schrödinger equation [
29
,
35
]. This formulation was later developed into an
effective modulation technique, which has been applied to analyse the singularity formation for
the nonlinear Schrödinger equation [
27
,
36
], compressible Euler equations [
1
], the nonlinear
heat equation [
37
], the generalized KdV equation [
34
], and other dispersive problems. Recently
this approach has been applied to prove singularity in various gCLM models [
3
,
4
,
8
] and in
Euler equations [
6
,
7
,
16
]. Our blowup analysis consists of several steps. First, we use the
dynamic rescaling formulation to link a self-similar singularity to the (stable) steady state of
the dynamic rescaling formulation. Secondly, we identify, either analytically or numerically,
an approximate steady state to the dynamic rescaling formulation. Thirdly, we perform energy
estimates using a singularly weighted norm to establish linear and nonlinear stability of the
approximate steady state. Finally, we establish exponential convergence to the steady state in
the rescaled time.
The crucial ingredient of the framework is the linear stability of the approximate steady
state, and we usually adopt a singularly weighted
L
2
-based estimate. To avoid an overestimate
in the linear stability analysis, we expand the perturbation in terms of the orthonormal basis
with respect to the weight
L
2
norm and reduce the linear stability estimate into an estimate
of a quadratic form for the Fourier coefficients. We further extract the damping effect of the
linearized operator by establishing a lower bound on the eigenvalues of an infinite-dimensional
symmetric matrix. We prove the positive-definiteness of this quadratic form by performing
an exact computation of the eigenvalues of a small number of Fourier modes with rigorous
computer-assisted bounds, and treat the high-frequency Fourier modes as a small perturbation
by using the asymptotic decay of the quadratic form in the high-frequency Fourier coefficients.
1.3. Organization of the paper and notations
In section
2
, we introduce our dynamic rescaling formulation and link the blowup of the phys-
ical equation to the steady state of the dynamic rescaling formulation. The linear stability of the
approximate steady state is established. In section
3
, we establish the nonlinear stability of the
approximate steady state and the exponential convergence to the steady state, which proves
theorem
1.2
and the blowup for the weak advection model. In section
4
, we prove theorem
1.4
and establish blowup for the original model with Hölder continuous data. In section
5
,
we prove theorem
1.7
by designing a special energy norm to estimate the viscous terms. We
provide the crucial linear damping estimates in the
appendix
using computer assistance.
Throughout the article, we use
(
·
,
·
)
to denote the inner product on
S
1
:
(
f
,
g
)=
́
π
−
π
fg
. We
use
C
to denote absolute constants, which may vary from line to line, and we use
C
(
k
) to denote
some constant that may depend on specific parameters
k
we choose. We use
A
≲
B
for positive
B
to denote that there exists an absolute constant
C
>
0 such that
A
⩽
CB
.
2. Dynamic rescaling formulation and linear estimates
2.1. Dynamic rescaling formulation
We will establish the singularity formation of the weak advection model by using the dynamic
rescaling formulation. We first consider the inviscid case with
ν
=
0. For solutions to the sys-
tem (
1.5
), we introduce
̃
u
(
x
,τ
)=
C
u
(
τ
)
u
(
x
,
t
(
τ
))
,
̃
ω
(
x
,τ
)=
C
u
(
τ
)
ω
(
x
,
t
(
τ
))
,
̃
ψ
(
x
,τ
)=
C
u
(
τ
)
ψ
(
x
,
t
(
τ
))
,
6
Nonlinearity
37
(2024) 035001
T Y Hou and Y Wang
where
C
u
(
τ
)=
exp
ˆ
τ
0
c
u
(
s
)
d
s
,
t
(
τ
)=
ˆ
τ
0
C
u
(
s
)
d
s
.
We can show that the rescaled variables solve the following dynamic rescaling equation
̃
u
τ
+
2
a
̃
ψ
̃
u
x
=
2
̃
u
̃
ψ
x
+
c
u
̃
u
,
̃
ω
τ
+
2
a
̃
ψ
̃
ω
x
=
̃
u
2
x
+
c
u
̃
ω,
−
̃
ψ
xx
= ̃
ω.
(2.1)
Remark 2.1.
We do not rescale the spatial variable
x
, since we are interested in a blowup
solution that is neither focusing nor expanding within a fixed period. The scaling factors for
u
,
ω
,
ψ
are thus the same.
When we establish a self-similar blowup, it suffices to show the dynamic stability of
equation (
2.1
) close to an approximate steady state with scaling parameter
c
u
<
−
ε <
0 uni-
formly in time for a small constant
ε
; see also [
8
]. In fact, it is easy to see that if
(
̃
u
,
̃
ω,
̃
ψ,
c
u
)
converges to a steady-state
(
u
∞
,ω
∞
,ψ
∞
,
c
u
,
∞
)
of (
2.1
), then
ω
(
x
,
t
)=
1
1
+
c
u
,
∞
t
ω
∞
,
u
(
x
,
t
)=
1
1
+
c
u
,
∞
t
u
∞
,ψ
(
x
,
t
)=
1
1
+
c
u
,
∞
t
ψ
∞
,
is a self-similar solution of (
1.5
).
From now on, we will primarily work in the dynamic rescaling formulation and use the
notations that
̃
u
=
̄
u
+
ˆ
u
, where
̄
u
is the approximate steady state that we perturb around and
ˆ
u
is the perturbation. Notations for variables
̃
ω
and
̃
ψ
are similar.
2.2. Equations governing the perturbation
We use the steady state corresponding to the case of
a
=
1 to construct an approximate steady
state for (
2.1
).
̄
ω
=
sin
x
,
̄
u
=
sin
x
,
̄
ψ
=
sin
x
,
̄
c
u
=
2
(
a
−
1
)
̄
ψ
x
(
0
)=
2
(
a
−
1
)
.
We consider odd perturbations
ˆ
u
,
ˆ
ω
,
ˆ
ψ
. The parities are preserved in time by equation (
2.1
).
We use the normalization condition as
c
u
=
2
(
a
−
1
)
ˆ
ψ
x
(
0
)
. This normalization ensures that
̄
u
x
(
0
)+
ˆ
u
x
(
0
)
is conserved in time.
To simplify our presentation, we will drop the
ˆ
in the perturbation
ˆ
u
and use
u
for
ˆ
u
,
ω
for
ˆ
ω
,
ψ
for
ˆ
ψ
. Now the perturbations satisfy the following system
u
τ
=
−
2
a
sin
xu
x
−
2
a
cos
x
ψ
+
2
u
cos
x
+
2sin
x
ψ
x
+
̄
c
u
u
+
c
u
̄
u
+
N
1
+
F
1
,
ω
τ
=
−
2
a
sin
x
ω
x
−
2
a
cos
x
ψ
+
2
u
cos
x
+
2sin
xu
x
+
̄
c
u
ω
+
c
u
̄
ω
+
N
2
+
F
2
,
−
ψ
xx
=
ω,
(2.2)
where
N
1
,
N
2
and
F
1
,
F
2
are the nonlinear terms and error terms defined below:
N
1
=(
c
u
+
2
ψ
x
)
u
−
2
a
ψ
u
x
,
N
2
=
c
u
ω
+
2
uu
x
−
2
a
ψω
x
,
F
1
=
̄
c
u
+
2
̄
ψ
x
̄
u
−
2
a
̄
ψ
̄
u
x
=
2
(
a
−
1
)
sin
x
(
1
−
cos
x
)
,
F
2
=
̄
c
u
̄
ω
+
2
̄
u
̄
u
x
−
2
a
̄
ψ
̄
ω
x
=
F
1
.
7
Nonlinearity
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T Y Hou and Y Wang
We further organize the system (
2.2
) into the main linearized term and a smaller term contain-
ing a factor of
a
−
1:
u
τ
=
L
1
+(
a
−
1
)
L
′
1
+
N
1
+
F
1
,
ω
τ
=
L
2
+(
a
−
1
)
L
′
2
+
N
2
+
F
2
,
−
ψ
xx
=
ω.
(2.3)
where
L
1
=
−
2sin
xu
x
−
2cos
x
ψ
+
2
u
cos
x
+
2sin
x
ψ
x
,
L
′
1
=
−
2sin
xu
x
−
2cos
x
ψ
+
2
u
+
2
ψ
x
(
0
)
sin
x
,
L
2
=
−
2sin
x
ω
x
−
2cos
x
ψ
+
2
u
cos
x
+
2sin
xu
x
,
L
′
2
=
−
2sin
x
ω
x
−
2cos
x
ψ
+
2
ω
+
2
ψ
x
(
0
)
sin
x
.
To show that the dynamic rescaling equation is stable and converges to a steady state, we
will perform a weighted-
L
2
estimate with a singular weight
ρ
and a weighted
L
2
norm
ρ
=
1
2
π
(
1
−
cos
x
)
,
∥
f
∥
ρ
=
f
2
,ρ
1
/
2
.
For initial perturbation with
u
x
(
0
,
0
)=
0, we have
u
x
(
0
,τ
)=
0 for all time and
E
2
(
τ
)=
1
2
u
2
x
,ρ
+
ω
2
,ρ
,
is well-defined. We will first show that the dominant parts
L
1
and
L
2
provide damping. The
following lemma is crucial and motivates the choice of
ρ
.
Lemma 2.2.
We have the following identity
(
sin
xf
x
,
f
ρ
)=
1
2
f
2
,ρ
,
which can be verified directly by using integration by parts.
2.3. Stability of the main parts in the linearized equation
In order to extract the maximal amount of damping, we will expand the perturbed solution in
the Fourier series and perform exact calculations. We first explore the orthonormal basis in
L
2
(
ρ
)
.
Lemma 2.3.
For the space of odd periodic functions on
[
0
,
2
π
]
, we describe a complete set of
orthonormal basis
{
o
k
}
in L
2
(
ρ
)
o
k
=
sin
(
kx
)
−
sin
((
k
−
1
)
x
)
,
k
=
1
,
2
,....
Similarly, for the space of even periodic functions that lie in L
2
(
ρ
)
, we describe a complete set
of orthonormal basis
{
e
k
}
e
k
=
cos
(
kx
)
−
cos
((
k
+
1
)
x
)
,
k
=
0
,
1
,....
Now we are now ready to establish linear stability.
8
Nonlinearity
37
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T Y Hou and Y Wang
Proposition 2.4.
The following energy estimate holds for the leading linearized operators
d
E
1
:=
(
L
1
)
x
,
u
x
ρ
+(
L
2
,ωρ
)
⩽
−
0
.
16
[(
u
x
,
u
x
ρ
)+(
ω,ωρ
)]
.
Proof.
Consider the expansion of
ω
,
u
x
, and
u
in the orthonormal basis
ω
=
X
k
⩾
1
a
k
o
k
,
u
=
X
k
⩾
1
b
k
o
k
,
u
x
=
X
k
⩾
1
c
k
e
k
.
Note the summation index for
u
x
satisfies
k
⩾
1 since we can easily see that
u
x
,
e
0
ρ
=
1
2
π
ˆ
2
π
0
u
x
=
0
.
We first express
b
k
in terms of
c
k
. If we insert the expression of the basis into
u
, take derivative
and compare the coefficients with the expansion of
u
x
, we get
c
i
=
i
X
k
=
1
b
k
−
ib
i
+
1
.
Therefore we can solve
b
i
+
1
=
b
1
−
i
−
1
X
k
=
1
c
k
k
(
k
+
1
)
−
c
i
i
.
Moreover, we have the compatibility condition
u
x
(
0
)=
P
k
⩾
0
b
k
=
0. Therefore we can solve
b
1
and obtain
b
i
=
X
k
⩾
i
c
k
k
(
k
+
1
)
−
c
i
−
1
i
,
(2.4)
where we define
c
0
=
0.
Now we write out the terms explicitly using the expansions
d
E
1
=
2
(
−
u
sin
x
−
u
xx
sin
x
,
u
x
ρ
)+
2
(
sin
x
ψ
+
sin
x
ψ
xx
,
u
x
ρ
)
+
2
(
−
sin
x
ω
x
−
cos
x
ψ,ωρ
)+
2
(
u
cos
x
+
sin
xu
x
,ωρ
)
=
−
[(
u
x
,
u
x
ρ
)+(
ω,ωρ
)+(
u
,
u
ρ
)]+
2
[
−
(
cos
x
ψ,ωρ
)+(
sin
x
ψ,
u
x
ρ
)+(
u
cos
x
,ωρ
)]
.
Here we use the crucial lemma
2.2
to extract damping on the local terms and the Biot–
Savart law
−
ψ
xx
=
ω
to cancel the effect of the nonlocal terms sin
x
ψ
xx
in
(
L
1
)
x
and sin
xu
x
in
L
2
. Next, we calculate the remaining nonlocal terms explicitly.
−
2
(
cos
x
ψ,ωρ
)=(
2
(
1
−
cos
x
)
ψ,ωρ
)
−
2
(
ψ,ωρ
)=
1
π
(
ψ,ω
)
−
2
(
ψ,ωρ
)
.
We express
ω
and
ψ
both in terms of orthonormal basis
o
k
corresponding to the weighted norm
and the canonical basis sin
(
kx
)
corresponding to the (normalized by
1
π
)
L
2
norm.
ω
=
X
k
⩾
1
(
a
k
−
a
k
+
1
)
sin
(
kx
)
,
9
Nonlinearity
37
(2024) 035001
T Y Hou and Y Wang
where we denote
a
0
=
0. Therefore
ψ
=
X
k
⩾
1
a
k
−
a
k
+
1
k
2
sin
(
kx
)
.
Furthermore, we collect
ψ
=
X
k
⩾
1
X
j
⩾
k
a
j
−
a
j
+
1
j
2
o
k
.
Therefore we can compute explicitly that
−
2
(
cos
x
ψ,ωρ
)=
X
k
⩾
1
(
a
k
−
a
k
+
1
)
2
k
2
−
2
X
k
⩾
1
a
k
X
j
⩾
k
a
j
−
a
j
+
1
j
2
.
We use integration by parts similar to lemma
2.2
to obtain
2
[(
sin
x
ψ,
u
x
ρ
)+(
u
cos
x
,ωρ
)]=
2
(
cos
x
ω
+
ψ
−
sin
x
ψ
x
,
u
ρ
):=
2
(
T
u
,
u
ρ
)
.
We further have
T
u
=
X
k
⩾
1
(
a
k
−
a
k
+
1
)
k
+
1
2
k
sin
((
k
−
1
)
x
)+
k
−
1
2
k
sin
((
k
+
1
)
x
)+
1
k
2
sin
(
kx
)
=
X
k
⩾
1
sin
(
kx
)
k
+
2
2
(
k
+
1
)
(
a
k
+
1
−
a
k
+
2
)+
a
k
−
a
k
+
1
k
2
+
k
−
2
2
(
k
−
1
)
(
a
k
−
1
−
a
k
)
=
X
k
⩾
1
"
k
−
2
2
(
k
−
1
)
a
k
−
1
+
1
2
k
(
k
−
1
)
+
1
k
2
a
k
+
k
+
1
2
k
+
1
(
k
+
1
)
2
−
1
k
2
!
a
k
+
1
+
X
j
>
k
+
1
1
j
2
−
1
(
j
−
1
)
2
!
a
j
3
5
o
k
,
where the terms involving
1
k
−
1
in the summand is regarded as 0 for
k
=
1. Therefore we collect
explicitly that
d
E
1
=
X
k
⩾
1
8
<
:
−
a
2
k
+
b
2
k
+
c
2
k
+
(
a
k
−
a
k
+
1
)
2
k
2
−
2
a
k
X
j
⩾
k
a
j
−
a
j
+
1
j
2
+
2
b
k
k
−
2
2
(
k
−
1
)
a
k
−
1
+
1
2
k
(
k
−
1
)
+
1
k
2
a
k
+
k
+
1
2
k
+
1
(
k
+
1
)
2
−
1
k
2
!
a
k
+
1
+
X
j
>
k
+
1
1
j
2
−
1
(
j
−
1
)
2
!
a
j
3
5
9
=
;
.
10
Nonlinearity
37
(2024) 035001
T Y Hou and Y Wang
Substituting (
2.4
) into the above and we can simplify
d
E
1
=
−
X
k
⩾
1
(
a
2
k
1
+
1
k
2
−
1
(
k
−
1
)
2
!
+
c
2
k
1
+
1
k
(
k
+
1
)
+
2
a
k
a
k
+
1
1
(
k
+
1
)
2
+
2
a
k
X
j
>
k
+
1
a
j
1
j
2
−
1
(
j
−
1
)
2
!
+
2
a
k
c
k
1
+
2
k
−
k
2
2
k
2
(
k
+
1
)
+
2
a
k
+
1
c
k
k
2
−
k
−
1
2
k
2
(
k
+
1
)
2
−
2
a
k
+
2
c
k
k
+
2
2
(
k
+
1
)
2
+
X
j
>
k
2
a
k
c
j
1
j
(
j
+
1
)
9
=
;
.
After this explicit computation, we notice that the damping estimate in proposition
2.4
can be
cast into an estimate of a quadratic form; see (
2.5
), which is equivalent to a lower bound on
the eigenvalues of an infinite-dimensional symmetric matrix.
F
(
a
,
c
):=
X
k
⩾
1
(
a
2
k
0
.
84
+
1
k
2
−
1
(
k
−
1
)
2
!
+
c
2
k
0
.
84
+
1
k
(
k
+
1
)
+
2
a
k
a
k
+
1
1
(
k
+
1
)
2
+
2
a
k
X
j
>
k
+
1
a
j
1
j
2
−
1
(
j
−
1
)
2
!
+
2
a
k
c
k
1
+
2
k
−
k
2
2
k
2
(
k
+
1
)
+
2
a
k
+
1
c
k
k
2
−
k
−
1
2
k
2
(
k
+
1
)
2
−
2
a
k
+
2
c
k
k
+
2
2
(
k
+
1
)
2
+
X
j
>
k
2
a
k
c
j
1
j
(
j
+
1
)
9
=
;
⩾
0
.
(2.5)
We notice that the entries decay fast. Therefore the strategy to prove (
2.5
) is to combine a
computer-assisted estimate of the eigenvalues of its finite truncation with a decay estimate of
the remaining part. We will defer the proof of (
2.5
) to the
appendix
, see lemma
A.1
and the
proof. Thereby we conclude the linear estimate.
3. Nonlinear estimates and convergence to self-similar profile
3.1. Nonlinear stability
By proposition
2.4
and equation (
2.3
), we have
1
2
d
d
τ
E
2
(
τ
)
⩽
−
0
.
16
E
2
(
τ
)+(
a
−
1
)
(
L
′
1
)
x
,
u
x
ρ
+(
L
′
2
,ωρ
)
+
(
N
1
)
x
,
u
x
ρ
+(
N
2
,ωρ
)+
(
F
1
)
x
,
u
x
ρ
+(
F
2
,ωρ
)
.
(3.1)
We first provide some estimates about the weighted
L
2
norm and
L
∞
norm of some lower-order
terms.
11
Nonlinearity
37
(2024) 035001
T Y Hou and Y Wang
Lemma 3.1.
The following estimates hold
(1)
Weighted L
2
norm:
∥
ψ
∥
ρ
,
∥
ψ
x
−
ψ
x
(
0
)
∥
ρ
,
∥
u
∥
ρ
≲
E
.
(2)
L
∞
norm:
∥
ψ
x
∥
∞
,
∥
ψ
sin
x
∥
∞
,
∥
u
∥
∞
≲
E
.
Proof.
For (1), we use the setting of the Fourier series approach as in the proof of proposition
2.4
and pick up the notation there.
∥
ψ
x
−
ψ
x
(
0
)
∥
2
ρ
=
X
k
⩾
1
0
@
X
j
⩾
k
a
j
−
a
j
+
1
j
1
A
2
⩽
X
k
⩾
1
X
j
⩾
k
a
2
j
0
@
1
k
2
+
X
j
>
k
1
j
−
1
j
+
1
2
1
A
≲
X
j
⩾
1
a
2
j
X
k
⩾
1
1
k
2
≲
∥
ω
∥
2
ρ
,
where we have used the Cauchy–Schwarz inequality. We can similarly estimate
∥
ψ
∥
ρ
. Then
we get
∥
u
∥
2
ρ
=
X
j
⩾
1
b
2
j
=
X
j
⩾
1
1
j
(
j
+
1
)
c
2
j
⩽
X
j
⩾
1
c
2
j
=
∥
u
x
∥
2
ρ
.
For (2), we first compute using Fourier series similar to (1)
ψ
x
(
0
)=
X
j
⩾
1
a
j
−
a
j
+
1
j
≲
∥
ω
∥
ρ
.
Next, we estimate
∥
ψ
x
−
ψ
x
(
0
)
∥
∞
≲
∥
ψ
xx
∥
1
≲
∥
ω
∥
ρ
.
Similarly, we obtain the estimate for
∥
u
∥
∞
. For
∥
ψ
sin
x
∥
∞
, since
ψ
is odd and periodic, we have
ψ
(
π
)=
ψ
(
0
)=
0 and only need to estimate this norm in
[
0
,π
]
. Since sin
x
⩾
2
π
min
{
x
,π
−
x
}
in
[
0
,π
]
, we have
∥
ψ
sin
x
∥
∞
≲
∥
ψ
x
∥
∞
by Lagrange’s mean value theorem.
Combined with the damping in lemma
2.2
, we further obtain
(
L
′
1
)
x
,
u
x
ρ
=
2
(
−
cos
xu
x
−
sin
xu
xx
+
sin
x
ψ
−
cos
x
ψ
x
+
u
x
+
ψ
x
(
0
)
cos
x
,
u
x
ρ
)
≲
E
2
+(
∥
ψ
∥
ρ
+
∥
ψ
x
−
ψ
x
(
0
)
∥
ρ
)
E
≲
E
2
,
(
L
′
2
,ωρ
)=
2
(
−
sin
x
ω
x
−
cos
x
ψ
+
ω
+
ψ
x
(
0
)
sin
x
,ωρ
)
≲
E
2
+(
∥
ψ
∥
ρ
+
|
ψ
x
(
0
)
|
)
E
≲
E
2
.
(
F
1
)
x
,
u
x
ρ
≲
|
a
−
1
|
E
,
(
F
2
,ωρ
)
≲
|
a
−
1
|
E
,
12
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(
N
1
)
x
,
u
x
ρ
=
2
(
−
ω
u
+(
1
−
a
)(
ψ
x
−
ψ
x
(
0
))
u
x
−
a
ψ
u
xx
,
u
x
ρ
)
≲
E
2
(
∥
ψ
x
∥
∞
+
∥
u
∥
∞
)+
|
u
2
x
,
(
ψρ
)
x
|
≲
E
2
∥
ψ
x
∥
∞
+
∥
u
∥
∞
+
∥
ψ
sin
x
∥
∞
≲
E
3
,
(
N
2
,ωρ
)=
2
((
a
−
1
)
ψ
x
(
0
)
ω
+
uu
x
−
a
ψω
x
,ωρ
)
≲
E
2
∥
ψ
x
∥
∞
+
∥
u
∥
∞
+
∥
ψ
sin
x
∥
∞
≲
E
3
.
Therefore we have
d
d
τ
E
(
τ
)
⩽
−
(
0
.
16
−
C
|
a
−
1
|
)
E
+
C
|
a
−
1
|
+
CE
2
.
(3.2)
We can perform the standard bootstrap argument to show that there exist absolute constants
δ,
C
>
0 such that if
|
a
−
1
|
< δ
and
E
(
0
)
<
C
|
a
−
1
|
, then we have
E
(
τ
)
<
C
|
a
−
1
|
for all
time. In particular
c
u
=
O
(
|
a
−
1
|
2
)
and
c
u
+
̄
c
u
<
0. Therefore we prove that the solution
blows up in finite time.
3.2. Estimates using a higher-order Sobolev norm
In order to establish convergence of the solution to a steady state, we need to estimate weighted
norms of
u
t
and
ω
t
. As was pointed out in [
4
], we need to provide stability estimates of the
equation in higher-order Sobolev norms to close the estimate. In particular, we choose
K
2
(
τ
)=
∥
D
x
u
x
∥
2
ρ
+
∥
D
x
ω
∥
2
ρ
,
where we denote
D
x
to be the operator sin
x
∂
x
.
Remark 3.2.
This choice of weighted norms is again motivated by the local linear damping
estimates. We recall that the leading order terms of the local terms in the linearized operators
(
L
1
)
x
,
L
2
are
−
2
D
x
u
x
and
−
2
D
x
w
, and we have
2
(
D
x
f
,
f
ρ
)=(
f
,
f
ρ
)
. Therefore in this new
weighted norm, the combined terms would again give damping
(
−
2
D
x
D
x
u
x
,
D
x
u
x
ρ
)+(
−
2
D
x
D
x
w
,
D
x
w
ρ
)=
−
K
2
.
(3.3)
We now obtain
1
2
d
d
τ
K
2
(
τ
)
⩽
D
x
(
L
1
)
x
,
D
x
u
x
ρ
+(
D
x
L
2
,
D
x
ωρ
)+
D
x
(
N
1
)
x
,
D
x
u
x
ρ
+(
D
x
N
2
,
D
x
ωρ
)+(
a
−
1
)
D
x
(
L
′
1
)
x
,
D
x
u
x
ρ
+(
D
x
L
′
2
,
D
x
ωρ
)
+
D
x
(
F
1
)
x
,
D
x
u
x
ρ
+(
D
x
F
2
,
D
x
ωρ
)
.
We will denote the terms that have
∥·∥
ρ
norm bounded by
E
as
l.o.t.
. The bound
∥
D
x
[
fg
]
∥
ρ
≲
(
∥
f
x
∥
2
+
∥
f
∥
2
)
for
g
=
1
,
cos
x
,
sin
x
combined with the oddness of
ψ
and
u
would imply that
D
x
[
fg
]
is
l.o.t.
for
f
=
ψ,ψ
x
,
u
and
g
=
sin
x
,
cos
x
,
1. Therefore combined with (
3.3
), we have the following estimate for the main
term
13
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d
K
1
:=
D
x
(
L
1
)
x
,
D
x
u
x
ρ
+(
D
x
L
2
,
D
x
ωρ
)
,
d
K
1
⩽
−
K
2
−
2
(
D
x
[
sin
x
ω
]
,
D
x
u
x
ρ
)+
2
(
D
x
D
x
u
,
D
x
ωρ
)+
CEK
=
−
K
2
−
(
sin2
x
ω,
D
x
u
x
ρ
)+(
sin2
xu
x
,
D
x
ωρ
)+
CEK
⩽
−
K
2
+
CEK
,
where we have again used a crucial cancellation in the equality, similar to that of d
E
1
in
section
2.3
. We estimate the rest of the terms similar to the nonlinear stability estimates
in (
3.1
).
D
x
(
L
′
1
)
x
,
D
x
u
x
ρ
+(
D
x
L
′
2
,
D
x
ωρ
)
≲
K
2
+
EK
,
D
x
(
F
1
)
x
,
D
x
u
x
ρ
+(
D
x
F
2
,
D
x
ωρ
)
≲
|
a
−
1
|
K
,
D
x
(
N
1
)
x
,
D
x
u
x
ρ
≲
EK
2
+
|
(
−
2
wD
x
u
+
2
(
1
−
a
)
D
x
ψ
x
u
x
−
2
a
ψ
D
x
u
xx
,
D
x
u
x
ρ
)
|
≲
EK
2
+
∥
sin
xu
x
∥
∞
EK
+
|
u
2
xx
,
ψ
sin
2
x
ρ
x
|
≲
EK
(
K
+
E
)+
K
2
∥
ψ
sin
x
∥
∞
≲
EK
(
K
+
E
)
,
(
D
x
N
2
,
D
x
ωρ
)
≲
EK
2
+
|
(
2
D
x
uu
x
−
2
a
ψ
D
x
ω
x
,
D
x
ωρ
)
|
≲
EK
(
K
+
E
)
,
where we have used integration by parts and the estimate
∥
sin
xu
x
∥
∞
≲
∥
sin
xu
xx
∥
1
+
∥
cos
xu
x
∥
1
≲
∥
D
x
u
x
∥
ρ
+
∥
u
x
∥
ρ
≲
E
+
K
.
We can finally prove that
d
d
τ
K
(
τ
)
⩽
−
(
1
−
C
|
a
−
1
|
)
K
+
CE
+
C
|
a
−
1
|
+
CE
(
E
+
K
)
.
Therefore combined with (
3.2
), we can find an absolute constant
μ>
1 such that
d
d
τ
(
K
+
μ
E
)
⩽
−
(
0
.
1
−
C
|
a
−
1
|
)(
K
+
μ
E
)+
C
|
a
−
1
|
+
C
(
K
+
μ
E
)
2
.
By using a standard bootstrap argument, there exist absolute constants
δ
0
< δ,
C
>
0, if
|
a
−
1
|
< δ
0
and
K
(
0
)+
μ
E
(
0
)
<
C
|
a
−
1
|
, then
K
(
τ
)+
μ
E
(
τ
)
<
C
|
a
−
1
|
for all time.
3.3. Convergence to the steady state
We estimate the weighted norm of
ω
τ
and
u
τ,
x
and then use the standard convergence in time
argument as in [
4
,
8
],
J
2
(
τ
)=
1
2
u
2
τ,
x
,ρ
+
ω
2
τ
,ρ
.
14
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Applying the estimates of
d
d
τ
E
to
d
d
τ
J
, we can get damping for the linear parts, and the small
error terms corresponding to
̄
ω
and
̄
u
vanishes. Therefore we yield
1
2
d
d
τ
J
2
⩽
−
(
0
.
16
−
C
|
a
−
1
|
)
J
2
+
(
N
1
)
τ,
x
,
u
τ,
x
ρ
+((
N
2
)
τ
,ω
t
ρ
)
.
Using estimates similar to lemma
3.1
and nonlinear estimates in (
3.1
), we get
(
N
1
)
τ,
x
,
u
τ,
x
ρ
≲
EJ
2
+(
ψ
τ
u
xx
,
u
τ,
x
ρ
)
≲
EJ
2
+
J
2
∥
u
xx
sin
x
∥
ρ
⩽
(
E
+
K
)
J
2
.
((
N
2
)
τ
,ω
τ
ρ
)
≲
EJ
2
+(
ψ
τ
ω
x
,ω
τ
ρ
)
≲
EJ
2
+
J
2
∥
ω
x
sin
x
∥
ρ
⩽
(
E
+
K
)
J
2
.
Combined with the
a priori
estimates on
E
+
K
, we can establish exponential convergence of
J
to zero. Then we can use the same argument as in [
4
,
8
] to establish exponential convergence
to the steady state and conclude the proof of theorem
1.2
.
4. Blowup of the original model with Hölder continuous data
In this section, we follow the strategy of the linear and nonlinear estimates of the weak advec-
tion model in sections
2
and
3
, and establish blowup of
C
α
data for the original model (
1.5
)
with
a
=
1. Here
α<
1 is close to 1. Many of the ideas are drawn from the paper [
3
] and we
only outline the most important steps. Intuitively,
C
α
regularity of the profile weakens the
advection and therefore contributes to a blowup in finite time.
4.1. Dynamic rescaling formulation around the approximate steady state
Before we start, we will solve the Biot–Savart law of recovering
ψ
from
ω
with odd symmetry.
Lemma 4.1.
Suppose that
ω
,
ψ
are odd and periodic on
[
−
π,π
]
, with
−
ψ
xx
=
ω
. Then we
solve
ψ
x
(
0
)=
−
1
2
π
́
2
π
0
y
ω
(
y
)
and obtain
ψ
=
ˆ
x
0
(
y
−
x
)
ω
(
y
)
d
y
+
x
ψ
x
(
0
)
.
(4.1)
The proof of this lemma is straightforward by integration in
x
.
We construct the following approximate steady state with
C
α
regularity for (
2.1
).
̄
ω
α
=
sgn
(
x
)
|
sin
x
|
α
,
̄
u
α
=
sgn
(
x
)
|
sin
x
|
1
+
α
2
,
̄
c
u
,α
=(
α
−
1
)
̄
ψ
α,
x
(
0
)
,
where
̄
ψ
α
is related to
̄
ω
α
via (
4.1
). We consider odd perturbations
u
,
ω
,
ψ
. The odd symmetry
of the solution is preserved in time by equation (
2.1
). We will use the normalization condition
as
c
u
=(
α
−
1
)
ψ
x
(
0
)
, which ensures that
u
vanishes to a higher order at all times so that we
can use the same singular weight
ρ
. In fact we compute using (
2.1
) and the normalization
conditions that
lim
x
→
0
̄
u
α
(
x
)+
u
(
x
,τ
)
x
1
+
α
2
=
lim
x
→
0
̄
u
α
(
x
)+
u
(
x
,
0
)
x
1
+
α
2
.
Therefore if we make the initial perturbation
u
(
x
,
0
)
vanish to order
1
+
α
around the origin,
u
(
x
,τ
)
will also vanish to order
1
+
α
for all time.
15
Nonlinearity
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Now similar to what we obtain in (
2.2
), the perturbations satisfy the following system
u
τ
=
L
1
+
R
1
,α
+
N
1
,α
+
F
1
,α
,
ω
τ
=
L
2
+
R
2
,α
+
N
2
,α
+
F
2
,α
,
−
ψ
xx
=
ω,
(4.2)
where we extract the same leading order linear parts as in (
2.3
), while the nonlinear and
error terms change and the residual error terms
R
1
,α
,
R
2
,α
model the discrepancy between our
approximate profile with
C
α
regularity and the steady state profile
̄
ω
=
̄
u
=
̄
ψ
=
sin
x
. Define
ψ
res
=
̄
ψ
α
−
̄
ψ,ω
res
= ̄
ω
α
−
̄
ω,
u
res
=
̄
u
α
−
̄
u
. We can express
R
i
,α
and
F
i
,α
as follows
R
1
,α
=
−
2
ψ
res
u
x
−
2
u
res
,
x
ψ
+
2
u
ψ
res
,
x
+
2
u
res
ψ
x
+
̄
c
u
,α
u
+
c
u
̄
u
α
.
R
2
,α
=
−
2
ψ
res
ω
x
−
2
ω
res
,
x
ψ
+
2
uu
res
,
x
+
2
u
res
u
x
+
̄
c
u
,α
ω
+
c
u
̄
ω
α
,
N
1
,α
=(
c
u
+
2
ψ
x
)
u
−
2
ψ
u
x
,
N
2
,α
=
c
u
ω
+
2
uu
x
−
2
ψω
x
,
F
1
,α
=
̄
c
u
,α
+
2
̄
ψ
α,
x
̄
u
α
−
2
̄
ψ
α
̄
u
α,
x
,
F
2
,α
=
̄
c
u
,α
̄
ω
α
+
2
̄
u
α
̄
u
α,
x
−
2
̄
ψ
α
̄
ω
α,
x
.
Before we perform our energy estimates, we will obtain some basic estimates of the residues.
Lemma 4.2.
The following estimates hold for
κ
=
7
8
<
9
10
< α <
1
.
(1)
Pointwise estimates of the residues:
|
∂
i
x
ω
res
|
+
|
∂
i
x
u
res
|
≲
|
α
−
1
||
sin
x
|
κ
−
i
,
i
=
0
,
1
,
2
,
3
,
∥
ψ
res
∥
∞
+
∥
ψ
res
,
x
∥
∞
≲
|
α
−
1
|
.
(2)
Refined estimates using cancellations:
α
−
1
2
̄
u
α,
x
−
sin
xu
res
,
xx
+
sin
x
∂
x
α
−
1
2
̄
u
α,
x
−
sin
xu
res
,
xx
≲
|
α
−
1
|
1
/
2
|
x
||
sin
x
|
α
−
1
.
Proof.
The first part of (1) and (2) can be proved by using direct calculations, and we refer to
lemma 6.1 in [
3
] for details. There seems to be a typo in (6.11) in [
3
] where
̄
ω
α
should have
been
̄
ω
α,
x
. Furthermore, by the expression in lemma
4.1
we get the second part of (1).
Similar to the weak advection case, we define the energy
E
2
(
τ
)=
1
2
((
u
2
x
,ρ
)+(
ω
2
,ρ
))
.
We
will estimate the growth of
E
(
τ
)
. The leading order linear estimates
L
1
,
L
2
can be obtained
in proposition
2.4
. The estimates for the nonlinear terms
N
1
,α
,
N
2
,α
follow almost exactly the
same as the weak advection case by using lemma
3.1
.
4.2. Nonlinear stability
By the computation in the previous subsection, we get
1
2
d
d
τ
E
2
(
τ
)
⩽
−
0
.
16
E
2
+
CE
3
+
(
R
1
,α
)
x
,
u
x
ρ
+(
R
2
,α
,ωρ
)+
(
F
1
,α
)
x
,
u
x
ρ
+(
F
2
,α
,ωρ
)
.
Further, we get
(
R
1
,α
)
x
,
u
x
ρ
⩽
(
−
2
ψ
res
u
xx
,
u
x
ρ
)+(
c
u
̄
u
α,
x
−
2
u
res
,
xx
ψ,
u
x
ρ
)
+(
∥
ω
res
∥
∞
+
∥
u
res
∥
∞
+
C
|
α
−
1
|
)
E
2
.
16
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For the first term, we can use integration by parts and lemma
3.1
to obtain
CE
2
∥
ψ
res
,
x
∥
∞
+
∥
ψ
res
sin
x
∥
∞
≲
∥
ψ
res
,
x
∥
∞
E
2
.
For the second term, we compute
c
u
̄
u
α,
x
−
2
u
res
,
xx
ψ
=
ψ
x
(
0
)[(
α
−
1
)
̄
u
α,
x
−
2sin
xu
res
,
xx
]+
2
u
res
,
xx
(
sin
x
ψ
x
(
0
)
−
ψ
)
.
Thus by lemmas
3.1
and
4.2
, we have
∥
c
u
̄
u
α,
x
−
2
u
res
,
xx
ψ
∥
ρ
≲
E
|
α
−
1
|
1
/
2
+
|
α
−
1
|
sin
x
ψ
x
(
0
)
−
ψ
|
sin
x
|
x
∞
.
(4.3)
Finally, for
|
x
|
⩾
π/
2, we have
sin
x
ψ
x
(
0
)
−
ψ
|
sin
x
|
x
≲
|
ψ
x
(
0
)
|
+
ψ
sin
x
∞
≲
E
.
For
|
x
|
< π/
2, we use lemma
4.1
and
|
sin
x
|
⩾
2
/π
|
x
|
to obtain
sin
x
ψ
x
(
0
)
−
ψ
|
sin
x
|
x
≲
sin
x
ψ
x
(
0
)
−
ψ
x
2
⩽
(
sin
x
−
x
)
ψ
x
(
0
)
x
2
+
́
x
0
(
y
−
x
)
ω
(
y
)
x
2
≲
E
,
where we have used the bound
∥
ω/
x
∥
1
≲
∥
ω/
x
∥
2
≲
∥
ω
∥
ρ
in the last inequality. Thus we yield
(
R
1
,α
)
x
,
u
x
ρ
≲
|
α
−
1
|
1
/
2
E
2
.
Similarly we get
(
R
2
,α
,ωρ
)
⩽
−
2
ψ
sin
x
ω
res
,
x
sin
x
,ωρ
+(
c
u
̄
ω
α
,ωρ
)+(
2
uu
res
,
x
,ωρ
)+
C
|
α
−
1
|
E
2
.
For the first two terms, we can estimate them using lemmas
3.1
and
4.2
. For the third term, we
use Hardy’s inequality to derive
∥
uu
res
,
x
∥
ρ
/
|
α
−
1
|
≲
∥
u
/
x
/
sin
x
∥
2
≲
∥
u
/
x
2
∥
2
+
∥
u
/
(
π
−
x
)
∥
2
≲
∥
u
x
/
x
∥
2
+
∥
u
x
∥
2
≲
E
.
Therefore we have
(
R
2
,α
,ωρ
)
≲
|
α
−
1
|
E
2
.
For the error terms, we can just perform standard norm estimates. We focus on the pointwise
estimates for
x
⩾
0 and the case
x
<
0 follows by using the odd symmetry of the solution.
F
2
,α
=(
α
−
1
)
̄
ψ
α,
x
(
0
)
sin
α
x
+(
α
+
1
)
cos
x
sin
α
x
−
2
α
(
ψ
res
+
sin
x
)
cos
x
sin
α
−
1
x
=(
α
−
1
)
̄
ψ
α,
x
(
0
)
−
cos
x
sin
α
x
−
2
α
ψ
res
sin
x
cos
x
sin
α
x
.
17