# 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.
Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We study the singularity formation of a quasi-exact 1D model proposed by Hou-
Li in [24]. 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 [15, 16] 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 if we just use singular
weights. 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\. 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 nonlinear 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) [14] to capture the effect of
nonlocal vortex stretching. The CLM model can be solved explicitly and can
develop a finite time singularity from smooth initial data. Later on, De
Gregorio incorporated the advection term into the CLM model to study the
competition between advection and vortex stretching [15, 16], see also [12,
24, 23] for a related study on the stabilizing effect of advection for the 3D
Euler and Navier-Stokes equations. In [39], Okamoto, Sakajo, and Wunsch
further introduced a parameter for the advection term to measure the relative
strength of the advection in the De Gregorio model. These simplified 1D models
have inspired many subsequent studies. Interested readers may consult the
excellent surveys [13, 19, 20, 27, 33] and the references therein. Very
recently, the authors in [25] established self-similar blowup for the whole
family of gCLM models with $a\leq 1$ 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.

In 2014, Luo-Hou [31, 30] presented convincing numerical evidence that the 3D
axisymmetric Euler equations with smooth initial data and boundary develop a
potential finite time singularity. In the same papers [31, 30], the authors
proposed the Hou-Luo model along the boundary $r=1$. This model captures many
essential features observed in the Hou-Luo blowup scenario for the
axisymmetric Euler equations. In [11], the authors proved the blowup of the
Hou-Luo model. In [10], Chen-Hou-Huang proved the asymptotically self-similar
blowup of the Hou-Luo model by extending the method of analysis for the finite
time blowup of the De Gregorio model by the same authors in [9]. Inspired by
Elgindi’s recent breakthrough for finite time singularity of the axisymmetric
Euler with no swirl and $C^{1,\alpha}$ velocity [17], Chen and Hou proved the
finite time blowup of the 2D Boussinesq and 3D Euler equations with
$C^{1,\alpha}$ initial velocity and boundary [7]. 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 [8].

In 2008, Hou and Li [24] 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 function are linear in
$r$. This model shares many intrinsic properties similar to those of the 3D
Navier-Stokes equations. Thus, it captures some essential nonlinear features
of the 3D Euler and Navier-Stokes equations and is very different from the
Hou-Luo model, which is defined along the boundary $r=1$, instead of the
symmetry axis $r=0$. In the same paper [24], the authors proved the global
regularity 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 $\nu=0$) with $C^{\alpha}$ initial
data. Inspired by the recent work of Chen [5] for the De Gregorio model, we
consider the case of $a<1$ and treat $1-a$ as a small parameter. For the
$C^{\alpha}$ initial data, we consider the original Hou-Li model with $a=1$
and $1-\alpha$ small. By using the dynamic rescaling formulation and analyzing
the stability of the linearized 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
[9, 7, 10]. 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 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
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 [24], Hou-Li introduced the following reformulation of the axisymmetric
Navier-Stokes equation:

(1.1) $\displaystyle
u_{1,t}+u^{r}u_{1,r}+u^{z}u_{1,z}=2u_{1}\psi_{1,z}+\nu\Delta u_{1}\,,$ (1.2)
$\displaystyle\omega_{1,t}+u^{r}\omega_{1,r}+u^{z}\omega_{1,z}=\left(u_{1}^{2}\right)_{z}+\nu\Delta\omega_{1}\,,$
(1.3)
$\displaystyle-\left[\partial_{r}^{2}+(3/r)\partial_{r}+\partial_{z}^{2}\right]\psi_{1}=\omega_{1}\,,$

where
$u_{1}=u^{\theta}/r,\;\omega_{1}=\omega^{\theta},\;\psi_{1}=\psi^{\theta}/r,$
and $u^{\theta}$, $\omega^{\theta}$, and $\psi^{\theta}$ are the angular
velocity, angular vorticity, and angular stream function, respectively. By the
well-known Caffarelli-Kohn-Nirenberg partial regularity result [3], 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 [24]
proposed the following 1D model along the symmetry axis $r=0$:

(1.4) $\displaystyle u_{1,t}+2\psi_{1}u_{1,z}$
$\displaystyle=2\psi_{1,z}u_{1}+\nu u_{1,zz}\,,$
$\displaystyle\omega_{1,t}+2\psi_{1}\omega_{1,z}$
$\displaystyle=\left(u_{1}^{2}\right)_{z}+\nu\omega_{1,zz}\,,$
$\displaystyle-\psi_{1,zz}$ $\displaystyle=\omega_{1}\,.$

Such a reduction is exact in the sense that if ($\omega_{1}$, $u_{1}$,
$\psi_{1}$) 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
$u^{\theta}=ru_{1}$, $\omega^{\theta}=r\omega_{1}$ are linear in $r$. We
assume that the solutions are periodic in $z$ on $[0,2\pi]$. We already know
from the original Hou-Li paper that this system is well-posed for $C^{m}$
initial data with $m\geq 1$. In [24], 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 [21, 22], the author presented new
numerical evidence 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
$u^{z}=2\psi_{1}+r\psi_{1,r}$ near the maximal point of $u_{1}$ is
significantly weaker than $2\psi_{1}$. This is due to the fact that $\psi_{1}$
reaches the maximum at a position $r=r_{\psi}$ that is smaller than the
position $r=r_{u}$ in which $u_{1}$ achieves its maximum, i.e.
$r_{\psi}<r_{u}$. Therefore $\psi_{1,r}$ is negative near the maximal position
of $u_{1}$. Thus the axial velocity $u^{z}$ is actually weaker than
$2\psi_{1}$, 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
behavior, we introduce the following 1D weak advection model.

(1.5) $\displaystyle u_{t}+2a\psi u_{z}$ $\displaystyle=2u\psi_{z}+\nu
u_{zz}\,,$ $\displaystyle\omega_{t}+2a\psi\omega_{z}$
$\displaystyle=\left(u^{2}\right)_{z}+\nu\omega_{zz}\,,$
$\displaystyle-\psi_{zz}$ $\displaystyle=\omega\,,$

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 $\nu=0$ resembles the
generalized Constantin-Lax-Majda model (gCLM) [39]

$\omega_{t}+au\omega_{x}=u_{x}\omega\,,\qquad u_{x}=H\omega\,,$

where

$H\omega(x)=\frac{1}{\pi}p.v.\int_{\mathbb{R}}\frac{\omega(y)}{x-y}\mathrm{~{}d}y$

is the Hilbert transform. They share similar structures of competition between
advection and vortex stretching. The case when $a=1$ corresponds to the De
Gregorio (DG) model. We obtain an explicit steady-state to the inviscid Hou-Li
model (1.4) $(\omega,u,\psi)=(\sin x,\sin x,\sin x)$, similar to the steady
state $(\omega,u)=(-\sin x,\sin x)$ 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 [5, 6].

###  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 $\nu=0$, there exists
a constant $\delta>0$ such that for $a\in(1-\delta,1)$, the weak advection
model (1.5) develops a finite time singularity for some $C^{\infty}$ initial
data. Moreover, there exists a self-similar profile
$({\omega}_{\infty},{\omega}_{\infty},{\omega}_{\infty})$ corresponding to a
blowup that is neither expanding nor focusing. More precisely, the blowup
solution to (1.5) has the form

$\omega(x,t)=\frac{1}{1+c_{u,\infty}t}{\omega}_{\infty}\,,u(x,t)=\frac{1}{1+c_{u,\infty}t}u_{\infty}\,,\psi(x,t)=\frac{1}{1+c_{u,\infty}t}{\psi}_{\infty}\,,$

for some negative constant $c_{u,\infty}$ with a blowup time given by
$T=\frac{-1}{c_{u,\infty}}$.

######  Remark 1.3.

Such self-similar blowup that is neither expanding nor focusing is observed
numerically for $a\in[0.6,0.9]$. See also a similar phenomenon observed for
the gCLM model in [32] for $a\in[0.68,0.95]$. The blowup result for the gCLM
model has been proved in [6] for $a$ sufficiently close to $1$. We remark that
for $a$ very close to $1$, since $c_{u,\infty}=O(|a-1|)$, the blowup time
becomes very large due to the very small coefficient $1-a$ 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
$C^{\alpha}$ initial data; for its proof see Section 4. In [18], the authors
made an important observation that advection can be weakened by $C^{\alpha}$
data. Intuitively if $u=O(x^{\alpha})$ in the origin, since $\psi$ is $C^{2}$,
we have that $\psi u_{x}\approx\alpha\psi_{x}u$ near the origin, the vortex
stretching term is stronger than the advection term if $\alpha<1$. See [9, 5]
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 $\nu=0$. For any
$\alpha<1$, (1.4) develops a finite time singularity for some $C^{\alpha}$
initial data. Moreover, there exists a $C^{\alpha}$ self-similar profile
corresponding to a blowup that is neither expanding nor focusing.

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 1.5. See also the ideas in [4]. We will provide more details of the
blowup analysis for the viscous case in Section 5.

######  Theorem 1.5.

Consider the weak advection model (1.5) with viscosity. There exists a
constant $\delta_{1}>0$ such that for $a\in(1-\delta_{1},1)$, the weak
advection model (1.5) develops a finite time singularity for some $C^{\infty}$
initial data.

We use the framework of the dynamic rescaling formulation to establish the
blowups. This formulation was first introduced by McLaughlin, Papanicolaou,
and co-workers in their study of self-similar blowup of the nonlinear
Schrödinger equation [35, 28]. This formulation was later developed into an
effective modulation technique, which has been applied to analyze the
singularity formation for the nonlinear Schrödinger equation [26, 36],
compressible Euler equations [2, 1], the nonlinear wave equation [38], 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 [9, 4, 5, 6], the 1D Hou-Luo model [10],
and in Euler equations [17, 7, 8]. 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 physical 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 1.5 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 $(\cdot,\cdot)$ to denote the inner product on
$S^{1}$: $(f,g)=\int_{-\pi}^{\pi}fg$. 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 $A\lesssim B$ for
positive $B$ to denote that there exists an absolute constant $C>0$ such that
$A\leq CB$.

##  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 $\nu=0$. For solutions to the system (1.5), we introduce

$\tilde{u}(x,\tau)=C_{u}(\tau)u(x,t(\tau))\,,\quad\tilde{\omega}(x,\tau)=C_{u}(\tau)\omega(x,t(\tau))\,,\quad\tilde{\psi}(x,\tau)=C_{u}(\tau)\psi(x,t(\tau))\,,$

where

$C_{u}(\tau)=\exp\left(\int_{0}^{\tau}c_{u}(s)ds\right)\,,\quad
t(\tau)=\int_{0}^{\tau}C_{u}(s)ds\,.$

We can show that the rescaled variables solve the following dynamic rescaling
equation

(2.1) $\displaystyle\tilde{u}_{\tau}+2a\tilde{\psi}\tilde{u}_{x}$
$\displaystyle=2\tilde{u}\tilde{\psi}_{x}+c_{u}\tilde{u}\,,$
$\displaystyle\tilde{\omega}_{\tau}+2a\tilde{\psi}\tilde{\omega}_{x}$
$\displaystyle=\left(\tilde{u}^{2}\right)_{x}+c_{u}\tilde{\omega}\,,$
$\displaystyle-\tilde{\psi}_{xx}$ $\displaystyle=\tilde{\omega}\,.$

######  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 $u$, $\omega$, $\psi$ 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 $c_{u}<-\epsilon<0$ uniformly in time for a small constant
$\epsilon$; see also [9]. In fact, it’s easy to see that if
$(\tilde{u},\tilde{\omega},\tilde{\psi},c_{u})$ converges to a steady-state
$({u}_{\infty},{\omega}_{\infty},{\psi}_{\infty},c_{u,\infty})$ of (2.1), then

$\omega(x,t)=\frac{1}{1+c_{u,\infty}t}{\omega}_{\infty}\,,u(x,t)=\frac{1}{1+c_{u,\infty}t}u_{\infty}\,,\psi(x,t)=\frac{1}{1+c_{u,\infty}t}{\psi}_{\infty}\,,$

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 $\tilde{u}=\bar{u}+\hat{u}$, where $\bar{u}$ is the
approximate steady state that we perturb around and $\tilde{u}$ is the
perturbation. Notations for variables $\tilde{\omega}$ and $\tilde{\psi}$ 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).

$\bar{\omega}=\sin x\,,\quad\bar{u}=\sin x\,,\quad\bar{\psi}=\sin
x\,,\quad\bar{c}_{u}=2(a-1)\bar{\psi}_{x}(0)=2(a-1)\,.$

We consider odd perturbations $\hat{u}$, $\hat{\omega}$, $\hat{\psi}$. The
parities are preserved in time by equation (2.1). We use the normalization
condition as $c_{u}=2(a-1)\hat{\psi}_{x}(0)$. This normalization ensures that
$\bar{u}_{x}(0)+\hat{u}_{x}(0)$ is conserved in time.

To simplify our presentation, we will drop the $\hat{}$ in the perturbation
$\hat{u}$ and use $u$ for $\hat{u}$, $\omega$ for $\hat{\omega}$, $\psi$ for
$\hat{\psi}$. We also use $t$ for the rescaled time $\tau$ variable. Now the
perturbations satisfy the following system

(2.2) $\displaystyle{u}_{t}$ $\displaystyle=-2a\sin xu_{x}-2a\cos x\psi+2u\cos
x+2\sin x\psi_{x}+\bar{c}_{u}u+c_{u}\bar{u}+N_{1}+F_{1}\,,$
$\displaystyle{\omega}_{t}$ $\displaystyle=-2a\sin x\omega_{x}-2a\cos
x\psi+2u\cos x+2\sin
xu_{x}+\bar{c}_{u}\omega+c_{u}\bar{\omega}+N_{2}+F_{2}\,,$
$\displaystyle-{\psi}_{xx}$ $\displaystyle={\omega}\,,$

where $N_{1},\;N_{2}$ and $F_{1},\;F_{2}$ are the nonlinear terms and error
terms defined below:

$N_{1}=(c_{u}+2\psi_{x})u-2a\psi u_{x}\,,\quad
N_{2}=c_{u}\omega+2uu_{x}-2a\psi\omega_{x}\,,$
$F_{1}=(\bar{c}_{u}+2\bar{\psi}_{x})\bar{u}-2a\bar{\psi}\bar{u}_{x}=2(a-1)\sin
x(1-\cos x)\,,\quad
F_{2}=\bar{c}_{u}\bar{\omega}+2\bar{u}\bar{u}_{x}-2a\bar{\psi}\bar{\omega}_{x}=F_{1}\,.$

We further organize the system (2.2) into the main linearized term and a
smaller term containing a factor of $a-1$:

(2.3) $\displaystyle{u}_{t}$
$\displaystyle=L_{1}+(a-1)L^{\prime}_{1}+N_{1}+F_{1}\,,$
$\displaystyle{\omega}_{t}$
$\displaystyle=L_{2}+(a-1)L^{\prime}_{2}+N_{2}+F_{2}\,,$
$\displaystyle-{\psi}_{xx}$ $\displaystyle={\omega}\,.$

where

$L_{1}=-2\sin xu_{x}-2\cos x\psi+2u\cos x+2\sin x\psi_{x}\,,$
$L^{\prime}_{1}=-2\sin xu_{x}-2\cos x\psi+2u+2\psi_{x}(0)\sin x\,,$
$L_{2}=-2\sin x\omega_{x}-2\cos x\psi+2u\cos{x}+2\sin xu_{x}\,,$
$L^{\prime}_{2}=-2\sin x\omega_{x}-2\cos x\psi+2\omega+2\psi_{x}(0)\sin x\,.$

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 $\rho$ and a weighted $L^{2}$ norm

$\rho=\frac{1}{2\pi(1-\cos x)}\,,\quad\|f\|_{\rho}=(f^{2},\rho)^{1/2}\,.$

For initial perturbation with $u_{x}(0,0)=0$, we have $u_{x}(0,t)=0$ for all
time and

$E^{2}(t)=\frac{1}{2}((u_{x}^{2},\rho)+(\omega^{2},\rho))\;,$

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 $\rho$.

######  Lemma 2.2.

We have the following identity

$(\sin xf_{x},f\rho)=\frac{1}{2}(f^{2},\rho)\,,$

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 $L^{2}(\rho)$.

######  Lemma 2.3.

For the space of odd periodic functions on $[0,2\pi]$, we describe a complete
set of orthonormal basis $\\{o^{k}\\}$ in $L^{2}(\rho)$

$o^{k}=\sin(kx)-\sin((k-1)x)\,,\quad k=1,2,\cdots\,.$

Similarly, for the space of even periodic functions that lie in $L^{2}(\rho)$,
we describe a complete set of orthonormal basis $\\{e^{k}\\}$

$e^{k}=\cos(kx)-\cos((k+1)x)\,,\quad k=0,1,\cdots\,.$

Now we are now ready to establish linear stability.

######  Proposition 2.4.

The following energy estimate holds for the leading linearized operators

$dE_{1}\coloneqq((L_{1})_{x},u_{x}\rho)+(L_{2},\omega\rho)\leq-0.16[(u_{x},u_{x}\rho)+(\omega,\omega\rho)]\,.$

###### Proof.

Consider the expansion of $\omega$, $u_{x}$, and $u$ in the orthonormal basis

$\omega=\sum_{k\geq 1}a_{k}o^{k}\,,\quad u=\sum_{k\geq 1}b_{k}o^{k}\,,\quad
u_{x}=\sum_{k\geq 1}c_{k}e^{k}\,.$

Note the summation index for $u_{x}$ satisfies $k\geq 1$ since we can easily
see that

$(u_{x},e^{0}\rho)=\frac{1}{2\pi}\int_{0}^{2\pi}u_{x}=0\,.$

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

$c_{i}=\sum_{k=1}^{i}b_{k}-ib_{i+1}\,.$

Therefore we can solve

$b_{i+1}=b_{1}-\sum_{k=1}^{i-1}\frac{c_{k}}{k(k+1)}-\frac{c_{i}}{i}\,.$

Moreover, we have the compatibility condition $u_{x}(0)=\sum_{k\geq
0}b_{k}=0$. Therefore we can solve $b_{1}$ and obtain

(2.4) $b_{i}=\sum_{k\geq i}\frac{c_{k}}{k(k+1)}-\frac{c_{i-1}}{i}\,,$

where we define $c_{0}=0$.

Now we write out the terms explicitly using the expansions

$\displaystyle dE_{1}$ $\displaystyle=2(-u\sin x-u_{xx}\sin
x,u_{x}\rho)+2(\sin x\psi+\sin x\psi_{xx},u_{x}\rho)$ $\displaystyle+2(-\sin
x\omega_{x}-\cos x\psi,\omega\rho)+2(u\cos{x}+\sin xu_{x},\omega\rho)$
$\displaystyle=-[(u_{x},u_{x}\rho)+(\omega,\omega\rho)+(u,u\rho)]+2[-(\cos
x\psi,\omega\rho)+(\sin x\psi,u_{x}\rho)+(u\cos{x},\omega\rho)]\,.$

Here we use the crucial Lemma 2.2 to extract damping on the local terms and
the Biot-Savart law $-\psi_{xx}=\omega$ to cancel the effect of the nonlocal
terms $\sin x\psi_{xx}$ in $(L_{1})_{x}$ and $\sin xu_{x}$ in $L_{2}$. Next,
we calculate the remaining nonlocal terms explicitly.

$-2(\cos x\psi,\omega\rho)=(2(1-\cos
x)\psi,\omega\rho)-2(\psi,\omega\rho)=\frac{1}{\pi}(\psi,\omega)-2(\psi,\omega\rho)\,.$

We express $\omega$ and $\psi$ both in terms of orthonormal basis $o^{k}$
corresponding to the weighted norm and the canonical basis $\sin(kx)$
corresponding to the (normalized by $\frac{1}{\pi}$) $L^{2}$ norm.

$\omega=\sum_{k\geq 1}(a_{k}-a_{k+1})\sin(kx)\,,$

where we denote $a_{0}=0$. Therefore

$\psi=\sum_{k\geq 1}\frac{a_{k}-a_{k+1}}{k^{2}}\sin(kx)\,.$

Furthermore, we collect

$\psi=\sum_{k\geq 1}\sum_{j\geq k}\frac{a_{j}-a_{j+1}}{j^{2}}o^{k}\,.$

Therefore we can compute explicitly that

$-2(\cos x\psi,\omega\rho)=\sum_{k\geq
1}\frac{(a_{k}-a_{k+1})^{2}}{k^{2}}-2\sum_{k\geq 1}a_{k}\sum_{j\geq
k}\frac{a_{j}-a_{j+1}}{j^{2}}\,.$

We use integration by parts similar to Lemma 2.2 to obtain

$2[(\sin x\psi,u_{x}\rho)+(u\cos{x},\omega\rho)]=2(\cos{x}\omega+\psi-\sin
x\psi_{x},u\rho)\coloneqq 2(T_{u},u\rho)\,.$

We further have

$\displaystyle T_{u}$ $\displaystyle=\sum_{k\geq
1}(a_{k}-a_{k+1})[\frac{k+1}{2k}\sin((k-1)x)+\frac{k-1}{2k}\sin((k+1)x)+\frac{1}{k^{2}}\sin(kx)]$
$\displaystyle=\sum_{k\geq
1}\sin(kx)[\frac{k+2}{2(k+1)}(a_{k+1}-a_{k+2})+\frac{a_{k}-a_{k+1}}{k^{2}}+\frac{k-2}{2(k-1)}(a_{k-1}-a_{k})]$
$\displaystyle=\sum_{k\geq
1}[\frac{k-2}{2(k-1)}a_{k-1}+(\frac{1}{2k(k-1)}+\frac{1}{k^{2}})a_{k}+(\frac{k+1}{2k}+\frac{1}{(k+1)^{2}}-\frac{1}{k^{2}})a_{k+1}$
$\displaystyle+\sum_{j>k+1}(\frac{1}{j^{2}}-\frac{1}{(j-1)^{2}})a_{j}]o^{k}\,,$

where the terms involving $\frac{1}{k-1}$ in the summand is regarded as $0$
for $k=1$. Therefore we collect explicitly that

$\displaystyle dE_{1}$ $\displaystyle=\sum_{k\geq
1}\\{-(a_{k}^{2}+b_{k}^{2}+c_{k}^{2})+\frac{(a_{k}-a_{k+1})^{2}}{k^{2}}-2a_{k}\sum_{j\geq
k}\frac{a_{j}-a_{j+1}}{j^{2}}+2b_{k}[\frac{k-2}{2(k-1)}a_{k-1}$
$\displaystyle+(\frac{1}{2k(k-1)}+\frac{1}{k^{2}})a_{k}+(\frac{k+1}{2k}+\frac{1}{(k+1)^{2}}-\frac{1}{k^{2}})a_{k+1}+\sum_{j>k+1}(\frac{1}{j^{2}}-\frac{1}{(j-1)^{2}})a_{j}]\\}\,.$

Substituting (2.4) into the above and we can simplify

$\displaystyle dE_{1}$ $\displaystyle=-\sum_{k\geq
1}\\{a_{k}^{2}(1+\frac{1}{k^{2}}-\frac{1}{(k-1)^{2}})+c_{k}^{2}(1+\frac{1}{k(k+1)})+2a_{k}a_{k+1}\frac{1}{(k+1)^{2}}$
$\displaystyle+2a_{k}\sum_{j>k+1}a_{j}(\frac{1}{j^{2}}-\frac{1}{(j-1)^{2}})+2a_{k}c_{k}\frac{1+2k-k^{2}}{2k^{2}(k+1)}+2a_{k+1}c_{k}\frac{k^{2}-k-1}{2k^{2}(k+1)^{2}}$
$\displaystyle-2a_{k+2}c_{k}\frac{k+2}{2(k+1)^{2}}+\sum_{j>k}2a_{k}c_{j}\frac{1}{j(j+1)}\\}\,.$

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.

(2.5) $\displaystyle F(a,c)$ $\displaystyle\coloneqq\sum_{k\geq
1}\\{a_{k}^{2}(0.84+\frac{1}{k^{2}}-\frac{1}{(k-1)^{2}})+c_{k}^{2}(0.84+\frac{1}{k(k+1)})+2a_{k}a_{k+1}\frac{1}{(k+1)^{2}}$
$\displaystyle+2a_{k}\sum_{j>k+1}a_{j}(\frac{1}{j^{2}}-\frac{1}{(j-1)^{2}})+2a_{k}c_{k}\frac{1+2k-k^{2}}{2k^{2}(k+1)}+2a_{k+1}c_{k}\frac{k^{2}-k-1}{2k^{2}(k+1)^{2}}$
$\displaystyle-2a_{k+2}c_{k}\frac{k+2}{2(k+1)^{2}}+\sum_{j>k}2a_{k}c_{j}\frac{1}{j(j+1)}\\}\geq
0\,.$

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 6.1 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

(3.1) $\displaystyle\frac{1}{2}\frac{d}{dt}E^{2}(t)$
$\displaystyle\leq-0.16E^{2}(t)+(a-1)[((L^{\prime}_{1})_{x},u_{x}\rho)+(L^{\prime}_{2},\omega\rho)]$
$\displaystyle+((N_{1})_{x},u_{x}\rho)+(N_{2},\omega\rho)+((F_{1})_{x},u_{x}\rho)+(F_{2},\omega\rho)\,.$

We first provide some estimates about the weighted $L^{2}$ norm and
$L^{\infty}$ norm of some lower-order terms.

######  Lemma 3.1.

The following estimates hold

  1. (1)

Weighted $L^{2}$ norm:

$\|\psi\|_{\rho},\|\psi_{x}-\psi_{x}(0)\|_{\rho},\|u\|_{\rho}\lesssim E\,.$

  2. (2)

$L^{\infty}$ norm:

$\|\psi_{x}\|_{\infty},\|\frac{\psi}{\sin x}\|_{\infty},\|u\|_{\infty}\lesssim
E\,.$

###### 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.

$\displaystyle\|\psi_{x}-\psi_{x}(0)\|_{\rho}^{2}$ $\displaystyle=\sum_{k\geq
1}(\sum_{j\geq k}\frac{a_{j}-a_{j+1}}{j})^{2}\leq\sum_{k\geq 1}\sum_{j\geq
k}a^{2}_{j}(\frac{1}{k^{2}}+\sum_{j>k}(\frac{1}{j}-\frac{1}{j+1})^{2})$
$\displaystyle\lesssim\sum_{j\geq 1}a_{j}^{2}\sum_{k\geq
1}\frac{1}{k^{2}}\lesssim\|\omega\|_{\rho}^{2}\,,$

where we have used the Cauchy-Schwarz inequality. We can similarly estimate
$\|\psi\|_{\rho}$. Then we get

$\|u\|_{\rho}^{2}=\sum_{j\geq 1}b_{j}^{2}=\sum_{j\geq
1}\frac{1}{j(j+1)}c_{j}^{2}\leq\sum_{j\geq
1}c_{j}^{2}=\|u_{x}\|_{\rho}^{2}\,.$

For (2), we first compute using Fourier series similar to (1)

$\psi_{x}(0)=\sum_{j\geq
1}\frac{a_{j}-a_{j+1}}{j}\lesssim\|\omega\|_{\rho}\,.$

Next, we estimate

$\|\psi_{x}-\psi_{x}(0)\|_{\infty}\lesssim\|\psi_{xx}\|_{1}\lesssim\|\omega\|_{\rho}\,.$

Similarly, we obtain the estimate for $\|u\|_{\infty}$. For
$\|\frac{\psi}{\sin x}\|_{\infty}$, since $\psi$ is odd and periodic, we have
$\psi(\pi)=\psi(0)=0$ and only need to estimate this norm in $[0,\pi]$. Since
$\sin x\geq\frac{2}{\pi}\min\\{x,\pi-x\\}$ in $[0,\pi]$, we have
$\|\frac{\psi}{\sin x}\|_{\infty}\lesssim\|\psi_{x}\|_{\infty}$ by Lagrange’s
mean value theorem. ∎

Combined with the damping in Lemma 2.2, we further obtain

$\displaystyle((L^{\prime}_{1})_{x},u_{x}\rho)$ $\displaystyle=2(-\cos
xu_{x}-\sin xu_{xx}+\sin x\psi-\cos x\psi_{x}+u_{x}+\psi_{x}(0)\cos
x,u_{x}\rho)$ $\displaystyle\lesssim
E^{2}+(\|\psi\|_{\rho}+\|\psi_{x}-\psi_{x}(0)\|_{\rho})E\lesssim E^{2}\,,$
$(L^{\prime}_{2},\omega\rho)=2(-\sin x\omega_{x}-\cos
x\psi+\omega+\psi_{x}(0)\sin x,\omega\rho)\lesssim
E^{2}+(\|\psi\|_{\rho}+|\psi_{x}(0)|)E\lesssim E^{2}\,.$
$((F_{1})_{x},u_{x}\rho)\lesssim|a-1|E\,,\quad(F_{2},\omega\rho)\lesssim|a-1|E\,,$
$\displaystyle((N_{1})_{x},u_{x}\rho)$ $\displaystyle=2(-\omega
u+(1-a)(\psi_{x}-\psi_{x}(0))u_{x}-a\psi u_{xx},u_{x}\rho)$
$\displaystyle\lesssim
E^{2}(\|\psi_{x}\|_{\infty}+\|u\|_{\infty})+|(u_{x}^{2},(\psi\rho)_{x})|$
$\displaystyle\lesssim
E^{2}(\|\psi_{x}\|_{\infty}+\|u\|_{\infty}+\|\frac{\psi}{\sin
x}\|_{\infty})\lesssim E^{3}\,,$
$(N_{2},\omega\rho)=2((a-1)\psi_{x}(0)\omega+uu_{x}-a\psi\omega_{x},\omega\rho)\lesssim
E^{2}(\|\psi_{x}\|_{\infty}+\|u\|_{\infty}+\|\frac{\psi}{\sin
x}\|_{\infty})\lesssim E^{3}\,.$

Therefore we have

(3.2) $\frac{d}{dt}E(t)\leq-(0.16-C|a-1|)E+C|a-1|+CE^{2}\,.$

We can perform the standard bootstrap argument to show that there exist
absolute constants $\delta,C>0$ such that if $|a-1|<\delta$ and $E(0)<C|a-1|$,
then we have $E(t)<C|a-1|$ for all time. In particular $c_{u}=O(|a-1|^{2})$
and $c_{u}+\bar{c}_{u}<0$. 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 $\omega_{t}$. As was pointed out in
[6], we need to provide stability estimates of the equation in higher-order
Sobolev norms to close the estimate. In particular, we choose

$K^{2}(t)=\|D_{x}u_{x}\|_{\rho}^{2}+\|D_{x}\omega\|_{\rho}^{2}\,,$

where we denote $D_{x}$ to be the operator $\sin x\partial x$.

######  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 $-2D_{x}u_{x}$ and $-2D_{x}w$,
and we have $2(D_{x}f,f\rho)=(f,f\rho)$. Therefore in this new weighted norm,
the combined terms would again give damping

(3.3)
$(-2D_{x}D_{x}u_{x},D_{x}u_{x}\rho)+(-2D_{x}D_{x}w,D_{x}w\rho)=-K^{2}\,.$

We now obtain

$\displaystyle\frac{1}{2}\frac{d}{dt}K^{2}(t)$
$\displaystyle\leq(D_{x}(L_{1})_{x},D_{x}u_{x}\rho)+(D_{x}L_{2},D_{x}\omega\rho)+(D_{x}(N_{1})_{x},D_{x}u_{x}\rho)$
$\displaystyle+(D_{x}N_{2},D_{x}\omega\rho)+(a-1)[(D_{x}(L^{\prime}_{1})_{x},D_{x}u_{x}\rho)+(D_{x}L^{\prime}_{2},D_{x}\omega\rho)]$
$\displaystyle+(D_{x}(F_{1})_{x},D_{x}u_{x}\rho)+(D_{x}F_{2},D_{x}\omega\rho)\,.$

We will denote the terms that have $\|\cdot\|_{\rho}$ norm bounded by $E$ as
l.o.t.. The bound

$\|D_{x}[fg]\|_{\rho}\lesssim(\|f_{x}\|_{2}+\|f\|_{2})\quad\text{for}\,g=1,\cos
x,\sin x$

combined with the oddness of $\psi$ and $u$ would imply that $D_{x}[fg]$ is
l.o.t. for $f=\psi,\psi_{x},u$ and $g=\sin x,\cos x,1$. Therefore combined
with (3.3), we have the following estimate for the main term

$dK_{1}\coloneqq(D_{x}(L_{1})_{x},D_{x}u_{x}\rho)+(D_{x}L_{2},D_{x}\omega\rho)\,,$
$\displaystyle dK_{1}$ $\displaystyle\leq-K^{2}-2(D_{x}[\sin
x\omega],D_{x}u_{x}\rho)+2(D_{x}D_{x}u,D_{x}\omega\rho)+CEK$
$\displaystyle=-K^{2}-(\sin 2x\omega,D_{x}u_{x}\rho)+(\sin
2xu_{x},D_{x}\omega\rho)+CEK$ $\displaystyle\leq-K^{2}+CEK\,,$

where we have again used a crucial cancellation in the equality, similar to
that of $dE_{1}$ in Subsection 2.3. We estimate the rest of the terms similar
to the nonlinear stability estimates in (3.1).

$(D_{x}(L^{\prime}_{1})_{x},D_{x}u_{x}\rho)+(D_{x}L^{\prime}_{2},D_{x}\omega\rho)\lesssim
K^{2}+EK\,,$
$(D_{x}(F_{1})_{x},D_{x}u_{x}\rho)+(D_{x}F_{2},D_{x}\omega\rho)\lesssim|a-1|K\,,$
$\displaystyle(D_{x}(N_{1})_{x},D_{x}u_{x}\rho)$ $\displaystyle\lesssim
EK^{2}+|(-2wD_{x}u+2(1-a)D_{x}\psi_{x}u_{x}-2a\psi
D_{x}u_{xx},D_{x}u_{x}\rho)|$ $\displaystyle\lesssim EK^{2}+\|\sin
xu_{x}\|_{\infty}EK+|(u^{2}_{xx},(\psi\sin^{2}x\rho)_{x})|$
$\displaystyle\lesssim EK(K+E)+K^{2}\|\frac{\psi}{\sin x}\|_{\infty}\lesssim
EK(K+E)\,,$ $(D_{x}N_{2},D_{x}\omega\rho)\lesssim EK^{2}+|(2D_{x}uu_{x}-2a\psi
D_{x}\omega_{x},D_{x}\omega\rho)|\lesssim EK(K+E)\,,$

where we have used integration by parts and the estimate

$\|\sin xu_{x}\|_{\infty}\lesssim\|\sin xu_{xx}\|_{1}+\|\cos
xu_{x}\|_{1}\lesssim\|D_{x}u_{x}\|_{\rho}+\|u_{x}\|_{\rho}\lesssim E+K\,.$

We can finally prove that

$\frac{d}{dt}K(t)\leq-(1-C|a-1|)K+CE+C|a-1|+CE(E+K)\,.$

Therefore combined with (3.2), we can find an absolute constant $\mu>1$ such
that

$\frac{d}{dt}(K+\mu E)\leq-(0.1-C|a-1|)(K+\mu E)+C|a-1|+C(K+\mu E)^{2}\,.$

By using a standard bootstrap argument, there exist absolute constants
$\delta_{0}<\delta,C>0$, if $|a-1|<\delta_{0}$ and $K(0)+\mu E(0)<C|a-1|$,
then $K(t)+\mu E(t)<C|a-1|$ for all time.

###  3.3. Convergence to the steady state

We estimate the weighted norm of $\omega_{t}$ and $u_{t,x}$ and then use the
standard convergence in time argument as in [9, 6].

$J^{2}(t)=\frac{1}{2}((u_{t,x}^{2},\rho)+(\omega_{t}^{2},\rho))\,.$

Applying the estimates of $\frac{d}{dt}E$ to $\frac{d}{dt}J$, we can get
damping for the linear parts, and the small error terms corresponding to
$\bar{\omega}$ and $\bar{u}$ vanishes. Therefore we yield

$\frac{1}{2}\frac{d}{dt}J^{2}\leq-(0.16-C|a-1|)J^{2}+((N_{1})_{t,x},u_{t,x}\rho)+((N_{2})_{t},\omega_{t}\rho)\,.$

Using estimates similar to Lemma 3.1 and nonlinear estimates in (3.1), we get

$((N_{1})_{t,x},u_{t,x}\rho)\lesssim
EJ^{2}+(\psi_{t}u_{xx},u_{t,x}\rho)\lesssim EJ^{2}+J^{2}\|u_{xx}\sin
x\|_{\rho}\leq(E+K)J^{2}\,.$ $((N_{2})_{t},\omega_{t}\rho)\lesssim
EJ^{2}+(\psi_{t}\omega_{x},\omega_{t}\rho)\lesssim
EJ^{2}+J^{2}\|\omega_{x}\sin x\|_{\rho}\leq(E+K)J^{2}\,.$

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 [9, 6] 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 advection model in Sections 2 and 3, and establish blowup of
$C^{\alpha}$ data for the original model (1.5) with $a=1$. Here $\alpha<1$ is
close to $1$. Many of the ideas are drawn from the paper [5] and we only
outline the most important steps. Intuitively, $C^{\alpha}$ 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 $\psi$ from
$\omega$ with odd symmetry.

######  Lemma 4.1.

Suppose that $\omega$, $\psi$ are odd and periodic on $[-\pi,\pi]$, with
$-\psi_{xx}=\omega$. Then we solve
$\psi_{x}(0)=-\frac{1}{2\pi}\int_{0}^{2\pi}y{\omega}(y)$ and obtain

(4.1) $\psi=\int_{0}^{x}(y-x){\omega}(y)dy+x\psi_{x}(0)\,.$

The proof of this lemma is straightforward by integration in $x$.

We construct the following approximate steady state with $C^{\alpha}$
regularity for (2.1).

$\bar{\omega}_{\alpha}=sgn(x)|\sin
x|^{\alpha}\,,\quad\bar{u}_{\alpha}=sgn(x)|\sin
x|^{\frac{1+\alpha}{2}}\,,\quad\bar{c}_{u,{\alpha}}=(\alpha-1)\bar{\psi}_{{\alpha},x}(0)\,,$

where $\bar{\psi}_{\alpha}$ is related to $\bar{\omega}_{\alpha}$ via (4.1).
We consider odd perturbations $u$, $\omega$, $\psi$. The odd symmetry of the
solution is preserved in time by equation (2.1). We will use the normalization
condition as $c_{u}=(\alpha-1){\psi}_{x}(0)$, which ensures that $u$ vanishes
to a higher order at all times so that we can use the same singular weight
$\rho$. In fact we compute using (2.1) and the normalization conditions that

$\lim_{x\to
0}\frac{\bar{u}_{{\alpha}}(x)+u(x,t)}{x^{\frac{1+\alpha}{2}}}=\lim_{x\to
0}\frac{\bar{u}_{{\alpha}}(x)+u(x,0)}{x^{\frac{1+\alpha}{2}}}\,.$

Therefore if we make the initial perturbation $u(x,0)$ vanish to order
$1+\alpha$ around the origin, $u(x,t)$ will also vanish to order $1+\alpha$
for all time.

Now similar to what we obtain in (2.2), the perturbations satisfy the
following system

(4.2) $\displaystyle{u}_{t}$
$\displaystyle=L_{1}+R_{1,\alpha}+N_{1,\alpha}+F_{1,\alpha}\,,$
$\displaystyle{\omega}_{t}$
$\displaystyle=L_{2}+R_{2,\alpha}+N_{2,\alpha}+F_{2,\alpha}\,,$
$\displaystyle-{\psi}_{xx}$ $\displaystyle={\omega}\,,$

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 $R_{1,\alpha}$,
$R_{2,\alpha}$ model the discrepancy between our approximate profile with
$C^{\alpha}$ regularity and the steady state profile
$\bar{\omega}=\bar{u}=\bar{\psi}=\sin x$. Define
$\psi_{\text{res}}=\bar{\psi}_{\alpha}-\bar{\psi},\omega_{\text{res}}=\bar{\omega}_{\alpha}-\bar{\omega},u_{\text{res}}=\bar{u}_{\alpha}-\bar{u}$.
We can express $R_{i,\alpha}$ and $F_{i,\alpha}$ as follows

$R_{1,\alpha}=-2\psi_{\text{res}}u_{x}-2u_{\text{res},x}\psi+2u\psi_{\text{res},x}+2u_{\text{res}}\psi_{x}+\bar{c}_{u,\alpha}u+c_{u}\bar{u}_{\alpha}\,.\,$
$R_{2,\alpha}=-2\psi_{\text{res}}\omega_{x}-2\omega_{\text{res},x}\psi+2uu_{\text{res},x}+2u_{\text{res}}u_{x}+\bar{c}_{u,\alpha}\omega+c_{u}\bar{\omega}_{\alpha}\,,$
$N_{1,\alpha}=(c_{u}+2\psi_{x})u-2\psi u_{x}\,,\quad
N_{2,\alpha}=c_{u}\omega+2uu_{x}-2\psi\omega_{x}\,,$
$F_{1,\alpha}=(\bar{c}_{u,\alpha}+2\bar{\psi}_{\alpha,x})\bar{u}_{\alpha}-2\bar{\psi}_{\alpha}\bar{u}_{\alpha,x}\,,\quad
F_{2,\alpha}=\bar{c}_{u,\alpha}\bar{\omega}_{\alpha}+2\bar{u}_{\alpha}\bar{u}_{\alpha,x}-2\bar{\psi}_{\alpha}\bar{\omega}_{\alpha,x}\,.$

Before we perform our energy estimates, we will obtain some basic estimates of
the residues.

######  Lemma 4.2.

The following estimates hold for $\kappa=\frac{7}{8}<\frac{9}{10}<\alpha<1$.

  1. (1)

Pointwise estimates of the residues:

$|\partial_{x}^{i}\omega_{\text{res}}|+|\partial_{x}^{i}u_{\text{res}}|\lesssim|\alpha-1||\sin
x|^{\kappa-i}\,,\quad i=0,1,2,3,$
$\|\psi_{\text{res}}\|_{\infty}+\|\psi_{\text{res},x}\|_{\infty}\lesssim|\alpha-1|\,.$

  2. (2)

Refined estimates using cancellations:

$|\frac{\alpha-1}{2}\bar{u}_{\alpha,x}-\sin xu_{\text{res},xx}|+|\sin
x\partial x[\frac{\alpha-1}{2}\bar{u}_{\alpha,x}-\sin
xu_{\text{res},xx}]|\lesssim|\alpha-1|^{1/2}|x||\sin x|^{\alpha-1}\,.$

###### Proof.

The first part of (1) and (2) can be proved by using direct calculations, and
we refer to Lemma 6.1 in [5] for details. There seems to be a typo in (6.11)
in [5] where $\bar{\omega}_{\alpha}$ should have been
$\bar{\omega}_{\alpha,x}$. 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
$E^{2}(t)=\frac{1}{2}((u_{x}^{2},\rho)+(\omega^{2},\rho))\,.$ We will estimate
the growth of $E(t)$. The leading order linear estimates $L_{1},L_{2}$ can be
obtained in Proposition 2.4. The estimates for the nonlinear terms
$N_{1,\alpha},N_{2,\alpha}$ 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

$\frac{1}{2}\frac{d}{dt}E^{2}(t)\leq-0.16E^{2}+CE^{3}+((R_{1,\alpha})_{x},u_{x}\rho)+(R_{2,\alpha},\omega\rho)+((F_{1,\alpha})_{x},u_{x}\rho)+(F_{2,\alpha},\omega\rho)\,.$

Further, we get

$\displaystyle((R_{1,\alpha})_{x},u_{x}\rho)$
$\displaystyle\leq(-2\psi_{\text{res}}u_{xx},u_{x}\rho)+(c_{u}\bar{u}_{\alpha,x}-2u_{\text{res},xx}\psi,u_{x}\rho)$
$\displaystyle+(\|\omega_{\text{res}}\|_{\infty}+\|u_{\text{res}}\|_{\infty}+C|\alpha-1|)E^{2}\,.$

For the first term, we can use integration by parts and Lemma 3.1 to obtain

$CE^{2}(\|\psi_{\text{res},x}\|_{\infty}+\|\frac{\psi_{\text{res}}}{\sin
x}\|_{\infty})\lesssim\|\psi_{\text{res},x}\|_{\infty}E^{2}\,.$

For the second term, we compute

$c_{u}\bar{u}_{\alpha,x}-2u_{\text{res},xx}\psi=\psi_{x}(0)[(\alpha-1)\bar{u}_{\alpha,x}-2\sin
xu_{\text{res},xx}]+2u_{\text{res},xx}(\sin x\psi_{x}(0)-\psi)\,.$

Thus by Lemmas 3.1 and 4.2, we have

(4.3) $\|c_{u}\bar{u}_{\alpha,x}-2u_{\text{res},xx}\psi\|_{\rho}\lesssim
E|\alpha-1|^{1/2}+|\alpha-1|\|\frac{\sin x\psi_{x}(0)-\psi}{|\sin
x|x}\|_{\infty}\,.$

Finally, for $|x|\geq\pi/2$, we have

$|\frac{\sin x\psi_{x}(0)-\psi}{|\sin
x|x}|\lesssim|\psi_{x}(0)|+\|\frac{\psi}{\sin x}\|_{\infty}\lesssim E\,.$

For $|x|<\pi/2$, we use Lemma 4.1 and $|\sin x|\geq 2/\pi|x|$ to obtain

$|\frac{\sin x\psi_{x}(0)-\psi}{|\sin x|x}|\lesssim|\frac{\sin
x\psi_{x}(0)-\psi}{x^{2}}|\leq|\frac{(\sin
x-x)\psi_{x}(0)}{x^{2}}|+|\frac{\int_{0}^{x}(y-x)\omega(y)}{x^{2}}|\lesssim
E\,,$

where we have used the bound
$\|\omega/x\|_{1}\lesssim\|\omega/x\|_{2}\lesssim\|\omega\|_{\rho}$ in the
last inequality. Thus we yield

$((R_{1,\alpha})_{x},u_{x}\rho)\lesssim|\alpha-1|^{1/2}E^{2}\,.$

Similarly we get

$(R_{2,\alpha},\omega\rho)\leq(-2\frac{\psi}{\sin x}\omega_{\text{res},x}\sin
x,\omega\rho)+(c_{u}\bar{\omega}_{\alpha},\omega\rho)+(2uu_{\text{res},x},\omega\rho)+C|\alpha-1|E^{2}\,.$

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

$\|uu_{\text{res},x}\|_{\rho}/|\alpha-1|\lesssim\|u/x/\sin
x\|_{2}\lesssim\|u/x^{2}\|_{2}+\|u/(\pi-x)\|_{2}\lesssim\|u_{x}/x\|_{2}+\|u_{x}\|_{2}\lesssim
E\,.$

Therefore we have

$(R_{2,\alpha},\omega\rho)\lesssim|\alpha-1|E^{2}\,.$

For the error terms, we can just perform standard norm estimates. We focus on
the pointwise estimates for $x\geq 0$ and the case $x<0$ follows by using the
odd symmetry of the solution.

$\displaystyle F_{2,\alpha}$
$\displaystyle=(\alpha-1)\bar{\psi}_{\alpha,x}(0)\sin^{\alpha}x+(\alpha+1)\cos
x\sin^{\alpha}x-2\alpha(\psi_{\text{res}}+\sin x)\cos x\sin^{\alpha-1}x$
$\displaystyle=(\alpha-1)(\bar{\psi}_{\alpha,x}(0)-\cos
x)\sin^{\alpha}x-2\alpha\frac{\psi_{\text{res}}}{\sin x}\cos
x\sin^{\alpha}x\,.$

Therefore, we obtain

$\|F_{2,\alpha}\|_{\rho}\lesssim|\alpha-1|+\|\frac{\psi_{res}}{\sin
x}\|_{\infty}\lesssim|\alpha-1|\,.$

Similarly, we have

$\displaystyle(F_{1,\alpha})_{x}$
$\displaystyle=\frac{\alpha^{2}-1}{2}\sin^{\frac{\alpha-1}{2}}x\cos
x[\bar{\psi}_{\alpha,x}(0)-\bar{\psi}_{\alpha}\frac{\cos x}{\sin
x}]+\sin^{\frac{\alpha+1}{2}}x[(\alpha+1)\bar{\psi}_{\alpha}-2\sin^{\alpha}x]\,.$

Further, we obtain the following estimate:

$\|(\alpha+1)\bar{\psi}_{\alpha}-2\sin^{\alpha}x\|_{\infty}\leq|\alpha-1|\|\bar{\psi}_{\alpha}\|_{\infty}+2\|\psi_{\text{res}}\|_{\infty}+2\|\sin
x-\sin^{\alpha}x\|_{\infty}\lesssim|\alpha-1|\,,$
$\displaystyle\bar{\psi}_{\alpha,x}(0)-\bar{\psi}_{\alpha}\frac{\cos x}{\sin
x}$ $\displaystyle=(1-\cos
x)+\psi_{\text{res},x}(0)-\psi_{\text{res}}\frac{\cos x}{\sin x}$
$\displaystyle=(1-\cos x)(1+\frac{\psi_{\text{res}}}{\sin
x})+\psi_{\text{res},x}(0)-\frac{\psi_{\text{res}}}{\sin x}\,.$

Combined with the estimate similar to that in (4.3), we get

$\|[\bar{\psi}_{\alpha,x}(0)-\bar{\psi}_{\alpha}\frac{\cos x}{\sin
x}]\rho^{1/2}\|_{\infty}\lesssim\|\frac{1-\cos
x}{x}(1+\frac{\psi_{\text{res}}}{\sin
x})\|_{\infty}+\|[\psi_{\text{res},x}(0)-\frac{\psi_{\text{res}}}{\sin
x}]/x\|_{\infty}\lesssim 1\,.$

Therefore we yield

$\|(F_{1,\alpha})_{x}\|_{\rho}\lesssim|\alpha-1|\,.$

Collecting all the estimates of the residues and the error terms, we arrive at

$\frac{d}{dt}E(t)\leq-(0.16-C|\alpha-1|^{1/2}|)E+CE^{2}+C|\alpha-1|\,.$

Similar to Subsection 3, we can perform the bootstrap argument to conclude
finite-time blowup.

###  4.3. Estimates in higher-order Sobolev norms and convergence to steady
state

Following the ideas in Subsection 3.2, we can perform estimates in higher-
order Sobolev norms and then close the estimates to establish convergence to a
steady state. We use the same energy $K$ and only sketch the main steps here.
We first have from Subsection 3.2 the estimates of $L_{i}$ and $N_{i}$ and
obtain

$\displaystyle\frac{1}{2}\frac{d}{dt}K^{2}(t)$
$\displaystyle\leq-K^{2}+CEK+CEK^{2}+((R_{1,\alpha})_{xx},\sin^{2}xu_{xx}\rho)$
$\displaystyle+((R_{2,\alpha})_{x},\sin^{2}x\omega_{x}\rho)+((F_{1,\alpha})_{xx},\sin^{2}xu_{xx}\rho)+((F_{2,\alpha})_{x},\sin^{2}x\omega_{x}\rho)\,.$

By Lemma 4.2, $\sin x\partial xu_{\text{res}},\sin x\partial
x\omega_{\text{res}}$ shares the same pointwise estimates as $\
u_{\text{res}},\omega_{\text{res}}$. We can obtain, similar to the estimates
in $E$, the estimates

$((R_{1,\alpha})_{xx},\sin^{2}xu_{xx}\rho)\lesssim|\alpha-1|(E+K)K+K\|[u_{\text{res},xx}(\psi-\sin
x\psi_{x}(0))]_{x}\sin x\|_{\rho}\,.$

We further obtain the following estimate

$\displaystyle\|[u_{\text{res},xx}(\psi-\sin x\psi_{x}(0))]_{x}\sin
x\|_{\rho}$
$\displaystyle\lesssim|\alpha-1|E+\|u_{\text{res},xx}(\psi_{x}-\cos
x\psi_{x}(0))\sin x\|_{\rho}$
$\displaystyle\lesssim|\alpha-1|E+|\alpha-1|\|(\psi_{x}-\psi_{x}(0))/x\|_{\infty}\,.$

Finally by the Cauchy-Schwarz inequality, we have

(4.4)
$|(\psi_{x}-\psi_{x}(0))/x|\leq\int_{0}^{x}|\omega(y)/y|dy\lesssim\|\omega/x\|_{2}\lesssim\|\omega\|_{\rho}\lesssim
E\,,$

and therefore we conclude

$((R_{1,\alpha})_{xx},\sin^{2}xu_{xx}\rho)\lesssim|\alpha-1|(E+K)K\,.$

By similar computations, we have

$((R_{2,\alpha})_{x},\sin^{2}x\omega_{x}\rho)\lesssim|\alpha-1|(E+K)K\,.$

And for the residue terms, we use similar estimates to obtain

$\|(F_{2,\alpha})_{x}\sin x\|_{\rho}\lesssim|\alpha-1|\,,$
$\|(F_{1,\alpha})_{xx}\sin
x\|_{\rho}\lesssim|\alpha-1|+|\alpha-1|\|(\frac{\psi_{\text{res}}}{\sin
x})_{x}\rho^{1/2}\sin x\|_{\infty}\;.$

We can use the triangular inequality to estimate

$|(\frac{\psi_{\text{res}}}{\sin x})_{x}\rho^{1/2}\sin
x|\leq|[\psi_{\text{res},x}(0)-\frac{\psi_{\text{res}}}{\sin
x}]/x|+|(\psi_{\text{res},x}-\psi_{\text{res},x}(0))/x|+|\frac{1-\cos
x}{x}\frac{\psi_{\text{res}}}{\sin x}|\,.$

Combined with the estimate similar to that in (4.3) and (4.4), we conclude

$\|(F_{1,\alpha})_{xx}\sin x\|_{\rho}\lesssim|\alpha-1|\,.$

We can finally obtain the same estimate as in Subsection 3.2

$\frac{d}{dt}K(t)\leq-(1-C|a-1|)K+CE+C|a-1|+CEK\,.$

We can again find an absolute constant $\mu_{\alpha}>1$ such that

$\frac{d}{dt}(K+\mu_{\alpha}E)\leq-(0.1-C|a-1|)(K+\mu_{\alpha}E)+C|a-1|+C(K+\mu_{\alpha}E)^{2}\,.$

And we can use the bootstrap argument on this higher-order energy
$K+\mu_{\alpha}E$ to conclude a priori estimate in this norm. Now we can
perform weighted estimates in time using the energy $J$ as in Subsection 3.3,
where the linear estimates follow from the linear estimates of $E$, the
nonlinear estimates follow from Subsection 3.3, and the error term vanishes
under time-differentiation. We can obtain the same estimates of $J$ and
establish exponential convergence of $J$ to zero. Then we use the argument in
[9, 6] to establish exponential convergence to the steady state and conclude
the proof of Theorem 1.4.

##  5\. Blowup of the viscous model with weak advection

In this section, we follow the strategy of the linear and nonlinear estimates
of the weak advection model and establish blowup for the weak advection
viscous model with $a<1$. Intuitively, the viscosity term is small in the
dynamic rescaling formulation and nonlinear stability can be closed using a
higher-order norm, where the viscosity term has a stability effect. Therefore
we expect that the viscous weak advection model develops a finite time
singularity as well.

###  5.1. Dynamic rescaling formulation

We recall the weak advection model with viscosity.

(5.1) $\displaystyle u_{t}+2a\psi u_{z}$ $\displaystyle=2u\psi_{z}+\nu
u_{zz}\,,$ $\displaystyle\omega_{t}+2a\psi\omega_{z}$
$\displaystyle=\left(u^{2}\right)_{z}+\nu\omega_{zz}\,,$
$\displaystyle-\psi_{zz}$ $\displaystyle=\omega\,.$

We use the rescaled variables

$\tilde{u}(x,\tau)=C_{u}(\tau)u(x,t(\tau))\,,\quad\tilde{\omega}(x,\tau)=C_{u}(\tau)\omega(x,t(\tau))\,,\quad\tilde{\psi}(x,\tau)=C_{u}(\tau)\psi(x,t(\tau))\,,$

where

$C_{u}(\tau)=C_{u}(0)\exp\left(\int_{0}^{\tau}c_{u}(s)ds\right)\,,\quad
t(\tau)=\int_{0}^{\tau}C_{u}(s)ds\,.$

For solutions to (5.1), the rescaled variables satisfy the dynamic rescaling
equation

(5.2) $\displaystyle\tilde{u}_{\tau}+2a\tilde{\psi}\tilde{u}_{x}$
$\displaystyle=2\tilde{u}\tilde{\psi}_{x}+c_{u}\tilde{u}+\nu
C_{u}(\tau)u_{xx}\,,$
$\displaystyle\tilde{\omega}_{\tau}+2a\tilde{\psi}\tilde{\omega}_{x}$
$\displaystyle=\left(\tilde{u}^{2}\right)_{x}+c_{u}\tilde{\omega}+\nu
C_{u}(\tau)\omega_{xx}\,,$ $\displaystyle-\tilde{\psi}_{xx}$
$\displaystyle=\tilde{\omega}\,.$

######  Remark 5.1.

Different from the rescaling in the inviscid case, we introduce an extra
degree of freedom: the constant $C_{u}(0)$. We will choose it later to ensure
that the viscous term has a relatively small scaling compared to the main
terms.

In order to establish a finite time blowup, it suffices to prove the dynamic
stability of (5.2) with scaling parameter $c_{u}<-\epsilon<0$ for all time;
see also [9]. As before, we will primarily work in the dynamic rescaling
formulation and use the notations $\tilde{u}=\bar{u}+u$, where $\bar{u}$ is an
approximation steady state and $u$ is the perturbation that we will control in
time. We still use $t$ to represent the rescaled time $\tau$.

We consider the following approximate steady state.

$\bar{\omega}=\bar{u}=\bar{\psi}=\sin
x\,,\quad\bar{c}_{u}(t)=2(a-1)\bar{\psi}_{x}(0)-\nu
C_{u}(t)\bar{u}_{xxx}(0)/\bar{u}_{x}(0)=2(a-1)+\nu C_{u}(t)\,.$

We consider odd perturbations $u$, $\omega$, $\psi$. The odd symmetry of the
solution is preserved in time by equation (5.2). We use the following
normalization condition: $c_{u}=2(a-1){\psi}_{x}(0)-\nu C_{u}(t)u_{xxx}(0)$.
This normalization ensures that $u_{x}(0)$ remains $0$ if the initial
perturbation satisfies $u_{x}(0,0)=0$. In fact, if $u_{x}(t,0)=0$, then we
obtain

$\displaystyle\frac{d}{dt}u_{x}(t,0)$
$\displaystyle=\frac{d}{dt}(u_{x}(t,0)+\bar{u}_{x}(t,0))=(2-2a)(u_{x}(t,0)+\bar{u}_{x}(t,0))(\psi_{x}(t,0)+\bar{\psi}_{x}(t,0))$
$\displaystyle+(c_{u}+\bar{c}_{u})(u_{x}(t,0)+\bar{u}_{x}(t,0))+\nu
C_{u}(t)(\bar{u}_{xxx}(0)+{u}_{xxx}(0))=0\,.$

This particular choice of the approximate steady state and the normalization
conditions ensure that $u_{x}(0,t)=0$ for all time provided that the initial
perturbation satisfies $u_{x}(0,0)=0$. We will perform the same weighted norm
estimate in the singular weight $\rho$ and the weighted norm $E$ as in the
inviscid case.

###  5.2. Estimates of the viscous terms

Now the perturbation satisfies

(5.3) $\displaystyle{u}_{t}$
$\displaystyle=L_{1}+(a-1)L^{\prime}_{1}+N_{1}+F_{1}+\nu C_{u}(t)V^{u}\,,$
$\displaystyle{\omega}_{t}$
$\displaystyle=L_{2}+(a-1)L^{\prime}_{2}+N_{2}+F_{2}+\nu
C_{u}(t)V^{\omega}\,,$ $\displaystyle-{\psi}_{xx}$ $\displaystyle={\omega}\,.$

Here the terms $V^{u}$ and $V^{\omega}$ correspond to all of the terms
containing the effect of the viscosity and we factor out explicitly the small
factor $\nu C_{u}(t)$ for a fixed $\nu$.

$V^{u}=u_{xx}+\bar{u}_{xx}+(1-u_{xxx}(0))(u+\bar{u})=u_{xx}-u_{xxx}(0)\sin
x+(1-u_{xxx}(0))u\,,$ $V^{\omega}=\omega_{xx}-\omega_{xxx}(0)\sin
x+(1-\omega_{xxx}(0))\omega\,.$

We invoke the nonlinear estimates in the inviscid case and obtain for the
viscous model:

(5.4)
$\frac{1}{2}\frac{d}{dt}E^{2}(t)\leq-(0.16-C|a-1|)E^{2}+C|a-1|E+CE^{3}+\nu
C_{u}(t)[((V^{u})_{x},u_{x}\rho)+(V^{\omega},\omega\rho)]\,.$

We estimate the viscous terms carefully since they involve singular weights.

$((V^{u})_{x},u_{x}\rho)=(u_{xxx}-u_{xxx}(0),u_{x}{\rho})+u_{xxx}(0)(1-\cos
x,u_{x}{\rho})+(1-u_{xxx}(0))\|u_{x}\|^{2}_{\rho}\,.$

Notice that

$|\rho_{x}|=\rho\left|\frac{-\sin x}{1-\cos
x}\right|\lesssim\rho|\frac{1}{x}|\,,\quad|\rho_{xx}|=\rho\left|\frac{-\cos
x}{1-\cos x}+\frac{2\sin^{2}x}{(1-\cos x)^{2}}\right|\lesssim\rho|x^{-2}|$

are singular near the origin and are smooth elsewhere. We can use integration
by parts twice to compute

$\displaystyle(u_{xxx}-u_{xxx}(0),u_{x}{\rho}$
$\displaystyle)=-(u_{xx}-xu_{xxx}(0),u_{xx}{\rho})-(u_{xx}-xu_{xxx}(0),\frac{x^{2}}{2}u_{xxx}(0){\rho_{x}})$
$\displaystyle-(u_{xx}-xu_{xxx}(0),(u_{x}-\frac{x^{2}}{2}u_{xxx}(0)){\rho_{x}})$
$\displaystyle\leq-\|u_{xx}\|^{2}_{\rho}+C[|u_{xxx}(0)|\|u_{xx}\|_{\rho}+|u_{xxx}(0)|^{2}+\|\frac{u_{x}}{x}-\frac{x}{2}u_{xxx}(0)\|_{\rho}^{2}]$
$\displaystyle\leq-\frac{1}{2}\|u_{xx}\|^{2}_{\rho}+C|u_{xxx}(0)|^{2}+C\|\frac{u_{x}}{x}\|_{\rho}^{2}\,,$

where for the last inequality we use the weighted AM-GM inequality
$ab\leq\epsilon a^{2}+\frac{1}{4\epsilon}b^{2}$ for a very small constant
$\epsilon$. Therefore we get

(5.5)
$((V^{u})_{x},u_{x}\rho)\leq-\frac{1}{2}\|u_{xx}\|^{2}_{\rho}+C[|u_{xxx}(0)|^{2}+\|\frac{u_{x}}{x}\|_{\rho}^{2}+(1+|u_{xxx}(0)|)E^{2}]\,.$

Similarly, we estimate via integration by parts

$\displaystyle(\omega_{xx},\omega{\rho})$
$\displaystyle=-(\omega_{x}-\omega_{x}(0),(\omega_{x}-\omega_{x}(0)){\rho})-(\omega_{x}-\omega_{x}(0),\omega_{x}(0){\rho})$
$\displaystyle-(\omega_{x}-\omega_{x}(0),x\omega_{x}(0){\rho_{x}})-(\omega_{x}-\omega_{x}(0),(\omega-x\omega_{x}(0)){\rho_{x}})$
$\displaystyle\leq-\|\omega_{x}-\omega_{x}(0)\|^{2}_{\rho}+C[|\omega_{x}(0)|\|\frac{\omega_{x}-\omega_{x}(0)}{x}\|_{\rho}+\|\frac{\omega}{x}-\omega_{x}(0)\|_{\rho}^{2}]\,.$

And we obtain

(5.6) $\displaystyle(V^{\omega},\omega\rho)$
$\displaystyle\leq-\|\omega_{x}-\omega_{x}(0)\|^{2}_{\rho}+C[|\omega_{x}(0)|\|\frac{\omega_{x}-\omega_{x}(0)}{x}\|_{\rho}$
$\displaystyle+\|\frac{\omega}{x}-\omega_{x}(0)\|_{\rho}^{2}+|\omega_{xxx}(0)|^{2}+(1+|\omega_{xxx}(0)|)E^{2}]\,.$

The essential difficulty for the viscous terms is that after integration by
parts, the singular weight produces various positive terms, on top of the
damping terms $-\|u_{xx}\|_{\rho}$; see (5.5), (5.6). Fortunately, the
positive terms contribute only to higher-order terms near the origin.

Consider the interval $I=[-\pi/2,\pi/2]$. $\rho$ and $|1/x|$ are upper bounded
by a positive constant outside of the interval and we have

$\|\frac{u_{x}}{x}\|_{\rho}^{2}\lesssim\|{u_{x}}\|_{\rho}^{2}+\|\frac{u_{x}}{x^{2}}\|_{L^{2}(I)}^{2}\lesssim
E^{2}+\|u_{xxx}\|_{L^{\infty}(I)}^{2}\,,$
$\|\frac{\omega}{x}-\omega_{x}(0)\|_{\rho}^{2}\lesssim\|\omega-x\omega_{x}(0)\|_{\rho}^{2}+\|\frac{\omega}{x^{2}}-\omega_{x}(0)/x\|_{L^{2}(I)}^{2}\lesssim
E^{2}+|\omega_{x}(0)|^{2}+\|\omega_{xx}\|_{L^{\infty}(I)}^{2}\,,$
$\|\frac{\omega_{x}-\omega_{x}(0)}{x}\|_{\rho}\lesssim\|{\omega_{x}-\omega_{x}(0)}\|_{\rho}+\|\frac{\omega_{x}-\omega_{x}(0)}{x^{2}}\|_{L^{2}(I)}\lesssim\|{\omega_{x}-\omega_{x}(0)}\|_{\rho}+\|\omega_{xxx}\|_{L^{\infty}(I)}\,.$

Plugging these estimates into the (5.5), (5.6) and using again the weighted
AM-GM inequality, we can yield

(5.7)
$(V^{\omega},\omega\rho)+((V^{u})_{x},u\rho)\leq-\frac{1}{2}(\|\omega_{x}-\omega_{x}(0)\|^{2}_{\rho}+\|u_{xx}\|^{2}_{\rho})+C[E_{V}^{2}+(1+E_{V})E^{2}]\,,$

where

$E_{V}=\|\omega_{xxx}\|_{L^{\infty}(I)}+\|u_{xxx}\|_{L^{\infty}(I)}+|\omega_{x}(0)|\,.$

###  5.3. Estimates in a higher-order norm

We will use a weighted higher-order norm to close the estimates. To have a
good estimate in this higher-order norm, it needs to satisfy three criteria.
First, we need to extract damping in the leading order linear term. Secondly,
we need to bound the terms like $\omega_{xxx}(0)$ using interpolation between
the lower and the higher-order norms via the Gagliardo-Nirenberg inequality;
therefore it needs to be at least as strong as a regular higher-order norm
near the origin. Thirdly, we need damping for the diffusion terms to close the
estimates. This motivates us to choose a combination of the $k-$th order
weighted norms for $k\geq 1$:

$E_{k}^{2}(t)=(u^{(k+1)},u^{(k+1)}\rho_{k})+(\omega^{(k)},\omega^{(k)}\rho_{k})\,,\quad\rho_{k}=(1+\cos
x)^{k}\;,$

where we use the notation that $f^{(k)}=\partial_{x}^{k}f$. We denote
$E_{0}=E$ and $\rho_{0}=\rho$.

######  Remark 5.2.

This weighted norm immediately satisfies criterion 2 and we will verify in the
linear estimates that it satisfies criterion 1. Finally, a clever combination
of the weighted norms can produce damping for the viscous terms and we make
the damping terms in the estimates of the $(k-1)$-th order norms greater than
the positive terms in the estimates of the $k$-th order norms. We will
elaborate on those points and establish the nonlinear estimates.

Now we can estimate $\frac{d}{dt}E_{k}(t)$ for $k>0$ as follows

$\displaystyle\frac{1}{2}\frac{d}{dt}E_{k}^{2}(t)=(L_{2}^{(k)}+(a-1)(L_{2}^{\prime})^{(k)}+N_{2}^{(k)}+F_{2}^{(k)}+\nu
C_{u}(t)(V^{\omega})^{(k)},\omega^{(k)}\rho_{k})$
$\displaystyle+(L_{1}^{(k+1)}+(a-1)(L_{1}^{\prime})^{(k+1)}+N_{1}^{(k+1)}+F_{1}^{(k+1)}+\nu
C_{u}(t)(V^{u})^{(k+1)},u^{(k+1)}\rho_{k})\,,$

where the parts $L_{i},L_{i}^{\prime},F_{i},N_{i}$ are defined exactly the
same as in the inviscid case.

We first look at the viscous terms. We have for example

$((V^{u})^{(k+1)},u^{(k+1)}(1+\cos x)^{k})\leq(u^{(k+3)},u^{(k+1)}(1+\cos
x)^{k})+CE_{V}E_{k}+(1+E_{V})E_{k}^{2}\,.$

We use integration by parts twice to obtain

$\displaystyle(u^{(k+3)},u^{(k+1)}(1+\cos x)^{k})=-(u^{(k+2)},u^{(k+2)}(1+\cos
x)^{k}-u^{(k+1)}k(1+\cos x)^{k-1}\sin x)$
$\displaystyle=-(u^{(k+2)},u^{(k+2)}(1+\cos
x)^{k})+\frac{1}{2}(u^{(k+1)},u^{(k+1)}k(1+\cos x)^{k-1}[(k-1)-k\cos x])$
$\displaystyle\leq-(u^{(k+2)},u^{(k+2)}(1+\cos
x)^{k})+C(k)(u^{(k+1)},u^{(k+1)}(1+\cos x)^{k-1})\,.$

We can also get a similar bound for $V^{\omega}$. Therefore combined with the
leading order estimate (5.7) and using the idea in Remark 5.2, we conclude
that for small enough constants $0<\mu<\mu_{0}({k_{0}})<1$, we have the
following viscous estimate

(5.8)
$\sum_{k=0}^{k_{0}}\mu^{k}[((V^{u})^{(k+1)},u^{(k+1)}\rho_{k})+((V^{\omega})^{(k)},\omega^{(k)}\rho_{k})]\leq
C(k)[E_{V}^{2}+(1+E_{V})\sum_{k=0}^{k_{0}}\mu^{k}E_{k}^{2}]\,.$

Here $\mu_{0}(k_{0})$ is a generic constant depending on $k_{0}$. We can
choose $k_{0}$ large enough later so that $E_{V}$ can be bounded using the
interpolation inequalities.

Now we look at the linear terms and extract damping. We denote the terms as
lower order terms (l.o.t. for short) if their $\rho_{k}$-weighted
$L^{2}$-norms are bounded by $\sum_{i=0}^{k-1}E_{i}$. For the terms of
intermediate order, since $\rho_{k}\leq C(k)\rho_{i}$ for $i<k$, combined with
the classical elliptic estimate, we can show that $u^{j}$, $\psi^{i}$ for
$0\leq j<k+1$ and $0\leq i<k+2$ are l.o.t. Using the l.o.t. notation, we keep
track only of the higher-order terms

$(L_{1}^{(k+1)},u^{(k+1)}\rho_{k})=(-2\sin xu^{(k+2)}-2k\cos xu^{(k+1)}+2\sin
x\psi^{(k+2)}+\text{l.o.t.},u^{(k+1)}\rho_{k})\,,$
$(L_{2}^{(k)},\omega^{(k)}\rho_{k})=(-2\sin x\omega^{(k+1)}-2k\cos
x\omega^{(k)}+2\sin xu^{(k+1)}+\text{l.o.t.},\omega^{(k)}\rho_{k})\,.$

Again we have a crucial cancellation of the cross terms and for the leading
order terms we use integration by parts to obtain for example

$\displaystyle(-2\sin xu^{(k+2)}-2k\cos xu^{(k+1)},u^{(k+1)}\rho_{k})$
$\displaystyle=(u^{(k+1)},u^{(k+1)}(-k-(k-1)\cos x)\rho_{k})$
$\displaystyle\leq-(u^{(k+1)},u^{(k+1)}\rho_{k})\,.$

Therefore we derive the following estimate

$(L_{1}^{(k+1)},u^{(k+1)}\rho_{k})+(L_{2}^{(k)},\omega^{(k)}\rho_{k})\leq-
E_{k}^{2}+C(k)\sum_{i=0}^{k-1}E_{i}E_{k}\leq-\frac{1}{2}E_{k}^{2}+C(k)\sum_{i=0}^{k-1}E_{i}^{2}\,.$

Similarly we have

$(L_{1}^{\prime(k+1)},u^{(k+1)}\rho_{k})+(L_{2}^{\prime(k)},\omega^{(k)}\rho_{k})\leq
2E_{k}^{2}+C(k)\sum_{i=0}^{k-1}E_{i}^{2}\,.$

We have the trivial bound for the error term

$(F_{1}^{(k+1)},u^{(k+1)}\rho_{k})+(F_{2}^{(k)},\omega^{(k)}\rho_{k})\leq
C(k)(a-1)E_{k}\,.$

The nonlinear terms are more subtle. We will show that

(5.9) $(N_{1}^{(k+1)},u^{(k+1)}\rho_{k})\leq
C(k)\sum_{i=0}^{k}E_{i}^{2}E_{k}\,,$

and we can have the same bound for $\omega$. In fact, for a canonical term in
$N_{1}^{(k+1)}$ and $N_{2}^{(k)}$, it is of the form $\psi^{(i)}u^{(k+2-i)}$
or $\psi^{(i)}\psi^{(k+3-i)}$ or $u^{(i)}u^{(k+1-i)}.$ For the terms $\psi
u^{(k+2)}$ and $\psi\omega^{(k+1)}$, we can use integration by parts and Lemma
3.1 to show that

$(\psi u^{(k+2)},u^{(k+1)}(1+\cos x)^{k})\leq
C(k)(\|\psi_{x}\|_{\infty}+\|\frac{\psi}{\sin x}\|_{\infty})E_{k}^{2}\leq
C(k)EE_{k}^{2}\,.$

The terms associated with $\psi_{x}u^{(k+1)}$, $\psi_{x}\omega^{(k)}$,
$uu^{(k+1)}$, and $u\omega^{(k)}$ have the same bound trivially. We can then
focus on controlling the weighted norms of $\omega^{(i)}u^{(k-i)}$,
$\omega^{(i)}\omega^{(k-1-i)}$, $u^{(i+1)}u^{(k-i)}$ for indices $0<i<k$ to
establish the bound (5.9). For example, we get

$\|\omega^{(i)}u^{(k-i)}(1+\cos x)^{k/2}\|_{2}\leq\|\omega^{(i)}(1+\cos
x)^{(i+1)/2}\|_{\infty}E_{k-1-i}\,.$

Finally, by the fundamental theorem of calculus, we can bound the
$L^{\infty}$-norm by

$\displaystyle C(K)[\|\omega^{(i+1)}(1+\cos
x)^{(i+1)/2}\|_{1}+\|\omega^{(i)}(1+\cos x)^{(i-1)/2}\sin x\|_{1}]$
$\displaystyle\leq C(K)[E_{i+1}+\|\omega^{(i)}(1+\cos x)^{(i)/2}\|_{1}]\leq
C(k)\sum_{i=0}^{k}E_{i}\,.$

Therefore we conclude that (5.9) holds.

###  5.4. Collection of norms and finite time blowup

We collect the bounds (5.8) (5.9) for viscous and nonlinear terms, along with
the linear bounds and the leading order estimate (5.4). For any fixed
${k_{0}}$, there exists a small enough constant
$0<\mu_{1}({k_{0}})<\mu_{0}({k_{0}})$, such that the following estimate holds

$\frac{d}{dt}I_{k_{0}}^{2}\leq-(0.1-C|a-1|)I_{k_{0}}^{2}+C|a-1|I_{k_{0}}+CI_{k_{0}}^{3}+C\nu
C_{u}(t)[E_{V}^{2}+(1+E_{V})I_{k_{0}}^{2}]\,,$

where the energy is defined as

$I_{k_{0}}^{2}=\sum_{k=0}^{k_{0}}\mu_{1}^{k}({k_{0}})E_{k}^{2}\,.$

Here the constants depend on $k_{0}$ and $\mu$ but once we first prescribe
$k_{0}$ then $\mu=\mu_{1}({k_{0}})$, they become just constants. We will later
make our $C_{u}(t)$ and $|a-1|$ small to close the argument.

Finally, by the Gagliardo-Nirenberg inequality, for $k=1,3$, we have

$\|\omega^{(k)}\|_{L^{\infty}(I)}\lesssim\|\omega^{(4)}\|^{\theta}_{L^{2}(I)}\|\omega\|^{1-\theta}_{L^{2}(I)}\,,\quad\theta=\frac{k+1/2}{4}\,.$

This is the classical Gagliardo-Nirenberg inequality applied to a bounded
domain, and we can just use the extension technique to prove it; see for
example [29]. We get similar bounds involving $u$ and conclude that
$E_{V}\lesssim I_{k_{0}}$, for any fixed $k_{0}\geq 4$. For example, we just
take $k_{0}=4$ and obtain

$\frac{d}{dt}I_{4}\leq-(0.1-C|a-1|)I_{4}+C|a-1|+CI_{4}^{2}+C\nu
C_{u}(t)(1+I_{4})I_{4}\,.$

Now we choose $C_{u}(0)=|a-1|^{2}$ for $|a-1|<\delta$ with a small enough
$\delta>0$. It is easy to check that the bootstrap argument for $I_{4}\leq
C|a-1|$ and $C_{u}(t)\leq C_{u}(0)\exp((a-1)t)\leq C_{u}(0)$ will hold for all
time provided that it holds initially. We again use the estimate for the
normalization constants

$c_{u}+\bar{c}_{u}=2(a-1)+\nu
C_{u}(t)(1-u_{xxx}(0))+2(a-1)\psi_{x}(0)<(a-1)<0\,.$

Thus we can obtain a blowup in finite time in the physical variables.

##  6\. Appendix

######  Lemma 6.1.

Assume $\sum_{k\geq 1}a_{k}^{2}+c_{k}^{2}<\infty$, then we have the following
inequality for (2.5)

$\displaystyle F(a,c)$ $\displaystyle\coloneqq\sum_{k\geq
1}\\{a_{k}^{2}(0.84+\frac{1}{k^{2}}-\frac{1}{(k-1)^{2}})+c_{k}^{2}(0.84+\frac{1}{k(k+1)})+2a_{k}a_{k+1}\frac{1}{(k+1)^{2}}$
$\displaystyle+2a_{k}\sum_{j>k+1}a_{j}(\frac{1}{j^{2}}-\frac{1}{(j-1)^{2}})+2a_{k}c_{k}\frac{1+2k-k^{2}}{2k^{2}(k+1)}+2a_{k+1}c_{k}\frac{k^{2}-k-1}{2k^{2}(k+1)^{2}}$
$\displaystyle-2a_{k+2}c_{k}\frac{k+2}{2(k+1)^{2}}+\sum_{j>k}2a_{k}c_{j}\frac{1}{j(j+1)}\\}\geq
0\,.$

###### Proof.

Denote the summation of terms in $F(a,c)$ that only involve $a_{i}$, $c_{j}$
for $i,j\leq N$ as $F_{N}(a,c)$. Here $N=200$. This quadratic form
$F_{N}(a,c)$ can be expressed as $a^{(N),T}F^{(N)}c^{(N)}$, where
$a^{(N)},c^{(N)}$ are two vectors with entries $a_{i}$,$c_{j}$ respectively
and $F^{(N)}$ is a symmetric matrix. We numerically verify using interval
arithmetic in Matlab that the smallest eigenvalue of $F^{(N)}$ is greater than
$0.01$; see remarks after the proof for details. Therefore we have

$F_{N}(a,c)\geq 0.01\sum_{k=1}^{N}(a_{k}^{2}+c_{k}^{2})\,.$

For the remainder $F(a,c)-F_{N}(a,c)$, we estimate it term by term via the
trivial bound $2ab\geq-(a^{2}+b^{2})$ and obtain

$\displaystyle
F(a,c)-F_{N}(a,c)\geq\sum_{k>N}[a_{k}^{2}(0.84+\frac{1}{k^{2}}-\frac{1}{(k-1)^{2}})+c_{k}^{2}(0.84+\frac{1}{k(k+1)})]$
$\displaystyle+\sum_{k=1}^{N}a_{k}^{2}(-\frac{2}{N^{2}}-\frac{1}{N+1})+\sum_{k=N-1}^{N}c_{k}^{2}(-\frac{N^{2}-N-1}{2N^{2}(N+1)^{2}}-\frac{N+1}{2N^{2}})$
$\displaystyle+\sum_{k>N}a_{k}^{2}(-\frac{3}{N^{2}}-N(\frac{1}{(N)^{2}}-\frac{1}{(N+1)^{2}})-\frac{N^{2}-N-1}{2N^{2}(N+1)^{2}}-\frac{N+1}{2N^{2}}-\frac{N^{2}-2N-1}{2N^{2}(N+1)}$
$\displaystyle-\frac{1}{N+2})+\sum_{k>N}c_{k}^{2}(-\frac{N^{2}-N-1}{2N^{2}(N+1)^{2}}-\frac{N+1}{2N^{2}}-\frac{N^{2}-2N-1}{2N^{2}(N+1)}-\frac{1}{N+2})\,.$

For $N=200$, we estimate all of the coefficients by a lower bound and obtain

$F(a,c)-F_{N}(a,c)\geq-\frac{2}{N}\sum_{k=1}^{N}(a_{k}^{2}+c_{k}^{2})+(0.84-\frac{3}{N})\sum_{k>N}(a_{k}^{2}+c_{k}^{2})\geq-\frac{2}{N}\sum_{k=1}^{N}(a_{k}^{2}+c_{k}^{2})\,.$

Therefore we conclude $F(a,c)\geq 0$. ∎

######  Remark 6.2.

We now explain how to verify that the smallest eigenvalue of the symmetric
matrix $F^{(200)}$ is greater than $0.01$. We proceed in three steps.

  1. (1)

We first use Matlab to perform an (approximate) SVD decomposition of

$F^{(200)}-0.011I\approx VDV^{\prime}\,.$

Here $D$ is the diagonal matrix consisting of (approximate) eigenvalues of
$F^{(200)}-0.011I$, and $V$ is the unitary matrix consisting of (approximate)
eigenvectors of $F^{(200)}-0.011I$.

  2. (2)

We use interval arithmetic to verify that the maximal absolute value of
entries of $F^{(200)}-0.011I-VDV^{\prime}$ is at most $10^{-10}$. Therefore
the spectral norm of $F^{(200)}-0.011I-VDV^{\prime}$, which is bounded by its
$1$-norm, is rigorously bounded from above by $200\times 10^{-10}$.

  3. (3)

Since $D$ has positive entries, we know that $VDV^{\prime}$ is positive
definite. We conclude that

$F^{(200)}-0.01I=VDV^{\prime}+0.001I+F^{(200)}-0.011I-VDV^{\prime}$

is positive definite.

Acknowledgments. The research was in part supported by NSF Grant DMS-2205590
and the Choi Family Gift Fund and the Graduate Fellowship Fund. We would like
to thank Dr. Jiajie Chen for some stimulating discussions, especially on the
design of the energy norm. We also thank Changhe Yang for some valuable
discussions in the early stage of this project.

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