# The singularities for a periodic transport equation

Yong Zhang, Fei Xu, Fengquan Li  School of Mathematical Sciences, Dalian
University of Technology, Dalian, 116024, China<EMAIL_ADDRESS>School of
Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China
School of Mathematical Sciences, Dalian University of Technology, Dalian,
116024, China

###### Abstract.

In this paper, we consider a 1D periodic transport equation with nonlocal flux
and fractional dissipation

$u_{t}-(Hu)_{x}u_{x}+\kappa\Lambda^{\alpha}u=0,\quad(t,x)\in R^{+}\times S,$

where $\kappa\geq 0$, $0<\alpha\leq 1$ and $S=[-\pi,\pi]$. We first establish
the local-in-time well-posedness for this transport equation in $H^{3}(S)$. In
the case of $\kappa=0$, we deduce that the solution, starting from the smooth
and odd initial data, will develop into singularity in finite time. If adding
a weak dissipation term $\kappa\Lambda^{\alpha}u$, we also prove that the
finite time blowup would occur.

Key Words: singularity; nonlocal flux; fractional dissipation; odd initial
data

##  1\. Introduction

We will discuss the following equations

$\left\\{\begin{array}[]{ll}u_{t}-(H^{a}\partial_{x}^{b}u)u_{x}+\kappa\Lambda^{\alpha}u=0,~{}~{}~{}(t,x)\in
R^{+}\times S,\\\ u(0,x)=\varphi(x),\end{array}\right.$ (1.1)

where $a,b\in\\{0,1\\}$, $Hu(x)$ is the Hilbert transform of $u(x)$ defined by
(see [13])

$Hu(x)=\frac{1}{2\pi}P.V.\int^{\pi}_{-\pi}\frac{u(x-y)}{tan\frac{y}{2}}dy$

and $H^{0}$ denotes the identity operator.
$\Lambda^{\alpha}u=(-\partial_{xx})^{\frac{\alpha}{2}}u$ is defined by the
Fourier transform

$\widehat{\Lambda^{\alpha}u}(\xi)=|\xi|^{\alpha}\hat{u}(\xi).$

Here $\kappa\geq 0$ is viscosity coefficient which controls the strength of
dissipation and $0<\alpha\leq 2$ controls the magnitude of the dissipation.

If $a=b=0$, (1.1) is transformed into the classical Brugers equation with
fractional dissipation. This equation has been investigated in [1, 2, 3]. The
authors in [1, 2] proved the occurrence of shock-type singularities in
supercritical case with responding to $0<\alpha<1$ and the global existence in
critical case corresponding with $\alpha=1$ and subcritical case corresponding
with $1<\alpha\leq 2$. In [3], the authors studied comprehensively the
existence, uniqueness, blow up and regularity properties of solutions.

If $a=1,b=0$, (1.1) can be written a famous nonlocal transport equation, which
was firstly considered by A.Córdoba et. in [4], where the authors concluded
that the solutions would develop into gradient blow-up as $\kappa=0$ and exist
globally as $\kappa>0,\alpha>1$. References [5, 6] are recommended for some
recent results on this transport equation.

If $a=0,b=1$, (1.1) can be reduced to

$\left\\{\begin{array}[]{ll}u_{t}-u^{2}_{x}+\kappa\Lambda^{\alpha}u=0,~{}~{}~{}~{}(t,x)\in
R^{+}\times S,\\\ u(0,x)=\varphi(x),\end{array}\right.$ (1.2)

which can be regarded as a fractional order heat equation with a special
nonlinearity term $u^{2}_{x}$. On the other hand, it also can be viewed as
another form of Burgers equation. For example, taking derivative to (1.2) with
respect to $x$ and letting $v=u_{x}$, then we obtain that
$v_{t}-2vv_{x}+\kappa\Lambda^{\alpha}v=0$. Moreover, we can deduce

$\frac{d}{dt}\|u(t)\|_{H^{1}[\pi,\pi]}\leq 0,$

which implies that
$\|u(t)\|_{L^{\infty}[-\pi,\pi]}\leq\|u(t)\|_{H^{1}[-\pi,\pi]}\leq\|\varphi\|_{H^{1}[-\pi,\pi]}$.
Hence the solution to (1.2) would not blow up.

In this paper, we mainly consider the case of $a=b=1$, that is to say

$\left\\{\begin{array}[]{ll}u_{t}-(Hu)_{x}u_{x}+\kappa\Lambda^{\alpha}u=0,~{}~{}~{}~{}(t,x)\in
R^{+}\times S,\\\ u(0,x)=\varphi(x),\end{array}\right.$ (1.3)

which is a simplified equation of a class of ill-posed problems arising in the
theory of vortex sheets (see [12, 14]). As before, if taking derivative to
(1.3) with respect to $x$ and letting $\theta=u_{x}$, we would get

$\left\\{\begin{array}[]{ll}\theta_{t}-(\theta
H\theta)_{x}+\kappa\Lambda^{\alpha}\theta=0,~{}~{}~{}~{}(t,x)\in R^{+}\times
S,\\\ \theta(0,x)=f(x),\end{array}\right.$

where $f(x)=\varphi_{x}(x)$. This is a 1D model of the quasi-geostrophic
equation (see [7, 8, 9, 10, 11]). In [9], the authors showed that there is no
$C^{1}([-\pi,\pi]\times[0,+\infty))$ solutions in the case of $\kappa=0$ or
$\kappa>0,\alpha=1$. In fact, our results in this paper are consistent with
the conclusion in [9]. Besides, we also make up the gap of
$\kappa>0,0<\alpha<1$ by using a different method. For the non-periodic case,
the global existence for strictly positive initial data $f(x)>0$ has been
proved in [10]. Other important results on this equation can be seen in [11].

The paper is organized as follows. In section 2, we first give the local well-
posedness result of equation (1.3), and then state the main theorems on blow-
up. In section 3, we try to reduce the equation (1.3) to a family of ordinary
differential equations under a class of given initial data. Section 4 is
devoted to the proof of the main results on blow-up.

##  2\. Local well-posedness and main results

Before stating to state the main results, we first discuss the local well-
posedness result of the problem (1.3). Combining a priori bound with the
compactness argument, one may be able to establish local well-posedness for
(1.3) in $H^{3}[-\pi,\pi]$. The details of proof are the same as that of [15].
Below we merely show how to obtain a priori bound for
$\|u\|_{H^{3}[-\pi,\pi]}$, and other parts of the proof are omitted.

######  Theorem 2.1.

If $\varphi\in H^{3}[-\pi,\pi]$, then there exists a $T>0$ such that the
problem (1.3) has a unique solution $u\in C([0,T),H^{3}[-\pi,\pi])\cap
C^{1}([0,T),H^{2}[-\pi,\pi])$.

###### Proof.

Taking three derivatives to equation (1.3) and multiplying by $u_{xxx}$ over
$[\pi,\pi]$, we have that

$\displaystyle\frac{1}{2}\frac{d}{dt}\|u_{xxx}\|_{L^{2}[\pi,\pi]}^{2}$
$\displaystyle=\int^{\pi}_{-\pi}[(Hu)_{x}u_{x}]_{xxx}u_{xxx}dx-\kappa\|\Lambda^{\frac{\alpha}{2}}u_{xxx}\|_{L^{2}[\pi,\pi]}^{2}$
$\displaystyle\leq\int^{\pi}_{-\pi}[(\Lambda u_{x})u_{x}+(\Lambda
u)u_{xx})]_{xx}u_{xxx}dx$ $\displaystyle\leq(\|\Lambda
u_{x}\|_{L^{\infty}[-\pi,\pi]}+\|u_{xx}\|_{L^{\infty}[-\pi,\pi]})\int^{\pi}_{-\pi}u^{2}_{xxx}+(\Lambda
u_{xx})u_{xxx}dx.$ (2.1)

Thanks to the domain is limited to $[-\pi,\pi]$, the Sobolev embedding theorem
and Plancherel formula can be used to show that

$\frac{d}{dt}\|u\|_{H^{3}[\pi,\pi]}^{2}\leq c\|u\|_{H^{3}[\pi,\pi]}^{3},$
(2.2)

where $c$ is a positive constant. Furthermore, a priori bound

$\|u\|_{H^{3}[\pi,\pi]}\leq\frac{\|\varphi\|_{H^{3}[\pi,\pi]}}{1-c\|\varphi\|_{H^{3}[\pi,\pi]}t}$
(2.3)

follows. ∎

We are now in a position to state the blow-up results. It’s known that the
smooth solution $u(t,x)$ of (1.3) can be expanded by

$u(t,x)=\sum_{n=-\infty}^{\infty}u_{n}(t)e^{inx},~{}~{}u_{n}(t)\in\mathbb{C}.$
(2.4)

Here we assume that the initial data $\varphi\in H^{3}[-\pi,\pi]$ is odd with
the form of

$\varphi(x)=\sum_{n=1}^{\infty}A_{n}sin(nx),$ (2.5)

where $A_{n}$ are constants. In the case $\kappa\geq 0$ and $0<\alpha\leq 1$,
we establish that

######  Theorem 2.2.

(singularity for $\kappa=0$) Assume there exists a positive number $\delta$
such that $A_{n}\geq\frac{2\delta}{n^{5}}$, then the solution $u(t,x)$ to
equation (1.3) with $\kappa=0$ would blow up at some finite time $T_{1}$, i.e.

$\|u(t,\cdot)\|_{L^{2}(-\pi,\pi]}\rightarrow\infty,~{}~{}~{}as~{}~{}~{}t\rightarrow
T_{1}.$

Moreover, there is a rough estimate

$T_{1}=(\frac{1}{2\delta})^{+}.$

######  Theorem 2.3.

(singularity for $\kappa>0,~{}\alpha=1$) Assume there exists a positive number
$\delta$ such that $A_{n}\geq\frac{2\delta}{n^{5}}$ and
$0<\kappa<\frac{2\delta}{2e-1}$, then the solution $u(t,x)$ to equation (1.3)
would blow up at $T_{2}=\frac{1}{\kappa}$, i.e.

$\|u(t,\cdot)\|_{L^{2}(-\pi,\pi]}\rightarrow\infty,~{}~{}~{}as~{}~{}~{}t\rightarrow
T_{2}=\frac{1}{\kappa}.$

######  Theorem 2.4.

(singularity for $\kappa>0,~{}0<\alpha<1$) Assume there exists a positive
number $\delta$ such that $A_{n}\geq\frac{2\delta}{n^{5}}$ and
$0<\kappa<\frac{2\delta}{2e-1}$, then the solution $u(t,x)$ to equation (1.3)
would blow up at $T_{2}=\frac{1}{\kappa}$, i.e.

$\|u(t,\cdot)\|_{L^{2}(-\pi,\pi]}\rightarrow\infty,~{}~{}~{}as~{}~{}~{}t\rightarrow
T_{2}=\frac{1}{\kappa}.$

######  Remark 2.5.

In Theorem 2.2–2.4, the assumption on initial data
$A_{n}\geq\frac{2\delta}{n^{5}}$ is required to be compatible with the
convergence of series (2.5) in $H^{3}[-\pi,\pi]$. (For instance, we can take
$A_{n}=\frac{2\delta+1}{n^{5}}$.) In addition, the exponential 5 at
denominator in $\frac{2\delta}{n^{5}}$ can be replaced by any larger real
number. Essentially, we need to restrict the Fourier coefficients $A_{n}$ of
initial data to be positive due to the technique taken here .

##  3\. The reduction of the equation (1.3)

Thanks to the reasonable assumptions of the initial data, the equation (1.3)
can be reduced to a family of ordinary differential equations (ODEs) in this
section. Moveover, we can write easily down its implicit solutions.

Firstly, let’s reduce the equation in (1.3). The Hilbert transform of periodic
smooth solution $u(t,x)$ in (2.4) can be expressed by

$Hu=-\sum_{n=-\infty}^{\infty}isgn(n)u_{n}(t)e^{inx}=-i\sum_{n=1}^{\infty}u_{n}(t)e^{inx}+i\sum_{n=1}^{\infty}u_{-n}(t)e^{-inx},~{}u_{n}(t)\in\mathbb{C}.$
(3.1)

Then (2.4) and (3.1) give

$\displaystyle(Hu)_{x}u_{x}$
$\displaystyle=(\sum_{n=1}^{\infty}nu_{n}(t)e^{inx}+\sum_{n=1}^{\infty}nu_{-n}(t)e^{-inx})(i\sum_{n=1}^{\infty}nu_{n}(t)e^{inx}-i\sum_{n=1}^{\infty}nu_{-n}(t)e^{-inx})$
$\displaystyle=i(\sum_{n=1}^{\infty}nu_{n}(t)e^{inx})^{2}-i(\sum_{n=1}^{\infty}nu_{-n}(t)e^{-inx})^{2}.$
(3.2)

Let

$a(t,x):=(\sum_{n=1}^{\infty}nu_{n}(t)e^{inx})^{2}=\sum_{n=1}^{\infty}a_{n}(t)e^{inx},$
(3.3)

where the coefficients $a_{n}(t)$ can be defined by

$\displaystyle a_{n}(t)$
$\displaystyle=\frac{1}{2\pi}\int^{\pi}_{-\pi}a(t,x)e^{-inx}dx$
$\displaystyle=\frac{1}{2\pi}\int^{\pi}_{-\pi}(\sum_{k=1}^{\infty}\sum_{l=1}^{\infty}lu_{l}e^{ilx}ku_{k}e^{ikx})e^{-inx}dx$
$\displaystyle=\frac{1}{2\pi}\int^{\pi}_{-\pi}\sum_{k=1}^{\infty}\sum_{l=1}^{\infty}lu_{l}ku_{k}e^{i(l+k-n)x}dx$
$\displaystyle=\sum_{k=1}^{\infty}\sum_{l=1}^{\infty}lu_{l}ku_{k}\delta(k+l-n)$
$\displaystyle=\sum_{l=1}^{n-1}lu_{l}(n-l)u_{n-l}.$ (3.4)

It follows from (3.3) and (3) that

$a(t,x)=\sum_{n=1}^{\infty}[\sum_{l=1}^{n-1}lu_{l}(n-l)u_{n-l}]e^{inx}.$ (3.5)

A similar argument shows that

$b(t,x):=(\sum_{n=1}^{\infty}nu_{-n}(t)e^{-inx})^{2}=\sum_{n=1}^{\infty}[\sum_{l=1}^{n-1}lu_{-l}(n-l)u_{-n+l}]e^{-inx}.$
(3.6)

Therefore, we can obtain

$\displaystyle(Hu)_{x}u_{x}$
$\displaystyle=i(\sum_{n=1}^{\infty}nu_{n}(t)e^{inx})^{2}-i(\sum_{n=1}^{\infty}nu_{-n}(t)e^{-inx})^{2}$
$\displaystyle=\sum_{n=1}^{\infty}[i\sum_{l=1}^{n-1}lu_{l}(n-l)u_{n-l}]e^{inx}-\sum_{n=1}^{\infty}[i\sum_{l=1}^{n-1}lu_{-l}(n-l)u_{-n+l}]e^{-inx}.$
(3.7)

Besides, the diffusion term $\Lambda^{\alpha}u$ also can be expressed by

$\Lambda^{\alpha}u(t,x)=\sum_{n=1}^{\infty}n^{\alpha}u_{n}(t)e^{inx}+\sum_{n=1}^{\infty}n^{\alpha}u_{-n}(t)e^{-inx}.$
(3.8)

Taking (2.4), (3) and (3.8) into (1.3) and equating coefficients of $e^{inx}$
and $e^{-inx}$ respectly, we can obtain the following ODEs for
$\\{u_{n}(t)\\}_{n\geq 1}$, $\\{u_{-n}(t)\\}_{n\geq 1}$ and $u_{0}(t)$

$\left\\{\begin{array}[]{ll}\frac{du_{n}(t)}{dt}-i\sum_{l=1}^{n-1}lu_{l}(t)(n-l)u_{n-l}(t)+\kappa
n^{\alpha}u_{n}(t)=0,\\\ \frac{du_{0}(t)}{dt}=0,\\\
\frac{du_{-n}(t)}{dt}+i\sum_{l=1}^{n-1}lu_{-l}(t)(n-l)u_{-n+l}(t)+\kappa
n^{\alpha}u_{-n}(t)=0.\end{array}\right.$ (3.9)

Secondly let’s reduce the initial data in (1.3). It’s easy to see that

$\displaystyle\varphi(x)$
$\displaystyle=u(0,x)=\sum_{n=-\infty}^{\infty}u_{n}(0)e^{inx}=\sum_{n=-\infty}^{\infty}[iu_{n}(0)sin(nx)+u_{n}(0)cos(nx)]$
$\displaystyle=\sum_{n=1}^{\infty}[i(u_{n}(0)-u_{-n}(0))sin(nx)]+u_{0}(0)+\sum_{n=1}^{\infty}[(u_{n}(0)+u_{-n}(0))cos(nx)].$
(3.10)

Comparing the coefficients of the initial data $\varphi(x)$ in (2.5) with ones
in (3), we find

$\left\\{\begin{array}[]{ll}A_{n}=i(u_{n}(0)-u_{-n}(0)),\\\ u_{0}(0)=0,\\\
u_{-n}(0)=-u_{n}(0),\end{array}\right.$ (3.11)

which implies

$u_{n}(0)=\left\\{\begin{array}[]{ll}0,~{}~{}~{}~{}n=0,\\\
\frac{A_{n}}{2i},~{}~{}~{}n\geq 1.\end{array}\right.$ (3.12)

Considering the ODEs (3.9) with initial condition (3.12), we can get that
$u_{0}(t)=0$. It’s worth noting that $\\{-u_{n}(t)\\}$ will solve the third
equation in (3.9) if $\\{u_{n}(t)\\}$ solve the first equation in (3.9). Hence
the classical local well-posedness theorem of ODEs would ensure that

$u_{-n}(t)=-u_{n}(t).$ (3.13)

Then the fact $u_{0}(t)=0$ and (3.13) allow us to write the solution $u(t,x)$
by

$u(t,x)=\sum_{n=1}^{\infty}2iu_{n}(t)sin(nx).$ (3.14)

On the other hand, $\\{u_{n}(t)\\}$ are coefficients of Fourier series (2.4),
which yields

$u_{-n}(t)=\overline{u_{n}(t)}.$ (3.15)

(3.13) and (3.15) imply that $\\{u_{n}(t)\\}$ are pure imaginary numbers with
respect to $x$. For convenience, in the following we will take the real-valued
functions $\\{iu_{n}(t)\\}$ as a whole, where $i$ is the imaginary unit. To
sum up, if let

$w_{n}(t)=iu_{n}(t),$

the ODEs (3.9) with initial condition (3.12) would be reduced to the following
equations only for $\\{w_{n}(t)\\}_{n\geq 1}$

$\left\\{\begin{array}[]{ll}\frac{dw_{n}(t)}{dt}-\sum_{l=1}^{n-1}lw_{l}(t)(n-l)w_{n-l}(t)+\kappa
n^{\alpha}w_{n}(t)=0,\\\ w_{n}(0)=\frac{A_{n}}{2}.\end{array}\right.$ (3.16)

The solutions to (3.16) can be written as

$w_{n}(t)=\frac{A_{n}}{2}e^{-n^{\alpha}\kappa t}+e^{-n^{\alpha}\kappa
t}\int^{t}_{0}e^{n^{\alpha}\kappa
s}\sum_{l=1}^{n-1}lw_{l}(s)(n-l)w_{n-l}(s)ds,~{}~{}~{}n\geq 1.$ (3.17)

##  4\. The proof of the main results

In this section, we mainly give the proofs of blow-up results. Relying on the
formal symmetry of solutions in (3.17), we will finish the proofs by the
mathematical induction method.

###  4.1. The proof of Theorem 2.2

###### Proof.

As $\kappa=0$, (3.17) gives

$nw_{n}(t)=\frac{nA_{n}}{2}+n\int^{t}_{0}\sum_{l=1}^{n-1}lw_{l}(s)(n-l)w_{n-l}(s)ds\geq
0,~{}~{}for~{}~{}n\geq 1.$ (4.1)

In addition, we claim that $\\{nw_{n}(t)\\}_{n\geq 1}$ satisfy

$nw_{n}(t)\geq\frac{\delta}{n^{4}}(\frac{1}{2}+\delta t)^{n-1}.$ (4.2)

Now we use the mathematical induction to prove this claim.

If $n=1$, then

$w_{1}(t)=\frac{A_{1}}{2}\geq\delta.$

As $1\leq l\leq n-1$ , we assume that

$lw_{l}(t)\geq\frac{\delta}{l^{4}}(\frac{1}{2}+\delta t)^{l-1}$ (4.3)

holds. (4.1) and (4.3) imply that

$\displaystyle nw_{n}(t)$
$\displaystyle=\frac{nA_{n}}{2}+n\int^{t}_{0}\sum_{l=1}^{n-1}lw_{l}(s)(n-l)w_{n-l}(s)ds$
$\displaystyle\geq\frac{nA_{n}}{2}+\delta^{2}n\sum_{l=1}^{n-1}\frac{1}{l^{4}(n-l)^{4}}\int^{t}_{0}(\frac{1}{2}+\delta
s)^{n-2}ds$
$\displaystyle\geq\frac{\delta}{n^{4}}+\delta^{2}\frac{n-1}{n^{4}}\int^{t}_{0}(\frac{1}{2}+\delta
s)^{n-2}ds$ $\displaystyle=\frac{\delta}{n^{4}}(\frac{1}{2}+\delta
t)^{n-1}+\frac{\delta}{n^{4}}(1-\frac{1}{2^{n-1}})$
$\displaystyle\geq\frac{\delta}{n^{4}}(\frac{1}{2}+\delta t)^{n-1},$

and (4.2) follows.

(3.14), (4.2) and Paserval equality show that

$\displaystyle\|u(t,\cdot)\|^{2}_{L^{2}(-\pi,\pi]}$
$\displaystyle=2\pi\sum_{n=1}^{\infty}|2w_{n}(t)|^{2}=8\pi\sum_{n=1}^{\infty}|\frac{1}{n}nw_{n}(t)|^{2}$
$\displaystyle\geq 8\pi\delta^{2}\sum_{n=1}^{\infty}\frac{[(\frac{1}{2}+\delta
t)^{2}]^{n-1}}{n^{10}}.$ (4.4)

Let $g(t):=\frac{1}{2}+\delta t$ be bigger than 1, then the series in (4.1)
would diverge by the basic fact of calculus. Thus, we only need to choose
$T_{1}$ is bigger than $\frac{1}{2\delta}$ and the proof is completed. ∎

###  4.2. The proof of Theorem 2.3

###### Proof.

In this case, we also have from (3.17)

$nw_{n}(t)=\frac{nA_{n}}{2}e^{-n\kappa t}+ne^{-n\kappa
t}\int^{t}_{0}e^{n\kappa
s}\sum_{l=1}^{n-1}lw_{l}(s)(n-l)w_{n-l}(s)ds,~{}~{}~{}n\geq 1.$ (4.5)

Similarly we can infer that the following estimate holds

$nw_{n}(t)\geq\frac{\delta}{n^{4}}(\frac{1}{2}+\delta t)^{n-1}e^{-n\kappa t}.$
(4.6)

Here the mathematical induction method is used again.

As $n=1$, we have

$w_{1}(t)=\frac{A_{1}}{2}e^{-\kappa t}\geq\delta e^{-\kappa t}.$

For $1\leq l\leq n-1$, we assume that

$lw_{l}(t)\geq\frac{\delta}{l^{4}}(\frac{1}{2}+\delta t)^{l-1}e^{-l\kappa t}.$
(4.7)

Hence we can obtain that from (4.5) and (4.7)

$\displaystyle nw_{n}(t)$ $\displaystyle=\frac{nA_{n}}{2}e^{-n\kappa
t}+ne^{-n\kappa t}\int^{t}_{0}e^{n\kappa
s}\sum_{l=1}^{n-1}lw_{l}(s)(n-l)w_{n-l}(s)ds$
$\displaystyle\geq\frac{nA_{n}}{2}e^{-n\kappa
t}+\delta^{2}n\sum_{l=1}^{n-1}\frac{1}{l^{4}(n-l)^{4}}e^{-n\kappa
t}\int^{t}_{0}(\frac{1}{2}+\delta s)^{n-2}ds$
$\displaystyle\geq\frac{\delta}{n^{4}}e^{-n\kappa
t}+\delta^{2}\frac{n-1}{n^{4}}e^{-n\kappa t}\int^{t}_{0}(\frac{1}{2}+\delta
s)^{n-2}ds$ $\displaystyle=\frac{\delta}{n^{4}}e^{-n\kappa
t}(\frac{1}{2}+\delta t)^{n-1}+\frac{\delta}{n^{4}}e^{-n\kappa
t}(1-\frac{1}{2^{n-1}})$
$\displaystyle\geq\frac{\delta}{n^{4}}(\frac{1}{2}+\delta t)^{n-1}e^{-n\kappa
t},$

and (4.6) follows. On the other hand, (3.14), (4.6) and Paserval equality
imply that

$\displaystyle\|u(t,\cdot)\|^{2}_{L^{2}(-\pi,\pi]}$
$\displaystyle=2\pi\sum_{n=1}^{\infty}|2w_{n}(t)|^{2}=8\pi\sum_{n=1}^{\infty}|\frac{1}{n}nw_{n}(t)|^{2}$
$\displaystyle\geq 8\pi\delta^{2}e^{-2\kappa
t}\sum_{n=1}^{\infty}\frac{([(\frac{1}{2}+\delta t)e^{-\kappa
t}]^{2})^{n-1}}{n^{10}}$ (4.8)

Similarly let $g(t)=(\frac{1}{2}+\delta t)e^{-\kappa t}>1$, then the series in
(4.2) fails to converge. Since $0<\kappa<\frac{2\delta}{2e-1}$, we only need
to choose $T_{2}=\frac{1}{\kappa}$ to deduce the occurrence of singularity. ∎

###  4.3. The proof of Theorem 2.4

###### Proof.

Similar to the proof of Theorem 2.3, it’s sufficient for us to estimate

$nw_{n}(t)\geq\frac{\delta}{n^{4}}(\frac{1}{2}+\delta t)^{n-1}e^{-n\kappa t}.$
(4.9)

In the case of $n=1$, it’s easy to see

$w_{1}(t)=\frac{A_{1}}{2}e^{-\kappa t}\geq\delta e^{-\kappa t}.$

Then we assume that (4.9) holds for $l=1,2,...,n-1$,

$lw_{l}(t)\geq\frac{\delta}{l^{4}}(\frac{1}{2}+\delta t)^{l-1}e^{-l\kappa t}.$
(4.10)

(3.17) and (4.10) yield that

$\displaystyle nw_{n}(t)$ $\displaystyle=\frac{nA_{n}}{2}e^{-n^{\alpha}\kappa
t}+ne^{-n^{\alpha}\kappa t}\int^{t}_{0}e^{n^{\alpha}\kappa
s}\sum_{l=1}^{n-1}lw_{l}(s)(n-l)w_{n-l}(s)ds$
$\displaystyle\geq\frac{nA_{n}}{2}e^{-n^{\alpha}\kappa
t}+\delta^{2}n\sum_{l=1}^{n-1}\frac{1}{l^{4}(n-l)^{4}}e^{-n^{\alpha}\kappa
t}\int^{t}_{0}e^{(n^{\alpha}-n)\kappa s}(\frac{1}{2}+\delta s)^{n-2}ds$
$\displaystyle\geq\frac{\delta}{n^{4}}e^{-n\kappa
t}+\delta^{2}\frac{n-1}{n^{4}}e^{-n^{\alpha}\kappa
t}\int^{t}_{0}e^{(n^{\alpha}-n)\kappa s}(\frac{1}{2}+\delta s)^{n-2}ds$
$\displaystyle=\frac{\delta}{n^{4}}e^{-n\kappa t}+e^{-n\kappa
t}\delta^{2}\frac{n-1}{n^{4}}e^{(n-n^{\alpha})\kappa
t}\int^{t}_{0}e^{(n^{\alpha}-n)\kappa s}(\frac{1}{2}+\delta s)^{n-2}ds$
$\displaystyle\geq\frac{\delta}{n^{4}}e^{-n\kappa t}+e^{-n\kappa
t}\delta^{2}\frac{n-1}{n^{4}}\int^{t}_{0}(\frac{1}{2}+\delta s)^{n-2}ds$
$\displaystyle\geq\frac{\delta}{n^{4}}(\frac{1}{2}+\delta t)^{n-1}e^{-n\kappa
t},$ (4.11)

where the second inequality in (4.3) uses $0<\alpha<1$ and the third
inequality in (4.3) follows from the fact

$e^{(n-n^{\alpha})\kappa t}\int^{t}_{0}e^{(n^{\alpha}-n)\kappa
s}(\frac{1}{2}+\delta s)^{n-2}ds\geq\int^{t}_{0}(\frac{1}{2}+\delta
s)^{n-2}ds,~{}~{}for~{}t\geq 0,~{}0<\alpha<1.$

Define

$h(t)=e^{(n-n^{\alpha})\kappa t}\int^{t}_{0}e^{(n^{\alpha}-n)\kappa
s}(\frac{1}{2}+\delta s)^{n-2}ds-\int^{t}_{0}(\frac{1}{2}+\delta s)^{n-2}ds.$

It’s obvious that $h(0)=0$, and a simple computation shows that

$h^{\prime}(t)=(n-n^{\alpha})\kappa e^{(n-n^{\alpha})\kappa
t}\int^{t}_{0}e^{(n^{\alpha}-n)\kappa s}(\frac{1}{2}+\delta
s)^{n-2}ds>0,~{}~{}for~{}t\geq 0,~{}0<\alpha<1,$

then $h(t)\geq 0,~{}~{}for~{}t\geq 0,~{}0<\alpha<1$. ∎

######  Remark 4.1.

The technology in our proof mainly relies on the formal symmetry of ODEs in
(3.16). In fact, if this symmetry is lost, we find that the approach also can
be used to deduce the singularity. For example, let’s consider following ODEs

$\frac{dw_{n}(t)}{dt}-\sum_{l=1}^{n-1}lw_{l}(t)w_{n-l}(t)+\kappa
n^{\alpha}w_{n}(t)=0.$ (4.12)

Based on the fact

$\displaystyle\sum_{l=1}^{n-1}lw_{l}(t)w_{n-l}(t)$
$\displaystyle=\sum_{l=1}^{n-1}\sqrt{l}w_{l}(t)\sqrt{l}w_{n-l}(t)$
$\displaystyle=w_{1}w_{n-1}+2w_{2}w_{n-2}+\cdot\cdot\cdot+(n-2)w_{n-2}w_{2}+(n-1)w_{n-1}w_{1}$
$\displaystyle=w_{1}w_{n-1}+(n-1)w_{n-1}w_{1}+2w_{2}w_{n-2}+(n-2)w_{n-2}w_{2}+\cdot\cdot\cdot$
$\displaystyle\geq
2(\sqrt{1}\sqrt{n-1})w_{1}w_{n-1}+2(\sqrt{2}\sqrt{n-2})w_{2}w_{n-2}+\cdot\cdot\cdot$
$\displaystyle=\sum_{l=1}^{n-1}\sqrt{l}w_{l}(t)\sqrt{n-l}w_{n-l}(t)$ (4.13)

and the ODEs’ comparison principle, we only need to deal with

$\frac{dw_{n}(t)}{dt}-\sum_{l=1}^{n-1}\sqrt{l}w_{l}(t)\sqrt{n-l}w_{n-l}(t)+\kappa
n^{\alpha}w_{n}(t)=0,$ (4.14)

which possess the construction of symmetry. Thus the similar argument in this
section indicates that the solutions to (4.14) would develop into
singularities in finite time, hence holds for (4.12).

######  Remark 4.2.

Based on these singularity results, it’s not difficult to find that smoothing
effects due to weak dissipation can’t prevent the formation of singularity,
thus the global existence is expected when with strong dissipation
($\alpha>1$) for this model. However, here we can’t give a super bound for
solutions due to the limitation of the spectral method.

## Acknowledgements

This project is supported by National Natural Science Foundation of China
(No:11571057).

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