# Global Regularity and instability for the incompressible non-viscous
Oldroyd-B model {\dagger}{\dagger}${\dagger}$

Zhi Chen1, Weikui Ye2, Zhaoyang Yin 3,
1\. School of Mathematics and Statistics, Central South University,
Changsha, 410083 Hunan, P. R. China
2\. Institute of Applied Physics and Computational Mathematics,
P.O. Box 8009, Beijing 100088, P. R. China
3\. Department of Mathematics, Sun Yat-sen University,
Guangzhou, 510275 Guangdong, P. R. China
E-mail<EMAIL_ADDRESS>(Z. Chen)E-mail<EMAIL_ADDRESS>(W.
Ye)E-mail<EMAIL_ADDRESS>(Z. Yin)

Abstract. In this paper, we consider the 2-dimensional non-viscous Oldroyd-B
model. In the case of the ratio equal 1 ($\alpha=0$), it is a difficult case
since the velocity field $u(t,x)$ is no longer decay. Fortunately, by
observing the exponential decay of the stress tensor $\tau(t,x)$, we succeeded
in proving the global existence for this system with some large initial data.
Moreover, we give an unsteady result: when the ratio is close to 1
($a\rightarrow 0$), the system is not steady for large time. This implies an
interesting physical phenomenon that the term $a\mathbb{D}u$ is a bridge
between the transformation of kinetic energy $u$ and elastic potential energy
$\tau$, but this process is transient for large time, which leads the
instability.

Keywords: Oldroyd-B model; exponential decay; global solutions; instability.

2010 Mathematics Subject Classification. 35Q35; 35A01; 35A02; 35B45

##  1 Introduction and main results

In this paper, we study the incompressible Oldroyd-B model of the non-
Newtonian fluid in ${\mathbb{R}}^{+}\times{\mathbb{R}}^{d}$

$\begin{cases}\partial_{t}u+(u\cdot\nabla)u-\nu\Delta u+\nabla
p=\mu_{1}\mathrm{div}(\tau),\\\
\partial_{t}\tau+(u\cdot\nabla)\tau-\eta\Delta\tau+\mu_{2}\tau+\mathrm{Q}(\nabla
u,\tau)=\alpha\mathbb{D}u,\\\ \mathrm{div}u=0,\\\
u(x,0)=u_{0}(x),~{}~{}~{}\tau(0,x)=\tau_{0}(x),\end{cases}$ (1.1)

where $u$ denotes the velocity, $\tau=\tau_{i,j}$ is the non-Newtonian part of
the stress tensor($\tau$ is a $d\times d$ symmetric matrix here) and $p$ is a
scalar pressure of fluid. $\mathrm{D}(u)$ is the symmetric part of the
velocity gradient,

$\mathrm{D}(u)=\frac{1}{2}(\nabla u+(\nabla u)^{T}).$

The $\mathrm{Q}$ above is a given bilinear form:

$\mathrm{Q}(\tau,\nabla
u)=\tau\Omega(u)-\Omega(u)\tau+b(\mathrm{D}(u)\tau+\tau\mathrm{D}(u)),$

where $b$ is a parameter in $[-1,1],$ $\Omega(u)$ is the skew-symmetric part
of $\nabla u,$ i.e.

$\Omega(u)=\frac{1}{2}(\nabla u-(\nabla u)^{T}).$

The parameters $\nu,\eta,\mu_{1},\alpha$ are non-negative and they are
specific to the characteristic of the considered material. In [24],
$\nu,\mu_{2}$ and $b$ correspond respectively to $\theta/Re,1/We$ and
$2(1-\theta)/(WeRe),$ where $Re$ is the Reynolds number, $\theta$ is the ratio
between the relaxation and retardation times and $We$ is the Weissenberg
number. In this paper, we will investigate the case $\alpha=0$. Following [7,
17],we claim that the limit model $\alpha=0$ occurs when $\theta$, the ratio
between the so called relaxation and retardation times, is converging to 1. We
specifically pointed out some different Oldroyd models with infinite
Weissenberg number (and so $\alpha,\mu_{2}=0$) have been investigated in [22].

The Oldroyd-B model describes the motion of some viscoelastic flows.
Formulations about viscoelastic flows of Oldroyd-B type are first established
by Oldroyd in [25]. For more detailed physical background and derivations
about this model, we refer the readers to [2, 11, 21, 25].

The well-posedness of the system (1.1) has been extensively and continuously
studied. Guillopé and Saut [18, 19] obtained the local solutions with large
initial data and showed that these solutions are global when the coupling and
the initial data are small enough. In the corotational case, i.e. $b=0,$ Lions
and Masmoudi established the existence of global weak solution in [24].

Many works have been devoted to find the global solution in the case of small
data after the work of F. Lin et al.[23]. Chemin and Masmoudi [7] first
obtained the local solutions and global small solutions in the critical Besov
spaces when $\nu>0,\mu_{1}>0,\alpha>0,$ and $\eta=0.$ They get the global
small solutions when the initial and coupling parametra is small,
i.e.($\mu_{1}\alpha\leq c\mu_{2}\nu$). The condition $\mu_{1}\alpha\leq
c\mu_{2}\nu$ means that coupling effect between the two equation is less
important than the viscosity. Inspired by the work [5, 9], Zi, Fang and Zhang
improved their results in the critical $L^{p}$ framework for the case of non-
small coupling parameters in [35].

Some recent results dealt with the case when there is only kinematic
dissipation or stress tensor dissipation,i.e. the systems do not have damping
term. Zhu [34] got small global smooth solutions of the $\mathrm{3D}$
Oldroyd-B model with $\eta=0,\mu_{2}=0$ by observing the linearization of the
system satisfies the damped wave equation. Inspired by the work of Zhu [34]
and Danchin in [13], Chen and Hao [8] extended this small data global solution
in Sobolev spaces to the critical Besov spaces. Wu and Zhao [29] and Zhai [33]
established the small data global well-posedness in critical Besov spaces for
fractional dissipation of velocity respectively. Moreover, Zhai [32]
constructs global solutions for a class of highly oscillating initial
velocities by observing the special structure of the system.

When the damping mechanism and the Laplacian dissipation for $\tau$ exist,
Elgindi and Rousset [16] established a global large solution in a certain
sense by building a new quantity to avoid singular operators. Later, Liu and
Elgindi extend these results to the 3D in [15]. Constantin, Wu, Zhao and Zhu
[12] established the small data global solution to the n-dimensional(n=2,3) in
the case of no damping mechanism and general tensor dissipation. Recently, Wu
and Zhao [28] investigated the small data global well-posedness in Besov
spaces for fractional tensor dissipation. Very recently, Chen, Liu, Qin and Ye
[10] improved their results in the critical $L^{p}$ framework by using
Lagrangian coordinates in the high frequency.

It is worth pointing that the exist of $\mathrm{D}u$ is key for the proof of
the above results. A nature question is that Whether the system has a global
solution when the $\alpha=0$ in (1.1)? Since the $\mathrm{D}u$ play a
important role by the equality

$\int_{{\mathbb{R}}^{2}}\mathrm{div}\tau\cdot
udx+\int_{{\mathbb{R}}^{2}}Du:\tau=0.$

Moreover, the term $\mathrm{D}u$ ensures the damping term of $\Gamma$ in paper
[16], in the work of [12], the $\mathrm{D}u$ guarantee the linearized system
satisfy the damped wave structure. In this paper, we will give a positive
answer by the Theorem 2. As far as we know, this is the first study the case
of $\alpha=0$ in (1.1) with any $b\in[-1,1]$ in $Q(\nabla u,\tau)$.

When $\alpha=0,$ the system will become interesting which do not enjoy the
special structure similar to [12]. In this paper, We can get a well-posedness
result in the case of $\alpha=0.$ We also study how the global strong
solutions of (1.1) behave when the ratio $\theta$ trends to 1, (i.e.
$a\rightarrow 0$). Without lose of generality, we let the
$\nu=0,\mu_{1}=1,\eta=1$ and $\mu_{2}=1$, then (1.1) becomes:

$\begin{cases}\partial_{t}u+(u\cdot\nabla)u+\nabla p=\mathrm{div}(\tau),\\\
\partial_{t}\tau+(u\cdot\nabla)\tau-\Delta\tau+\tau+\mathrm{Q}(\nabla
u,\tau)=a\mathbb{D}u,\\\ \mathrm{div}u=0,\\\
u(x,0)=u_{0}(x),~{}~{}~{}\tau(0,x)=\tau_{0}(x),\end{cases}$ (1.2)

In [16], Elgindi and Rousset established a global large solution by building a
new quantity $\Gamma=w-\frac{curldiv}{\Delta}\tau$ and finding a $H^{1}$
estimation for small initial data:

$a\|u(t)\|_{L^{\infty}_{t}(H^{1})}+\|\tau(t)\|_{L^{\infty}_{t}(H^{1})}+\|\tau\|_{L^{2}_{t}(H^{2})}\leq
C\|(u_{0},\tau_{0})\|_{H^{1}}$

Recently, Deng, Luo and Yin [14] obtain the global small solutions for (1.2)
with $\mathrm{Q}(\tau,\nabla
u)=\tau\Omega(u)-\Omega(u)\tau+b(\mathrm{D}(u)\tau+\tau\mathrm{D}(u)),~{}b=0$,
and give an estimation of the $H^{1}$ decay for the global solutions with any
$b\in[-1,1]$ . Combining their results, and after some calculation, one can
get the following theorem:

######  Theorem 1.1.

[16, 14] Let $(u_{0},\tau_{0})\in H^{s}(\mathbb{R}^{2}),s>2$. Assume that

$\|u_{0}\|_{H^{1}}\leq\epsilon_{0},~{}\|\tau_{0}\|_{H^{1}}\leq
a\epsilon_{0}~{}~{}and~{}~{}\|w_{0}\|_{{{B}^{0}_{\infty,1}}}+\|\tau_{0}\|_{{{B}^{0}_{\infty,1}}}\leq\varepsilon_{0},$

then the systems (1.2) exists a unique global solution $(u,\tau)\in
C([0,+\infty),H^{s})$. Moreover, if $(u_{0},\tau_{0})\in L^{1}$, then we have

$\|(u,\tau)(t)\|_{H^{1}}\leq
C\|(u_{0},\tau_{0})\|_{H^{1}}[a(1+t)]^{-\frac{1}{2}}.$

In this paper, we consider the global existence for (1.2) with $a=0$ in
critical Besov space. The main difficulty is that the ratio is 1 ($a=0$).
Since [12] tells us that the term $a\|\mathbb{D}u\|_{H^{s}}$ has disspative
effect, which leads to the polynomial decay of
$\|\mathbb{D}u\|_{H^{s}},~{}s\geq 0.$ This implies that when $a=0$,
$\|\mathbb{D}u\|_{H^{s}}$ may grow! That causes the main difficulty to obtain
the global existence of (1.2). However, we find that $\|\tau(t)\|_{L^{2}\cap
L^{\infty}}$ is exponential decay for small initial data, which can offset the
exponential growth effect due to the Gronwall inequality. Then, combing the
critical estimation of the transport equation in Besov space and the bootstrap
argument, we obtain the global existence of (1.2). Compared with the results
in [16, 14], we needn’t the small condition of
$\|u_{0},\tau_{0}\|_{\dot{H}^{1}}$. Finally, we obtain the large time behavior
such that $\|\tau(t)\|_{B^{\frac{d}{p}}_{p,1}}$ and
$\|u(t)\|_{B^{1+\frac{d}{p}}_{p,1}}$ are double exponential growth (see
(3.14)-(3.15) ). Here is our first result:

######  Theorem 1.2.

Let
$(u_{0},\tau_{0})\in(B^{1+\frac{2}{p}}_{p,1},B^{\frac{2}{p}}_{p,1}),~{}1\leq
p\leq\infty$. If

$\|(u_{0},\tau_{0})\|_{L^{2}}+\|w_{0}\|_{{{B}^{0}_{\infty,1}}}+\|\tau_{0}\|_{{{B}^{0}_{\infty,1}}}\leq
4\varepsilon_{0},~{}~{}w_{0}=\mathrm{curl}u_{0},~{}\varepsilon_{0}=\frac{1}{64(C^{4}+1)},$
(1.3)

then there exists a unique global solution $(u,\tau)\in
C_{T}(B^{1+\frac{d}{p}}_{p,1})\times C_{T}(B^{\frac{d}{p}}_{p,1})\cup
L^{1}_{T}(B^{2+\frac{d}{p}}_{p,1})$ for system (1.2) with $a=0$. Moreover, we
have

$\|\tau\|_{L^{2}\cap L^{\infty}}\leq
Ce^{-\frac{1}{72}t}\|\tau_{0}\|_{L^{2}\cap L^{\infty}}.$

######  Remark 1.1.

In [3, 4], Bourgain and Li employed a combination of Lagrangian and Eulerian
techniques to obtain strong local ill-posed results of the Euler equation in
$B^{\frac{d}{p}+1}_{p,r}$ with $p\in[1,\infty),~{}r\in(1,\infty],~{}d=2,3$.
Recently, Guo, Li and Yin [20] proved the Euler equation is well-posed in
$B^{\frac{d}{p}+1}_{p,1}$ with $p\in[1,\infty]$, which means that
$B^{\frac{d}{p}+1}_{p,1}$ is the critical Besov space for the well-posedness
of the Euler equation. Thus, since (1.2) is the Euler equation when
$\tau_{0}=0$, we conclude that
$C([0,T];{B}^{\frac{d}{p}+1}_{p,1}(\mathbb{R}^{d}))\times\Big{(}C([0,T];{B}^{\frac{d}{p}}_{p,1}(\mathbb{R}^{d}))\cap
L^{1}([0,T];{B}^{\frac{d}{p}+2}_{p,1}(\mathbb{R}^{d}))\Big{)}$ is also the
critical Besov space for (1.2) with $d=2,3,~{}p\in[1,\infty)$.

######  Remark 1.2.

Since $B^{\frac{2}{p}}_{p,1}\hookrightarrow B^{0}_{\infty,1},~{}p<\infty$. By
(1.3) we claim that our result includes some large initial data. For example,
choose $\varphi$ be a smooth, radial and non-negative function in
$\mathbb{R}^{2}$ such that

$\phi=\begin{cases}1,~{}~{}for~{}|\xi|\leq 1,\\\ 0,~{}~{}for~{}|\xi|\geq
2.\end{cases}$ (1.4)

Let $(u_{0},\tau_{0}):=\frac{1}{N}(\psi,\varphi)$, where $\psi,\varphi\in
S(\mathbb{R}^{2})$, $div\psi=0$ and
$F(\varphi)=(\phi(\xi-2^{N}e),\phi(\xi-2^{N}e))$ with
$e=(1,1),N\in\mathbb{N}^{+}$. Then, one can easily deduce that

$\Delta_{j}\varphi=\varphi~{}~{}when~{}j=N;~{}~{}~{}\Delta_{j}\varphi=0~{}~{}when~{}j\neq
N.$

So for sufficient large $N$ and $p<\infty$, we have

$\|\tau_{0}\|_{H^{1}}\approx\frac{2^{N}}{N},~{}~{}\|\tau_{0}\|_{B^{\frac{2}{p}}_{p,1}}\approx\frac{2^{\frac{2}{p}N}}{N},~{}~{}but~{}~{}\|\tau_{0}\|_{B^{0}_{\infty,1}}+\|\tau_{0}\|_{L^{2}}\leq\frac{C}{N}.$

This implies the global existence for some large initial data. Combining
Remark 1.1, we obtain the global large solutions for (1.2) with $a=0$ in
critical Besov spaces, which implies some additions and improvements to [16,
14].

Next, by Theorem 1.1 ($a>0$) and Theorem 1.2 ($a=0$), we obtain the global
existence for the system (1.2). Furthermore, one will ask that whether these
two solutions are close to each other when $a\rightarrow 0$? For local time
the answer is true. That is

$\lim_{a\rightarrow
0}\|u^{0}-u^{a}\|_{L^{\infty}_{T}(B^{2}_{2,1})}+\|\tau^{0}-\tau^{a}\|_{L^{\infty}_{T}(B^{1}_{2,1})}=0,~{}~{}~{}\forall
t\in[0,T].$

However, for large time, the above equality is no longer valid. Indeed,
$\|u\|_{L^{2}}~{}({B^{2}_{2,1}}\hookrightarrow{L^{2}})$ will have a jump when
$T$ is sufficiently large. This implies the system (1.2) is not steady for
$a\rightarrow 0$ for large time, while for local time (1.2) is steady.
Therefore, whether the ratio is 1 ($a=0$) seems to have a fundamental effect
on the nature of the global solutions for (1.2).

In order to give the Theorem 1.3, we introduced a simple notation.

Set
$\mathbb{A}:=\\{(u_{0},\tau_{0})\in(B^{2}_{2,1}(\mathbb{R}^{2}),B^{1}_{2,1}(\mathbb{R}^{2}))|~{}\eqref{gs2222}~{}has~{}a~{}unique~{}solution~{}for~{}any~{}fixed~{}a\\}.$
Here is our second result.

######  Theorem 1.3.

Let
$(u_{0},\tau_{0})\in(B^{2}_{2,1}(\mathbb{R}^{2}),B^{1}_{2,1}(\mathbb{R}^{2}))$
as initial data of systems (1.2), $(u^{a},\tau^{a})$ is the corresponding
solution for (1.2) with every $a\geq 0$.
(1) Then there exists a lifespan $T(u_{0},\tau_{0})$ independent of $a$ such
that

$\displaystyle\lim_{a\rightarrow
0}\|u^{0}-u^{a}\|_{L^{\infty}_{T}(B^{2}_{2,1})}+\|\tau^{0}-\tau^{a}\|_{L^{\infty}_{T}(B^{1}_{2,1})}=0.$
(1.5)

(2) However, (1.5) false for large time. Assume

$0<a\leq\varepsilon_{0}.$

Then there exists a time $T(a)$ and a sequence $(u_{0},\tau_{0})\in\mathbb{A}$
as initial data of systems (1.2) such that when $t\geq T(a)$,

$\|(u^{0}-u^{a})(t)\|_{L^{2}}\geq\frac{\varepsilon_{0}}{8},$

where $\varepsilon_{0}=\frac{1}{64(C^{4}+1)}$, a fixed constant.

######  Remark 1.3.

When $a=0$, by Theorem 1.2, we deduce that $\|u\|_{L^{2}}$ is bounded, while
$\|\tau\|_{L^{2}}$ is exponential decay. However, when $a>0$, by Theorem 1.1,
we obtain the polynomial decay of $\|u\|_{L^{2}}$ and $\|\tau\|_{L^{2}}$. This
interesting fact implies that the ratio has two sides to the effect. It passes
the decay of $\tau$ to $u$, while $\tau$ decays from exponential attenuation
to polynomial attenuation. This shows that the energy of the velocity field
and the stress tensor are converted to each other, but the process of
transformation is transient for large time (see Theorem 1.3).

In the case $\alpha=0,$ we were surprised to find that the $\tau$ have a
exponential decay by iteration and reasonable assumption of bootstrap, using
the integral equation of the heat equation and bootstrap assumption, we can
estimate the $L^{\infty}$ norm of $\tau$. Combining the prior estimate of
transport equation in Besov space and subtle analytical skills, we complete
the proof of the Theorem 1.2. Moreover, on the basis of Theorem 1.2, we give
the proof of the Theorem 1.3 by boothstrap argument.

Notation Throughout the paper, we denote the norms of usual Lebesgue space
$L^{p}({\mathbb{R}}^{3})$ by
$\|u\|^{p}_{L^{p}}=\int_{\Omega}|u|^{p}dx,~{}\hbox{for}~{}1\leq p<\infty$.
$C_{i}$ and $C$ denote different positive constants in different places.

The bootstrap argument will be employed in our proof. A rigorous statement of
the abstract bootstrap principle can be found in T. Tao’s book (see [26]).

The paper is organized as follows. In section 2, we will give the
tools(Littlewood-Paley decomposition and paradifferential calculus) and some
useful prior estimate in Besov spaces. In section 3, we give the proof of The
theorem 1.2. In last section, we complete the proof of unstability results.

##  2 Preliminaries

In this section, we will recall some properties about the Littlewood-Paley
decomposition and Besov spaces. For more details, we refer the readers to the
[1].

######  Proposition 2.1.

[1] Let $\mathcal{C}$ be the annulus
$\\{\xi\in\mathbb{R}^{d}:\frac{3}{4}\leq|\xi|\leq\frac{8}{3}\\}$. There exist
radial functions $\chi$ and $\varphi$, valued in the interval $[0,1]$,
belonging respectively to $\mathcal{D}(B(0,\frac{4}{3}))$ and
$\mathcal{D}(\mathcal{C})$, and such that

$\forall\xi\in\mathbb{R}^{d},\ \chi(\xi)+\sum_{j\geq 0}\varphi(2^{-j}\xi)=1,$
$\forall\xi\in\mathbb{R}^{d}\backslash\\{0\\},\
\sum_{j\in\mathbb{Z}}\varphi(2^{-j}\xi)=1,$ $|j-j^{\prime}|\geq
2\Rightarrow\mathrm{Supp}\ \varphi(2^{-j}\cdot)\cap\mathrm{Supp}\
\varphi(2^{-j^{\prime}}\cdot)=\emptyset,$ $j\geq 1\Rightarrow\mathrm{Supp}\
\chi(\cdot)\cap\mathrm{Supp}\ \varphi(2^{-j}\cdot)=\emptyset.$

The set $\widetilde{\mathcal{C}}=B(0,\frac{2}{3})+\mathcal{C}$ is an annulus,
and we have

$|j-j^{\prime}|\geq 5\Rightarrow 2^{j}\mathcal{C}\cap
2^{j^{\prime}}\widetilde{\mathcal{C}}=\emptyset.$

Further, one has

$\forall\xi\in\mathbb{R}^{d},\ \frac{1}{2}\leq\chi^{2}(\xi)+\sum_{j\geq
0}\varphi^{2}(2^{-j}\xi)\leq 1,$
$\forall\xi\in\mathbb{R}^{d}\backslash\\{0\\},\
\frac{1}{2}\leq\sum_{j\in\mathbb{Z}}\varphi^{2}(2^{-j}\xi)\leq 1.$

######  Definition 2.1.

[1] Let $u$ be a tempered distribution in
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$ and $\mathcal{F}$ be the Fourier
transform and $\mathcal{F}^{-1}$ be its inverse. For all $j\in\mathbb{Z}$,
define

$\Delta_{j}u=0\,\ \text{if}\,\
j\leq-2,\quad\Delta_{-1}u=\mathcal{F}^{-1}(\chi\mathcal{F}u),\quad\Delta_{j}u=\mathcal{F}^{-1}(\varphi(2^{-j}\cdot)\mathcal{F}u)\,\
\text{if}\,\ j\geq 0,\quad S_{j}u=\sum_{j^{\prime}<j}\Delta_{j^{\prime}}u.$

Then the nonhomogeneous Littlewood-Paley decomposition is given as follows:

$\displaystyle u=\sum_{j\in\mathbb{Z}}\Delta_{j}u\quad\text{in}\
\mathcal{S}^{\prime}(\mathbb{R}^{d}).$

Let $s\in\mathbb{R},\ 1\leq p,r\leq\infty.$ The nonhomogeneous Besov space
$B^{s}_{p,r}(\mathbb{R}^{d})$ is defined by

$B^{s}_{p,r}=B^{s}_{p,r}(\mathbb{R}^{d})=\\{u\in
S^{\prime}(\mathbb{R}^{d}):\|u\|_{B^{s}_{p,r}(\mathbb{R}^{d})}=\Big{\|}(2^{js}\|\Delta_{j}u\|_{L^{p}(\mathbb{S}^{d})})_{j}\Big{\|}_{l^{r}(\mathbb{Z})}<\infty\\}.$

######  Definition 2.2.

[1] The homogeneous dyadic blocks $\dot{\Delta}_{j}$ are defined on the
tempered distributions by

$\dot{\Delta}_{j}u=\varphi(2^{-j}D)u:=\mathcal{F}^{-1}(\varphi(2^{-j\cdot})\hat{u}).$
$\dot{S}_{j}u=\sum_{j^{\prime}\leq j-1}\dot{\Delta}_{j^{\prime}}u.$

######  Definition 2.3.

We denote by $S^{\prime}_{h}$ the space of tempered distributions $u$ such
that

$\lim_{j\rightarrow-\infty}\dot{S}_{j}u=0~{}~{}\text{in}~{}~{}S^{\prime}.$

The homogeneous Littlewood-Paley decomposition is defined as

$u=\sum_{j\in\mathbb{Z}}\dot{\Delta}_{j}u,~{}~{}~{}~{}\text{for}~{}~{}u\in
S^{\prime}_{h}.$

######  Definition 2.4.

For $s\in{\mathbb{R}},1\leq p\leq\infty,$ the homogeneous Besov space
$\dot{B}^{s}_{p,r}$ is defined as

$\dot{B}^{s}_{p,r}:=\\{u\in
S^{\prime}_{h},\|u\|_{\dot{B}^{s}_{p,r}}<\infty\\},$

where the homogeneous Besov norm is given by

$\|u\|_{\dot{B}^{s}_{p,r}}:=\|\\{2^{js}\|\dot{\Delta}_{j}u\|_{L^{p}}\\}_{j}\|_{l^{r}}.$

In this paper, we use the ”time-space” Besov spaces or Chemin-Lerner space
first introduced by Chemin and Lerner in [6].

Definition 2.3. Let $s\in{\mathbb{R}}$ and $0<T\leq+\infty.$ We define

$\|u\|_{\tilde{L}^{q}_{T}({B}^{s}_{p,1})}:=\sum_{j\in\mathbb{Z}}2^{js}\bigg{(}\int^{T}_{0}\|{\Delta}_{j}u(t)\|^{q}_{L^{p}}dt\bigg{)}^{\frac{1}{q}},$

for $p,q\in[1,\infty)$ and with the standard modification for $p,q=\infty.$

By the Minkowski’s inequality, it is easy to verify that

$\|u\|_{\tilde{L}^{\lambda}_{T}({B}^{s}_{p,r}))}\leq\|u\|_{L^{\lambda}_{T}({B}^{s}_{p,r}))}~{}~{}~{}\text{if}~{}~{}\lambda\leq
r,$

and

$\|u\|_{\tilde{L}^{\lambda}_{T}({B}^{s}_{p,r}))}\geq\|u\|_{L^{\lambda}_{T}({B}^{s}_{p,r}))}~{}~{}~{}\text{if}~{}~{}\lambda\geq
r.$

The following Bernstein’s lemma will be repeatedly used in this paper.

######  Lemma 2.1.

Let $\mathcal{B}$ is a ball and $\mathcal{C}$ is a ring of ${\mathbb{R}}^{d}.$
There exists constant $C$ such that for any positive $\lambda,$ any non-
negative integer $k,$ any smooth homogeneous function $\sigma$ of degree $m,$
any couple $(p,q)\in[1,\infty]^{2}$ with $q\geq p\geq 1,$ and any function
$u\in L^{p},$ there holds

$\mathrm{supp}\hat{u}\subset\lambda\mathcal{B}\Rightarrow\sup_{|\alpha=k|}\|\partial^{\alpha}u\|_{L^{q}}\leq
C^{k+1}\lambda^{k+d(\frac{1}{p}-\frac{1}{q})}\|u\|_{L^{p}},$
$\mathrm{supp}\hat{u}\subset\lambda\mathcal{C}\Rightarrow
C^{-k-1}\lambda^{k}\|u\|_{L^{p}}\leq\sup_{|\alpha=k|}\|\partial^{\alpha}u\|_{L^{p}}\leq
C^{k+1}\lambda^{k}\|u\|_{L^{p}},$
$\mathrm{supp}\hat{u}\subset\lambda\mathcal{C}\Rightarrow\sup_{|\alpha=k|}\|\sigma(D)u\|_{L^{p}}\leq
C_{\sigma,m}\lambda^{m+d(\frac{1}{p}-\frac{1}{q})}\|u\|_{L^{p}}.$

Next, we will give the paraproducts and product estimates in Besov spaces.
Recalling the paraproduct decomposition

$uv={T}_{u}v+{T}_{v}u+{R}(u,v),$

where

${T}_{u}v:=\sum_{q}{S}_{q-1}u{\Delta}_{v},~{}~{}~{}{R}(u,v):=\sum_{q}{\Delta}_{q}u{\tilde{\Delta}}_{q}v,~{}~{}\text{and}~{}~{}{\tilde{\Delta}}_{q}={\Delta}_{q-1}+{\Delta}_{q}+{\Delta}_{q+1}.$

The paraproduct ${T}$ and the remainder ${R}$ operators satisfy the following
continuous properties.

######  Proposition 2.2.

For all $s\in{\mathbb{R}},\sigma>0,$ and $1\leq
p,p_{1},p_{2},r,r_{1},r_{2}\leq\infty,$ the paraproduct ${T}$ is a bilinear,
continuous operator from $L^{\infty}\times{B}^{s}_{p,r}$ to ${B}^{s}_{p,r}$
and from ${B}^{-\sigma}_{p_{1},r_{1}}\times\dot{B}^{s}_{p_{2},r_{2}}$ to
${B}^{s-\sigma}_{p,r}$ with
$\frac{1}{r}=\min\\{1,\frac{1}{r_{1}}+\frac{1}{r_{2}}\\},\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}.$
The remainder ${R}$ is bilinear continuous from
${B}^{s_{1}}_{p_{1},r_{1}}\times{B}^{s_{2}}_{p_{2},r_{2}}$ to
${B}^{s_{1}+s_{2}}_{p,r}$ with
$s_{1}+s_{2}>0,\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}\leq 1,$ and
$\frac{1}{r}=\frac{1}{r_{1}}+\frac{1}{r_{2}}\leq 1.$ In particular, if
$r=\infty,$ the continuous property for the remainder ${R}$ also holds for the
case $s_{1}+s_{2}=0,r=\infty,\frac{1}{r_{1}}+\frac{1}{r_{2}}=1.$

Combining the above proposition with Lemma 2.1 yields the following product
estimates:

######  Corollary 2.1.

Let $a$ and $b$ be in $L^{\infty}\cap{B}^{s}_{p,r}$ for some $s>0$ and
$(p,r)\in[1,\infty]^{2}.$ Then there exists a constant $C$ depending only on
$d,p$ and such that

$\|ab\|_{{B}^{s}_{p,r}}\leq
C(\|a\|_{L^{\infty}}\|b\|_{{B}^{s}_{p,r}}+\|b\|_{L^{\infty}}\|a\|_{{B}^{s}_{p,r}}).$

We note that the above Proposition and Corollary is valid in the homogeneous
framework (i.e. with $\dot{\Delta}_{j}$ instead of $\Delta_{j}$ and with
homogeneous Besov norms instead of nonhomogeneous ones), the readers can refer
the [1]. Finally, we intruduce some useful results about the following heat
conductive equation and the transport equation

$\left\\{\begin{array}[]{l}u_{t}-\Delta u+\beta u=G,\ x\in\mathbb{R}^{d},\
\beta\geq 0,~{}t>0,\\\ u(0,x)=u_{0}(x),\
x\in\mathbb{R}^{d},\end{array}\right.$ (2.1)
$\left\\{\begin{array}[]{l}f_{t}+v\cdot\nabla f=g,\ x\in\mathbb{R}^{d},\
t>0,\\\ f(0,x)=f_{0}(x),\ x\in\mathbb{R}^{d},\end{array}\right.$ (2.2)

which are crucial to the proof of our main theorem later.

######  Lemma 2.2.

[27] Let $1\leq p\leq q\leq\infty$ and $k\geq 0,$ it holds that

$\|\nabla^{k}e^{t\Delta}f\|_{L^{q}}\leq
Ct^{-\frac{k}{2}-\frac{1}{p}+\frac{1}{q}}\|f\|_{L^{p}}.$

######  Lemma 2.3.

[1] Let $s\in\mathbb{R},~{}\beta\geq 0,1\leq q,q_{1},p,r\leq\infty$ with
$q_{1}\leq q$. Assume $u_{0}$ in ${B}^{s}_{p,r}$, and $G$ in
$\widetilde{L}^{q_{1}}_{T}(^{s}_{p,r})$. Then (2.1) with $\beta=0$ has a
unique solution $u$ in $\widetilde{L}^{q}_{T}({B}^{s+\frac{2}{q}}_{p,r})$ and
satisfies

$\displaystyle\|u\|_{\widetilde{L}^{q}_{T}({B}^{s+\frac{2}{q}}_{p,r})}\leq
C_{1}\Big{(}\|u_{0}\|_{{B}^{s}_{p,r}}+(1+T^{1+\frac{1}{q}-\frac{1}{q_{1}}})\|G\|_{\widetilde{L}^{q_{1}}_{T}({B}^{s+\frac{2}{q_{1}}-2}_{p,r})}\Big{)}.$
(2.3)

Moreover, if $\beta>0$, without loss of generality we set $\beta=1$, one have

$\displaystyle\|u\|_{\widetilde{L}^{q}_{T}({B}^{s+\frac{2}{q}}_{p,r})}\leq
C_{1}\Big{(}\|u_{0}\|_{{B}^{s}_{p,r}}+\|G\|_{\widetilde{L}^{q_{1}}_{T}({B}^{s+\frac{2}{q_{1}}-2}_{p,r})}\Big{)}.$
(2.4)

###### Proof.

(2.3) can be founded in [1], we should only prove (2.4). Indeed, since

$\Delta_{j}u=e^{-t}e^{t\Delta}\Delta_{j}u_{0}+\int_{0}^{t}e^{-(t-s)}e^{(t-s)\Delta}\Delta_{j}Gds,$

when $j\geq 0$, by $\|e^{t\Delta}\Delta_{j}u\|_{L^{p}}\leq
Ce^{-2^{2j}t}\|\Delta_{j}u\|_{L^{p}}$ one can easily get

$\|2^{j(s+\frac{2}{q})}\|\Delta_{j}u\|_{L^{q}_{T}L^{p}}\|_{1_{j\geq
0}l^{r}}\leq
C_{1}\Big{(}\|u_{0}\|_{{B}^{s}_{p,r}}+\|G\|_{\widetilde{L}^{q_{1}}_{T}({B}^{s+\frac{2}{q_{1}}-2}_{p,r})}\Big{)}.$

Indeed, we just estimate the difficult term

$\sum_{j\geq
0}2^{j(s+\frac{2}{q})}(\int^{T}_{0}|\|\int^{t}_{0}e^{-(t-s)}e^{(t-s)\Delta}\Delta_{j}Gds\|_{L^{p}}|^{q}dt)^{\frac{1}{q}},$

Combining the Minkowski’s inequality and the Young inequality, one has

$\displaystyle\sum_{j\geq 0}$ $\displaystyle
2^{j(s+\frac{2}{q})}(\int^{T}_{0}|\int^{t}_{0}e^{-(t-s)}\|e^{(t-s)\Delta}\Delta_{j}G\|_{L^{p}}ds|^{q}dt)^{\frac{1}{q}}$
$\displaystyle\leq\sum_{j\geq
0}2^{j(s+\frac{2}{q})}(\int^{T}_{0}|e^{-(2^{2j}+1)\cdot}*\|\Delta_{j}G(\cdot)\|_{L^{p}}|^{q})^{\frac{1}{q}}$
$\displaystyle\leq\sum_{j\geq
0}2^{j(s+\frac{2}{q})}(\int^{T}_{0}|e^{-(2^{2j}+1)t}|^{\frac{1}{1+\frac{1}{q}-\frac{1}{q_{1}}}}dt)^{1+\frac{1}{q}-\frac{1}{q_{1}}}\|\Delta_{j}G\|_{L^{q_{1}}_{T}(L^{p})}$
$\displaystyle\leq C\sum_{j\geq
0}2^{j(s+\frac{2}{q}-2(1+\frac{1}{q}-\frac{1}{q_{1}}))}\|\Delta_{j}G\|_{L^{q_{1}}_{T}(L^{p})}$
$\displaystyle=C\sum_{j\geq
0}2^{j(s-2+\frac{2}{q_{1}})}\|\Delta_{j}G\|_{L^{q_{1}}_{T}(L^{p})}.$

When $j=-1$, by $\|e^{t\Delta}\Delta_{-1}u\|_{L^{p}}\leq
C\|\Delta_{-1}u\|_{L^{p}}$ we have

$\|\Delta_{-1}u\|_{L^{q}_{T}L^{p}}\leq
C_{1}\Big{(}\|\Delta_{-1}u_{0}\|_{L^{p}}+\|\Delta_{-1}G\|_{{L}^{q_{1}}_{T}(L^{p})}\Big{)}.$

Combining the above two inequality, we obtain (2.4). ∎

######  Lemma 2.4.

[1] Let
$s\in[\max\\{-\frac{d}{p},-\frac{d}{p^{\prime}}\\},\frac{d}{p}+1](s=1+\frac{d}{p},r=1;s=\max\\{-\frac{d}{p},-\frac{d}{p^{\prime}}\\},r=\infty).$
There exists a constant $C$ such that for all solutions $f\in
L^{\infty}([0,T];{B}^{s}_{p,r})$ of (2.2) with initial data $f_{0}$ in
${B}^{s}_{p,r}$, and $g$ in $L^{1}([0,T];{B}^{s}_{p,r})$, we have, for a.e.
$t\in[0,T]$,

$\displaystyle\|f(t)\|_{{B}^{s}_{p,r}}\leq$ $\displaystyle
C\Big{(}\|f_{0}\|_{{B}^{s}_{p,r}}+\int_{0}^{t}V^{\prime}(t^{\prime})\|f(t^{\prime})\|_{{B}^{s}_{p,r}}+\|g(t^{\prime})\|_{{B}^{s}_{p,r}}dt^{\prime}\Big{)}$
(2.5) $\displaystyle\leq$ $\displaystyle
e^{C_{2}V(t)}\Big{(}\|f_{0}\|_{{B}^{s}_{p,r}}+\int_{0}^{t}e^{-C_{2}V(t^{\prime})}\|g(t^{\prime})\|_{{B}^{s}_{p,r}}dt^{\prime}\Big{)},$

where $V(t)=\int_{0}^{t}\|\nabla v\|_{{B}^{\frac{d}{p}}_{p,r}\cap
L^{\infty}}ds($if $s=1+\frac{1}{p},r=1$, $V^{\prime}(t)=\int_{0}^{t}\|\nabla
v\|_{{B}^{\frac{d}{p}}_{p,1}}ds).$

######  Remark 2.1.

[1] If ${\rm{div}}v=0$, we can get the same result with a better indicator:
$\max\\{-\frac{d}{p},-\frac{d}{p^{\prime}}\\}-1<s<\frac{d}{p}+1($or
$s=\max\\{-\frac{d}{p},-\frac{d}{p^{\prime}}\\}-1,r=\infty).$

######  Lemma 2.5.

[1] Let ${\rm{div}}v=0$. There exists a constant $C$ such that for solutions
$f\in L^{\infty}([0,T];{B}^{0}_{p,r})$ of (2.2) with initial data $f_{0}$ in
${B}^{0}_{p,r}$, and $g$ in $L^{1}([0,T];{B}^{s}_{p,r})$, we have, for all
$1\leq p,r\leq\infty$ and $t\in[0,T]$,

$\displaystyle\|f(t)\|_{{B}^{s}_{p,r}}\leq$ $\displaystyle
C(1+\int_{0}^{t}V^{\prime}(t^{\prime})dt^{\prime})(\|f_{0}\|_{{B}^{s}_{p,r}}+\int_{0}^{t}\|g(t^{\prime})\|_{{B}^{s}_{p,r}}dt^{\prime}),$
(2.6)

where $V^{\prime}(t)=\int_{0}^{t}\|\nabla v(t)\|_{L^{\infty}}ds.$

##  3 Global existence

The proof of Theorem 1.2:

###### Proof.

The proof of the local well-posedness of the system 1.2 is standard, we should
only need to prove the global existence. We now use the bootstrap argument to
prove this Theorem. Let $T^{*}$ be the maximal existence time of the solution.
Assume that for any $t\leq T<T^{*}$,

$\displaystyle\|\tau\|_{L^{\infty}_{T}(B^{0}_{\infty,1})\cap
L^{1}_{T}(B^{2}_{\infty,1})}\leq\delta:=\frac{1}{C^{2}+1},~{}~{}\|u\|^{2}_{L^{\infty}_{T}(L^{2})}\leq\delta^{2}.$
(3.1)

where $C>10$ is a fixed constant. Let the sufficient small initial data such
that

$\displaystyle\|u_{0}\|_{B^{1}_{\infty,1}({\mathbb{R}}^{2})}+\|b_{0}\|_{B^{0}_{\infty,1}({\mathbb{R}}^{2})}\leq\epsilon_{0}:=\frac{1}{64(C^{4}+1)},~{}~{}\|u_{0}\|_{L^{2}}+\|\tau_{0}\|_{L^{2}}\leq\epsilon_{0}.$
(3.2)

The proof can be divided into 3 parts:

(1) First, we give the decay estimation of $\|\tau(t)\|_{L^{2}}$.

Recall the system:

$\begin{cases}\partial_{t}u+(u\cdot\nabla)u=\mathrm{div}(\tau),\\\
\partial_{t}\tau+(u\cdot\nabla)\tau-\Delta\tau+\tau+\mathrm{Q}(\nabla
u,\tau)=0,\\\ \mathrm{div}u=0,\\\
u(x,0)=u_{0}(x),~{}~{}~{}\tau(0,x)=\tau_{0}(x).\end{cases}$ (3.3)

Taking the curl of the first equation in (3.3), we get:

$\begin{cases}\partial_{t}w+(u\cdot\nabla)w=\mathrm{curl}\mathrm{div}(\tau),\\\
w_{0}=\mathrm{curl}u_{0}.\end{cases}$ (3.4)

It is easy to verify that

$\|w\|_{L^{\infty}}\leq\|w_{0}\|_{L^{\infty}}+\int^{t}_{0}\|\tau\|_{B^{2}_{\infty,1}}ds\leq
C(\epsilon_{0}+\delta).$ (3.5)

From the second equation of (3.3), we get

$\partial_{t}e^{t}\tau+(u\cdot\nabla)e^{t}\tau-\Delta e^{t}\tau=-Q(\nabla
u,e^{t}\tau).$

Taking the $L^{2}$ inner product with $e^{t}\tau,$ one has

$\frac{1}{2}\frac{d}{dt}\|e^{t}\tau\|_{L^{2}}^{2}+(\nabla e^{t}\tau,\nabla
e^{t}\tau)=-(Q(\nabla u,e^{t}\tau),e^{t}\tau),$

and using the integration by parts, H$\ddot{o}$lder’s inequalities and Young’s
inequalities,

$\displaystyle\|e^{t}\tau\|^{2}_{L^{2}}+2\int^{t}_{0}\|e^{s}\tau\|^{2}_{\dot{H}_{1}}ds$
$\displaystyle\leq\|\tau_{0}\|^{2}_{L^{2}}+\int^{t}_{0}\|u\|_{L^{\infty}}(\|\nabla\tau\|_{L^{2}}\|\tau\|_{L^{2}})e^{2s}ds$
(3.6)
$\displaystyle\leq\|\tau_{0}\|^{2}_{L^{2}}+\int^{t}_{0}(\|w\|_{L^{\infty}}+\|u\|_{L^{2}})\|e^{s}\tau\|^{2}_{\dot{H}_{1}}$
$\displaystyle~{}~{}+C(\|w\|_{L^{\infty}}+\|u\|_{L^{2}})\|e^{s}\tau\|^{2}_{L^{2}}ds$
$\displaystyle\leq\epsilon_{0}^{2}+C\int^{t}_{0}(\epsilon_{0}+\delta)\|e^{s}\tau\|^{2}_{\dot{H}_{1}}$
$\displaystyle~{}~{}+C(\epsilon_{0}+\delta)\|e^{s}\tau\|^{2}_{L^{2}}ds$

where the last inequality holds by (3.1) and (3.5). Combining the Gronwall’s
inequality with (3.2), one has

$\|e^{t}\tau\|^{2}_{L^{2}}+\int^{t}_{0}\|e^{s}\tau\|^{2}_{\dot{H}_{1}}ds\leq
e^{\frac{1}{2}t}\|\tau_{0}\|^{2}_{L^{2}}.$ (3.7)

Taking the $L^{2}$ inner product again with $e^{\frac{1}{2}t}\tau,$ we have

$\frac{1}{2}\frac{d}{dt}\|e^{\frac{1}{2}t}\tau\|_{L^{2}}^{2}+(e^{\frac{1}{2}t}\tau,e^{\frac{1}{2}t}\tau)+(\nabla
e^{\frac{1}{2}t}\tau,\nabla e^{\frac{1}{2}t}\tau)=-(Q(\nabla
u,e^{\frac{1}{2}t}\tau),e^{\frac{1}{2}t}\tau),$

Similar to the inequality (3.6), it is easy to see that

$\displaystyle\|e^{\frac{1}{2}t}\tau\|^{2}_{L^{2}}+\int^{t}_{0}\|e^{\frac{1}{2}s}\tau\|^{2}_{\dot{H}_{1}}ds$
$\displaystyle\leq\|\tau_{0}\|^{2}_{L^{2}}+\int^{t}_{0}C(\|w\|_{L^{\infty}}+\|u\|_{L^{2}})\|\tau\|^{2}_{L^{2}}e^{s}ds$
(3.8)

Noting the inequality (3.7), it implies that,

$\|\tau\|^{2}_{L^{2}}\leq e^{-\frac{3}{2}t}\|\tau_{0}\|^{2}_{L^{2}}.$ (3.9)

Using the inequality (3.8) and (3.9), one has

$\|e^{\frac{1}{2}t}\tau\|^{2}_{L^{2}}+\int^{t}_{0}\|e^{\frac{1}{2}s}\tau\|^{2}_{\dot{H}_{1}}ds\leq
C\|\tau_{0}\|^{2}_{L^{2}}$ (3.10)

(2) Then, we give the estimation of
$\|u(t)\|_{L^{\infty}_{T}(L^{2})}\leq\frac{1}{2}\delta.$

Taking the $L^{2}$ inner product with $u$ for the first equation of (1.2), we
get

$\displaystyle\|u\|^{2}_{L^{2}}$
$\displaystyle\leq\|u_{0}\|^{2}_{L^{2}}+\int^{t}_{0}\|div\tau\|_{L^{2}}\|u\|_{L^{2}}ds$
$\displaystyle\leq\|u_{0}\|^{2}_{L^{2}}+\int^{t}_{0}e^{\frac{1}{2}s}\|div\tau\|_{L^{2}}e^{-\frac{1}{2}s}\|u\|_{L^{2}}ds$
$\displaystyle\leq\|u_{0}\|^{2}_{L^{2}}+(\int^{t}_{0}e^{s}\|div\tau\|^{2}_{L^{2}})^{\frac{1}{2}}(\int^{t}_{0}e^{-s}\|u\|^{2}_{L^{2}})^{\frac{1}{2}}$
$\displaystyle\leq\epsilon_{0}^{2}+C\epsilon_{0}\delta,$

where we have used the fact the inequality
$\|u\|^{2}_{L^{\infty}_{T}(L^{2})}\leq\delta^{2}$ and (3.10). We can set
$\epsilon_{0}$ is small enough in (3.2) such that

$\|u\|_{L^{2}}\leq\frac{1}{2}\delta.$ (3.11)

(3) Next, we estimate $\|w(t)\|_{B^{0}_{\infty,1}}$ and
$\|\tau\|_{L^{\infty}}.$

Applying the Lemma 2.5 to the first equation of (3.4), one has

$\displaystyle\|w\|_{L^{\infty}_{T}({B^{0}_{\infty,1}})}$
$\displaystyle\leq(\|w_{0}\|_{B^{0}_{\infty,1}}+C\|\tau\|_{L^{1}_{T}(B^{2}_{\infty,1})})(1+\int_{0}^{T}\|\nabla
u\|_{L^{\infty}}ds)$ (3.12) $\displaystyle\leq
C(\|w_{0}\|_{B^{0}_{\infty,1}}+\|\tau\|_{L^{1}(B^{2}_{\infty,1})})(1+\int_{0}^{T}\|w\|_{L^{\infty}_{T}({B^{0}_{\infty,1}})}+\|u\|_{L^{2}}ds)$
$\displaystyle\leq C(\epsilon+\delta)+\delta
T+\int_{0}^{T}C(\epsilon+\delta)\|w\|_{L^{\infty}_{T}({B^{0}_{\infty,1}})}ds$
$\displaystyle\leq C(c_{small}+\delta T)e^{c_{small}T}$

where the last inequality holds by the Gronwall’s inequality, and
$c_{small}:=C(\epsilon+\delta)$.

Then, we estimate the $\|u\|_{L^{\infty}}.$ we can rewrite the second equation
(3.3) into the following equality:

$\partial_{t}(e^{t}\tau)+u\cdot\nabla(e^{t}\tau)-\Delta(e^{t}\tau)=-Q(\nabla
u,e^{t}\tau).$

We use the following integral equation

$\|e^{t}\tau\|_{L^{\infty}}=e^{t\Delta}\tau_{0}-\int^{t}_{0}e^{(t-s)\Delta}(u\cdot\nabla(e^{s}\tau))ds+\int^{t}_{0}e^{(t-s)\Delta}Q(\nabla
u,e^{s}\tau)ds.$

By Lemma 2.2 and H$\ddot{o}$lder’s inequalities, let $b$ be a fixed small
constant such that $8c_{small}<<\frac{1}{36}\leq b<\frac{1}{6}$, we have

$\displaystyle\|\tau\|_{L^{\infty}}$ $\displaystyle\leq
e^{-t}\|\tau_{0}\|_{L^{\infty}}+\int^{t}_{0}e^{-(t-s)}\|u\nabla\tau\|_{L^{\infty}}ds+\int_{0}^{t}e^{-(t-s)}(t-s)^{-\frac{1}{2}}\|\tau\nabla
u\|_{L^{2}}ds$ $\displaystyle\leq
e^{-t}\|\tau_{0}\|_{L^{\infty}}+\int^{t}_{0}e^{-(t-s)}\|\tau\|^{\frac{1}{3}}_{L^{2}}e^{\frac{1}{6}s}\|\tau\|^{\frac{2}{3}}_{B^{2}_{\infty,1}}e^{-\frac{1}{6}s}\delta
ds+\int_{0}^{t}e^{-(t-s)}(t-s)^{-\frac{1}{2}}\|\tau\nabla u\|_{L^{2}}ds$
$\displaystyle\leq
e^{-t}\|\tau_{0}\|_{L^{\infty}}+e^{-tb}\int^{t}_{0}e^{sb}\|\tau\|^{\frac{1}{3}}_{L^{2}}e^{\frac{1}{6}s}\|\tau\|^{\frac{2}{3}}_{B^{2}_{\infty,1}}e^{-\frac{1}{6}s}\delta
ds+\int_{0}^{t}e^{-b(t-s)}(t-s)^{-\frac{1}{2}}\|\tau\|_{L^{2}}\|\nabla
u\|_{L^{\infty}}ds$ $\displaystyle\leq
e^{-t}\epsilon_{0}+e^{-tb}C\|\tau_{0}\|_{L^{2}}^{\frac{1}{3}}\|\tau\|^{\frac{2}{3}}_{L^{1}_{B^{2}_{\infty,1}}}(\int^{t}_{0}|e^{sb}e^{-\frac{1}{6}s}|^{3}ds)^{\frac{1}{3}}\delta$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+e^{-tb}\int^{t}_{0}e^{sb}(t-s)^{-\frac{1}{2}}(c_{small}+\delta
s)e^{c_{small}s}\epsilon_{0}e^{-\frac{3}{4}s}ds$ $\displaystyle\leq
e^{-t}\epsilon_{0}+C(t+1)e^{-tb}\delta$ $\displaystyle\leq
C(1+t)e^{-tb}\delta\leq Ce^{-\frac{b}{2}t}\delta,$

where the second inequality holds by the interpolation inequality:

$\displaystyle\|u\nabla\tau\|_{L^{\infty}}$
$\displaystyle\leq\|\nabla\tau\|_{L^{\infty}}\|u\|_{L^{\infty}}$
$\displaystyle\leq\|\tau\|^{\frac{1}{3}}_{B^{-1}_{\infty,\infty}}\|\tau\|^{\frac{2}{3}}_{B^{2}_{\infty,\infty}}(\|u\|_{L^{2}}+\|w\|_{L^{\infty}})$
$\displaystyle\leq
C\|\tau\|^{\frac{1}{3}}_{B^{-1}_{\infty,1}}\|\tau\|^{\frac{2}{3}}_{B^{2}_{\infty,1}}\delta$
$\displaystyle\leq
C\|\tau\|^{\frac{1}{3}}_{L^{2}}\|\tau\|^{\frac{2}{3}}_{B^{2}_{\infty,1}}\delta.$

(4) Finally, to complete the bootstrap argument , we estimate
$\|\tau\|_{L^{\infty}_{T}(B^{0}_{\infty,1})\cap L^{1}_{T}(B^{2}_{\infty,1})}$
.

Using the inequality (2.4) of the Lemma (2.3) to the second equation of (3.4),
one has

$\displaystyle\|\tau\|_{L^{\infty}_{T}(B^{0}_{\infty,1})\cap
L^{1}_{T}(B^{2}_{\infty,1})}$
$\displaystyle\leq\|\tau_{0}\|_{B^{0}_{\infty,1}}+\int^{t}_{0}\|u\nabla\tau\|_{B^{0}_{\infty,1}}+\|Q(\nabla
u,\tau)\|_{B^{0}_{\infty,1}}ds$
$\displaystyle\leq\epsilon_{0}+\int^{t}_{0}\|u\|_{L^{2}}\|\tau\|_{B^{2}_{\infty,1}}+\|\nabla
u\|_{B^{0}_{\infty,1}}\|\tau\|_{L^{\infty}}ds$
$\displaystyle\leq\epsilon_{0}+\frac{1}{2}\delta\int^{t}_{0}\|\tau\|_{B^{2}_{\infty,1}}ds+\int^{t}_{0}[(c_{small}+\delta
s)e^{c_{small}s}+\delta]Ce^{-\frac{b}{2}s}\delta ds$
$\displaystyle\leq\epsilon_{0}+\frac{1}{2}\delta^{2}+C(c_{small}+\delta)\delta\int^{t}_{0}(1+s)e^{c_{small}s}e^{-\frac{b}{2}s}ds$
$\displaystyle\leq\frac{1}{2}\delta$

where we have used the fact that

$\displaystyle\|u\cdot\nabla\tau\|_{B^{0}_{\infty,1}}+\|Q(\nabla
u,\tau)\|_{B^{0}_{\infty,1}}\leq\|u\|_{L^{2}}\|\tau\|_{B^{2}_{\infty,1}}+\|\nabla
u\|_{B^{0}_{\infty,1}}\|\tau\|_{L^{\infty}},$

and the last inequality holds by (3.2) and $b\geq\frac{1}{36}>>8c_{small}$.

Combining (3.11), for any $t<T^{*}$ we have

$\displaystyle\|u\|^{2}_{L^{\infty}_{T}(L^{2})}\leq\delta^{2}~{}~{}and~{}~{}\|\tau\|_{L^{\infty}_{T}(B^{0}_{\infty,1})\cap
L^{1}_{T}(B^{2}_{\infty,1})}\leq\delta.$ (3.13)

Therefore, one can obtain the global existence of $(u,\tau)$ in
$C([0,\infty);{B}^{1+\frac{2}{p}}_{p,1})\times\Big{(}C([0,\infty);{B}^{\frac{2}{p}}_{p,1})\cap
L^{1}\big{(}[0,\infty);{B}^{\frac{2}{p}+2}_{p,1}\big{)}\Big{)}$ easily, since
(3.13) can be the blow-up criteria for (1.2). Indeed, applying Lemma 2.3–2.4
to (1.2), for any $t<T^{*}$ we have

$\displaystyle\|u\|_{L^{\infty}_{t}({B^{1+\frac{2}{p}}_{p,1}})}\leq\|u_{0}\|_{{B^{1+\frac{2}{p}}_{p,1}}}+C\int_{0}^{t}\|u\|_{B^{1}_{\infty,1}}\|u\|_{B^{1+\frac{2}{p}}_{p,1}}+\|\tau\|_{B^{1+\frac{2}{p}}_{p,1}}ds\leq
Ce^{e^{Ct}},$ (3.14)

and

$\displaystyle\|\tau\|_{L^{\infty}_{t}(B^{\frac{2}{p}}_{p,1})\cap
L^{1}_{t}(B^{2+\frac{2}{p}}_{p,1})}\leq$
$\displaystyle\|\tau_{0}\|_{B^{\frac{2}{p}}_{p,1}}+C\int_{0}^{t}\|u\|_{L^{2}}\|\tau\|_{B^{2+\frac{2}{p}}_{p,1}}+\|\tau\|_{L^{\infty}}\|u\|_{B^{1+\frac{2}{p}}_{p,1}}dt\leq
Ce^{e^{Ct}},$ (3.15)

where we have used (3.12) and Gronwall inequality. This complete the proof of
Theorem 1.2.

∎

##  4  Unstability of the solutions

To prove Theorem 1.3, we firstly give the following local-wellposedness
theorem:

######  Theorem 4.1.

Let $(u_{0},\tau_{0})\in B^{s+1}_{2,1}(\mathbb{R}^{2})\times
B^{s}_{2,1}(\mathbb{R}^{2})$ with $s\geq 1$. Then there exists a lifespan
$T(u_{0},\tau_{0})$ (which is independent on $\alpha$) such that the system
(1.2) with $\alpha\geq 0$ has a unique solution
$(u,\tau)\in(C_{T}(B^{s+1}_{2,1}(\mathbb{R}^{2})),C_{T}(B^{s}_{2,1}(\mathbb{R}^{2}))\cup
L^{1}_{T}(B^{s+2}_{2,1}(\mathbb{R}^{2})))$. Moreover, if there exists an
initial sequence satisfying

$\lim_{n\rightarrow\infty}\|u^{n}_{0}-u_{0}\|_{B^{s+1}_{2,1}}+\|\tau^{n}_{0}-\tau_{0}\|_{B^{s}_{2,1}}=0,$

then for any $t\in[0,T]$ we have

$\lim_{n\rightarrow\infty}\|u^{n}-u\|_{L^{\infty}_{t}({B^{s+1}_{2,1}})}+\|\tau^{n}-\tau\|_{L^{\infty}_{t}({B^{s}_{2,1}})}=0.$

######  Lemma 4.1.

Let $(u_{0},\tau_{0})\in{B}^{s+1}_{2,1}\times{B}^{s}_{2,1}$ be the initial
data of (1.2) with $s\geq 1$ and $a\geq 0$. If there exists an initial
sequence $(u^{j}_{0},\tau^{j}_{0})\in{B}^{s+1}_{2,1}\times{B}^{s}_{2,1}$ such
that
$\|u^{j}_{0}-u_{0}\|_{{B}^{s+1}_{2,1}}+\|\tau^{j}_{0}-\tau_{0}\|_{{B}^{s}_{2,1}}\rightarrow
0\ (j\rightarrow\infty)$, then one can construct a lifespan $T^{j}$
corresponding to $(u^{j}_{0},\tau^{j}_{0})$ such that

$T^{j}\rightarrow T,\quad\quad j\rightarrow\infty,$

where the lifespan $T$ corresponds to $(u_{0},\tau_{0})$. This implies that
for these initial data
$(u^{j}_{0},b^{j}_{0}),~{}j\in\mathbb{N}\cup\\{\infty\\}$, there exists a
common lifespan $\bar{T}:=\frac{1}{2}T(u_{0},b_{0})$ , which is independent of
$j$ when $j$ is large.

One can refer to [30, 31] and take the similar operators to obtain the above
two theorem. we omit them here.

The proof of Theorem 1.3:

###### Proof.

(1) To prove

$\lim_{a\rightarrow
0}\|u^{a}-u^{0}\|_{L^{\infty}_{T}(B^{2}_{2,1})}+\|\tau^{a}-\tau^{0}\|_{L^{\infty}_{T}(B^{1}_{2,1})}=0,$

where $T$ is the local common lifespan in Theorem 4.1 and Lemma 4.1. Our main
idea is to estimate

$\displaystyle\|u^{a}-u^{0}\|_{L^{\infty}_{T}(B^{2}_{2,1})}+\|\tau^{a}-\tau^{0}\|_{L^{\infty}_{T}(B^{1}_{2,1})}$
$\displaystyle\leq\|u^{a}-u^{a}_{j}\|_{L^{\infty}_{T}(B^{2}_{2,1})}+\|\tau^{a}-\tau^{a}_{j}\|_{L^{\infty}_{T}(B^{1}_{2,1})}$
$\displaystyle\quad+\|u^{a}_{j}-u^{0}_{j}\|_{L^{\infty}_{T}(B^{2}_{2,1})}+\|\tau^{a}_{j}-\tau^{0}_{j}\|_{L^{\infty}_{T}(B^{1}_{2,1})}$
$\displaystyle+\|u^{0}_{j}-u^{0}\|_{L^{\infty}_{T}(B^{2}_{2,1})}+\|\tau^{0}_{j}-\tau^{0}\|_{L^{\infty}_{T}(B^{1}_{2,1})},$
(4.1)

where $(u^{a}_{j},\tau^{a}_{j})$ is the local solution for (1.2) with the
initial data $(u^{a}_{j},\tau^{a}_{j})$ for every $a\geq 0$
($j\in\mathbb{N}\cup\\{\infty\\}$), and
$(u^{a}_{\infty},\tau^{a}_{\infty}):=(u^{a},\tau^{a})$.

Firstly, we estimate
$\|u^{a}_{j}-u^{0}_{j}\|_{L^{\infty}_{T}(B^{2}_{2,1})}+\|\tau^{a}_{j}-\tau^{0}_{j}\|_{L^{\infty}_{T}(B^{1}_{2,1})}$
with fix $j$. Recall the system:

$\begin{cases}\partial_{t}u^{a}_{j}+(u^{a}_{j}\cdot\nabla)u^{a}_{j}+\nabla
p^{a}_{j}=\mathrm{div}(\tau^{a}_{j}),\\\
\partial_{t}\tau^{a}_{j}+(u^{a}_{j}\cdot\nabla)\tau^{a}_{j}-\Delta\tau^{a}_{j}+\tau^{a}_{j}+\mathrm{Q}(\nabla
u^{a}_{j},\tau^{a}_{j})=\alpha\mathrm{D}u^{a}_{j},\\\
u^{a}_{j}(0,x)=S_{j}u_{0}(x),~{}~{}~{}\tau^{a}_{j}(0,x)=S_{j}\tau_{0}(x).\end{cases}$
(4.2)

By the local well-posedness in Theorem 4.1 and Lemma 4.1 , for any $a,j\geq
0$, there exists a local lifespan $T(u_{0},\tau_{0})$ independent of $a$ and
$j$ such that

$\displaystyle\|u^{a}_{j}\|_{L^{\infty}_{T}(B^{3}_{2,1})}+\|\tau^{a}_{j}\|_{L^{\infty}_{T}(B^{2}_{2,1})}\leq
C(\|S_{j}u_{0}\|_{B^{3}_{2,1}}+\|S_{j}\tau_{0}\|_{L^{\infty}_{T}(B^{2}_{2,1})})\leq
C2^{j}(\|u_{0}\|_{B^{2}_{2,1}}+\|\tau_{0}\|_{B^{1}_{2,1}}).$ (4.3)

Then, we give the equation of
$(u^{a}_{j}-u^{0}_{j},\tau^{a}_{j}-\tau^{0}_{j})$:

$\displaystyle\left\\{\begin{array}[]{ll}(u^{a}_{j}-u^{0}_{j})_{t}+u^{a}_{j}\nabla(u^{a}_{j}-u^{0}_{j})+(u^{a}_{j}-u^{0}_{j})\nabla
u^{0}_{j}+\nabla(P^{a}_{j}-P^{0}_{j})=div(\tau^{a}_{j}-\tau^{0}_{j}),\\\\[4.30554pt]
(\tau^{a}_{j}-\tau^{0}_{j})_{t}+(\tau^{a}_{j}-\tau^{0}_{j})-\Delta(\tau^{a}_{j}-\tau^{0}_{j})+u^{a}_{j}\nabla(\tau^{a}_{j}-\tau^{0}_{j})+(u^{a}_{j}-u^{0}_{j})\nabla\tau^{0}_{j}\\\\[4.30554pt]
+Q(\nabla(u^{a}_{j}-u^{0}_{j}),~{}\tau^{0}_{j})+Q(\nabla
u^{a}_{j},~{}(\tau^{a}_{j}-\tau^{0}_{j}))=\alpha\mathbb{D}u^{a}_{j}\\\\[4.30554pt]
(u^{a}_{j}-u^{0}_{j})(0,x)=0,~{}(\tau^{a}_{j}-\tau^{0}_{j})(0,x)=0\\\\[4.30554pt]
\end{array}\right.$ (4.8)

Applying Lemma 2.3–2.4 and the Gronwall inequality, we obtain

$\displaystyle\|u^{a}_{j}-u^{0}_{j}\|_{L^{\infty}_{T}(B^{1}_{2,1})}+\|\tau^{a}_{j}-\tau^{\infty}_{j}\|_{L^{\infty}_{T}(B^{1}_{2,1})}$
$\displaystyle\leq
2^{j}C_{u_{0},\tau_{0}}(\int_{0}^{T}\|u^{a}_{j}-u^{0}_{j}\|_{B^{2}_{2,1}}$
(4.9)
$\displaystyle~{}~{}~{}~{}+\|\tau^{a}_{j}-\tau^{0}_{j}\|_{B^{1}_{2,1}}+a\|u_{0}\|_{B^{2}_{2,1}}dt)$
$\displaystyle\leq C_{u_{0},\tau_{0}}e^{2^{j}C_{u_{0},\tau_{0}}}a,$
$\displaystyle\rightarrow
0,~{}~{}~{}a\rightarrow\infty,~{}for~{}any~{}fixed~{}j.$ (4.10)

Next, we estimate
$\|u^{a}-u^{0}\|_{L^{\infty}_{T}(B^{2}_{2,1})}+\|\tau^{a}-\tau^{0}\|_{L^{\infty}_{T}(B^{1}_{2,1})}$.
By Theorem 4.1, we see that system (1.2) is locally steady for any initial
data. Since
$\|u_{0}-S_{j}u_{0}\|_{B^{2}_{2,1}}+\|\tau_{0}-S_{j}\tau_{0}\|_{B^{1}_{2,1}}\rightarrow
0,~{}j\rightarrow\infty$, so we have

$\displaystyle\|u^{a}-u^{a}_{j}\|_{L^{\infty}_{T}(B^{2}_{2,1})}+\|\tau^{a}-\tau^{a}_{j}\|_{L^{\infty}_{T}(B^{1}_{2,1})}\rightarrow
0,~{}j\rightarrow\infty,~{}~{}for~{}any~{}a\geq 0.$ (4.11)

Finally, combining (4.9) and (4.11), we deduce that

$\lim_{a\rightarrow
0}\|u^{a}-u^{0}\|_{L^{\infty}_{T}(B^{1}_{2,1})}+\|\tau^{a}-\tau^{0}\|_{L^{\infty}_{T}(B^{1}_{2,1})}=0.$

Indeed, for any $\frac{\epsilon}{3}$, by (4.11), there exists a constant
$N(\epsilon)$ such that when $j\geq N$, we have

$\|u^{a}-u^{a}_{j}\|_{L^{\infty}_{T}(B^{2}_{2,1})}+\|\tau^{a}-\tau^{a}_{j}\|_{L^{\infty}_{T}(B^{1}_{2,1})}\leq\frac{\epsilon}{3},~{}for~{}any~{}a\geq
0.$

For this $j$, by (4.9), there exists a constant $\delta$ ($\delta$ is actually
dependent on $\epsilon$) such that when $0\leq a\leq\delta$, one have

$\|u^{a}_{j}-u^{0}_{j}\|_{L^{\infty}_{T}(B^{2}_{2,1})}+\|\tau^{a}_{j}-\tau^{0}_{j}\|_{L^{\infty}_{T}(B^{1}_{2,1})}\leq\frac{\epsilon}{3}.$

Therefore, by (4) we obtain

$\displaystyle\|u^{a}-u^{0}\|_{L^{\infty}_{T}(B^{2}_{2,1})}+\|\tau^{a}-\tau^{0}\|_{L^{\infty}_{T}(B^{1}_{2,1})}$
$\displaystyle\leq\|u^{a}-u^{a}_{j}\|_{L^{\infty}_{T}(B^{2}_{2,1})}+\|\tau^{a}-\tau^{a}_{j}\|_{L^{\infty}_{T}(B^{1}_{2,1})}$
$\displaystyle\quad+\|u^{a}_{j}-u^{0}_{j}\|_{L^{\infty}_{T}(B^{2}_{2,1})}+\|\tau^{a}_{j}-\tau^{0}_{j}\|_{L^{\infty}_{T}(B^{1}_{2,1})}$
$\displaystyle\quad+\|u^{0}_{j}-u^{0}\|_{L^{\infty}_{T}(B^{2}_{2,1})}+\|\tau^{0}_{j}-\tau^{0}\|_{L^{\infty}_{T}(B^{1}_{2,1})}$
$\displaystyle\leq\frac{\epsilon}{3}+\frac{\epsilon}{3}+\frac{\epsilon}{3}\leq\epsilon.$
(4.12)

(2) Let $\epsilon_{0}=\frac{1}{64(C^{4}+1)}$ be the fixed small constant in
Theorem 1.1 and Theorem 1.2. Recall the system:

$\begin{cases}\partial_{t}u+(u\cdot\nabla)u+\nabla p=\mathrm{div}(\tau),\\\
\partial_{t}\tau+(u\cdot\nabla)\tau-\Delta\tau+\tau+\mathrm{Q}(\nabla
u,\tau)=a\mathbb{D}u,~{}~{}a\in(0,\epsilon_{0}],\\\ \mathrm{div}u=0,\\\
u(x,0)=u_{0}(x),~{}~{}~{}\tau(0,x)=\tau_{0}(x),\end{cases}$ (4.13)

and

$\begin{cases}\partial_{t}u+(u\cdot\nabla)u+\nabla p=\mathrm{div}(\tau),\\\
\partial_{t}\tau+(u\cdot\nabla)\tau-\Delta\tau+\tau+\mathrm{Q}(\nabla
u,\tau)=0,\\\ \mathrm{div}u=0,\\\
u(x,0)=u_{0}(x),~{}~{}~{}\tau(0,x)=\tau_{0}(x),\end{cases}$ (4.14)

To prove the instability between (4.13) and (4.14) as $a\rightarrow 0$, we
first give the definition of the global stability:

$\lim_{a\rightarrow
0}\|u^{0}-u^{a}\|_{L^{\infty}_{t}(B^{2}_{2,1})}+\|\tau^{0}-\tau^{a}\|_{L^{\infty}_{T}(B^{1}_{2,1})}=0,~{}~{}\forall(u_{0},\tau_{0})\in\mathbb{A}~{}and~{}\forall
t\in[0,\infty),$

where
$\mathbb{A}:=\\{(u_{0},\tau_{0})\in(B^{2}_{2,1}(\mathbb{R}^{2}),B^{1}_{2,1}(\mathbb{R}^{2}))|~{}\eqref{th4-2}~{}and~{}\eqref{th4-1}~{}has~{}a~{}unique~{}solution~{}for~{}any~{}fixed~{}a\\}.$

In order to prove the instability in large time, we should prove that for any
$a\in[0,\epsilon_{0}]$, there exists a common initial sequence
$(u_{0},\tau_{0})(a)\in\mathbb{A}$ and a $T(a)$ such that, when $t\geq T(a)$,
we have

$\|(u^{0}-u^{a})(t)\|_{B^{2}_{2,1}}\geq\frac{\epsilon_{0}}{8},$

Now, let $\phi\in\mathbb{S}^{2}$ with $\|\phi\|_{L^{2}}=1$ and $div\phi=0$.
Set the initial data

$(u_{0},\tau_{0})(a)=\frac{\epsilon_{0}a}{2}(\phi(ax),a^{2}\phi(ax)),$

For any $0<a\leq\epsilon_{0}$, we have

$\|u_{0}\|_{L^{2}}=\frac{\epsilon_{0}}{2},~{}\|\tau_{0}\|_{H^{3}}\leq
C\epsilon_{0}a^{2}~{}~{}and~{}~{}\|w_{0}\|_{B^{0}_{\infty,1}}\leq\|w_{0}\|_{H^{2}}\leq
C\epsilon^{2}_{0},~{}~{}w_{0}=curlu_{0}.$

These satisfy the conditions in Theorem 1.1 and Theorem 1.2, which means
$(u_{0},\tau_{0})\in\mathbb{A}$.

On one hand, by Theorem 1.1, (4.13) has an unique global strong solution
$(u^{a},\tau^{a})$ with the initial data $(u_{0},\tau_{0})(a)$. We also obtain
the $L^{2}$ decay such that:

$\displaystyle\|(u^{a},\tau^{a})(t)\|_{L^{2}}\leq
C(\|u_{0}\|_{H^{1}}+\|\tau_{0}\|_{H^{1}})[a(1+t)]^{-\frac{1}{2}}.$ (4.15)

On the other hand, for the system (4.14), by Theorem 1.2 we also obtain a
unique global solution $(u^{0},\tau^{0})$ with the same initial data
$(u_{0},\tau_{0})(a)$. Although coefficients of the system (4.14) are
independent of $a$, one can still look for the initial data which is dependent
of $a$. Then, using the first equation of (4.14), We have

$\displaystyle\|u^{0}(t)\|^{2}_{L^{2}}\geq\|u_{0}\|^{2}_{L^{2}}-\int_{0}^{t}\|div\tau\|_{L^{2}}\|u\|_{L^{2}}.$
$\displaystyle\geq\|u_{0}\|^{2}_{L^{2}}-\|e^{-\frac{s}{2}}\|_{L^{2}_{t}}\|e^{\frac{s}{2}}div\tau\|_{L^{2}_{t}(L^{2})}\|u\|_{L^{\infty}_{t}(L^{2})}$
$\displaystyle\geq\frac{\epsilon^{2}_{0}}{4}-C\|\tau_{0}\|_{L^{2}}(\|u_{0}\|_{L^{2}}+\|\tau_{0}\|_{L^{2}})$
$\displaystyle\geq\frac{\epsilon^{2}_{0}}{4}-C\frac{\epsilon^{3}_{0}}{2}$
$\displaystyle=\frac{\epsilon^{2}_{0}}{16},$ (4.16)

where we use (3.10) for the third inequality.

Therefore, there exists a $T(a):=\frac{1}{a^{2}}$ such that when $t\geq T(a)$,
we have

$\displaystyle\|u^{0}(t)-u^{\alpha}(t)\|_{L^{2}}$
$\displaystyle\geq\|u^{0}(t)\|_{L^{2}}-\|u^{\alpha}(t)\|_{L^{2}}$
$\displaystyle\geq\frac{\epsilon_{0}}{4}-C(\|u_{0}\|_{H^{1}}+\|\tau_{0}\|_{H^{1}})[a(1+t)]^{-\frac{1}{2}}$
$\displaystyle\geq\frac{\epsilon_{0}}{4}-C\epsilon_{0}[a(1+t)]^{-\frac{1}{2}}$
$\displaystyle\geq\frac{\epsilon_{0}}{4}-\frac{\epsilon_{0}}{8}$
$\displaystyle=\frac{\epsilon_{0}}{8},$ (4.17)

where the second inequality is based on (4.15). This complete the proof
Theorem 1.3.

∎

## Acknowledgments

This work is partially supported by the National Natural Science Foundation of
China (Nos. 11801574, 11971485), Natural Science Foundation of Hunan Province
(No. 2019JJ50788), Central South University Innovation-Driven Project for
Young Scholars (No. 2019CX022) and Fundamental Research Funds for the Central
Universities of Central South University, China (Nos. 2020zzts038,
2021zzts0041).

## References

  * [1] H. Bahouri, J.-Y. Chemin, and R. Danchin, Fourier analysis and nonlinear partial differential equations, vol. 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Heidelberg, 2011.
  * [2] R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of polymeric liquids: v. 1 fluid mechanics, (1987).
  * [3] J. Bourgain and D. Li, Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces, Invent. Math., 201 (2015), pp. 97–157.
  * [4] , Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces, Geom. Funct. Anal., 25 (2015), pp. 1–86.
  * [5] F. Charve and R. Danchin, A global existence result for the compressible Navier-Stokes equations in the critical $L^{p}$ framework, Arch. Ration. Mech. Anal., 198 (2010), pp. 233–271.
  * [6] J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, J. Differential Equations, 121 (1995), pp. 314–328.
  * [7] J.-Y. Chemin and N. Masmoudi, About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal., 33 (2001), pp. 84–112.
  * [8] Q. Chen and X. Hao, Global well-posedness in the critical Besov spaces for the incompressible Oldroyd-B model without damping mechanism, J. Math. Fluid Mech., 21 (2019), pp. Paper No. 42, 23.
  * [9] Q. Chen, C. Miao, and Z. Zhang, Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity, Comm. Pure Appl. Math., 63 (2010), pp. 1173–1224.
  * [10] Z. Chen, L. Q. Liu, and Y. W. K. Qin, Dong Dong, Global Regularity for the incompressible Oldroyd-B model without damping mechanism, NON-105700, (2021).
  * [11] P. Constantin and M. Kliegl, Note on global regularity for two-dimensional Oldroyd-B fluids with diffusive stress, Arch. Ration. Mech. Anal., 206 (2012), pp. 725–740.
  * [12] W. J. Z. J. Constantin, Peter and Z. Yi, High reynolds number and high weissenberg number oldroyd-b model with dissipation, J. Evol. Equ., 21 (2021).
  * [13] R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), pp. 579–614.
  * [14] W. Deng, Z. Luo, and Z. Yin, Global solutions and large time behavior for some Oldroyd-B type models in $R^{2}$, arXiv e-prints, (2021), p. arXiv:2107.12029v1.
  * [15] T. M. Elgindi and J. Liu, Global wellposedness to the generalized Oldroyd type models in $\mathbb{R}^{3}$, J. Differential Equations, 259 (2015), pp. 1958–1966.
  * [16] T. M. Elgindi and F. Rousset, Global regularity for some Oldroyd-B type models, Comm. Pure Appl. Math., 68 (2015), pp. 2005–2021.
  * [17] E. Fernández-Cara, F. Guillén, and R. R. Ortega, Some theoretical results concerning non-Newtonian fluids of the Oldroyd kind, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), pp. 1–29.
  * [18] C. Guillopé and J.-C. Saut, Existence results for the flow of viscoelastic fluids with a differential constitutive law, Nonlinear Anal., 15 (1990), pp. 849–869.
  * [19] , Global existence and one-dimensional nonlinear stability of shearing motions of viscoelastic fluids of Oldroyd type, RAIRO Modél. Math. Anal. Numér., 24 (1990), pp. 369–401.
  * [20] Z. Guo, J. Li, and Z. Yin, Local well-posedness of the incompressible Euler equations in $b^{\infty}_{\infty,1}$ and the inviscid limit of the Navier–Stokes equations, J. Funct. Anal., 276 (2018), pp. 1361–1406.
  * [21] D. Hu and T. Lelièvre, New entropy estimates for Oldroyd-B and related models, Commun. Math. Sci., 5 (2007), pp. 909–916.
  * [22] X. Huo and W. Yong, Global existence for viscoelastic fluids with infinite Weissenberg number, Commun. Math. Sci., 15 (2017), pp. 1129–1140.
  * [23] F.-H. Lin, C. Liu, and P. Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math., 58 (2005), pp. 1437–1471.
  * [24] P. L. Lions and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows, Chinese Ann. Math. Ser. B, 21 (2000), pp. 131–146.
  * [25] J. G. Oldroyd, Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids, Proc. Roy. Soc. London Ser. A, 245 (1958), pp. 278–297.
  * [26] T. Tao, Nonlinear dispersive equations, vol. 106 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006.  Local and global analysis.
  * [27] D. Wei and Z. Zhang, Global well-posedness for the 2-D MHD equations with magnetic diffusion, Commun. Math. Res., 36 (2020), pp. 377–389.
  * [28] Z. J. F. Wu, Jiahong, Global regularity for the generalized incompressible Oldroyd-B model with only stress tentor dissipation in critical besov spaces, preprint, (2021).
  * [29] , Global regularity for the generalized incompressible Oldroyd-B model with only velocity dissipation and no stress tensor damping, preprint, (2021).
  * [30] W. Ye, W. Luo, and Z. Yin, The estimate of lifespan and local well-posedness for the non-resistive MHD equations in homogeneous Besov spaces, (arXiv:2012.03489v1).
  * [31] W. Ye and Z. Yin, Global well-posedness for the non-viscous MHD equations with magnetic diffusion in critical Besov spaces, (arXiv:2012.03489).
  * [32] X. Zhai, Global solutions to the $n$-dimensional incompressible Oldroyd-B model without damping mechanism, J. Math. Phys., 62 (2021), pp. 021503, 17.
  * [33] X. Zhai, Y. Dan, and Y. Li, Global well-posedness and inviscid limits of the generalized Oldroyd type models, arXiv e-prints, (2021), p. arXiv:2106.14785.
  * [34] Y. Zhu, Global small solutions of 3D incompressible Oldroyd-B model without damping mechanism, J. Funct. Anal., 274 (2018), pp. 2039–2060.
  * [35] R. Zi, D. Fang, and T. Zhang, Global solution to the incompressible Oldroyd-B model in the critical $L^{p}$ framework: the case of the non-small coupling parameter, Arch. Ration. Mech. Anal., 213 (2014), pp. 651–687.


challenge lies in harnessing superoscillations effectively, akin to their
application in material physics, to probe the underlying principles of Nature.

Acknowledgements. AA would like to thank Massimo Bianchi, Salvatore
Capozziello, Giampiero Esposito and Andrea Marini for valuable comments and
suggestions. AA work is supported by the National Science Foundation of China
(NSFC) through the grant No. 12350410358; the Talent Scientific Research
Program of College of Physics, Sichuan University, Grant No. 1082204112427;
the Fostering Program in Disciplines Possessing Novel Features for Natural
Science of Sichuan University, Grant No.2020SCUNL209 and 1000 Talent program
of Sichuan province 2021.

## Appendix A: Gupta-Bleuler quantization

In this section, we will show how to quantize superoscillating fields in a
Lorentz invariant fashion with the Lorentz Gauge $\partial_{\mu}A^{\mu}=0$ and
a general lagrangian

$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2\alpha}(\partial_{\mu}A^{\mu})^{2}\,.$
(IV.202)

with $\alpha$ an arbitrary constant ($\alpha=1$ corresponds to Feynman’s gauge
and $\alpha=0$ to Landau’s gauge). The corresponding equations of motion is

$\Box A^{\nu}=0\,.$ (IV.203)

These equations have superoscillating solutions with corresponding
superoscillating quantum fields as follows:

$A^{\mu}(t,\boldsymbol{x})=\left.\sum_{k}c_{k}\int\frac{d^{3}\boldsymbol{p}}{(2\pi)^{3}}\frac{1}{2\left(1-\frac{2k}{n}\right)^{2}E}\sum_{r=0}^{3}\left\\{b^{r}(\boldsymbol{p})\epsilon_{r}^{\mu}(\boldsymbol{p})e^{-i\left(1-\frac{2k}{n}\right)px}+b^{r*}(\boldsymbol{p})\epsilon_{r}^{*\mu}(\boldsymbol{p})e^{i\left(1-\frac{2k}{n}\right)px}\right\\}\right|_{E=|\boldsymbol{p}|}$
(IV.204)

with conjugate momenta $\pi^{\mu}(t,\boldsymbol{x})=\partial_{t}A^{\mu}$,

$\pi^{\mu}(t,\boldsymbol{x})=i\sum_{k}c_{k}\int\frac{d^{3}\boldsymbol{p}}{(2\pi)^{3}}\frac{1}{2\left(1-\frac{2k}{n}\right)}\sum_{r=0}^{3}\left\\{-b^{r}(\boldsymbol{p})\epsilon_{r}^{\mu}(\boldsymbol{p})e^{-i\left(1-\frac{2k}{n}\right)px}+b^{r*}(\boldsymbol{p})\epsilon_{r}^{*\mu}(\boldsymbol{p})e^{i\left(1-\frac{2k}{n}\right)px}\right\\}\,,$
(IV.205)

where both $\hat{\pi}^{0}=\partial\mathcal{L}/\partial\dot{\hat{A}}_{0}$ and
$\hat{\pi}^{i}=\partial\mathcal{L}/\partial\dot{\hat{A}}_{i}$ are treated as
dynamical. After the quantization, the $\partial_{\mu}\hat{A}^{\mu}=0$
constraint will reduce the degrees of freedom in the Fock space of the theory.
Above, we define 4-polarization vectors $\epsilon_{\mu}^{\lambda}$ with
$\lambda=0,1,2,3,4$ instead of 3 as in Coulomb gauge. The polarization
4-vectors are normalized as
$\epsilon_{\mu}^{\lambda}\epsilon_{\nu}^{\lambda^{\prime}}\eta_{\lambda\lambda^{\prime}}=\eta_{\mu\nu}$.

We define properly normalised annihilation/creation operators as

$\displaystyle\hat{b}^{r}(\mathbf{p},k,n,a)$
$\displaystyle=\sqrt{n+1}\frac{c_{k}(n,a)}{\left(1-\frac{2k}{n}\right)^{2}\sqrt{2E}}\hat{b}^{r}(\mathbf{p})$
(IV.206) $\displaystyle\hat{b}^{r\dagger}(\mathbf{p},k,n,a)$
$\displaystyle=\sqrt{n+1}\frac{c_{k}(n,a)}{\left(1-\frac{2k}{n}\right)^{2}\sqrt{2E}}\hat{b}^{r\dagger}(\mathbf{p}).$

and promote the corresponding classical fields to quantum operators as

$\hat{A}^{\mu}(t,\boldsymbol{x})=\sum_{k}\int\frac{d^{3}\boldsymbol{p}}{(2\pi)^{3}}\frac{1}{\sqrt{2(n+1)E}}$
$\times\sum_{r=0}^{3}\left\\{\hat{b}^{r}(\boldsymbol{p},k,n,a)\epsilon_{r}^{\mu}(\boldsymbol{p})e^{-i\left(1-\frac{2k}{n}\right)px}+\hat{b}^{r\dagger}(\boldsymbol{p},k,n,a)\epsilon_{r}^{*\mu}(\boldsymbol{p})e^{i\left(1-\frac{2k}{n}\right)px}\right\\}$
(IV.207)
$\hat{\pi}^{\mu}(t,\boldsymbol{x})=i\sum_{k}\int\frac{d^{3}\boldsymbol{p}}{(2\pi)^{3}}\frac{\left(1-\frac{2k}{n}\right)\sqrt{E}}{\sqrt{2(n+1)}}$
$\times\sum_{r=0}^{3}\left\\{-\hat{b}^{r}(\boldsymbol{p},k,n,a)\epsilon_{r}^{\mu}(\boldsymbol{p})e^{-i\left(1-\frac{2k}{n}\right)px}+\hat{b}^{r\dagger}(\boldsymbol{p},k,n,a)\epsilon_{r}^{*\mu}(\boldsymbol{p})e^{i\left(1-\frac{2k}{n}\right)px}\right\\}\,.$
(IV.208)

We impose the quantization conditions as

$[\hat{A}_{\mu}({\bf x}),\hat{A}_{\nu}({\bf y})]=[\hat{\pi}_{\mu}({\bf
x}),\hat{\pi}_{\nu}({\bf y})]=0\,,$ $[\hat{A}_{\mu}({\bf
x}),\hat{\pi}_{\nu}({\bf y})]=i\eta_{\mu\nu}\delta^{(3)}({\bf x}-{\bf y})\,.$
(IV.209)

corresponding to

$[\hat{b}^{\lambda}({\bf p},k,n,a),\hat{b}^{\dagger\lambda^{\prime}}({\bf
p^{\prime}},k^{\prime},n,a)]=(2\pi)^{3}\delta^{\lambda\lambda^{\prime}}\delta_{k,k^{\prime}}\delta^{(3)}({\bf
p}-{\bf p^{\prime}})\,,$ (IV.210)

with $\lambda,\lambda^{\prime}=1,2,3$, and

$[\hat{b}^{0}({\bf p},k,n,a),\hat{b}^{\dagger 0}({\bf
p^{\prime}},k^{\prime},n,a)]=-(2\pi)^{3}\delta_{k,k^{\prime}}\delta^{(3)}({\bf
p}-{\bf p^{\prime}})\,.$ (IV.211)

The Lorentz constraint on the Fock space
$\langle\Psi|\partial^{\mu}\hat{A}_{\mu}|\Psi\rangle=0$ can be implemented
through the condition

$\partial^{\mu}\hat{A}_{\mu}^{+}|\Psi\rangle=0\,,$ (IV.212)

where $\hat{A}_{\mu}=\hat{A}_{\mu}^{+}+\hat{A}_{\mu}^{-}$ and

$\hat{A}^{+\mu}(t,\boldsymbol{x})=\sum_{k}\int\frac{d^{3}\boldsymbol{p}}{(2\pi)^{3}}\frac{1}{\sqrt{2(n+1)E}}\sum_{r=0}^{3}\hat{b}^{r\dagger}(\boldsymbol{p})\epsilon_{r}^{*\mu}(\boldsymbol{p})e^{i\left(1-\frac{2k}{n}\right)px}\,.$
(IV.213)

Eq.IV.212 is the Gupta-Bleuler condition for superoscillating fields. Such a
constraint demands that

$(\hat{b}^{3}({\bf p},k,n,a)-\hat{b}^{0}({\bf p},k,n,a))|\Psi\rangle=0\,,$
(IV.214)

where $|\Psi\rangle=|\Psi_{T}\rangle|\Phi\rangle$ with $|\Psi_{T}\rangle$
containing only transverse photons and $|\Phi\rangle$ time-like and
longitudinal photons. Such a condition decouples the negative norm states from
positive ones in the L-Fock space.

## References

  * (1) M. Berry et al, J. Opt. 21 (2019) 053002.
  * (2) Y. Aharonov, J. Anandan, S. Popescu, L. Vaidman, Phys. Rev. Lett. 64 2965 (1990).
  * (3) M. V. Berry, in Proc. Intl. Conf. on Fund. Aspects of Quantum Theory, Columbia, SC, USA, 10-12 Dec. 1992 Eds. J.S. Anandan, J. L. Safko, World Scientific, Singapore (1995).
  * (4) M. V. Berry, J. Phys. A 27 L391 (1994)
  * (5) Y. Aharonov, B. Reznik, A. Stern, Phys. Rev. Lett. 81 2190 (1998).
  * (6) R. Buniy, F. Colombo, I. Sabadini and D.C. Struppa, J. Math. Phys. 55 113511 (2014) .
  * (7) F. Colombo, J. Gantner and D.C. Struppa, J. Math. Phys. 58 092013 (2017).
  * (8) M.R. Dennis, A.C. Hamilton, J. Courtial, Opt. Lett. 33 2976–8 (2018).
  * (9) R. Remez, A. Arie, Optica 2 472–5 (2015).
  * (10) A.M.H. Wong, G.V. Eleftheriades, IEEE Trans. Antennas Propag. 59 4766–76 (2011).
  * (11) Aharonov Y, Popescu S and Rohrlich D 1991 How can an infrared photon behave as a gamma ray? Tel-Aviv University Preprint TAUP 1847-90.
  * (12) Aharonov, Y., Popescu, S., Rohrlich, D. On conservation laws in quantum mechanics, arXiv e-printsarXiv: 1609.05041 [quant-ph] (2016).
  * (13) V. Fock, Z. Phys. , 75 (1932) pp. 622–647.
  * (14) F.A. Berezin, Acad. Press (1966) (Translated from Russian) (Revised (augmented) second edition: Kluwer, 1989).
  * (15) D. M. Eardley and S. B. Giddings, Phys. Rev. D 66, 044011 (2002).
  * (16) E. Kohlprath and G. Veneziano, JHEP 0206 (2002) 057 [gr-qc/0203093].
  * (17) G. Amelino-Camelia, Nature 398, 216 (1999).
  * (18) J Aasi, B P Abbott, R Abbott, T Abbott, M R Abernathy, K Ackley, C Adams, T Adams, P Addesso, R X Adhikari, et al., Classical and Quantum Gravity, 32(7):074001, mar 2015.
  * (19) B.P. Abbott et al., Phys. Rev. Lett., 116(13):131103, 2016.
  * (20) Amaro-Seoane, Pau, et al., "Laser interferometer space antenna." arXiv preprint arXiv:1702.00786 (2017).
  * (21) A. M. Cruise. Class. Quant. Grav., 29:095003, 2012.
  * (22) N. Aggarwal, O. D. Aguiar, A. Bauswein, G. Cella, S. Clesse, A. M. Cruise, V. Domcke, D. G. Figueroa, A. Geraci and M. Goryachev, et al. Living Rev. Rel. 24 (2021) no.1, 4 doi:10.1007/s41114-021-00032-5 [arXiv:2011.12414 [gr-qc]].
  * (23) M. Bianchi, H. Elvang and D. Z. Freedman, JHEP 09 (2008), 063 doi:10.1088/1126-6708/2008/09/063 [arXiv:0805.0757 [hep-th]].
  * (24) C. Marletto and V. Vedral, Phys. Rev. Lett. 119 (2017) no.24, 240402 doi:10.1103/PhysRevLett.119.240402 [arXiv:1707.06036 [quant-ph]].
  * (25) S. Bose, A. Mazumdar, G. W. Morley, H. Ulbricht, M. Toroš, M. Paternostro, A. Geraci, P. Barker, M. S. Kim and G. Milburn, Phys. Rev. Lett. 119 (2017) no.24, 240401 doi:10.1103/PhysRevLett.119.240401 [arXiv:1707.06050 [quant-ph]].
  * (26) T. W. van de Kamp, R. J. Marshman, S. Bose and A. Mazumdar, Phys. Rev. A 102 (2020) no.6, 062807 doi:10.1103/PhysRevA.102.062807 [arXiv:2006.06931 [quant-ph]].
  * (27) M. Tse et al. (LIGO collaboration), Phys. Rev. Lett. 123, 231107, https://doi.org/10.1103/PhysRevLett.123.231107.
  * (28) D. Walls, Nature 306, 141-146 (1983), https://doi.org/10.1038/306141a0.
  * (29) M. Punturo et al., Quant. Grav. 27 (2010) 194002.
  * (30) S. Hild et al., Class. Quant. Grav. 28 (2011) 094013, arXiv:1012.0908 [gr-qc].
  * (31) D. Reitze et al., Bull. Am. Astron. Soc. 51 no. 7, (2019) 035, arXiv:1907.04833 [astro-ph.IM].
  * (32) M. Evans et al., “A Horizon Study for Cosmic Explorer: Science, Observatories, and Community,” arXiv:2109.09882 [astro-ph.IM].
  * (33) V. F. Mukhanov, "Quantum Theory of Gauge Invariant Cosmological Perturbations," Soviet Physics JETP, vol. 67, no. 7, 1985, pp. 1297-1302.
  * (34) M. Sasaki, "Gauge Invariant Scalar Perturbations in the New Inflationary Universe," Progress of Theoretical Physics, vol. 70, no. 2, 1983, pp. 394-411.
  * (35) M.C. Gonzalez-Garcia, M. Yokoyama, 14. Neutrino Masses, Mixings and Oscillations, Particle Data Group, https://pdg.lbl.gov/2020/reviews/rpp2020-rev-neutrino-mixing.pdf.
  * (36) J.R. Ellis, J.S. Hagelin, D.V. Nanopoulos and M. Srednicki, Search for violations of quantum mechanics, Nucl. Phys. B 241 (1984) 381.
  * (37) G. Amelino-Camelia and F. Buccella, Mod. Phys. Lett. A 15 (2000), 2119-2128 doi:10.1142/S0217732300002474 [arXiv:hep-ph/0001305 [hep-ph]].
  * (38) A. Addazi, P. Belli, R. Bernabei and A. Marciano, Chin. Phys. C 42 (2018) no.9, 094001 doi:10.1088/1674-1137/42/9/094001 [arXiv:1712.08082 [hep-th]].
  * (39) A. Addazi, J. Alvarez-Muniz, R. Alves Batista, G. Amelino-Camelia, V. Antonelli, M. Arzano, M. Asorey, J. L. Atteia, S. Bahamonde and F. Bajardi, et al. Prog. Part. Nucl. Phys. 125 (2022), 103948 doi:10.1016/j.ppnp.2022.103948 [arXiv:2111.05659 [hep-ph]].
  * (40) L. Maiani, G. Pancheri, N. Paver, eds., The Second DA$\Phi$NE Physics Handbook, INFN, Laboratori Nazionali di Frascati, 1995.
  * (41) A. Di Domenico, “Kaon Interferometry: a Quantum Test of the Equivalence Principle,” Foundations of Physics, Vol. 40, Issue 8, pp. 852-866 (2010).
  * (42) A. Angelopoulos et al, $K_{0}-\bar{K}_{0}$ mass and decay-width differences: CPLEAR evaluation, report CERN-EP/99-150, Phys. Lett. B 471 (1999) 332-8.
  * (43) R. Carosi et al, Phys. Lett. B237 (1990) 303.
  * (44) M. Woods et al, Phys. Rev. Lett. 60 (1988) 1695; J.R. Patterson et al, Phys. Rev. Lett. 64 (1990) 1491.
  * (45) KLOE-2 Collaboration, “Precision tests of discrete symmetries in the $K^{0}$ system with the KLOE detector,” The European Physical Journal C (2014) 74: 3198.
  * (46) A. Addazi, Int. J. Geom. Meth. Mod. Phys. 14 (2016) no.01, 1750012 doi:10.1142/S0219887817500128 [arXiv:1607.02593 [hep-th]].
  * (47) A.M. Steinberg, 1995, Phys. Rev. A 52, 32.
  * (48) A.M. Steinberg, 1995, Phys. Rev. Lett. 74, 2405.
  * (49) V. S.Olkhovsky, E. Recami, G. Salesi, Eur. Phys. Letters, 57 (2002) 879-884, https://doi.org/10.48550/arXiv.quant-ph/0002022, arXiv:quant-ph/0002022.
  * (50) Y. Aharonov, N. Erez, B. Reznik, Phys. Rev. A 65, 052124 (2002), https://doi.org/10.48550/arXiv.quant-ph/0110104 [arXiv:quant-ph/0110104]
  * (51) S.W. Hawking, Commun. Math. Phys. 43 (1975) 199.
  * (52) S. W. Hawking, Phys. Rev. D14, 2460-2473 (1976)
  * (53) J. Pinochet, Phys. Educ. 51 (2016) 015010.
  * (54) E. Keski-Vakkuri and P. Kraus, Phys. Rev. D54 (1996) 7407.
  * (55) P. Kraus and F. Wilczek, Nucl. Phys. B433 (1995) 403.
  * (56) M.K. Parikh and F. Wilczek, Phys. Rev. Lett. 85 (2000) 5042.
  * (57) L. Susskind, L. Thorlacius and J. Uglum, Phys. Rev. D 48 (1993), 3743-3761 doi:10.1103/PhysRevD.48.3743 [arXiv:hep-th/9306069 [hep-th]].
  * (58) S. L. Braunstein, S. Pirandola and K. Życzkowski, Phys. Rev. Lett. 110 (2013) no.10, 101301 doi:10.1103/PhysRevLett.110.101301 [arXiv:0907.1190 [quant-ph]].
  * (59) A. Almheiri, D. Marolf, J. Polchinski and J. Sully, JHEP 02 (2013), 062 doi:10.1007/JHEP02(2013)062 [arXiv:1207.3123 [hep-th]].
  * (60) Don N. Page, Phys. Rev. Lett. 71, 3743–3746 (1993), arXiv:hep-th/9306083 [hep-th]
  * (61) A. Kempf J. Math. Phys. 41 2360 (2000).
  * (62) H. Rosu, Nuovo Cim. 112B 131 (1997), gr-qc/9606070.
  * (63) B. Reznik, Phys. Rev. D55 2152 (1997)


# A Short Note on Modeling 2D Taut Ropes with Visibility Decompositions

Adem Berat Dalkılıç

###### Abstract

The problem of modeling ropes arises in many applications, including providing
haptic feedback to surgeons who are using surgical robots to realign the
distal and proximal ends of split bones. Here, we consider a simplified, 2D
variant of the haptic feedback estimation problem and discuss how visibility
decompositions greatly simplify the problem. Then, we introduce an efficient,
concise algorithm for modeling the dynamics of 2D ropes around polygonal
obstacles in O($n$) time, where $n$ is the number of line segment obstacles.

##  1 PROBLEM STATEMENT

We start by providing a brief definition of our 2D rope problem. We define a
constrained space as an open subset of $\mathbb{R}^{2}$, $C$, and the free
space as $\mathbb{R}^{2}$ \- $C$, or $F$. Two points in $\mathbb{R}^{2}$ are
said to be visible if the open line segment connecting them never lies in $C$.
In this simplified version of the problem we define obstacles as open line
segments made up of two points. The open line segment obstacles constitute $C$
entirely.

Consider a taut rope with two endpoints, $A$ and $B$, where $A$ and $B$ are
initially visible to one another. Consider the trace created when $A$ is moved
about such that it always remains in $F$. We parameterize this trace, such
that $A(t)$ defines the positions of $A$ in $\mathbb{R}^{2}$ as a function of
$t$. Because the trace is described as a set of connected line segments, the
generation of such traces requires that for all $t$, $A(t-1)$ and $A(t+1)$
must both be visible to $A(t)$; if this condition was not true, then the rope
would be passing through obstacle line segments, which is impossible. We
define this requirement as the small update condition. We also define the
essential single-cut condition in the next section.

If the shortest path is generated between $A$ and $B$ for each parameter $t$,
then, excluding the cases where $A$ is visible to $B$, the path will contain
at least one point of an obstacle line segment, which we call a wrapping
point. We can define the rope at parameter $t$, $R(t)$, as an ordered set of
wrapping points contained by the trace between $A$ and $B$ at time $t$ if that
trace were to be made taut; $R(t)$ is ordered by recency: the least recent
wrapping point that was appended to $R(t)$ is the first element, and the most
recently appended wrapping point is the last element.

To limit technicality, a very informal problem definition is presented. Given
a trace, $A(t)$, which fulfills the small update and single-cut conditions, we
wish to compute for all $t$ the taut rope made when the two ends of the trace
are tightened without the locations of the endpoints of the trace actually
changing.

Figure 1: Visual depiction of the algorithm. $B$, the fixed end of the rope
(filled circle) is shown, the rays of the GD that were computed until an
intersection was found are shown (dash-dot lines), the unwrapping ray is shown
(dotted), the rope is shown (dashed), and the obstacle line segments are shown
(straight) for all $t$ of $A(t)$ (hollow circle).

##  2 A SIMPLE SOLUTION

The solution presented here requires that a partial visibility decomposition
of $\mathbb{R}^{2}$ is computed. We call this pseudo-visibility decomposition
a Gazan decomposition (GD). The GD of $\mathbb{R}^{2}$ is the collection of
all rays made by an observer point and all obstacle points; specifically, the
rays begin from each obstacle point and extend away from the observer point
infinitely with a slope equal to that of the line formed between the observer
and obstacle point. The GD is useful because it provides a quantitative metric
to determine when a new wrapping point should be added or removed from the
rope. Specifically, when a ray of the GD is passed through by a movement of
the trace, the source vertex of that ray must be appended to the rope. Because
there are $n$ obstacle line segments, for every $t$, $n+1$ ray intersection
tests must be performed between two entities: (1) the line segment defined by
$A(t)$ and $A(t+1)$; (2) all rays of the GD with the most recent element of
$R(t)$ as the observer point. To facilitate unwrapping, it is required to also
include the most recently intersected ray as part of the rays to be checked
for intersections, even if that ray is not part of the current GD. This is why
there are $n+1$ intersection tests that must be performed instead of $n$. If
it is shown that the normal from $A(t)$ to $A(t+1)$ passes through a ray of
the GD of $R(t)$, then, the source vertex of that ray must be added to
$R(t+1)$. However, if the ray is the same as the most recently passed through
ray, then the source vertex of that ray must be removed from $R(t+1)$. The
single-cut condition requires that, for all $t$, the norm created by the
points $A(t)$ and $A(t+1)$ intersect at most one ray of not only the current
GD, but also all the other GDs that are created when all other points are
taken as observer points. This condition is important for preventing ambiguity
when multiple rays are intersected simultaneously. Pythonic pseudocode for
this algorithm is shown in listing 1.

Listing 1: O($n$) Time Algorithm for 2D Rope Modeling

⬇

obstacleSpace = [((Px1, Py1), (Px2, Py2)), . . .,

((Px(f-1), Py(f-1)), (Pxf, Pyf))]

obstaclePoints = list(sum(obstacleSpace, ())) # flattens the list

rope = [B]

As = [(A1x, A1y), . . ., (Afx, Afy)]

A0 = As[0]

unwrappingRay = [(0,0), (0,0)]

for idx, A in enumerate(As[1:]):

rays = []

for obstaclePoint in obstaclePoints:

if obstaclePoint == rope[-1]:

continue

ray = (obstaclePoint, (obstaclePoint[0] + Inf *

(obstaclePoint[0] - rope[-1][0]),

obstaclePoint[1] + Inf *

(obstaclePoint[1] - rope[-1][1])))

rays.append(ray)

if intersect(A0, A, ray[0], ray[1]):

unwrappingRay = ray

rope.append(ray[0])

A0 = A

break

if not len(rope) < 2:

if intersect(A0, A, unwrappingRay[0], unwrappingRay[1]):

rope.pop()

if len(rope) != 1:

unwrappingRay = (rope[-1], (rope[-1][0] + Inf *

(rope[-1][0] - rope[-2][0]),

rope[-1][1] + Inf *

(rope[-1][1] - rope[-2][1])))

else:

unwrappingRay = ((0,0),(0,0))

A0 = A

break

A0 = A

Figure 2: Failure case caused by colinearity. $B$, the fixed end of the rope
(filled circle) is shown, the rope is shown (dashed), and the obstacle line
segments are shown (straight) for all $t$ of $A(t)$ (hollow circle).

##  3 A SMALL LIMITATION

If an obstacle space contains collinear line segments, then there is a
possibility of ambiguity when determining which segments were intersected. For
example, consider the case shown in figure 2. It is possible that the next
movement would pass through 2 rays simultaneously, making the correct vertex
to be chosen ambiguous. For this reason, this algorithm requires that there be
no collinear obstacle line segments. However, this limitation can be resolved
without increasing the time complexity: for all intersected rays, perform a
linear search to identify the ray whose source is furthest from the most
recent wrapping point of the rope; the furthest source should be the only
appended vertex to the rope. Angles could also have been used as an
alternative instead of rays when computing the GD, but rays are more
convenient for reasons that the reader can discover on their own.

##  4 AN OVERVIEW OF RELATED WORK FOR FURTHER READING

The core of this algorithm is derived from the visibility decomposition
technique, which was extensively explored in Boland’s thesis [3]. Visibility
decompositions can be thought of as a type of space partitioning in which
visibility to some set of vertices in the scene is the defining feature of
each partitioned region [5]. There have been many applications of visibility
decompositions for solving 2D visibility query problems over different
polygonal regions, especially for the visibility polygon problem. Bose et al.
was probably the first to use the visibility decomposition to generate a data
structure that facilitates O(log$n$) visibility query time for a query point
inside or outside a single, simple polygon [4]. The same decomposition had
been described by Aronov et al. [1], who used the decomposition to partition
the interior of a polygon into cells; a point lying in any cell would have a
unique visibility polygon. Zarei et al. [7] presented an extension of the
visibility decomposition and provided a data structure for solving the query
problem over a polygonal domain with holes. Inkulu and Kappor provided two
improved algorithms that used the visibility decomposition to solve the query
problem. Lu, provided alternative algorithms that incorporated aspects of the
visibility decomposition technique [6]. Shortly after, Baygi and Ghodsi
developed another data structure that incorporated visibility decompositions
to compute visibility polygons [2]. The previous techniques that utilize the
visibility decomposition focused on creating data structures that facilitate
O(log$n$) online query time by precomputing visibility decompositions. Here,
however, our technique iteratively computes a lighter form of the visibility
decomposition, the Gazan decomposition, facilitating linear online time
queries without any convoluted precomputation. Most importantly, we extend the
visibility decomposition technique from traditional visibility querying to
that of modeling taut ropes.

## References

  * [1] Aronov, Guibas, Teichmann, and Zhang.  Visibility queries and maintenance in simple polygons.  Discrete & Computational Geometry, 27:461–483, 2002.
  * [2] Mostafa Nouri Baygi and Mohammad Ghodsi.  Space/query-time tradeoff for computing the visibility polygon.  Computational Geometry, 46(3):371–381, 2013.
  * [3] Ralph Patrick Boland.  Polygon visibility decompositions with applications. University of Ottawa (Canada), 2002.
  * [4] Prosenjit Bose, Anna Lubiw, and J Ian Munro.  Efficient visibility queries in simple polygons.  Computational Geometry, 23(3):313–335, 2002.
  * [5] Ziv Gigus and Jitendra Malik.  Computing the aspect graph for line drawings of polyhedral objects.  IEEE Transactions on Pattern Analysis and Machine Intelligence, 12(2):113–122, 1990.
  * [6] Lin Lu, Chenglei Yang, and Jiaye Wang.  Point visibility computing in polygons with holes.  JOURNAL OF INFORMATION &COMPUTATIONAL SCIENCE, 8(16):4165–4173, 2011.
  * [7] Alireza Zarei and Mohammad Ghodsi.  Query point visibility computation in polygons with holes.  Computational Geometry, 39(2):78–90, 2008.