# Ill-posedness for the Euler equations in Besov spaces

Jinlu Li1, Yanghai Yu2,111E-mail<EMAIL_ADDRESS>yuyanghai214@sina.com(Corresponding author<EMAIL_ADDRESS>and Weipeng
Zhu3
1 School of Mathematics and Computer Sciences, Gannan Normal University,
Ganzhou 341000, China
2 School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002,
China
3 School of Mathematics and Big Data, Foshan University, Foshan, Guangdong
528000, China

Abstract: In the paper, we consider the Cauchy problem to the Euler equations
in $\mathbb{R}^{d}$ with $d\geq 2$. We construct an initial data $u_{0}\in
B^{\sigma}_{p,\infty}$ showing that the corresponding solution map of the
Euler equations starting from $u_{0}$ is discontinuous at $t=0$ in the metric
of $B^{\sigma}_{p,\infty}$, which implies the ill-posedness for this equation
in $B^{\sigma}_{p,\infty}$. We generalize the periodic result of Cheskidov and
Shvydkoy [4] (Proc. Amer. Math. Soc. 138 (2010), 1059–1067).

Keywords: Euler equations, Ill-posedness, Besov spaces

MSC (2010): 76D03; 35Q30

##  1 Introduction

In this article, we consider the Euler equations governing the motion of an
incompressible fluid in $\mathbb{R}^{d}$ with $d\geq 2$

$\displaystyle\begin{cases}\partial_{t}u+u\cdot\nabla u+\nabla
P=0,&\quad(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{d},\\\
\mathrm{div\,}u=0,&\quad(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{d},\\\
u(0,x)=u_{0}(x),&\quad x\in\mathbb{R}^{d},\end{cases}$ (1.1)

where the vector field
$u(t,x):[0,\infty)\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ stands for the
velocity of the fluid, the quantity
$P(t,x):[0,\infty)\times{\mathbb{R}}^{d}\to{\mathbb{R}}$ denotes the scalar
pressure, and $\mathrm{div\,}u=0$ means that the fluid is incompressible. The
mathematical study of the Euler equations of ideal hydrodynamics has a long
and distinguished history. We do not detail the literature since it is huge
and refer the readers to see the monographs of Bahouri–Chemin–Danchin [3] and
Majda–Bertozzi [13].

We say that the Cauchy problem (1.1) is Hadamard (locally) well-posed in a
Banach space $X$ if for any data $u_{0}\in X$ there exists (at least for a
short time) $T>0$ and a unique solution in the space $\mathcal{C}([0,T),X)$
which depends continuously on the data. In particular, we say that the
solution map is continuous if for any $u_{0}\in X$, there exists a
neighborhood $B\subset X$ of $u_{0}$ such that for every $u\in B$ the map
$u\mapsto U$ from $B$ to $\mathcal{C}([0,T];X)$ is continuous, where $U$
denotes the solution to (1.1) with initial data $u_{0}$. Our goal of this
paper is to consider the ill-posedness of (1.1) in
$B^{\sigma}_{p,\infty}(\mathbb{R}^{d})$ and generalize the periodic result of
Cheskidov and Shvydkoy [4]. Our main result of this paper reads as follows:

###### Theorem 1.1

Let $d\geq 2$. Assume that

$\sigma>1+\frac{d}{p}\quad\text{with}\quad 1\leq p\leq\infty.$

There exits $u_{0}\in B^{\sigma}_{p,\infty}(\mathbb{R}^{d})$ and a positive
constant $\varepsilon_{0}$ such that the data-to-solution map
$u_{0}\mapsto\mathbf{S}_{t}(u_{0})$ of the Cauchy problem (1.1) satisfies

$\displaystyle\limsup_{t\to
0^{+}}\|\mathbf{S}_{t}(u_{0})-u_{0}\|_{B^{\sigma}_{p,\infty}}\geq\varepsilon_{0}.$

###### Remark 1.1

Theorem 1.1 demonstrates the ill-posedness of the Euler equations in
$B^{\sigma}_{p,\infty}$. More precisely, there exists $u_{0}\in
B^{\sigma}_{p,\infty}$ such that the corresponding solution to the Euler
equations that starts from $u_{0}$ does not converge back to $u_{0}$ in the
metric of $B^{\sigma}_{p,\infty}$ as time goes to zero.

###### Remark 1.2

The method we used in proving Theorem 1.1 is completely different from that
[4].

###### Remark 1.3

Due to the incompressible condition $\eqref{e}_{2}$, the pressure can be
eliminated from (1.1). In fact, applying the Leray operator $\mathcal{P}$ to
$\eqref{e}_{1}$, then we have

$\displaystyle\partial_{t}u=-\mathcal{P}(u\cdot\nabla u)=:\mathbf{P}(u).$

Thus, we transform (1.1) equivalently into the nonlocal form

$\displaystyle\begin{cases}\partial_{t}u=\mathbf{P}(u),\\\
\mathrm{div\,}u=0,\\\ u(0,x)=u_{0}(x).\end{cases}$ (1.2)

Concerning the well-posedness of the Euler equations we refer to see [3, 5,
13]. Next, we recall some results about the properties of the data-to-solution
map. The first results of this type was proved by Kato [9] who showed that the
solution operator for the (inviscid) Burgers equation is not Hölder continuous
in the $H^{s}(\mathbb{T})$-norm $(s>3/2)$ for any Hölder exponent. Kato–Ponce
[11] obtained the continuity results for the solution map of the Euler
equations in $W^{s}_{p}$ (see also [10]). Pak–Park [17] established existence
and uniqueness of solutions of the Euler equations in $B^{1}_{\infty,1}$ and
showed that the solution map is in fact Lipschitz continuous when viewed as a
map between $B^{0}_{\infty,1}$ and $\mathcal{C}([0,\infty);B^{0}_{\infty,1})$.
Later, Cheskidov–Shvydkoy [4] proved that the solution of the Euler equations
cannot be continuous as a function of the time variable at $t=0$ in the spaces
$B^{s}_{r,\infty}(\mathbb{T}^{d})$ where $s>0$ if $2<r\leq\infty$ and
$s>d(2/r-1)$ if $1\leq r\leq 2$. Furthermore, Bourgain–Li [1, 2] proved the
strong local ill-posedness of the Euler equations in borderline Besov spaces
$B^{d/p+1}_{p,r}$ with $(p,r)\in[1,\infty)\times(1,\infty]$ when $d=2,3$.
Subsequently, Misiołek–Yoneda [14] studied the borderline cases and showed
that the 2D Euler equations are not locally well-posed in the sense of
Hardamard in the $C^{1}$ space and in the Besov space $B^{1}_{\infty,1}$.
Recently, Misiołek–Yoneda [15] showed that the solution map for the Euler
equations is not even continuous in the space of Hölder continuous functions
and thus not locally Hadamard well-posed in $B^{1+\sigma}_{\infty,\infty}$
with any $\sigma\in(0,1)$. Concerning the non-uniform continuity of the data-
to-solution map, we refer the readers to see [6, 7, 8, 16, 12].

##  2 Littlewood-Paley Decomposition and Besov Spaces

We will use the following notations throughout this paper. The notation
$A\lesssim B$ (resp., $A\gtrsim B$) means that there exists a harmless
positive constant $c$ such that $A\leq cB$ (resp., $A\geq cB$). For $X$ a
Banach space and $I\subset\mathbb{R}$, we denote by $\mathcal{C}(I;X)$ the set
of continuous functions on $I$ with values in $X$. Let us recall that for all
$u\in\mathcal{S}^{\prime}$, the Fourier transform $\mathcal{F}u$, also denoted
by $\widehat{u}$, is defined by

$\mathcal{F}u(\xi)=\widehat{u}(\xi)=\int_{\mathbb{R}^{d}}e^{-ix\cdot\xi}u(x)\mathrm{d}x\quad\text{for
any}\;\xi\in\mathbb{R}^{d}.$

Due to the Holdge decomposition, we know that any vector field
$f=(f_{1},...,f_{d})$ with components in $S_{h}^{\prime}(\mathbb{R}^{d})$ may
be decomposed into one potential part $\mathcal{Q}f$ and one divergence-free
part $\mathcal{P}f$, where the projectors $\mathcal{P}$ and $\mathcal{Q}$ are
defined by

$\displaystyle\mathcal{P}=\mathbb{I}+(-\Delta)^{-1}\nabla{\rm{div}}\quad\mbox{and}\quad\mathcal{Q}=-(-\Delta)^{-1}\nabla{\rm{div}}.$
(2.3)

Next, we will recall some facts about the Littlewood-Paley decomposition, the
nonhomogeneous Besov spaces and their some useful properties.

######  Proposition 2.1 (Littlewood-Paley decomposition, See [3])

Let $\mathcal{B}:=\\{\xi\in\mathbb{R}^{d}:|\xi|\leq\frac{4}{3}\\}$ and
$\mathcal{C}:=\\{\xi\in\mathbb{R}^{d}:\frac{3}{4}\leq|\xi|\leq\frac{8}{3}\\}.$
There exist two radial functions $\chi\in C_{c}^{\infty}(\mathcal{B})$ and
$\varphi\in C_{c}^{\infty}(\mathcal{C})$ both taking values in $[0,1]$ such
that

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

Moreover, we can deduce that $\varphi\equiv 1$ for
$\frac{4}{3}\leq|\xi|\leq\frac{3}{2}$. This basic fact will be used in the
sequel.

For every $u\in\mathcal{S^{\prime}}(\mathbb{R}^{d})$, the inhomogeneous dyadic
blocks ${\Delta}_{j}$ are defined as follows

$\displaystyle\Delta_{j}u=$ $\displaystyle 0,$ if $j\leq-2$;
$\displaystyle\Delta_{j}u=$
$\displaystyle\chi(D)u=\mathcal{F}^{-1}(\chi\mathcal{F}u),$ if $j=-1$;
$\displaystyle\Delta_{j}u=$
$\displaystyle\varphi(2^{-j}D)u=\mathcal{F}^{-1}\big{(}\varphi(2^{-j}\cdot)\mathcal{F}u\big{)},$
if $j\geq 0$.

In the inhomogeneous case, the following Littlewood-Paley decomposition makes
sense

$u=\sum_{j\geq-1}{\Delta}_{j}u\quad\text{for
any}\;u\in\mathcal{S^{\prime}}(\mathbb{R}^{d}).$

We turn to the definition of the Besov Spaces and norms which will come into
play in our paper.

######  Definition 2.1 (Besov Space [3])

Let $s\in\mathbb{R}$ and $(p,r)\in[1,\infty]^{2}$. The nonhomogeneous Besov
space $B^{s}_{p,r}(\mathbb{R}^{d})$ consists of all tempered distribution $f$
such that

$\displaystyle
B^{s}_{p,r}(\mathbb{R}^{d}):=\Big{\\{}f\in\mathcal{S}^{\prime}(\mathbb{R}):\;\|f\|_{B^{s}_{p,r}(\mathbb{R}^{d})}<\infty\Big{\\}},$

where

$\displaystyle\|f\|_{B^{s}_{p,r}(\mathbb{R}^{d})}:=$
$\displaystyle\left(\sum_{j\geq-1}2^{sjr}\|\Delta_{j}f\|^{r}_{L^{p}(\mathbb{R}^{d})}\right)^{1/r},$
if $1\leq r<\infty$, $\displaystyle\|f\|_{B^{s}_{p,r}(\mathbb{R}^{d})}:=$
$\displaystyle\sup_{j\geq-1}2^{sj}\|\Delta_{j}f\|_{L^{p}(\mathbb{R}^{d})},$ if
$r=\infty$.

###### Remark 2.1

It should be emphasized that the following facts will be often used implicity:

  * •

$B^{s}_{p,q}(\mathbb{R})\hookrightarrow
B^{t}_{p,r}(\mathbb{R})\quad\text{for}\;s>t\quad\text{or}\quad s=t,1\leq q\leq
r\leq\infty.$

  * •

For $1\leq p\leq\infty$ and $s>\frac{d}{p}+1$,
$B^{s}_{p,\infty}(\mathbb{R}^{d})$ is a Banach algebra.

##  3 Proof of Theorem 1.1

In this section, we prove Theorem 1.1 only for the case $d=2$ since the case
$d\geq 3$ requires very little changes (for more details see [12]).

###  3.1 Construction of Initial Data

Firstly, we need to introduce smooth, radial cut-off functions to localize the
frequency region. Let $\widehat{\phi}\in\mathcal{C}^{\infty}_{0}(\mathbb{R})$
be an even, real-valued and non-negative function on $\mathbb{R}$ and satisfy

$\displaystyle\widehat{\phi}(\xi)=$ $\displaystyle 1,$ if
$|\xi|\leq\frac{1}{16}$,  $\displaystyle\widehat{\phi}(\xi)=$ $\displaystyle
0,$ if $|\xi|\geq\frac{1}{4}$.

We set the initial data $u_{0}(x)$ as follows:

$\displaystyle u_{0}(x):=\sum\limits^{\infty}_{j=0}f_{kj}(x),\quad
x\in\mathbb{R}^{2},\;k\in\mathbb{N}^{+},$ (3.4)

where

$\displaystyle f_{m}(x):=2^{-m(\sigma+1)}\begin{pmatrix}-\partial_{2}\\\
\partial_{1}\end{pmatrix}g_{m}\quad\text{with}\quad
g_{m}(x):=\phi(x_{1})\cos\Big{(}\frac{17}{12}2^{m}x_{1}\Big{)}\phi(x_{2}),\quad
m\geq 0.$

###  3.2 A Key Lemma

We establish the following crucial Lemmas which involves the properties of the
initial data.

###### Lemma 3.1

Define the initial data $u_{0}(x)$ by (3.4) above. Then there exist some
sufficiently large $k$ and $n$ such that

(1)

$\mathord{{\rm div}}u_{0}=0$;

(2)

$\|u_{0}\|_{B^{\sigma}_{p,\infty}}\leq C$;

(3)

$\|\Delta_{kn}(u_{0}\cdot\nabla u_{0})\|_{L^{p}}\geq c2^{kn}2^{-kn\sigma}$,

where $C$ and $c$ are uniform constants.

Proof. A direct calculation yields (1).

Easy computations give that

$\displaystyle\widehat{f}_{m}(\xi)=2^{-{m}(\sigma+1)-1}\begin{pmatrix}-i\xi_{2}\\\
i\xi_{1}\end{pmatrix}\Big{[}\widehat{\phi}\Big{(}\xi_{1}-\frac{17}{12}2^{m}\Big{)}+\widehat{\phi}\Big{(}\xi_{1}+\frac{17}{12}2^{m}\Big{)}\Big{]}\widehat{\phi}(\xi_{2}),$

which implies

$\displaystyle\mathrm{supp}\ \widehat{f}_{m}$
$\displaystyle\subset\Big{\\{}\xi\in\mathbb{R}^{2}:\
\frac{17}{12}2^{m}-\frac{1}{2}\leq|\xi|\leq\frac{17}{12}2^{m}+\frac{1}{2}\Big{\\}}$
$\displaystyle\subset\Big{\\{}\xi\in\mathbb{R}^{2}:\
\frac{33}{24}2^{m}\leq|\xi|\leq\frac{35}{24}2^{m}\Big{\\}}\quad\text{for}\quad
m\geq 4.$

Notice that

$\displaystyle\varphi(2^{-j}\xi)\equiv
1\quad\text{in}\;\Big{\\{}\xi\in\mathbb{R}^{2}:\
\frac{4}{3}2^{j}\leq|\xi|\leq\frac{3}{2}2^{j}\Big{\\}},$

and

$\displaystyle\mathcal{F}\big{(}\Delta_{j}(f_{m})\big{)}=\varphi(2^{-j}\cdot)\widehat{f}_{m},$

then we deduce for $m\geq 4$

$\displaystyle\Delta_{j}(f_{m})=\begin{cases}f_{m},&if\;\;j=m,\\\
0,&if\;\;j\neq m.\end{cases}$ (3.5)

We should mention that (3.5) also holds if cos were replaced by sin in the
definition of $g_{m}$.

Thus, we deduce from (3.5) that

$\displaystyle\|u_{0}\|_{B^{\sigma}_{p,\infty}(\mathbb{R}^{2})}$
$\displaystyle=\sup_{\ell\geq-1}2^{\ell\sigma}\|\Delta_{\ell}u_{0}\|_{L^{p}(\mathbb{R}^{2})}\leq
C.$

This gives (2).

To prove (3), we write

$\displaystyle u_{0}\cdot\nabla u_{0}=\sum^{\infty}_{j=0}f_{kj}\cdot\nabla
f_{kj}+\sum^{\infty}_{j=1}\sum^{j-1}_{i=0}(f_{kj}\cdot\nabla
f_{ki}+f_{ki}\cdot\nabla f_{kj}).$ (3.6)

On the one hand, we have for $\ell=1,2$

$\displaystyle\big{(}f_{kj}\cdot\nabla f_{kj}\big{)}_{\ell}$
$\displaystyle=2^{-2kj(\sigma+1)}\Big{[}\Phi_{1}(x_{1},x_{2})+\Phi_{2}(x_{1},x_{2})\cos\Big{(}\frac{17}{12}2^{kj+1}x_{1}\Big{)}+\Phi_{3}(x_{1},x_{2})\sin\Big{(}\frac{17}{12}2^{kj+1}x_{1}\Big{)}\Big{]},$

where

$\mathrm{supp}\ \widehat{\Phi_{1}},\;\mathrm{supp}\
\widehat{\Phi_{2}},\;\mathrm{supp}\
\widehat{\Phi_{3}}\subset\big{\\{}\xi\in\mathbb{R}^{2}:\ |\xi|\leq
2\big{\\}}.$

Then we have

$\displaystyle\Delta_{kn}\left((f_{kj}\cdot\nabla f_{kj})_{\ell}\right)\equiv
0\quad\text{for}\quad j\geq 0,$

which implies that

$\displaystyle\Delta_{kn}\left(\sum^{\infty}_{j=0}f_{kj}\cdot\nabla
f_{kj}\right)\equiv 0.$ (3.7)

On the other hand, notice that

$\displaystyle\mathrm{supp}\ \mathcal{F}\left(f_{kj}\cdot\nabla f_{ki}\right)$
$\displaystyle\subset\left\\{\xi\in\mathbb{R}^{2}:\
\frac{17}{12}2^{kj}-\frac{17}{12}2^{ki}-\frac{1}{2}\leq|\xi|\leq\frac{17}{12}2^{kj}+\frac{17}{12}2^{ki}+\frac{1}{2}\right\\}$
$\displaystyle\subset\left\\{\xi\in\mathbb{R}^{2}:\
\frac{33}{24}2^{kj}\leq|\xi|\leq\frac{35}{24}2^{kj}\right\\},$

and (3.5), we have

$\displaystyle\Delta_{kn}\left(\sum^{\infty}_{j=1}\sum^{j-1}_{i=0}(f_{kj}\cdot\nabla
f_{ki}+f_{ki}\cdot\nabla f_{kj})\right)=\sum^{n-1}_{i=0}(f_{kn}\cdot\nabla
f_{ki}+f_{ki}\cdot\nabla f_{kn}).$ (3.8)

Thus, we deduce from (3.7) and (3.8) that

$\displaystyle\Delta_{kn}(u_{0}\cdot\nabla u_{0})$
$\displaystyle=\sum^{n-1}_{i=0}f_{kn}\cdot\nabla
f_{ki}+\sum^{n-1}_{i=1}f_{ki}\cdot\nabla f_{kn}+f_{0}\cdot\nabla f_{kn}$
$\displaystyle=:\mathbf{I}_{1}+\mathbf{I}_{2}+\mathbf{I}_{3}.$

The first two terms can be estimated as follows:

$\displaystyle\|\mathbf{I}_{1}\|_{L^{p}}$
$\displaystyle\leq\sum^{n-1}_{i=0}\|f_{kn}\cdot\nabla f_{ki}\|_{L^{p}}$
$\displaystyle\leq\|f_{kn}\|_{L^{p}}\sum^{n-1}_{i=0}\|\nabla
f_{ki}\|_{L^{\infty}}$ $\displaystyle\leq
C2^{-kn\sigma}\sum^{n-1}_{i=0}2^{-ki(\sigma-1)}$ $\displaystyle\leq
C2^{-kn\sigma}$ (3.9)

and

$\displaystyle\|\mathbf{I}_{2}\|_{L^{p}}$
$\displaystyle\leq\sum^{n-1}_{i=1}\|f_{ki}\cdot\nabla f_{kn}\|_{L^{p}}$
$\displaystyle\leq\|\nabla
f_{kn}\|_{L^{p}}\sum^{n-1}_{i=1}\|f_{ki}\|_{L^{\infty}}$ $\displaystyle\leq
C2^{kn}2^{-kn\sigma}\sum^{n-1}_{i=1}2^{-k\sigma i}$ $\displaystyle\leq
C2^{kn}2^{-kn\sigma}2^{-k\sigma}.$ (3.10)

For the last term $f_{0}\cdot\nabla f_{kn}$, we compute the two components:

$\displaystyle\big{(}f_{0}\cdot\nabla f_{kn})_{1}$
$\displaystyle=-2^{-kn(\sigma+1)}\big{[}(f_{0})_{1}\partial_{1}\partial_{2}g_{kn}+(f_{0})_{2}\partial_{2}^{2}g_{kn}\big{]}$
$\displaystyle=2^{-kn(\sigma+1)}(\partial_{2}g_{0}\partial_{1}\partial_{2}g_{kn}-\partial_{1}g_{0}\partial_{2}^{2}g_{kn})$
$\displaystyle=2^{-kn(\sigma+1)}(\partial_{2}g_{0}h_{2}-\partial_{1}g_{0}h_{1})=:\mathbf{I}_{3,1}$

and

$\displaystyle\big{(}f_{0}\cdot\nabla f_{kn})_{2}$
$\displaystyle=2^{-kn(\sigma+1)}\big{[}(f_{0})_{1}\partial^{2}_{1}g_{kn}+(f_{0})_{2}\partial_{1}\partial_{2}g_{kn}\big{]}$
$\displaystyle=2^{-kn(\sigma+1)}(\partial_{1}g_{0}h_{2}-\partial_{2}g_{0}h_{3})+\Big{(}\frac{17}{12}\Big{)}^{2}2^{-kn(\sigma-1)}h_{4}$
$\displaystyle=:\mathbf{I}_{3,2}+\mathbf{I}_{3,3},$

where

$\displaystyle h_{1}$
$\displaystyle:=\phi(x_{1})\cos\Big{(}\frac{17}{12}2^{kn}x_{1}\Big{)}\phi^{\prime\prime}(x_{2}),$
$\displaystyle h_{2}$
$\displaystyle:=\phi^{\prime}(x_{1})\cos\Big{(}\frac{17}{12}2^{kn}x_{1}\Big{)}\phi^{\prime}(x_{2})-\frac{17}{12}2^{kn}\phi(x_{1})\sin\Big{(}\frac{17}{12}2^{kn}x_{1}\Big{)}\phi^{\prime}(x_{2}),$
$\displaystyle h_{3}$
$\displaystyle:=\phi^{\prime\prime}(x_{1})\cos\Big{(}\frac{17}{12}2^{kn}x_{1}\Big{)}\phi(x_{2})-\frac{17}{6}2^{kn}\phi^{\prime}(x_{1})\sin\Big{(}\frac{17}{12}2^{kn}x_{1}\Big{)}\phi(x_{2}),$
$\displaystyle h_{4}$
$\displaystyle:=\phi^{2}(x_{1})\cos\Big{(}\frac{17}{12}x_{1}\Big{)}\cos\Big{(}\frac{17}{12}2^{kn}x_{1}\Big{)}\phi(x_{2})\phi^{\prime}(x_{2}).$

By simple computations, one has

$\displaystyle\|\mathbf{I}_{3,1}\|_{L^{p}}+\|\mathbf{I}_{3,2}\|_{L^{p}}$
$\displaystyle\leq
2^{-kn(\sigma+1)}\|\partial_{1}g_{0},\partial_{2}g_{0}\|_{L^{p}}\|h_{1},h_{2},h_{3}\|_{L^{\infty}}\leq
C2^{-kn\sigma}$

and

$\displaystyle\|\mathbf{I}_{3,3}\|_{L^{p}}=\Big{(}\frac{17}{12}\Big{)}^{2}2^{-kn(\sigma-1)}\|h_{4}\|_{L^{p}}\geq
c2^{-kn(\sigma-1)},$

where we have used the simple fact for sufficiently large $n$

$\|h_{4}\|_{L^{p}}=\Big{\|}\phi^{2}(x_{1})\cos\Big{(}\frac{17}{12}x_{1}\Big{)}\cos\Big{(}\frac{17}{12}2^{n}x_{1}\Big{)}\Big{\|}_{L^{p}(\mathbb{R})}\Big{\|}\phi(x_{2})\phi^{\prime}(x_{2})\Big{\|}_{L^{p}(\mathbb{R})}\geq
c_{0}.$

This gives

$\displaystyle\|\mathbf{I}_{3}\|_{L^{p}}\geq
c_{0}2^{-kn(\sigma-1)}-C2^{-kn\sigma}.$ (3.11)

Combining (3.9)–(3.11) yields

$\displaystyle\|\Delta_{kn}(u_{0}\cdot\nabla u_{0})\|_{L^{p}}\geq c_{0}\cdot
2^{kn}2^{-kn\sigma}-C2^{kn}2^{-kn\sigma}2^{-k\sigma}-C2^{-kn\sigma}.$

We take sufficiently large $k$ and $n$ and obtain the desired (3). Thus, we
complete the proof of Lemma 3.1.

###  3.3 Error Estimates

###### Proposition 3.1

Let $u_{0}\in B^{\sigma}_{p,\infty}$. Assume that $u\in
L^{\infty}_{T}B^{\sigma}_{p,\infty}$ be the solution of the Cauchy problem
(1.2), we have

$\displaystyle\|\mathbf{w}(t,u_{0})\|_{\dot{B}^{\sigma-2}_{p,\infty}}\leq
Ct^{2}\|u_{0}\|^{3}_{B^{\sigma}_{p,\infty}},$

here and in what follows we denote

$\mathbf{w}(t,u_{0}):=\mathbf{S}_{t}(u_{0})-u_{0}-t\mathbf{P}(u_{0}).$

Proof. For simplicity, we denote $u(t):=\mathbf{S}_{t}(u_{0})$ here and in
what follows. Due to the fact $B^{\sigma}_{p,\infty}\hookrightarrow\rm Lip$,
we know that there exists a positive time
$T=T(\|u_{0}\|_{B^{\sigma}_{p,\infty}})$ such that

$\displaystyle\|u(t)\|_{L^{\infty}_{T}B^{\sigma}_{p,\infty}}\leq
C\|u_{0}\|_{B^{\sigma}_{p,\infty}}\leq C.$ (3.12)

Moreover, for $\gamma\geq\sigma-1$, we have

$\displaystyle\|u(t)\|_{L^{\infty}_{T}B^{\gamma}_{p,\infty}}\leq
C\|u_{0}\|_{B^{\gamma}_{p,\infty}}.$ (3.13)

By the Mean Value Theorem, we obtain from (1.2) that

$\displaystyle\|u(t)-u_{0}\|_{B^{\sigma-1}_{p,\infty}}$
$\displaystyle\leq\int^{t}_{0}\|\partial_{\tau}u\|_{B^{\sigma-1}_{p,\infty}}\mathrm{d}\tau$
$\displaystyle\leq\int^{t}_{0}\|\mathbf{P}(u)\|_{B^{\sigma-1}_{p,\infty}}\mathrm{d}\tau$
$\displaystyle\lesssim\int^{t}_{0}\|\mathbf{P}(u)\|_{\dot{B}^{\sigma-1}_{p,\infty}\cap\dot{B}^{0}_{p,1}}\mathrm{d}\tau$
$\displaystyle\lesssim\int^{t}_{0}\|u\otimes
u\|_{\dot{B}^{\sigma}_{p,\infty}\cap\dot{B}^{1}_{p,1}}\mathrm{d}\tau$
$\displaystyle\lesssim\int^{t}_{0}\|u\otimes
u\|_{B^{\sigma}_{p,\infty}}\mathrm{d}\tau$ $\displaystyle\lesssim
t\|u_{0}\|_{B^{\sigma}_{p,\infty}}^{2},$ (3.14)

where we have used the fact that $B^{\sigma}_{p,\infty}$ is a Banach algebra.

By the Mean Value Theorem and (1.2), we obtain

$\displaystyle\|\mathbf{w}(t,u_{0})\|_{\dot{B}^{\sigma-2}_{p,\infty}}$
$\displaystyle\leq\int^{t}_{0}\|\partial_{\tau}u-\mathbf{P}(u_{0})\|_{\dot{B}^{\sigma-2}_{p,\infty}}\mathrm{d}\tau$
$\displaystyle\leq\int^{t}_{0}\|\mathbf{P}(u)-\mathbf{P}(u_{0})\|_{\dot{B}^{\sigma-2}_{p,\infty}}\mathrm{d}\tau$
$\displaystyle\lesssim\int^{t}_{0}\|u\otimes u-u_{0}\otimes
u_{0}\|_{\dot{B}^{\sigma-1}_{p,\infty}}\mathrm{d}\tau$
$\displaystyle\lesssim\int^{t}_{0}\|u\otimes u-u_{0}\otimes
u_{0}\|_{{B}^{\sigma-1}_{p,\infty}}\mathrm{d}\tau$
$\displaystyle\lesssim\int^{t}_{0}\|u(\tau)-u_{0}\|_{B^{\sigma-1}_{p,\infty}}\|u_{0},u(\tau)\|_{B^{\sigma}_{p,\infty}}\mathrm{d}\tau$
$\displaystyle\lesssim t^{2}\|u_{0}\|^{3}_{B^{\sigma}_{p,\infty}},$

where we have used (3.3) in the last step.

Thus, we complete the proof of Proposition 3.1.

Now we present the proof of Theorem 1.1.
Proof of Theorem 1.1. Using Proposition 3.1 and Lemma 3.1, we have

$\displaystyle\|\mathbf{S}_{t}(u_{0})-u_{0}\|_{B^{\sigma}_{p,\infty}}$
$\displaystyle\geq
2^{{kn\sigma}}\big{\|}\Delta_{kn}\big{(}\mathbf{S}_{t}(u_{0})-u_{0}\big{)}\big{\|}_{L^{p}}$
$\displaystyle=2^{{kn\sigma}}\big{\|}\Delta_{kn}\big{(}t\mathbf{P}(u_{0})+\mathbf{w}(t,u_{0})\big{)}\big{\|}_{L^{p}}$
$\displaystyle\geq
t2^{{kn\sigma}}\big{\|}\Delta_{kn}\big{(}\mathcal{P}(u_{0}\cdot\nabla
u_{0})\big{)}\big{\|}_{L^{p}}-2^{{kn\sigma}}\big{\|}\Delta_{kn}\big{(}\mathbf{w}(t,u_{0})\big{)}\big{\|}_{L^{p}}$
$\displaystyle\geq t2^{{kn\sigma}}\|\Delta_{kn}\big{(}u_{0}\cdot\nabla
u_{0}\big{)}\|_{L^{p}}-t2^{{kn\sigma}}\|\Delta_{kn}\mathcal{Q}(u_{0}\cdot\nabla
u_{0})\|_{L^{p}}$
$\displaystyle~{}~{}~{}~{}-2^{{2kn}}2^{{kn(\sigma-2)}}\big{\|}\Delta_{kn}\big{(}\mathbf{w}(t,u_{0})\big{)}\big{\|}_{L^{p}}$
$\displaystyle\geq t2^{{kn\sigma}}\|\Delta_{kn}\big{(}u_{0}\cdot\nabla
u_{0}\big{)}\|_{L^{p}}-Ct\|\mathcal{Q}(u_{0}\cdot\nabla
u_{0})\|_{B^{\sigma}_{p,\infty}}$
$\displaystyle~{}~{}~{}~{}-C2^{2{kn}}\|\mathbf{w}(t,u_{0})\|_{\dot{B}^{\sigma-2}_{p,\infty}}$
$\displaystyle\geq ct2^{{kn}}-Ct-C2^{2{kn}}t^{2},$

where we have used the fact:

$\displaystyle\|\mathcal{Q}(u_{0}\cdot\nabla
u_{0})\|_{B^{\sigma}_{p,\infty}}=\|(-\Delta)^{-1}\nabla(\partial_{i}u^{j}_{0}\partial_{j}u_{0}^{i})\|_{B^{\sigma}_{p,\infty}}\lesssim\|\partial_{i}u^{j}_{0}\partial_{j}u_{0}^{i}\|_{B^{\sigma-1}_{p,\infty}}\lesssim\|u_{0}\|^{2}_{B^{\sigma}_{p,\infty}}.$

Thus, picking $t2^{kn}\approx\varepsilon$ with small $\varepsilon$, we have

$\displaystyle\|\mathbf{S}_{t}(u_{0})-u_{0}\|_{B^{\sigma}_{p,\infty}}\geq
c\varepsilon-C\varepsilon^{2}\geq\tilde{c}\varepsilon.$

This completes the proof of Theorem 1.1.

## Acknowledgements

J. Li is supported by the National Natural Science Foundation of China
(11801090 and 12161004) and Jiangxi Provincial Natural Science Foundation
(20212BAB211004). Y. Yu is supported by the National Natural Science
Foundation of China (12101011) and Natural Science Foundation of Anhui
Province (1908085QA05). W. Zhu is supported by the Guangdong Basic and Applied
Basic Research Foundation (2021A1515111018).

## Data Availability

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

## Conflict of interest

The authors declare that they have no conflict of interest.

## References

  * [1] J. Bourgain, D. Li, Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces, Invent. Math., 201 (1) (2015) 97–157.
  * [2] J. Bourgain, D. Li, Strong ill-posedness of the incompressible Euler equation in integer $C^{m}$ spaces, Geom. Funct. Anal., 25 (1) (2015) 1–86.
  * [3] H. Bahouri, J. Y. Chemin, R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 343, Springer, Heidelberg, 2011.
  * [4] A. Cheskidov, Shvydkoy, Ill-posedness of the basic equations of fluid dynamics in Besov spaces, Proc. Amer. Math. Soc. 138 (2010), 1059–1067.
  * [5] Z. Guo, J. Li, Z. Yin, Local well-posedness of the incompressible Euler equations in $B_{\infty,1}^{1}$ and the inviscid limit of the Navier-Stokes equations, J. Funct. Anal. 276 (2019), 2821–2830.
  * [6] A. Himonas, G. Misiołek, Non-uniform dependence on initial data of solutions to the Euler equations of hydrodynamics, Commun. Math. Phys. 296 (2010), 285–301.
  * [7] A. Himonas, C. Kenig, Non-uniform dependence on initial data for the CH equation on the line, Differ. Integral Equ. 22 (2009), 201–224.
  * [8] A. Himonas, C. Kenig, G. Misiołek, Non-uniform dependence for the periodic CH equation, Commun. Partial Differ. Equ. 35 (2010), 1145–1162.
  * [9] T. Kato, The cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rat. Mech. Anal. 58 (1975), 181–205.
  * [10] T. Kato, C. Lai, Nonlinear evolution equations and the Euler flow, J. Funct. Anal. 56 (1984) 15–28.
  * [11] T. Kato, G. Ponce, On nonstationary flows of viscous and ideal fluids in $L^{p}_{s}$ $(\mathbb{R}^{2})$, Duke Math. J. 55 (1987) 487–499.
  * [12] J. Li, Y. Yu, W. Zhu, Non-uniform dependence on initial data for the Euler equations in Besov spaces, arXiv:2001.03301.
  * [13] A. Majda, A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2001.
  * [14] G. Misiołek, T. Yoneda, Local ill-posedness of the incompresssible Euler equations in $C^{1}$ and $B^{1}_{\infty,1}$, Math. Ann. 363 (2016), 243–268.
  * [15] G. Misiołek, T. Yoneda, Continuity of the solution map of the Euler equations in Hölder spaces and weak norm inflation in Besov spaces, Trans. Am. Math. Soc. 370 (7) (2018) 4709–4730.
  * [16] J. Pastrana, Non-uniform dependence of Euler equations in Besov spaces, J. Differ. Equ. 273 (2021), 40–57.
  * [17] H.C. Pak, Y.J. Park, Existence of solutions for the Euler equations in a critical Besov space $B^{1}_{\infty,1}(\mathbb{R}^{n})$, Commun. Partial Differ. Equ. 29 (7-8) (2004) 1149–1166.