# An alternative solution to the $\gamma$-ray Gradient problem

D. Gaggero  INFN Pisa and Pisa University, Largo B. Pontecorvo 3, I-56127
Pisa, Italy  C. Evoli  Institut für Theoretische Physik, Universität Hamburg,
Luruper Chaussee 149, D-22761 Hamburg, Germany.  D. Grasso  INFN Pisa, Largo
B. Pontecorvo 3, I-56127 Pisa, Italy  L. Maccione  Ludwig-Maximilians-
Universität, Fakultät für Physik, Theresienstraße 37, D-80333 München, Germany
Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, Föhringer Ring 6,
D-80805 München, Germany

###### Abstract

The Fermi-LAT collaboration recently confirmed EGRET finding of a discrepancy
between the observed longitudinal profile of $\gamma$-ray diffuse emission
from the Galaxy and that computed with GALPROP assuming that cosmic rays are
produced by Galactic supernova remnants. The accurate Fermi-LAT measurements
make this anomaly hardly explainable in terms of conventional diffusion
schemes. Here we use DRAGON numerical diffusion code to implement a physically
motivated scenario in which the diffusion coefficient is spatially correlated
to the source density. We show that under those conditions we are able to
reproduce the observed flat emissivity profile in the outer Galaxy with no
need to change the source term, the diffusion halo height, or the CO-${\rm
H_{2}}$ conversion factor (${\rm X_{CO}}$) with respect to their preferred
values/distributions. We also show that our models are compatible with gamma-
ray longitudinal profiles measured by Fermi-LAT, and still provide a
satisfactory fit of all observed secondary-to-primary ratios, such as B/C and
antiprotons/protons.

##  I Introduction

It has been known since the EGRET era that, if one computes the cosmic ray
(CR) Galactocentric radial distribution adopting a source function deduced
from pulsar or supernova remnant (SNR) catalogues, the result appears much
steeper than the profile inferred from the $\gamma$-ray diffuse emission along
the Galactic plane: the latter appears flatter, with a high contribution from
large Galactic radii. This discrepancy is known as the $\gamma$-ray gradient
problem. A sharp rise of the conversion factor between CO emissivity and ${\rm
H_{2}}$ density (the so called ${\rm X_{CO}}$) with the Galactocentric radius
was invoked at the time to fix the problem (Strong et al., 2004): a larger gas
density at large radii compensates for the decreasing CR population and is
able to explain the $\gamma$-ray flux detected at high Galactic longitudes.

Fermi-LAT confirmed the existence of such a problem (Ackermann et al., 2011).
Moreover, the high spatial resolution of the LAT permitted to disentangle the
emission coming from the interaction of CRs with the molecular gas (whose
modelling is strongly affected by the uncertainty on the ${\rm X_{CO}}$) from
the emission originated by the interaction of the Galactic CRs with the atomic
gas (whose density is better known from its 21 cm radio emission). An analysis
based on $\gamma$-ray maps of the third Galactic quadrant (Ackermann et al.,
2011) pointed out that the $\gamma$-ray emissivity from neutral gas (tracing
the actual CR density) is indeed flatter than the predicted one confirming the
gradient problem independently of the ${\rm X_{CO}}$. This result led the
authors of (Ackermann et al., 2011) to look for alternative explanations of
the problem, e.g. invoking a thick CR diffusion halo or a source term that
becomes flatter at large radii. Both solutions, however, do not appear
completely satisfactory: a thick halo is disfavoured both from 10Be/9Be and
synchrotron data; a smooth source distribution is in contrast with SNR
catalogues.

Here we consider a different interpretation based on relaxing the
approximation of isotropic and spatially uniform diffusion.

##  II Inhomogeneous and anisotropic diffusion

Nearly all CR diffusion models presented in the literature adopt an isotropic
and spatially uniform diffusion coefficient throughout all the Galaxy. This is
the case, for example, of GALPROP numerical package on which the predictions
of (Ackermann et al., 2011) are based. It is reasonable, however, to expect
that CR diffusion is not isotropic in the Galaxy. This could be the
consequence either of Galactic winds (Gebauer and de Boer, 2009) or just of
the anisotropy of the regular component of the Galactic magnetic field which
is oriented almost azimutally along the Galactic plane. The former possibility
was suggested as a possible solution of the $\gamma$-ray gradient problem
originated by EGRET observations (Breitschwerdt et al., 2002). Here we
consider the latter option and extend also to the recent Fermi-LAT data the
arguments we developed in [Evoli et al. 2008] to interpret earlier EGRET
measurements. Our approach is based on the consideration that, for geometrical
reasons, CRs should escape from the Galaxy almost perpendicularly to the
Galactic plane: their density, therefore, should be determined by the
perpendicular component of the diffusion coefficient $D_{\perp}$. We know –
both from quasi linear diffusion theory and from more realistic numerical
simulations (DeMarco et al., 2007) – that $D_{\perp}$ should increase with
increasing strength of the Galactic magnetic field turbulent component; from a
physical point of view, such behaviour can be understood in terms of magnetic
field line random walk becoming stronger when the turbulence strength
increases. As a consequence, the regions where CR injection is more intense
should also be those characterized by a stronger MHD turbulence and hence a
faster CR escape along the $z$ axis: this should smooth the CR gradient, and
hence the $\gamma$-ray profile, in a rather natural way.

In the next section we will show this effect by means of dedicated numerical
computations.

##  III Our method and results

Figure 1: Two different CR proton distribution maps in arbitrary units
computed with DRAGON at $10$ GeV are shown as functions of the Galactic
cylindrical coordinates $R$ and $z$. Panel a) The proton distribution is
computed with no radial dependence of diffusion coefficient. Panel b) Here the
diffusion coefficient is correlated to the source term: $D\propto Q^{\tau}$,
with $\tau=0.8$. The model shows a significant flattening in the CR profile
along $R$. The normalization is fixed at $R_{\rm Sun}=8.5$ kpc in both cases;
notice how the maximum proton density is reduced by a factor $\simeq 2$ in the
second panel.

Figure 2: Two $gamma$-ray longitudinal profiles along the Galactic plane
computed with DRAGON and GammaSky and compared to preliminary Fermi-LAT data
extracted from the talk by A.W.Strong at the Workshop on Indirect Dark Matter
Searches, DESY, Hamburg, June 2011 (http://www.mpe.mpg.de/ aws/talks/). Data
are integrated over the latitude interval $-5^{\circ}<b<+5^{\circ}$ and in
energy between 1104 and 1442 MeV. Red line: IC. Green line: Bremsstrahlung.
Blue line: $\pi^{0}$ decay. Purple line: contribution from unresolved sources.
Grey line: $\pi^{0}$ \+ IC + Bremsstrahlung. Black line: total. Panel a) The
profile is computed with no radial dependence of diffusion coefficient. Panel
b) Here the diffusion coefficient follows the source term: $D\propto
Q^{\tau}$, with $\tau=0.8$.  Figure 3: Here the effect of the parameter $\tau$
defined by equation 1 is explored. Dotted line: no radial dependence of
diffusion coefficient ($\tau=0$). Dot-dashed line: $\tau=0.2$. Dashed line:
$\tau=0.5$. Solid lines: $\tau=0.7$ – $0.8$ – $1.0$. The values corresponding
to the solid lines within the grey band match the observed gradient.

In this section we use DRAGON numerical diffusion package111The DRAGON code
for cosmic-ray transport and diffuse emission production is available online
at http://www.desy.de/~maccione/DRAGON/ to solve the diffusion equation in the
presence of a diffusion coefficient spatially correlated to the CR source term
$Q(r,z)$. We perform our analysis using a Plain Diffusion (PD) setup with no
convection and no reacceleration in order to better highlight the effects of
inhomogeneous diffusion; similar results may be obtained with different
choices of the diffusion parameters, and a more detailed study on the effects
of another setup will be performed in a forthcoming paper. The CR propagation
model adopted here is basically the same as the PD model described in (Di
Bernardo et al., 2011); the astrophysical parameters (in particular the source
term, gas distribution and ${\rm X_{CO}}$) are also the same used in that
analysis. Only the normalization of the proton injection spectrum is slightly
tuned to match the recently released proton spectrum measured by the PAMELA
collaboration (Adriani et al., 2011). The model is also compatible with most
other CR data sets. Only a little excess in the antiproton spectrum must be
pointed out which, however, is still compatible with data if astrophysical and
particle physics uncertainties are taken into account. As we mentioned, the CR
distribution is computed with DRAGON: this numerical package is suitable for
our purpose since, differently from GALPROP, it implements the possibility to
vary the diffusion coefficient through the Galaxy. The CR distributions is
then used as an input to compute the $\gamma$-ray longitude profile along the
Galactic plane; the $\gamma$-ray map is evaluated with a separate package
called GammaSky.

The result of a combined DRAGON and GammaSky computation in the case of a
uniform diffusion coefficient and a PD setup is shown in Fig. 2 (panel a). It
is clear from that plot that the predicted longitude profile is too steep
compared to the observations: in the Galactic center region the model
prediction overshoots the data and in the anti-center region the model is
lower than the observations by several $\sigma$. Tuning the ${\rm X_{CO}}(R)$
could help in principle: assuming a lower value of this parameter in the bulge
and a high value at large $R$ could smooth the $\gamma$-ray profile (as done
in several previous works such as (Strong et al., 2004)). Unfortunately, as
pointed out in the introduction, the gradient problem is present especially in
the emissivity profile, and this quantity is independent of the molecular gas:
it only traces the actual CR distribution222The emissivity is the number of
$\gamma$ photons emitted by each gas atom per unit time and unit energy.

So we apply our previous considerations and adopt a diffusion coefficient
correlated to the radial dependence of the source term $Q(R)$ by the following
expression:

$D(R)\,\propto\,Q(R)^{\tau}$ (1)

This is the parametrization we already used in (Evoli et al., 2008) to
interpret EGRET data. The parameter $\tau$ is tuned against data. In Fig. 3 we
show the emissivity profile for different values of $\tau$ in the range
${\rm[0\div 1]}$. It is evident from that figure that an increasing value of
$\tau$ yields a much smoother behaviour of the emissivity as function of $R$.
Values in the range ${\rm[0.7\div 1]}$ allow a good match of Fermi-LAT data
((Ackermann et al., 2011), (Abdo et al., 2010)).

With this result at hand, we considered a modified version of the Plain
Diffusion CR propagation setup with $D(R)=Q^{\tau}$ and $\tau=0.8$. The
smoothing in the CR distribution corresponding to such a value of $\tau$ is
shown in Fig. 1. As shown in Fig. 2 (panel b), the $\gamma$ ray longitude
profile along the Galactic plane is nicely reproduced with no tuning at all of
the ${\rm X_{CO}}$. It is remarkable that a simple CR propagation setup, with
only the addition of the radial dependence of $D$ and no ad hoc tuning,
permits to reproduce the $\gamma$-ray profile with such accuracy. Noticeably,
the modified model is still compatible with most relevant CR data set, most
importantly the B/C. Furthermore, we checked that the $\gamma$-ray spectrum
measured by Fermi-LAT along the Galactic plane is also correctly reproduced
under those conditions (see Fig. 4).

Figure 4: The gamma-ray spectrum corresponding to the plain diffusion model
with varying diffusion coefficient described in the text ($\tau=0.8$). The
spectrum was computed with DRAGON and GammaSky. The data points measured by
Fermi-LAT are taken from the same reference as Fig. 2

##  IV Conclusions

In this paper we presented an alternative solution to the well known
$\gamma$-ray gradient problem. Our approach is based on the physically
motivated hypothesis that the CR diffusion coefficient is spatially correlated
to the source density: regions in which star, hence SNR, formation is stronger
are expected to present a stronger turbulence level and therefore a larger
value of the perpendicular diffusion coefficient. This effect favours CR
escape from most active regions helping to smooth their density through the
Galaxy hence also the $\gamma$-ray gradient. We used DRAGON package to
implement this scenario and to check that CR data are still reproduced under
those conditions. In spite of being purely phenomenological (as a self-
consistent theory/computation of non-linear CR - MHD turbulence interaction in
the Galaxy is far from being developed) our approach provides a remarkably
good description of the spectrum and longitude distribution of the diffuse
$\gamma$-ray emission measured by the Fermi-LAT collaboration.

###### Acknowledgements.

D. Gaggero would like to thank the LAPTH (Laboratoire d’Annecy-le-Vieux de
Physique Théorique) for hosting him during the last part of the work presented
in this paper.

## References

  * Strong et al. (2004) A. W. Strong, I. V. Moskalenko, O. Reimer, S. Digel, and R. Diehl, Astronomy and Astrophysics 422, L47 (2004), eprint http://arxiv.org/abs/astro-ph/0405275.
  * Ackermann et al. (2011) M. Ackermann, M. Ajello, L. Baldini, J. Ballet, G. Barbiellini, D. Bastieri, K. Bechtol, R. Bellazzini, B. Berenji, E. D. Bloom, et al., Astrophysical Journal 726, 81 (2011), eprint http://arxiv.org/abs/1011.0816.
  * Gebauer and de Boer (2009) I. Gebauer and W. de Boer (2009), * Brief entry *, eprint http://arxiv.org/abs/0910.2027.
  * Breitschwerdt et al. (2002) D. Breitschwerdt, V. A. Dogiel, and H. J. Völk, Astronomy and Astrophysics 385, 216 (2002), eprint http://arxiv.org/abs/astro-ph/0201345.
  * DeMarco et al. (2007) D. DeMarco, P. Blasi, and T. Stanev, Journal of Cosmology and Astroparticle Physics 6, 27 (2007), eprint http://arxiv.org/abs/0705.1972.
  * Di Bernardo et al. (2011) G. Di Bernardo, C. Evoli, D. Gaggero, D. Grasso, L. Maccione, and M. N. Mazziotta, Astroparticle Physics 34, 528 (2011), eprint http://arxiv.org/pdf/1010.0174v2.
  * Adriani et al. (2011) O. Adriani, G. C. Barbarino, G. A. Bazilevskaya, R. Bellotti, M. Boezio, E. A. Bogomolov, L. Bonechi, M. Bongi, V. Bonvicini, S. Borisov, et al., Science 332, 69 (2011), eprint http://arxiv.org/abs/1103.4055.
  * Evoli et al. (2008) C. Evoli, D. Gaggero, D. Grasso, and L. Maccione, Journal of Cosmology and Astroparticle Physics 10, 18 (2008), eprint http://arxiv.org/abs/0807.4730.
  * Abdo et al. (2010) A. A. Abdo, M. Ackermann, M. Ajello, L. Baldini, J. Ballet, G. Barbiellini, D. Bastieri, B. M. Baughman, K. Bechtol, R. Bellazzini, et al., Astrophysical Journal 710, 133 (2010), eprint http://arxiv.org/abs/0912.3618.


# Global existence of large solutions for the parabolic-elliptic Keller–Segel
system in Besov type spaces ††thanks: This work is partially supported by the
National Natural Science Foundation of China (no. 11961030, 12361034), the
Natural Science Foundation of Shaanxi Province (no. 2022JM-034).

Jihong Zhao
School of Mathematics and Information Science, Baoji University of Arts and
Sciences,
Baoji, Shaanxi 721013, China E-mail address<EMAIL_ADDRESS>
###### Abstract

In this paper, we investigate global existence of large solutions for the
parabolic-elliptic Keller–Segel system in the homogeneous Besov type spaces. A
class of initial data was presented, generating a global smooth solution
although the $\dot{B}^{-2}_{\infty,\infty}$-norm of the initial data may be
chosen arbitrarily large.

Keywords: Keller–Segel system; global existence; Besov spaces.

2020 AMS Subject Classification: 35A01; 35M11; 92C17

##  1 Introduction

In this paper, we consider the global existence of large solutions for the
Cauchy problem of the parabolic-elliptic Keller–Segel system:

$\begin{cases}\partial_{t}u-\Delta u+\nabla\cdot(u\nabla v)=0,\\\ -\Delta
v=u,\\\ u(x,0)=u_{0}(x),\end{cases}$ (1.1)

where two unknown functions $u$ and $v$ representing the cell density and the
concentration of the chemical attractant, respectively, and both are unknown
functions of the space variable $x\in\mathbb{R}^{d}$ and of the time variable
$t\in\mathbb{R}^{+}$.

The system (1.1) is a simplified version of the mathematical model of
chemotaxis, which was first introduced by E.F. Keller and L.A. Segel [13] in
1970. This basic model was used to describe the collective movement of
bacteria possibly leading to cell aggregation by chemotactic effect. It is
also connected with astrophysical models of gravitational self-interaction of
massive particles in a cloud or a nebula, see [4, 3]. We refer to [2, 6, 7, 8,
11, 14, 15, 16, 18, 19, 20, 21] for more details and information concerning
about the existence and nonexistence (blow-up) of global in time solutions to
the different Keller–Segel models studied since the 1970s.

Our aim here is to show the existence of global solutions of the system (1.1)
in critical Besov type spaces. Karch [12] established global well-posedness
with small initial data in critical Besov space
$\dot{B}^{-2+\frac{d}{p}}_{p,\infty}(\mathbb{R}^{d})$ with $\frac{d}{2}<p<d$.
Subsequently, Iwabuchi [9] proved local well-posedness and global well-
posedness for small initial data in critical Besov spaces
$\dot{B}^{-2+\frac{d}{p}}_{p,q}(\mathbb{R}^{d})$ and Fourier–Herz spaces
$\dot{\mathcal{B}}^{-2}_{2}(\mathbb{R}^{d})$ with $1\leq p<\infty$ and $1\leq
q\leq\infty$, moreover, the ill-posedness is also established in critical
Besov spaces $\dot{B}^{-2}_{\infty,q}(\mathbb{R}^{d})$ with $2<q\leq\infty$;
Finally, Iwabuchi–Nakamura [10] showed global existence of solutions for small
initial data in $\dot{F}^{-2}_{\infty,2}(\mathbb{R}^{d})=BMO^{-2}$ through the
Triebel–Lizorkin spaces, which combining the well-posed results in [9], we
know that, the simplified Keller–Segel system (1.1) is well-posed in
$\dot{B}^{-2+\frac{d}{p}}_{p,\infty}(\mathbb{R}^{d})$ and
$\dot{F}^{-2}_{\infty,2}(\mathbb{R}^{d})$ with
$\max\\{1,\frac{d}{2}\\}<p<\infty$, but ill-posed in
$\dot{B}^{-2}_{\infty,q}(\mathbb{R}^{d})$ with $2<q\leq\infty$.

Before we state the main results, let us first introduce the definition of the
homogeneous Besov spaces.

###### Definition 1.1

Let $\varphi\in\mathcal{S}(\mathbb{R}^{d})$ be a positive radial function such
that $\varphi$ is supported in the shell
$\mathcal{C}=\\{\xi\in\mathbb{R}^{d},\
\frac{3}{4}\leq|\xi|\leq\frac{8}{3}\\}$, and
$\sum_{j\in\mathbb{Z}}\varphi(2^{-j}\xi)=1\ \ \text{for}\ \
\xi\in\mathbb{R}^{d}\backslash\\{0\\}.$ Let $h=\mathcal{F}^{-1}\varphi$. Then
for any $f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$, we define the homogeneous
dyadic block $\Delta_{j}$ as

$\displaystyle\Delta_{j}f(x):=2^{dj}\int_{\mathbb{R}^{d}}h(2^{j}y)f(x-y)dy.$

Then for any $s\in\mathbb{R}$, $1\leq p,r\leq\infty$, we define the
homogeneous Besov space $\dot{B}^{s}_{p,r}(\mathbb{R}^{d})$ by

$\dot{B}^{s}_{p,r}(\mathbb{R}^{d}):=\Big{\\{}f\in\mathcal{S}(\mathbb{R}^{d})/\mathcal{P}:\
\ \|f\|_{\dot{B}^{s}_{p,r}}<\infty\Big{\\}},$
$\|f\|_{\dot{B}^{s}_{p,r}}:=\begin{cases}\left(\sum_{j\in\mathbb{Z}}2^{srj}\|\Delta_{j}f\|_{L^{p}}^{r}\right)^{\frac{1}{r}}\
\ &\text{for}\ \ 1\leq r<\infty,\\\
\sup_{j\in\mathbb{Z}}2^{sj}\|\Delta_{j}f\|_{L^{p}}\ \ &\text{for}\ \
r=\infty,\end{cases}$

where $\mathcal{P}$ denotes the set of all polynomials.

Next, we recall the definition of the so-called Chemin–Lerner mixed time-space
spaces.

###### Definition 1.2

For $0<T\leq\infty$, $s\in\mathbb{R}$ and $1\leq p,r,\rho\leq\infty$. We
define the mixed time-space
$\mathcal{L}^{\rho}(0,T;\dot{B}^{s}_{p,r}(\mathbb{R}^{d}))$ as the completion
of $\mathcal{C}([0,T];\mathcal{S}(\mathbb{R}^{d}))$ by the norm

$\|f\|_{\mathcal{L}^{\rho}_{T}(\dot{B}^{s}_{p,r})}:=\left(\sum_{j\in\mathbb{Z}}2^{srj}\left(\int_{0}^{T}\|\Delta_{j}f(\cdot,t)\|_{L^{p}}^{\rho}dt\right)^{\frac{r}{\rho}}\right)^{\frac{1}{r}}<\infty$

with the usual change if $\rho=\infty$ or $r=\infty$.

Now we are ready to state our main results. The first result we shall prove is
a new global existence result, under a nonlinear smallness assumption on the
initial data.

###### Theorem 1.3

For any $u_{0}\in\dot{B}^{-2+\frac{d}{p}}_{p,1}(\mathbb{R}^{d})$ with
$1<p<\infty$, there exist two positive constants $c_{0}$ and $C_{0}$ such that
if the initial data $u_{0}$ satisfies

$\displaystyle
C_{0}\|e^{t\Delta}u_{0}\nabla(-\Delta)^{-1}e^{t\Delta}u_{0}\|_{L^{1}(\mathbb{R}^{+};\dot{B}^{-1+\frac{d}{p}}_{p,1})}\exp\\{C_{0}\|u_{0}\|_{\dot{B}^{-2+\frac{d}{p}}_{p,1}}\\}\leq
c_{0},$ (1.2)

then the system (1.1) admits a unique global solution associated with $u_{0}$,
satisfying

$u\in\mathcal{L}^{\infty}(\mathbb{R}^{+};\dot{B}^{-2+\frac{d}{p}}_{p,1}(\mathbb{R}^{d}))\cap
L^{1}(\mathbb{R}^{+};\dot{B}^{\frac{d}{p}}_{p,1}(\mathbb{R}^{d})).$ (1.3)

Moreover, there exists a positive constant $\kappa$ such that

$\|u-e^{t\Delta}u_{0}\|_{\mathcal{L}^{\infty}(\mathbb{R}^{+};\dot{B}^{-2+\frac{d}{p}}_{p,1})}+\frac{\kappa}{2}\|u-e^{t\Delta}u_{0}\|_{L^{1}(\mathbb{R}^{+};\dot{B}^{\frac{d}{p}}_{p,1})}\leq\frac{c_{0}}{2}.$
(1.4)

The proof of Theorem 1.3 is given in Section 2 below. It consists in writing
the solution $u$ (which exists for a short time at least), as
$u=e^{t\Delta}u_{0}+\bar{u}$ and in proving a global existence result for the
perturbed Keller–Segel system satisfied by $\bar{u}$, under assumption (1.2).
Moreover, the second result of this paper is to exhibit examples of
applications of Theorem 1.3 which go beyond the assumption of smallness of
$\dot{B}^{-2}_{\infty,\infty}$-norm. For the sake of simplicity, we take $d=3$
to construct such an initial data.

###### Theorem 1.4

Let $\phi\in\mathcal{S}(\mathbb{R}^{3})$ be a given function, and consider two
real numbers $\varepsilon$, $\alpha\in(0,1)$. Define

$\varphi_{\varepsilon}(x):=\frac{(-\log\varepsilon)^{\frac{1}{5}}}{\varepsilon^{1-\alpha}}\cos(\frac{x_{3}}{\varepsilon})\phi(x_{1},\frac{x_{2}}{\varepsilon^{\alpha}},x_{3}).$

There is a constant $C>0$ such that for $\varepsilon$ small enough, the smooth
function $u_{0,\varepsilon}:=\nabla\cdot w_{0,\varepsilon}$ satisfies

$\|u_{0,\varepsilon}\|_{\dot{B}^{-2}_{\infty,\infty}}=\|\nabla\cdot
w_{0,\varepsilon}\|_{\dot{B}^{-2}_{\infty,\infty}}\approx\|w_{0,\varepsilon}\|_{\dot{B}^{-1}_{\infty,\infty}}\geq
C^{-1}(-\log\varepsilon)^{\frac{1}{5}},$ (1.5)

where $w_{0,\varepsilon}$ is defined as
$w_{0,\varepsilon}:=(\partial_{2}\varphi_{\varepsilon},\partial_{1}\varphi_{\varepsilon},0).$
Moreover, we obtain

$\|e^{t\Delta}u_{0,\varepsilon}\nabla(-\Delta)^{-1}e^{t\Delta}u_{0,\varepsilon}\|_{L^{1}(\mathbb{R}^{+};\dot{B}^{-1+\frac{d}{p}}_{p,1})}\leq
C\varepsilon^{\frac{4}{3}\alpha}(-\log\varepsilon)^{\frac{2}{5}}.$ (1.6)

The proof of Theorem 1.4 is the purpose of Section 3. Throughout the paper,
$C$ stands for a real positive constant which may be different in each
occurrence.

##  2 Proof of Theorem 1.3

Let us denote $u_{L}:=e^{t\Delta}u_{0}$, $\bar{u}:=u-u_{L}$. Then we can
rewrite system (1.1) as

$\begin{cases}\partial_{t}\bar{u}-\Delta\bar{u}=-\nabla\cdot[(u_{L}+\bar{u})\nabla(-\Delta)^{-1}(u_{L}+\bar{u})],\\\
\bar{u}(x,0)=0.\end{cases}$ (2.1)

Moreover, by using the Duhamel’s principle, the system (2.1) can be further
reduced into the following equivalent integral equations:

$\bar{u}(t)=-\int_{0}^{t}e^{(t-s)\Delta}\nabla\cdot[(u_{L}+\bar{u})\nabla(-\Delta)^{-1}(u_{L}+\bar{u})]ds.$
(2.2)

Since $u_{L}$ is a solution of the heat equation with initial data $u_{0}$,
one can easily obtain that for any $0<T\leq\infty$,

$\|u_{L}\|_{L^{1}_{T}(\dot{B}^{-2+\frac{d}{p}}_{p,1})}\leq\|u_{0}\|_{\dot{B}^{-2+\frac{d}{p}}_{p,1}}.$
(2.3)

Before giving the desired estimate of $\bar{u}$, we denote
$f(t):=\|u_{L}(t)\|_{\dot{B}^{\frac{d}{p}}_{p,1}}$, and for any positive real
number $\lambda$, we introduce the following weighted function:

$\bar{u}_{\lambda,f}(x,t):=\bar{u}(x,t)\exp\\{-\lambda\int_{0}^{t}f(\tau)d\tau\\}.$

Back to (2.2), it holds that

$\displaystyle\bar{u}_{\lambda,f}$
$\displaystyle=-\int_{0}^{t}e^{(t-s)\Delta}e^{-\lambda\int_{0}^{t}f(\tau)d\tau}\nabla\cdot(u_{L}\nabla(-\Delta)^{-1}u_{L})ds$
$\displaystyle-\int_{0}^{t}e^{(t-s)\Delta}e^{-\lambda\int_{s}^{t}f(\tau)d\tau}\nabla\cdot\big{[}u_{L}\nabla(-\Delta)^{-1}\bar{u}_{\lambda,f}+\bar{u}_{\lambda,f}\nabla(-\Delta)^{-1}u_{L}+\bar{u}\nabla(-\Delta)^{-1}\bar{u}_{\lambda,f}\big{]}ds.$

We know from the solvability of heat equation in the framework of the
homogeneous Besov spaces that there exists a positive constant $\kappa$ such
that

$\displaystyle\|$
$\displaystyle\bar{u}_{\lambda,f}\|_{\mathcal{L}^{\infty}_{t}(\dot{B}^{-2+\frac{d}{p}}_{p,1})}+\kappa\|\bar{u}_{\lambda,f}\|_{L^{1}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}\leq\|e^{-\lambda\int_{0}^{t}f(\tau)d\tau}u_{L}\nabla(-\Delta)^{-1}u_{L}\|_{L^{1}_{t}(\dot{B}^{-1+\frac{d}{p}}_{p,1})}$
$\displaystyle+\|e^{-\lambda\int_{s}^{t}f(\tau)d\tau}\big{[}u_{L}\nabla(-\Delta)^{-1}\bar{u}_{\lambda,f}+\bar{u}_{\lambda,f}\nabla(-\Delta)^{-1}u_{L}+\bar{u}\nabla(-\Delta)^{-1}\bar{u}_{\lambda,f}\big{]}\|_{L^{1}_{t}(\dot{B}^{-1+\frac{d}{p}}_{p,1})}$
$\displaystyle:=I_{1}+I_{2}+I_{3}+I_{4}.$ (2.4)

For $I_{1}$, one obtains that

$\displaystyle I_{1}\leq
C\|u_{L}\nabla(-\Delta)^{-1}u_{L}\|_{L^{1}_{t}(\dot{B}^{-1+\frac{d}{p}}_{p,1})}.$
(2.5)

For $I_{2}$ and $I_{3}$, bu using the following symmetric structure:

$u\nabla(-\Delta)^{-1}v+v\nabla(-\Delta)^{-1}u=-\nabla\cdot\big{(}\nabla(-\Delta)^{-1}u\nabla(-\Delta)^{-1}v\big{)},$
(2.6)

we can deduce that

$\displaystyle I_{2}+I_{3}$ $\displaystyle\leq
C\int_{0}^{t}e^{-\lambda\int_{s}^{t}f(\tau)d\tau}\|\nabla\cdot(\nabla(-\Delta)^{-1}u_{L}\nabla(-\Delta)^{-1}\bar{u}_{\lambda,f})\|_{\dot{B}^{-1+\frac{d}{p}}_{p,1}}ds$
$\displaystyle\leq
C\int_{0}^{t}e^{-\lambda\int_{s}^{t}f(\tau)d\tau}\|\nabla(-\Delta)^{-1}u_{L}\nabla(-\Delta)^{-1}\bar{u}_{\lambda,f}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}ds$
$\displaystyle\leq
C\int_{0}^{t}e^{-\lambda\int_{s}^{t}f(\tau)d\tau}\|\nabla(-\Delta)^{-1}u_{L}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}\|\nabla(-\Delta)^{-1}\bar{u}_{\lambda,f}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}ds$
$\displaystyle\leq
C\int_{0}^{t}e^{-\lambda\int_{s}^{t}f(\tau)d\tau}\|u_{L}\|_{\dot{B}^{-1+\frac{d}{p}}_{p,1}}\|\bar{u}_{\lambda,f}\|_{\dot{B}^{-1+\frac{d}{p}}_{p,1}}ds$
$\displaystyle\leq
C\int_{0}^{t}e^{-\lambda\int_{s}^{t}f(\tau)d\tau}\|u_{L}\|_{\dot{B}^{-2+\frac{d}{p}}_{p,1}}^{\frac{1}{2}}\|u_{L}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}^{\frac{1}{2}}\|\bar{u}_{\lambda,f}\|_{\dot{B}^{-2+\frac{d}{p}}_{p,1}}^{\frac{1}{2}}\|\bar{u}_{\lambda,f}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}^{\frac{1}{2}}ds$
$\displaystyle\leq
C\int_{0}^{t}e^{-\lambda\int_{s}^{t}f(\tau)d\tau}\big{(}\varepsilon\|\bar{u}_{\lambda,f}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}+\|u_{L}\|_{\dot{B}^{-2+\frac{d}{p}}_{p,1}}\|u_{L}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}\|\bar{u}_{\lambda,f}\|_{\dot{B}^{-2+\frac{d}{p}}_{p,1}}\big{)}ds$
$\displaystyle\leq
C\varepsilon\|\bar{u}_{\lambda,f}\|_{L^{1}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}+\|u_{0}\|_{\dot{B}^{-2+\frac{d}{p}}_{p,1}}\int_{0}^{t}e^{-\lambda\int_{s}^{t}f(\tau)d\tau}\|u_{L}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}\|\bar{u}_{\lambda,f}\|_{\dot{B}^{-2+\frac{d}{p}}_{p,1}}ds,$
(2.7)

where we have used the fact that

$\|u_{L}\|_{L^{\infty}_{t}(\dot{B}^{-2+\frac{d}{p}}_{p,1})}\leq\|u_{L}\|_{\mathcal{L}^{\infty}_{t}(\dot{B}^{-2+\frac{d}{p}}_{p,1})}\leq\|u_{0}\|_{\dot{B}^{-2+\frac{d}{p}}_{p,1}}.$

For $I_{4}$, since it has a nice structure as

$\bar{u}\nabla(-\Delta)^{-1}\bar{u}_{\lambda,f}=-\frac{1}{2}\nabla\cdot(\nabla(-\Delta)^{-1}\bar{u}\nabla(-\Delta)^{-1}\bar{u}_{\lambda,f}),$

we can proceed the same procedure as $I_{2}+I_{3}$ to obtain that

$\displaystyle I_{4}$ $\displaystyle\leq
C\int_{0}^{t}e^{-\lambda\int_{s}^{t}f(\tau)d\tau}\|\nabla(-\Delta)^{-1}\bar{u}\nabla(-\Delta)^{-1}\bar{u}_{\lambda,f}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}ds$
$\displaystyle\leq
C\int_{0}^{t}\|\bar{u}\|_{\dot{B}^{-1+\frac{d}{p}}_{p,1}}\|\bar{u}_{\lambda,f}\|_{\dot{B}^{-1+\frac{d}{p}}_{p,1}}ds$
$\displaystyle\leq
C\int_{0}^{t}\|\bar{u}\|_{\dot{B}^{-2+\frac{d}{p}}_{p,1}}^{\frac{1}{2}}\|\bar{u}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}^{\frac{1}{2}}\|\bar{u}_{\lambda,f}\|_{\dot{B}^{-2+\frac{d}{p}}_{p,1}}^{\frac{1}{2}}\|\bar{u}_{\lambda,f}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}^{\frac{1}{2}}ds$
$\displaystyle\leq
C\int_{0}^{t}\|\bar{u}\|_{\dot{B}^{-2+\frac{d}{p}}_{p,1}}\|\bar{u}_{\lambda,f}\|_{\dot{B}^{\frac{d}{p}}_{p,1}}ds$
$\displaystyle\leq
C\|\bar{u}\|_{\mathcal{L}^{\infty}_{t}(\dot{B}^{-2+\frac{d}{p}}_{p,1})}\|\bar{u}_{\lambda,f}\|_{L^{1}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}.$
(2.8)

Notice that

$\displaystyle\int_{0}^{t}e^{-\lambda\int_{s}^{t}f(\tau)d\tau}\|u_{L}(s)\|_{\dot{B}^{\frac{d}{p}}_{p,1}}ds=\int_{0}^{t}e^{-\lambda\int_{s}^{t}f(\tau)d\tau}f(s)ds=\frac{1}{\lambda}\int_{0}^{t}d\Big{(}e^{-\lambda\int_{s}^{t}f(\tau)d\tau}\Big{)}\leq\frac{1}{\lambda}.$

Thus taking all above estimates (2.5)–(2) into (2), there exists a positive
constant $C$ such that

$\displaystyle\|$
$\displaystyle\bar{u}_{\lambda,f}\|_{\mathcal{L}^{\infty}_{t}(\dot{B}^{-2+\frac{d}{p}}_{p,1})}+\kappa\|\bar{u}_{\lambda,f}\|_{L^{1}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}\leq
C\Big{(}\|u_{L}\nabla(-\Delta)^{-1}u_{L}\|_{L^{1}_{t}(\dot{B}^{-1+\frac{d}{p}}_{p,1})}$
$\displaystyle+\frac{1}{\lambda}\|u_{0}\|_{\dot{B}^{-2+\frac{d}{p}}_{p,1}}\|\bar{u}_{\lambda,f}\|_{\mathcal{L}^{\infty}_{t}(\dot{B}^{-2+\frac{d}{p}}_{p,1})}+\varepsilon\|\bar{u}_{\lambda,f}\|_{L^{1}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}+\|\bar{u}\|_{\mathcal{L}^{\infty}_{t}(\dot{B}^{-2+\frac{d}{p}}_{p,1})}\|\bar{u}_{\lambda,f}\|_{L^{1}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}\Big{)}.$
(2.9)

Now we complete the proof of Theorem 1.3. We know from [9] that there exists
$T>0$ such that the system (1.1) admits a unique local solution
$u\in\mathcal{L}^{\infty}(0,T;\dot{B}^{-2+\frac{d}{p}}_{p,1}(\mathbb{R}^{d}))\cap
L^{1}(0,T;\dot{B}^{\frac{d}{p}}_{p,1}(\mathbb{R}^{d}))$. Let us denote by
$T_{*}$ the maximal existence time of this solution. Then to prove Theorem
1.3, it suffices to prove $T_{*}=\infty$ under the initial condition (1.2),
and (1.4) holds. We prove it by contradiction. Assume that $T_{*}<\infty$,
based on the estimates (2), let $\eta$ be a small enough positive constant
which the exact value will be determined later, we define $T_{\eta}$ as

$\displaystyle T_{\eta}:=\max\Big{\\{}t\in[0,T_{*}):\
\|\bar{u}\|_{\mathcal{L}^{\infty}_{t}(\dot{B}^{-2+\frac{d}{p}}_{p,1})}+\kappa\|\bar{u}\|_{L^{1}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}\leq\eta\Big{\\}}.$
(2.10)

Taking $\lambda$ large enough, $\varepsilon$ and $\eta$ small enough such that

$\lambda\geq 2C\|u_{0}\|_{\dot{B}^{-2+\frac{d}{p}}_{p,1}},\ \ \
\varepsilon<\frac{\kappa}{4C}\ \ \text{and}\ \ \
\eta\leq\min\\{1,\frac{\kappa}{4C}\\},$

we can deduce from (2) to get that

$\displaystyle\|\bar{u}_{\lambda,f}\|_{\mathcal{L}^{\infty}_{t}(\dot{B}^{-2+\frac{d}{p}}_{p,1})}+\kappa\|\bar{u}_{\lambda,f}\|_{L^{1}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}$
$\displaystyle\leq
2C\|u_{L}\nabla(-\Delta)^{-1}u_{L}\|_{L^{1}_{t}(\dot{B}^{-1+\frac{d}{p}}_{p,1})}.$
(2.11)

As a consequence, for all $t\leq T_{\eta}$, it holds that

$\displaystyle\|\bar{u}\|_{\mathcal{L}^{\infty}_{t}(\dot{B}^{-2+\frac{d}{p}}_{p,1})}+\kappa\|\bar{u}\|_{L^{1}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}\leq
2C\|u_{L}\nabla(-\Delta)^{-1}u_{L}\|_{L^{1}_{t}(\dot{B}^{-1+\frac{d}{p}}_{p,1})}\exp\Big{\\{}\lambda\int_{0}^{t}\|u_{L}(\tau)\|_{\dot{B}^{\frac{d}{p}}_{p,1}}d\tau\Big{\\}}.$
(2.12)

Based on (2.3), we obtain that there exists a positive constant $C$ such that
for all $t\leq T_{\eta}$, it holds that

$\displaystyle\|\bar{u}\|_{\mathcal{L}^{\infty}_{t}(\dot{B}^{-2+\frac{d}{p}}_{p,1})}+\kappa\|\bar{u}\|_{L^{1}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}\leq
C\|u_{L}\nabla(-\Delta)^{-1}u_{L}\|_{L^{1}_{t}(\dot{B}^{-1+\frac{d}{p}}_{p,1})}\times\exp\Big{\\{}C\|u_{0}\|_{\dot{B}^{-2+\frac{d}{p}}_{p,1}}\Big{\\}}.$
(2.13)

We finally conclude that if we take $C_{0}$ large enough and $c_{0}$ small
enough in (1.2), then it follows from (2.13) that

$\displaystyle\|\bar{u}\|_{\mathcal{L}^{\infty}_{t}(\dot{B}^{-2+\frac{d}{p}}_{p,1})}+\kappa\|\bar{u}\|_{L^{1}_{t}(\dot{B}^{\frac{d}{p}}_{p,1})}\leq\frac{\eta}{2}$

for all $t<T_{\eta}$, which contradicts with the maximality of $T_{\eta}$,
thus $T^{*}=\infty$, and (1.4) holds true. We complete the proof of Theorem
1.3.

##  3 Proof of Theorem 1.4

In this section we give the proof of Theorem 1.4. We shall check that the
initial data $u_{0,\varepsilon}$ introduced in Theorem 1.4 satisfies the
nonlinear smallness assumption (1.2), but its $\dot{B}^{-2}_{\infty,\infty}$
norm is greater than $(-\log\varepsilon)^{\frac{1}{5}}$.

###### Lemma 3.1

(see [5]) Let $f\in\mathcal{S}(\mathbb{R}^{3})$ be given and
$\sigma\in(0,3-\frac{3}{p})$. Then there exists a constant $C$ such that for
any $\varepsilon,\alpha\in(0,1)$, the function
$f_{\varepsilon}(x):=e^{i\frac{x_{3}}{\varepsilon}}f(x_{1},\frac{x_{2}}{\varepsilon^{\alpha}},x_{3})$
satisfies, for all $1\leq p\leq\infty$,

$\|f_{\varepsilon}\|_{\dot{B}^{-\sigma}_{p,1}}\leq
C\varepsilon^{\sigma+\frac{\alpha}{p}}\ \ \text{and}\ \
\|f_{\varepsilon}\|_{\dot{B}^{-\sigma}_{\infty,\infty}}\geq
C^{-1}\varepsilon^{\sigma}.$ (3.1)

###### Lemma 3.2

(see [17]) Let $f$ and $g$ be in
$\dot{B}^{-1}_{2p,1}(\mathbb{R}^{3})\cap\dot{H}^{-1}(\mathbb{R}^{3})$. Then
for all $3<p\leq\infty$, we have

$\|e^{t\Delta}fe^{t\Delta}g\|_{L^{1}(\mathbb{R}^{+};\dot{B}^{-1+\frac{3}{p}}_{p,1})}\leq
C\big{(}\|f\|_{\dot{B}^{-1}_{2p,1}}\|g\|_{\dot{B}^{-1}_{2p,1}}\big{)}^{\frac{2p}{3p-3}}\big{(}\|f\|_{\dot{H}^{-1}}\|g\|_{\dot{H}^{-1}}\big{)}^{\frac{p-3}{3p-3}}.$
(3.2)

Define a smooth function as $u_{0,\varepsilon}:=\nabla\cdot
w_{0,\varepsilon}$, where $w_{0,\varepsilon}$ is defined as
$w_{0,\varepsilon}:=(\partial_{2}\varphi_{\varepsilon},\partial_{1}\varphi_{\varepsilon},0)$
and

$\varphi_{\varepsilon}(x):=\frac{(-\log\varepsilon)^{\frac{1}{5}}}{\varepsilon^{1-\alpha}}\cos(\frac{x_{3}}{\varepsilon})\phi(x_{1},\frac{x_{2}}{\varepsilon^{\alpha}},x_{3}).$

Then we have the following upper and lower bound of $w_{0,\varepsilon}$ in
Besov type space.

###### Lemma 3.3

(see [5]) A constant $C$ exists such that, for all $p\geq 1$, we have

$\|w_{0,\varepsilon}\|_{\dot{B}^{-1}_{p,1}}\leq
C\varepsilon^{\frac{\alpha}{p}}(-\log\varepsilon)^{\frac{1}{5}}\ \ \text{and}\
\ \|w_{0,\varepsilon}\|_{\dot{B}^{-1}_{\infty,\infty}}\geq
C^{-1}(-\log\varepsilon)^{\frac{1}{5}}.$ (3.3)

By using Lemma 3.1, we know that there exists a positive constant $C$ such
that

$\|u_{0,\varepsilon}\|_{\dot{B}^{-2}_{\infty,\infty}}=\|\nabla\cdot
w_{0,\varepsilon}\|_{\dot{B}^{-2}_{\infty,\infty}}\approx\|w_{0,\varepsilon}\|_{\dot{B}^{-1}_{\infty,\infty}}\geq
C^{-1}(-\log\varepsilon)^{\frac{1}{5}}.$

Then for all $3<p<\infty$, it follows the Sobolev embedding result that

$\|u_{0,\varepsilon}\|_{\dot{B}^{-2+\frac{3}{p}}_{p,1}}\geq\|u_{0,\varepsilon}\|_{\dot{B}^{-2}_{\infty,\infty}}\geq
C^{-1}(-\log\varepsilon)^{\frac{1}{5}}.$ (3.4)

On the other hand, it can easily calculate

$\displaystyle\nabla\cdot
w_{0,\varepsilon}=\frac{2}{\varepsilon}(-\log\varepsilon)^{\frac{1}{5}}\cos(\frac{x_{3}}{\varepsilon})\partial_{1}\partial_{2}\phi(x_{1},\frac{x_{2}}{\varepsilon^{\alpha}},x_{3}),$

which yields that

$e^{t\Delta}w_{0,\varepsilon}\nabla\cdot
e^{t\Delta}w_{0,\varepsilon}=\frac{2}{\varepsilon^{2-\alpha}}(-\log\varepsilon)^{\frac{2}{5}}e^{t\Delta}f_{\varepsilon}e^{t\Delta}g_{\varepsilon},$
(3.5)

where $f_{\varepsilon},g_{\varepsilon}$ are smooth functions. By using Lemmas
3.1 and 3.2, it follows that for all $3<p\leq\infty$, we have

$\displaystyle\|e^{t\Delta}f_{\varepsilon}e^{t\Delta}g_{\varepsilon}\|_{L^{1}(\mathbb{R}^{+};\dot{B}^{-1+\frac{3}{p}}_{p,1})}$
$\displaystyle\leq
C\big{(}\|f_{\varepsilon}\|_{\dot{B}^{-1}_{2p,1}}\|g_{\varepsilon}\|_{\dot{B}^{-1}_{2p,1}}\big{)}^{\frac{2p}{3p-3}}\big{(}\|f_{\varepsilon}\|_{\dot{H}^{-1}}\|g_{\varepsilon}\|_{\dot{H}^{-1}}\big{)}^{\frac{p-3}{3p-3}}$
$\displaystyle\leq
C\big{(}\|f_{\varepsilon}\|_{\dot{B}^{-1}_{2p,1}}\|g_{\varepsilon}\|_{\dot{B}^{-1}_{2p,1}}\big{)}^{\frac{2p}{3p-3}}\big{(}\|f_{\varepsilon}\|_{\dot{B}^{-1}_{2,1}}\|g_{\varepsilon}\|_{\dot{B}^{-1}_{2,1}}\big{)}^{\frac{p-3}{3p-3}}$
$\displaystyle\leq C\varepsilon^{2+\frac{\alpha}{3}}.$ (3.6)

Combining (3.5) and (3), we immediately obtain

$\|e^{t\Delta}w_{0,\varepsilon}\nabla\cdot
e^{t\Delta}w_{0,\varepsilon}\|_{L^{1}(\mathbb{R}^{+};\dot{B}^{-1+\frac{3}{p}}_{p,1})}\leq
C\varepsilon^{\frac{4\alpha}{3}}(-\log\varepsilon)^{\frac{2}{5}}.$

Notice that $u_{0,\varepsilon}=\nabla\cdot w_{0,\varepsilon}$, which gives us
to $w_{0,\varepsilon}=-\nabla(-\Delta)^{-1}u_{0,\varepsilon}$, thus we finally
obtain

$\|e^{t\Delta}u_{0,\varepsilon}\nabla(-\Delta)^{-1}e^{t\Delta}u_{0,\varepsilon}\|_{L^{1}(\mathbb{R}^{+};\dot{B}^{-1+\frac{3}{p}}_{p,1})}\leq
C\varepsilon^{\frac{4\alpha}{3}}(-\log\varepsilon)^{\frac{2}{5}}.$

We complete the proof of Theorem 1.4.

## References

  * [1]
  * [2] P. Biler, Local and global solvability of parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl. 8 (1998) 715–743.
  * [3] P. Biler, M. Cannone, I.A. Guerra, G. Karch, Global regular and singular solutions for a model of gravitating particles, Math. Ann. 330 (2004) 693–708.
  * [4] P. Biler, D. Hilhorst, T. Nadzieja, Existence and nonexistence of solutions for a model gravitational of particles, Colloq. Math. 67 (1994) 297–308.
  * [5] J.-Y. Chemin, I. Gallagher, Well-posedness and stability results for the Navier–Stokes equations in $\mathbb{R}^{3}$, Ann. Inst. H. Poincaré Anal. Non Linéaie 26 (2009) 599–624.
  * [6] L.C.F. Ferreira, J.C. Precioso, Existence and asymptotic behaviour for the parabolic-parabolic Keller–Segel system with singular data, Nonlinearity 24 (2011) 1433–1449
  * [7] H. Gajewski, K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr. 195 (1998) 77–114.
  * [8] M.A. Herrero, J.J.L. Velázquez, Chemotaxis collapse for the Keller–Segel model, J. Math. Biol. 35 (1996) 177–194.
  * [9] T. Iwabuchi, Global well-posedness for Keller–Segel system in Besov type spaces, J. Math. Anal. Appl. 379 (2011) 930–948.
  * [10] T. Iwabuchi, M. Nakamura, Small solutions for nonlinear heat equations, the Navier–Stokes equation and the Keller–Segel system in Besov and Triebel–Lizorkin spaces, Adv. Differential Equations 18 (2013) 687–736.
  * [11] W. Jäger, S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc. 329(2) (1992) 819–824.
  * [12] G. Karch, Scaling in nonlinear parabolic equations, J. Math. Anal. Appl. 234 (1999) 534–558.
  * [13] E.F. Keller, L.A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol. 26 (1970) 399–415.
  * [14] H. Kozono, Y. Sugiyama, Local existence and finite time blow-up of solutions in the 2-D Keller–Segel system, J. Evol. Equ. 8 (2008) 353–378.
  * [15] P.G. Lemarié-Rieusset, Small data in an optimal Banach space for the parabolic-parabolic and parabolic-elliptic Keller–Segel equations in the whole space, Adv. Differential Equations 18 (2013) 1189–1208.
  * [16] Y. Li, W. Wang, Finite-time blow-up and boundedness in a 2D Keller–Segel system with rotation, Nonlinearity 36(1) (2023) 287–318.
  * [17] Q. Liu, Global well-posedness and temporal decay estimates for the 3D nematic liquid crystal flows, J. Math. Fluid Mech. 20(4) (2018) 1459–1485.
  * [18] T. Ogawa, S. Shimizu, End-point maximal regularity and wellposedness of the two dimensional Keller–Segel system in a critical Besov space, Math. Z. 264 (2010) 601–628.
  * [19] M. Rascle, C. Ziti, Finite time blow-up in some models of chemotaxis, J. Math. Biol. 33 (1995) 388–414.
  * [20] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model, J. Differential Equations 248 (2010) 2889–2905.
  * [21] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller–Segel system, J. Math. Pures Appl. 100 (2013) 748–767.